Quantitative trading - page 16

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Financial Engineering Course: Lecture 7/14, part 2/2, (Swaptions and Negative Interest Rates)

Financial Engineering Course: Lecture 7/14, part 2/2, (Swaptions and Negative Interest Rates)

The video lecture delves into the intricacies of pricing swaptions under a negative interest rate environment. The instructor introduces the algorithm proposed by Farshid Jamshidian in 1989, which facilitates the transformation of the problem of calculating the maximum of a sum into a sum of specific maximums, given certain conditions. A crucial requirement for this approach is that the function psi_k(x) must be monotone increasing or decreasing in order to achieve accurate calculations. The lecture concludes by assigning homework and providing a Python exercise that focuses on numerical computation techniques.

The speaker emphasizes the significance of determining the value of x_star, which corresponds to the maximum summation of psi equating to zero. Finding this value enables the substitution of the summation size, k, into the equation. The speaker then explores how this condition, along with the use of monotone increasing functions, allows for the elimination of the maximum from the outermost to the innermost part of the equation. Additionally, an exercise is presented that involves computing the expectation of a maximum using both brute force and James' junction streak techniques.

The speaker proceeds to share a personal exercise involving the evaluation of the summation of all psi_i terms for i ranging from 0 to 14. They also touch on the use of Monte Carlo simulation for pricing, employing the Jump Diffusion trick to determine the optimal x value, which significantly influences the summation outcome. The speaker iterates through all the terms for each strike to identify the maximum and subsequently applies the Jamshidian trick by taking the expectation of the maximum and summing the maximum values. However, it is important to recognize certain limitations associated with this technique, such as its inapplicability to high-dimensional factors and the need for careful consideration of underlying assumptions.

Next, the lecture delves into the pricing equation for solutions using the whole white model. This involves defining a zero coupon bond within the whole white model framework, with explicit functions A and B expressed in terms of model parameters. The speaker explains how the function Theta can be represented in terms of zero coupon bonds, which can then be substituted for forward rates. The key takeaway is that, compared to the Black-Scholes equation utilized for pricing swaptions under the annuity measure, it is more advantageous to transition to the measure associated with discounting, which necessitates simulating a short rate process. By employing the Jamshidian trick, it becomes possible to search for R_star and obtain a summation comprising two components: one related to optimization and the other related to zero coupon bonds with specific weights.

The lecture progresses to discuss the pricing of swaptions using Jamshidian's trick, showcasing how this approach facilitates the calculation of implied volatility. The pricing of a swaption can be expressed as a weighted sum of options on zero-coupon bonds, where the weights c_k represent the proportions of options and the zero-coupon bond options are adjusted put options. The pricing of these zero-coupon bond options follows a straightforward process based on previously covered material. The implementation of this approach is relatively straightforward as it involves analyzing monotonic functions during the computation of implied volatility or the pricing of swaptions.

Moving forward, the lecturer explains the sequence of economic events that led to negative interest rates, highlighting the distinction between real and nominal interest rates. They discuss how a lack of trust and deflationary events can impact trading activity and the overall economy. The lecturer acknowledges the interventions carried out by central banks to stimulate the monetary supply and regain trust during the Great Recession, including lowering interest rates to encourage investment and economic activity. However, they also acknowledge the potential drawbacks and unfairness associated with the situation, particularly in terms of purchasing power if inflation exceeds nominal rates.

The lecture delves into the use of negative interest rates as an unconventional measure to incentivize investors to borrow money and invest in the market. The goal is to stimulate the economy by encouraging major financial institutions to purchase assets or engage in market activities. The concept of negative interest rates can work effectively when there is no inflation present. However, if inflation occurs and surpasses the expectations of central banks, the rates may need to be increased to compensate. This can pose a risk to companies and investors with low-rate debts, potentially leading to bankruptcy. These developments highlight the existence of both long economic cycles spanning up to 100 years and shorter-term cycles lasting around 10 years. The lecturer also touches upon the concept of inflation and emphasizes the importance of understanding how the inflation market operates in order to be prepared for any inflation-related phenomena.

Furthermore, the instructor delves into the issue of negative interest rates, which have become more prevalent in the current economic environment. A comparison of European rates between 2008 and 2017 demonstrates that short-term investments now yield negative rates, providing little incentive for saving. The instructor also discusses the challenges posed by negative interest rates when it comes to calculating volatilities and dealing with float rate bonds. Consequently, there is a need for new and alternative models to address these issues effectively. Additionally, the instructor mentions that banks often attempt to mitigate the adverse consequences of negative interest rates by incorporating maximums or waiving coupon payments for clients.

The video lecture proceeds by exploring strategies for dealing with negative interest rates and determining implied volatility for pricing options. This is crucial because, in a scenario where interest rates become negative, trading activities for derivatives can come to a halt. When using the traditional Black-Scholes model to calculate implied volatilities, the output may be "NaN" (not a number). One approach to tackle this challenge is to utilize shifted implied volatilities. This involves incorporating an additional shift parameter in the Black-Scholes model to account for the maximum negative interest rate. However, it is important to monitor this shift parameter closely. If it approaches the negative forward, the issue arises once again.

The speaker further discusses the use of the shifted variant of the LIBOR for pricing swaptions, highlighting how it resolves the problem of negative interest rates. By introducing an extra shift parameter, even if the considered strike is negative, it does not affect the pricing outcome. This is because the shifted model guarantees that rates remain above the negative range, given the log-normal nature of the model. Moreover, it is crucial to associate the shift parameter with the expiry and tenor of the underlying asset. To illustrate these concepts, the speaker provides visual representations of the log-normal distribution and showcases option prices under different shift parameters.

Expanding on the notion of shifting within the Black-Scholes formula, the lecture delves into the impact of shift parameters on volatilities and distribution shapes. A code implementation is presented for pricing, utilizing both Monte Carlo simulation and analytical expressions. The simulation involves generating paths for the shifted Geometric Brownian Motion (GBM) and calculating the average price. The code also adjusts initial points, generates densities for the local model with a shift for theta, and plots log-normal densities for different shift parameters. The importance of keeping the shift parameter as close to zero as possible is emphasized, as higher shift parameters can significantly affect the distribution and volatility.

The professor emphasizes the crucial aspect of accurately accounting for shift parameters when pricing swaptions, highlighting that even a small mistake can lead to significant pricing errors. The lecture consolidates the concepts covered, including the pricing of caplets and floors, interest rate swaps, pricing of swaptions using the Black model, negative interest rates, and the application of Jamshidian's trick in swaption pricing under the Hull-White model. To conclude, the professor assigns homework to students, encouraging them to apply the concepts learned in the lecture to calculate implied volatilities and price options.

In the final section of the video, the speaker discusses how to price an option under the whole line model by combining two blocks together. The objective is to compare the results with Monte Carlo simulation to ensure the code is free from bugs and errors. The lecture concludes with the instructor encouraging students to enjoy their assignments and delve further into the topics covered.

Tthe video lecture provides a comprehensive exploration of negative interest rates, pricing swaptions, and the application of various mathematical techniques and models. It emphasizes the importance of understanding concepts such as Jamshidian's trick, shifted implied volatilities, and the influence of shift parameters on pricing and distribution shapes. By equipping students with these tools and insights, the lecture prepares them to navigate the complexities of the financial world, make informed decisions, and accurately price options and swaptions under challenging market conditions.

• 00:00:00 In this section, the video covers the concept of negative interest rates and pricing swaptions under a negative interest rate environment. The lecture also discusses an algorithm presented in 1989 by Farshid Jamshidian, which allows the problem of an uncalculating maximum of a sum to be transformed into a sum of certain maximums under certain conditions. The most important requirement is that the function psi k of x has to be a monotone increasing or monotone decreasing function in x for the calculation to be possible. The lecture concludes with a homework assignment and a Python exercise on how to perform the numerical computation.
  
  
• 00:05:00 In this section of the lecture, the speaker explains the importance of finding the x star value in the maximum summation of psi. By finding this value for which the expression is equal to zero, the summation of size is equal to k, which can be substituted in the equation. The speaker then goes on to discuss how this condition and monotone increasing functions can help remove the maximum from outside to inside of the equation. They also provide an exercise involving the computation of the expectation of a maximum using brute force and James' junction streak techniques.
  
  
• 00:10:00 In this section, the speaker explains his personal exercise of evaluating the summation of all psi i's for each i from 0 to 14. He also mentions about the Monte Carlo simulation for pricing and using the Jump Diffusion trick for finding the optimal x, which is important because this will affect the summation. He then iterates over all the terms for each strike to find the maximum, following which he applies the Jamshidian trick by taking the expectation of the maximum and performing summation of maximums. However, there are some limitations to this technique such as its inability to work with high dimensional factors and assumptions need to be carefully considered while using this trick.
  
  
• 00:15:00 In this section of the Financial Engineering Course, the pricing equation for solutions using the whole white model is discussed. This includes the definition of a zero coupon bond under the whole white model, where functions A and B are given explicitly in terms of model parameters. The section explains how the function Theta is expressed in terms of zero coupon bonds, which can be substituted for forward rates. The key takeaway is that, compared to the Black-Scholes equation used for pricing swaptions under the annuity measure, it is more beneficial to switch to the measure corresponding to discounting, which involves simulating a short rate process. By using the obsidian trick, it is possible to sort for R star and obtain a sum involving two sums: one corresponding to optimization and the other corresponding to zero coupon bonds with special weights.
  
  
• 00:20:00 In this section, the lecturer discusses the pricing of swaptions using the Jamshidian's trick and shows how this approach allows the calculation of implied volatility. The pricing of a swaption can be expressed as a weighted sum of options on zero-coupon bonds. The weights c_k represent the proportions of options, and the zero-coupon bond options are put options with adjusted strikes. The pricing of these zero-coupon bond options is straightforward and is based on previously covered material. The implementation of this approach is trivial as monotonic functions are being analyzed in the calculation of implied volatility or the pricing of swaptions.
  
  
• 00:25:00 In this section, the lecturer explains the sequence of economic events that led to negative interest rates, the difference between real and nominal interest rates, and how a lack of trust and deflationary events can affect trading activity and the economy. He also mentions how central banks intervened to stimulate the monetary supply and to recover trust amidst the Great Recession, including lowering interest rates to encourage investment and activity. However, he acknowledges the potential unfairness of the situation and the negative impact on purchasing power if inflation is higher than nominal rates.
  
  
• 00:30:00 In this section, the lecturer discusses the use of negative interest rates as an unconventional way to inspire investors to borrow money and invest in the market. The intention is to encourage big financial institutions to buy houses or invest in the market to stimulate the economy. The concept of negative interest rates can work under the assumption of no inflation. However, if there is inflation and inflation is higher than central bank expectations, the rates may be increased to compensate, putting many companies and investors with low-rate debts at risk of bankruptcy. This development showcases a cycle where there are long economic cycles of up to 100 years and short-term cycles of 10 years. The lecturer also touches on the concept of inflation and the need to understand how the inflation market works to be ready for any inflation phenomena.
  
  
• 00:35:00 In this section, the instructor discusses the issue of negative interest rates, which are becoming more common in the current economic environment. He presents a comparison of European rates in 2008 versus 2017, showing that rates for short-term investments are now negative, providing little encouragement for saving. The instructor also discusses the problems with negative interest rates when it comes to calculating volatilities and dealing with float rate bonds. He highlights the need for new and alternative models to fix these issues. Finally, the instructor mentions that banks typically try to avoid the negative consequences of negative interest rates by including maximums or not charging clients for coupon payments.
  
  
• 00:40:00 In this section, the video discusses how to deal with negative interest rates and how to find volatility in order to price an option. This is important because if interest rates become negative, trading activities for those derivatives will freeze, and if you use the existing Black-Scholes model to calculate implied volatilities, you will get "NaN." One approach is to use shifted implied volatilities. This relies on a Black-Scholes model with an additional shift parameter to determine the maximum negative interest rate. However, this shift parameter must be monitored closely, and if it is close to the negative forward, then the problem arises again.
  
  
• 00:45:00 In this section of the lecture, the speaker discusses using the shifted variant of the LIBOR for the pricing of swaptions and how it solves the problem of negative interest rates. By adding an extra shift parameter, even if the strike considered is negative, it does not affect the pricing since it is log-normal and the shift guarantees that it stays above the negative rates. Additionally, it is important to keep in mind that the shift is always associated with the expiry and the tenor of the underlying asset. Lastly, the speaker provides an illustration of the log-normal distribution and shows the option prices depending on different shift parameters.
  
  
• 00:50:00 In this section, the concept of shifting in Black-Scholes formula is explored further with a focus on the impact of shift parameters on volatilities and distribution shapes. A code is presented for pricing, using Monte Carlo simulation and analytical expression. The simulation involves generating paths for shifted gbm and calculating the average of the price. The code also adjusts initial points, generating densities for the local model with a shift for theta, and plots log-normal density for different shift parameters. The importance of keeping the shift parameter as close to zero as possible is emphasized due to the impact of higher shift parameters on distribution and volatility.
  
  
• 00:55:00 In this section of the lecture, the professor discusses the importance of correctly accounting for shift parameters when pricing swaptions, as even a small mistake could lead to significant errors in pricing. The lecture also covers the use of the shifted Black-Scholes model in generating different prices for varying shift parameters. The professor then summarizes the concepts addressed in the lecture, which included pricing of caplets and floors, interest rate swaps, pricing of swaptions using the Black model, negative interest rates, and the use of Jamshidian's trick in pricing swaptions under the Hull-White model. The lecture ends with the professor assigning homework to students to apply the concepts discussed in the lecture to calculate implied volatilities and price options.
  
  
• 01:00:00 In this section, the speaker discusses how to price an option under the whole line model by combining two blocks together, with the objective of comparing the results with Monte Carlo simulation. The goal is to ensure that the code is bug-free, and the lecture ends with encouragement for students to enjoy their assignments.

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Financial Engineering Course: Lecture 8/14, part 1/4, (Mortgages and Prepayments)

Financial Engineering Course: Lecture 8/14, part 1/4, (Mortgages and Prepayments)

In the lecture, the concept of pricing mortgages is thoroughly discussed, highlighting the complex nature of this task from a financial engineering perspective. The main challenge lies in managing the risks associated with client prepayments and additional payments made on top of regular monthly installments. Two types of mortgages are specifically focused on: bullet mortgages and annuity mortgages.

A bullet mortgage entails clients paying only the interest rate and outstanding notional at the end of the contract, while an annuity mortgage involves gradual reduction of the notional until no outstanding notional remains at the contract's conclusion. Prepayments, pipeline risks, and the inclusion of people's behavior and incentives in financial contract pricing are also addressed in the lecture.

It is emphasized that risks related to prepayments are minimized for floating rate mortgages since clients have no optimal incentive to make prepayments. The constant prepayment rate is discussed in relation to portfolio management. Assessing the repayment profile of a mortgage portfolio requires considering prepayment risks based on the overall repayment profile rather than individual clients.

The lecture delves into the index amortizing swap and how it can be utilized to match prepayment risks within the portfolio. Furthermore, the behavioral aspect of prepayments is explored, taking into account refinancing incentives and individuals' rational or irrational decision-making when deciding to allocate extra funds towards their mortgage.

The risks faced by banks and other financial institutions are also highlighted, particularly regarding mortgage cash flows and the uncertainty surrounding them. This includes the potential for client defaults and the need for banks to resell houses, sometimes at a loss. The lecture emphasizes the importance of pricing and risk management in issuing mortgages, specifically addressing pipeline risk and prepayment risk. Pipeline risk arises due to the time delay between agreeing to a mortgage and signing the contract, which leaves room for interest rate changes during that period.

The risks associated with mortgages, such as pipeline risk and prepayment risk, are further elaborated upon. Pipeline risk refers to the risk that a client may opt for a lower interest rate, which occurs when a client has the optionality to execute a contract at a lower rate. On the other hand, prepayment risk pertains to a client's desire to modify the contract and the associated risk of prepayments. Financial institutions that enter into contracts with clients face unhatched positions that introduce additional risks in derivative pricing. Mortgages possess an embedded option that enables the mortgagee to pay off their mortgage faster than the agreed schedule, resulting in prepayment risk. The lecture highlights that it is logical for a mortgagee to prioritize paying off their mortgage rather than keeping savings in an account with negative or no interest rates.

While pricing mortgages under the risk-neutral measure is important, the lecture emphasizes that consumer incentives to take out or prepay mortgages may not be solely driven by market circumstances. Factors such as age or financial freedom can influence the incentive to prepay mortgages and avoid monthly payments. The lecture explores the connection between pricing under the risk-neutral measure and the behavioral aspects involved in pricing prepayments. It also delves into two types of amortization schedules: annuity mortgages and bullet mortgages, which ensure borrowers ultimately repay the initial borrowed sum for purchasing the house along with additional amounts representing loan costs.

The video explains the relationship between mortgages, prepayments, and the risks faced by financial institutions. Prepayments made by borrowers, exceeding their scheduled payments, require the bank to adjust their hedge, leading to additional costs. Large prepayments can also decrease the bank's incoming cash flow and contract duration. However, a significant number of sudden prepayments generate prepayment risk that needs to be analyzed and mitigated. To manage these risks, banks create mortgage portfolios and utilize swaps to offset fixed-rate payments.

The lecturer discusses the risks and profits associated with mortgages and prepayments. Mortgages are priced at the portfolio level, with hedges consisting of significantly larger notionals. The profitability for a bank in a mortgage depends on factors such as the notional amount, duration of the loan, and the interest rate. Prepayments, however, pose a potential loss for the bank. Other risks associated with mortgages include pipeline risk, tax risk, default risk, and the risk of a housing market crash. The lecture emphasizes that the amortization plan chosen for a mortgage can impact the amount of interest accrued.

The lecturer provides a detailed exploration of different types of mortgages and their associated amortization schedules. One such type is the bullet mortgage, which involves a single lump sum payment at the end of the mortgage term. While this simplifies payment obligations throughout the term, it carries the risk of a substantial payment due at the end. The lecturer suggests that a bullet mortgage may be suitable for individuals who have alternative investment opportunities, such as a savings account with a higher interest rate than the mortgage. The lecture also offers an overview of monthly payments and accrual periods, providing a comprehensive understanding of mortgage payment structures.

Constant prepayment rates associated with mortgages are discussed in detail. These rates represent fixed amounts that homeowners choose to prepay towards their mortgages. The prepayment rate is typically estimated based on a large portfolio of mortgages, and it affects the notional value over the amortization period. Legal constraints on prepayment amounts are also mentioned. The lecturer calculates the total amount of interest paid on a mortgage using a prepayment rate and emphasizes the importance of considering prepayments in mortgage pricing. Numerical experiments and exercises are presented to illustrate the concepts, and a Python plot and code are used to analyze cash flows and amortization schedules effectively.

The lecture emphasizes the impact of prepayment rates on the amortization of a mortgage over time. An example is provided for a 10-year fixed-rate mortgage at a 3% interest rate, which the bank needs to hedge using a swap. The experiment compares scenarios with and without prepayments, demonstrating how prepayments gradually decrease over time as the outstanding notional decreases. The results highlight that prepayments can significantly reduce the amount of interest paid, but a substantial lump sum payment is still required at the end. The lecturer also notes that in practice, mortgages may be combined with savings accounts or derivatives that offer higher returns, while also minimizing taxation on the outstanding notional.

Furthermore, the lecture dives into the construction of an amortization schedule for a bullet mortgage using Python code. The code allows for the calculation of payment schedules based on given interest rates and prepayment rates. It provides a matrix array that outlines the required payments throughout the mortgage's lifetime. The prepayment rates can be expressed as percentages, making it convenient for analyzing a large portfolio of mortgages. The payment schedule is affected when prepayments are introduced, showcasing the flexibility and usefulness of the Python code for analyzing payment structures.

The speaker explains the columns of a mortgage payment matrix. Time is represented in the first column, followed by the outstanding notion in the second column. Prepayment, repayment, and notional quote are defined in the subsequent columns. The prepayment column indicates the fraction of the notional that will be prepaid and is determined by the constant prepayment rate (CPR). Repayment, in the fourth column, signifies the reduction in the outstanding notion each month with regular payments. The fifth column represents interest payments, while the last column displays the monthly installments required. The lecturer showcases the model using a 30-year bullet mortgage example with no prepayment.

In summary, the lecture provides an extensive exploration of mortgage pricing, prepayment risks, and their impact on financial institutions. It covers various types of mortgages, including bullet mortgages and annuity mortgages, and emphasizes the importance of considering client behavior and incentives in mortgage pricing. The lecture delves into the risks faced by financial institutions, such as pipeline risk and prepayment risk, and discusses strategies for mitigating these risks through portfolio management and the use of financial derivatives like swaps. The lecture also highlights the uncertainty surrounding mortgage cash flows, including the possibility of client defaults and the need for banks to resell houses at a potential loss.

Moreover, the lecture acknowledges that pricing mortgages solely under a risk-neutral measure may not capture the full range of consumer incentives and behaviors. Factors such as age, financial freedom, and personal preferences can significantly influence clients' decisions to prepay or refinance their mortgages. Therefore, the lecture emphasizes the importance of integrating behavioral aspects into mortgage pricing models, considering the motivations and rational/irrational decision-making of borrowers.

The lecturer explores the concept of constant prepayment rates and their relationship to portfolio management. Instead of analyzing prepayment risks on an individual client level, the lecture stresses the need to assess the overall repayment profile of a mortgage portfolio. By considering the aggregate prepayment behavior, banks can better manage the associated risks and use tools like index amortizing swaps to match and hedge prepayment risks effectively.

Furthermore, the lecture delves into the risks faced by financial institutions due to mortgages and prepayments. When borrowers make significant prepayments, it necessitates adjustments to the bank's hedging strategy, resulting in additional costs and potential disruptions to cash flow and contract duration. The sudden prepayment of a significant number of clients creates prepayment risk, which must be carefully analyzed and hedged to mitigate its impact on the bank's portfolio. The lecturer highlights that banks create mortgage portfolios and utilize swaps to offset fixed-rate payments, thereby reducing risks.

The lecture concludes with a discussion on the valuation of mortgage securities, noting that it depends on market observable quantities. Although this aspect is briefly mentioned, the lecture implies that a more in-depth exploration of these quantities will be covered in subsequent parts of the course.

The lecture provides a comprehensive understanding of mortgage pricing, prepayment risks, and their implications for financial institutions. It addresses various types of mortgages, behavioral aspects, portfolio management techniques, and risk mitigation strategies. By considering the complex dynamics of mortgage cash flows, prepayments, and client behavior, the lecture equips viewers with the knowledge and tools necessary to navigate the challenges of pricing and managing mortgage portfolios effectively.

• 00:00:00 In this section of the lecture, the concept of pricing mortgages is discussed, which is a non-trivial task from a financial engineering perspective due to the risks associated with clients prepaying or giving extra amounts on top of regular monthly instalments. The lecture focuses on two types of mortgages: bullet mortgage, where clients only pay the interest rate and outstanding notion at the end of the contract, and annuity mortgage, where clients gradually decrease the notional of the mortgage until there is no outstanding notion at the end. The lecture also covers prepayments, pipeline risks, and the inclusion of people's behavior and incentives in the pricing of financial contracts. Finally, it is noted that the risks associated with prepayments are minimized for floating rate mortgages as there is no optimality for clients to perform prepayments.
  
  
• 00:05:00 In this section, the lecturer discusses the concept of a constant prepayment rate and how it relates to portfolio management. They explain that when assessing the repayment profile of a mortgage portfolio, prepayment risks must be taken into account based on the overall repayment profile, rather than just individual clients. They also address the index amortizing swap and how it can be used to match prepayment risks in the portfolio. The lecturer further delves into the behavioral aspect of prepayments, including how to account for refinancing incentives and people's rational/irrational decision-making when it comes to putting extra money on their mortgage. Lastly, they touch upon the risk that banks and other financial institutions face and offer a preview of the upcoming Python experiment.
  
  
• 00:10:00 In this section of the lecture, the professor discusses mortgage cash flows and the uncertainty surrounding them, with the potential for clients to default and banks having to deal with reselling the house at a possible loss. Banks may not be interested in selling houses and may sell them at a discount to other parties, resulting in higher mortgage rates. The lecture focuses on pricing and risk associated with issuing mortgages, particularly pipeline risk and prepayment risk. Pipeline risk arises due to the time delay between agreeing to a mortgage and signing the contract, with the possibility of interest rate changes during that period.
  
  
• 00:15:00 In this section, the lecturer discusses the risks associated with mortgages, including pipeline risk and prepayment risk. Pipeline risk refers to the risk that a client may choose a lower rate and occurs when a client has the optionality to execute a contract at a lower rate. Prepayment risk is associated with a client who wants to change the contract and refers to the risk of prepayments. The lecturer notes that financial institutions that sign contracts with clients have unhatched positions that lead to additional risks in derivative pricing. The section also explains that mortgages have an embedded option that allows the mortgagee to pay their mortgage faster than the agreed schedule, which creates the prepayment risk. The lecturer notes that it's logical for a mortgagee to pay off their mortgage instead of keeping their savings in an account with negative or no interest rate.
  
  
• 00:20:00 In this section of the lecture, the focus is on the pricing of mortgages and prepayment risks. While pricing under risk-neutral measure is important, the incentives of consumers to take out or prepay mortgages may not be purely driven by market circumstances. For example, younger or older people may have more incentive to prepay their mortgages to avoid monthly payments and have more financial freedom. The lecture discusses connecting these two elements, pricing under risk-neutral measure and behavioral aspects involved in pricing prepayments. Additionally, it explores two types of amortization schedules: annuity mortgages and bullet mortgages, which guarantee that borrowers ultimately pay back the initial sum borrowed to buy the house plus an extra amount representing loan costs.
  
  
• 00:25:00 In this section, the video explains mortgages and prepayments in relation to the risk for financial institutions. When borrowers make prepayments, meaning they pay more than their scheduled payments, it causes the bank to adjust their hedge and incur additional costs. A large prepayment can also reduce the bank's incoming cash flow and contract duration. However, when a significant number of clients suddenly prepay, it produces prepayment risk that needs to be analyzed and hedged. Overall, banks create a portfolio of mortgages and use swaps to offset fixed-rate payments to mitigate risks.
  
  
• 00:30:00 In this section of the video, the speaker discusses the risks and profits associated with mortgages and prepayments. Mortgages are priced at the portfolio level, with hedges consisting of much larger notionals. Notional, the duration of the loan, and the interest rate all represent the profits generated for the bank from a mortgage. If there are prepayments, this would be a loss for the bank. Different risks include pipeline risk, tax risk, default risk, and the risk of a housing market crash. Mortgages can be classified according to the amortization plan, with annuities and bullets being two typical ones. The choice of the amortization plan can influence the amount of interest raised.
  
  
• 00:35:00 In this section, the lecturer discusses the different types of mortgages and their associated amortization schedules, explaining how these impact the pricing of mortgage portfolios. The simplest mortgage is a bullet mortgage, where only one payment is made at the end of the mortgage. This is associated with a constant notional and interest payments but carries the risk of having a large lump sum payment at the end. The lecturer notes that this may be a good option for those who have alternative ways of investing their money, such as a savings account with a higher interest rate than the mortgage. They also provide an overview of monthly payments and accrual periods.
  
  
• 00:40:00 In this section of the Financial Engineering Course, the lecturer discusses constant prepayment rates associated with mortgages. The constant prepayment rate is a fixed amount that homeowners prepay towards their mortgage. The prepayment rate is typically estimated based on a large portfolio of mortgages, and it changes the notional value over the amortization period. The lecturer also mentions legal constraints on prepayment amounts, and calculates the total amount of interest paid on a mortgage using a prepayment rate. The lecture includes numerical experiments and exercises, and the lecturer also demonstrates how to use a Python plot and code to analyze cash flows and amortization schedules.
  
  
• 00:45:00 In this section of the lecture, the instructor discusses how prepayment rates impact the amortization of a mortgage over time. The example given is for a 10-year fixed-rate mortgage at a 3% interest rate, which the bank will need to hedge using a swap. The experiment compares scenarios with and without prepayments, with prepayments decreasing over time as the outstanding notional decreases. The results show that prepayments can significantly reduce the amount of interest paid but still require a substantial lump sum payment at the end. The instructor also notes that in practice, such mortgages may be combined with savings accounts or derivatives that provide a higher return, while also reducing taxation on the outstanding notional.
  
  
• 00:50:00 In this section, the lecturer discusses the construction of an amortization schedule for a bullet mortgage using Python code, which describes the payment schedule of a mortgage with only one lump sum at the end. The code's output is a matrix array that describes each payment required over the mortgage's lifetime, and it can evaluate the schedule for given interest rates and prepayment rates. The prepayment rates can be taken as a percentage, allowing for easy application to a large portfolio of mortgages, and the payment schedule is affected if prepayments are introduced. Overall, the Python code enables the analysis of how the payment schedule would look like, given the interest rates and prepayments.
  
  
• 00:55:00 In this section, the speaker defines the columns of a mortgage payment matrix. The first column represents time, while the second column represents outstanding notion. Prepayment, repayment, and notional quote are also defined. The third column represents prepayment, which indicates the fraction of the notional that will be pre-paid and is defined by the CPR. The fourth column is repayment, which is the reduction in outstanding notion every month upon making monthly payments, and the fifth column represents interest payments. Finally, the last column in the matrix represents monthly installments that must be paid. The matrix is then run through a model for a 30-year bullet mortgage with no prepayment.
  
  
• 01:00:00 In this section of the lecture, the concept of mortgages and prepayments is explored. Without prepayments, the monthly or yearly payments will be fixed except for the last one where the full amount will need to be repaid. However, with prepayments, the outstanding notional is reduced, resulting in a decreasing prepayment amount over time. The prepayment rate can also be linked to market observable quantities, making it a stochastic quantity. The effect of prepayment on the curvature of the outstanding notion profile is also discussed.
  
  
• 01:05:00 In this section, the speaker briefly mentions that the valuation of mortgage securities depends on market observable quantities and that such quantities will be discussed in detail later in the lecture.
  

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Financial Engineering Course: Lecture 8/14, part 2/4, (Mortgages and Prepayments)

Financial Engineering Course: Lecture 8/14, part 2/4, (Mortgages and Prepayments)

In addition to the topics covered so far, the lecture further explores the concept of annuity mortgages and their essential characteristics. An annuity mortgage is a type of mortgage where the outstanding notion gradually decreases over time due to regular repayments. The monthly payments for annuity mortgages comprise two components: interest rate payments and contractual repayment schedules denoted by "q." These repayments are structured in a way that the outstanding notional is reduced with each payment until the final payment covers the remaining balance.

The instructor explains that annuity mortgages have fixed installment payments throughout the contract's duration, ensuring a balance between the interest rate and principal portions. This balance results in a constant sum on each payment date. As the outstanding notional decreases, both the repayments and interest rate payments follow opposite trends. The interest compounded on the remaining notional diminishes over time. To calculate the correct installment amount, the discounted future cash flows of the mortgage must be equal to the value of the outstanding notional. Any prepayments made should adjust the constant payment amount accordingly.

The lecture delves into the calculation of constant payments or annuities. The value of an annuity is determined by summing all future cash flows discounted to the present day. By utilizing the formula for geometric sums, one can derive an analytical expression for the annuity. However, if prepayments are made, the constant payment amount will change, necessitating a recalculation. The lecturer also explains how to calculate interest rate payments and principal payments, as well as how to adjust the outstanding notional after prepayments are made.

Furthermore, the lecturer emphasizes the notion of time and its impact on mortgages, repayments, and prepayments. As repayments and prepayments are made, the outstanding notional of a mortgage decreases, leading to a corresponding decrease in the monthly payments. The prepayment rate can be seen as a reformulation of the interest rate payment and is included in the interest rate component. When a borrower decides to prepay an installment, the remaining payment schedule is adjusted to reflect the updated outstanding notional. Graphs are presented to illustrate the impact of varying prepayment levels on the constantly reducing notional, considering scenarios with zero percent and 12 percent prepayment rates. The lecture concludes that larger prepayment rates can hinder the reduction in the outstanding notional.

The lecture also delves into the structure of annuity mortgages and their repayment mechanism. An annuity mortgage consists of fixed monthly payments that encompass both repayment and interest rate components. These fixed payments ensure a balanced repayment structure over the lifetime of the mortgage. The lecturer explores the impact of prepayments on monthly payments and explains how the constant payment amount (c) needs to be recalculated when prepayments are made. Additionally, the notional amount of the mortgage gradually decreases until there is no outstanding notional remaining. By the end of the mortgage period, all payments reach zero, facilitating a smooth transition in the presence of prepayment rates. The lecturer provides Python code for the repayment schedule and explains its significance.

Furthermore, the lecture discusses the steps involved in calculating the new notional after a repayment or prepayment takes place in a mortgage. This process is iterative and considers the previous notional, repayment, prepayment rates, and interest rate payments over the lifetime of the contract. If the prepayment is time-dependent or stochastic, adjustments need to be made in the calculations. Additionally, the lecture highlights that prepayments reduce monthly costs, while a zero prepayment rate leads to constant installments throughout the mortgage's lifetime. It is explained that if prepayment occurs only on a specific date, the installments will remain constant until that date, after which everything will be recomputed.

The lecturer then moves on to explain how prepayment rates for mortgages are estimated from a portfolio management perspective. The prepayment rate, represented by the lambda coefficient, is a crucial factor in portfolio management as it affects the performance and risk of the portfolio. Estimating the prepayment rate involves considering historical data and analyzing various factors that influence a borrower's decision to prepay their mortgage. These factors may include interest rates, individuals' financial goals, and market conditions. The lecture explores how market observable quantities impact the prepayment rate and discusses methods for estimating it based on a portfolio of mortgages.

Next, the lecture delves into the concept of refinancing incentive and its relationship to prepayment models for mortgages. Borrowers are more likely to prepay their mortgage when they observe a lower interest rate compared to the rate of their current mortgage. This refinancing incentive is a key driver in any prepayment model and is closely linked to market rates. Additionally, the type of mortgage, its maturity, and the collateral associated with it can affect mortgage rates. The lecturer emphasizes that the attractiveness of the collateral influences the interest rate offered by banks. Other factors that can impact prepayment rates include the age of the mortgage, the month of the year, tax considerations, and burnout.

The lecture discusses factors that affect prepayment rates, considering both the market situation and individual client profiles. The interest rate incentive is identified as the most significant factor influencing prepayment rates. Determining the prepayment incentive involves evaluating market observable quantities. The lecture suggests that the most reasonable benchmark for pricing a mortgage is a swap rate, which banks use to derive the mortgage rate for new clients. The liquidity risk factor determines an additional spread for the mortgage rate. Prepayments are viewed as a cost for banks as they reduce the hedging position, and determining the mortgage rate involves assessing associated risks and profits.

The focus then shifts to the incentive function of mortgage prepayments. The swap rate is dependent on prepayment amounts, which are directly related to the initial mortgage rate of a fixed-rate mortgage and the rate associated with refinancing. The liquidity risk coefficient and the bank's profit margin further contribute to determining the new mortgage rate. However, the lecture acknowledges that people do not always behave logically or rationally when deciding to prepay their mortgage. For example, individuals may choose to prepay when it is not necessarily optimal, such as when they come into extra money. The incentive function is defined as the difference between the current mortgage rate and the new mortgage rate, and it is used to assess whether it makes sense to refinance or prepay a mortgage.

The instructor emphasizes the importance of understanding the shape of the incentive function in different market circumstances. The graph representing the incentive function exhibits breakpoints and a sigmoid shape, which reflects both the incentive function and the non-rational behavior of borrowers. The lecture highlights the significance of considering small details when implementing incentive functions, as even subtle variations can have a crucial impact.

The lecture concludes with the speaker discussing the concept of prepayments on mortgages. As the swap rate decreases or reaches zero, the incentive for prepayment diminishes. In cases where swap rates become negative, the incentive may reach its maximum level. The shape of the incentive function graph is further explored, with particular attention given to the difference between the old mortgage rate and swap values. It is underscored that although the shape is generally decreasing, it is essential to pay attention to small details when implementing incentive functions.

The lecture provides a comprehensive understanding of annuity mortgages, their repayment mechanisms, the calculation of constant payments, the impact of prepayments, estimation of prepayment rates, refinancing incentives, and the factors influencing prepayment behavior. By considering these aspects, individuals can make informed decisions regarding their mortgages and understand the dynamics of the mortgage market.

• 00:00:00 In this section of the video, the concept of annuity mortgages is discussed. An annuity mortgage is a type of mortgage where the outstanding notion is either zero or decreasing in time due to repayments. The monthly payments for annuity mortgages consist of two elements: interest rate payments and contractual repayment schedules, denoted by q. With annuity mortgages, the repayments are scheduled to decrease the notional to the extent that the final payment covers the last outstanding notion. Additionally, the prepayment determinants are examined, which are the factors that impact a client's decision to prepay or extract the reduction of the outstanding notion of the mortgage before its scheduled time.
  
  
• 00:05:00 In this section of the Financial Engineering Course, the instructor discusses annuity mortgages and their essential characteristics. Annuity mortgages have fixed installment payments throughout the lifetime of the contract, which balances the interest rate and principal parts, ensuring that the sum is constant on each payment date. Both repayments and interest rate payments follow opposite trends as the notion progressively decreases, whereby the interest compounds on the notion will diminish. To calculate the correct installment amount, discounted future cash flows of the mortgage must be equal to the value of the outstanding notional, and any prepayments would need to adjust the constant amount.
  
  
• 00:10:00 In this section of the lecture, the instructor discusses the calculation of constant payments or annuities. The value of this annuity is equal to the sum of all future cash flows discounted to the present day. By using the formula for geometric sums, the analytical expression for the annuity can be found. If prepayments are made, this will change the constant payment amount, so a new one must be calculated. The instructor also explains how to calculate interest rate payments and principal payments, as well as how to adjust the notional outstanding after prepayments.
  
  
• 00:15:00 In this section of the Financial Engineering course, the lecturer discusses the notion of time and the impact of repayments and prepayments on mortgages. The outstanding notional of a mortgage decreases with repayments and prepayments, and the monthly payments will also decrease accordingly. The prepayment rate can be interpreted as a reformulation of the interest rate payment and is included in the interest rate part. When a mortgager decides to prepay an installment, the remaining payment schedule is rebalanced according to the updated outstanding notion. The lecturer presents graphs showing the impact of varying prepayment levels on constantly notional for scenarios with zero percent and 12 percent prepayment rates, and concludes that larger prepayment rates can degrade the reduction in the outstanding notional.
  
  
• 00:20:00 In this section, the lecturer discusses the structure of an annuity mortgage and its repayment mechanism. The mortgage consists of fixed monthly payments that have both repayment and interest rate components. These fixed payments make it possible to have a balanced repayment structure over the mortgage's lifetime. The lecturer also explores prepayments' impact on monthly payments and recalculates the constant size c when prepayments are made. Additionally, the notional amount of the mortgage reduces until no outstanding notional amount is left. Ultimately, by the end of the mortgage period, all the payments reach zero, and there is a smooth transition concerning prepayment rates. The lecturer provides some python code for the repayment schedule and explains the code's meaning.
  
  
• 00:25:00 In this section, the lecture discusses the steps involved in calculating the new notional after a repayment and prepayment takes place in a mortgage. The new notional is calculated using the previous notional, repayment, and prepayment rates along with interest rate payments. The process is iterative and goes over the lifetime of the contract. If the prepayment is time-dependent or stochastic, adjustments need to be made in the calculations. Additionally, a prepayment reduces monthly costs, whereas a zero prepayment rate leads to constant installments over the lifetime of the mortgage. The lecture explains that if prepayment takes place only at a given date, the installments will remain constant until the date of prepayment after which everything will be recomputed.
  
  
• 00:30:00 In this section, the lecturer explains how prepayment rates for mortgages are estimated from the perspective of portfolio management. The prepayment rate, represented by the lambda coefficient, is a key element in this process as it affects the portfolio's performance and risk. The prepayment rate is estimated historically based on people's behavior and various factors that may influence a person's incentive to prepay their mortgage, such as interest rates and individuals targeting financial independence. The lecturer also discusses how market observable quantities impact the prepayment rate and how one can estimate it from a portfolio of mortgages.
  
  
• 00:35:00 In this section, the concept of refinancing incentive and its relation to prepayment models for mortgages is discussed. When borrowers observe a lower interest rate than the rate of their mortgage, they are more likely to prepay. This is due to a primary driver in any prepayment model, the refinancing incentive, and its relationship to market rates. Additionally, several other factors can affect mortgage rates, such as the type of mortgage, the maturity of the mortgage, and the collateral for the mortgage. The more attractive the collateral for the bank's mortgage, the lower the interest rate they will offer. Other factors that can affect prepayment rates include mortgage age, month of the year, tax reasons, and burnout.
  
  
• 00:40:00 In this section, the lecturer discusses factors that affect prepayment rates for mortgages, including market situation and individual profiles of clients. The interest rate incentive is the most significant factor affecting prepayment rates, and a suitable definition for the prepayment incentive involves determining the market observable quantities. The consensus is that the most reasonable benchmark for the price of a mortgage is a swap rate, which banks use to derive the under money mortgage rate for new clients, and the liquidity risk factor determines the additional spread for a mortgage rate. Prepayments are considered a cost for banks since they reduce the hedging position, and there are associated risks and profits involved in determining the mortgage rate.
  
  
• 00:45:00 In this section, the focus is on the incentive function of mortgage prepayments. The swap rate will depend on the prepayment amounts, which is directly linked to the fixed-rate mortgage's initial mortgage rate, as well as the rate associated with refinancing the mortgage. The liquid risk coefficient and the bank’s profit margin further determine the new mortgage rate. People do not always behave logically and rationally, and they may prepay when it is not optimal, such as when they come into extra money. The incentive function is defined as the difference between the current mortgage rate and the new mortgage rate, and it is this function that is used to determine whether it makes sense to refinance or prepay a mortgage.
  
  
• 00:50:00 In this section of the lecture, the instructor discusses the rational behavior and incentives of prepaying mortgages based on swap rates and mortgage rates. He explains that the graph depicting prepayment has breakpoints and a sigmoid shape, representing the incentive function and non-rational behavior of clients. He emphasizes the importance of understanding the shape of the incentive function in different market circumstances, depending on whether the incentive is derived from rates or the difference between the old and new mortgages. The instructor also provides a code for visualizing the incentive function and determining prepayment rates.
  
  
• 00:55:00 In this section of the Financial Engineering lecture, the concept of prepayments on mortgages is discussed. The speaker mentions that as the swap rate decreases or reaches zero, the incentive for prepayment decreases and if swap rates become negative, then the incentive may reach its maximum amount. The shape of the graph for incentive functions is also discussed, with a focus on the difference between the old mortgage and swap values. It is emphasized that even though the shape is mostly decreasing, it is important to keep in mind that it is a difference function and that small details are crucial when implementing incentive functions.

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Financial Engineering Course: Lecture 8/14, part 3/4, (Mortgages and Prepayments)

Financial Engineering Course: Lecture 8/14, part 3/4, (Mortgages and Prepayments)

In today's lecture, we aim to establish a strong connection between refinancing incentives, prepayments, and various types of mortgages. We begin by examining the concept of a constant payment rate and its relationship to mortgages as amortizing swaps without uncertainty. Building upon this foundation, we introduce the concept of an index amortizing swap, which incorporates clients' willingness to prepay or refinance based on market conditions. This further leads us to link refinancing incentives and the benchmark swap rate in derivative pricing, specifically applied to a mortgage portfolio that amortizes over time.

To better understand the dynamics involved, we explore both deterministic and stochastic functions of amortization schedules. While a deterministic function suffices in simpler cases, the more advanced scenario introduces stochasticity, primarily driven by the swap rate. This stochasticity captures the irrational behavior of clients, which is important to consider when observing market rates and incorporating them into the pricing of an amortizing swap. However, pricing a stochastic notion poses challenges, and a standard approach may not suffice, necessitating the involvement of advanced counterparties to create such derivatives.

We delve into the impact of stochastic factors, such as the swap rate and volatility, on mortgage pricing and prepayment risk. Employing Ito's lemma becomes essential to ascertain whether observed quantities adhere to martingale properties, particularly when the factor being observed is a function of Libor. It is noteworthy that prepayment risk only exists in fixed-rate mortgages, as floating-rate mortgages lack the incentive for prepayment. By understanding the principles behind index amortizing swaps, we can effectively manage prepayment risk and reduce interest rate risk.

Expanding our knowledge, we introduce the concept of an index amortizing swap—an over-the-counter interest rate swap that combines a plain vanilla swap with partial absorption. Typically designed for sophisticated investors due to its large notionals, this exotic derivative is not commonly included in XVA evaluations. Nevertheless, exploring the pricing of mortgages and their connection to prepayment behavior, refinancing incentives, and market observations holds significant value. Deterministic amortization schemes serve as commonly traded instruments, facilitating their processing and integration into the framework of an index amortizing swap, which inherently carries embedded optionality.

Our focus now shifts to the modeling of the notional of an index amortizing swap, which encapsulates the possibility of stochastic amortization via a complex function tied to the type of mortgage. The prepayment rate, in turn, becomes a function dependent on the swap rate, while the refinancing incentive relies on historical estimations derived from various factors such as age, income, wealth, and taxes. Estimating the coefficients involved in these prepayment models requires historical data and detailed analysis. As each bank's portfolio of clients differs, determining these coefficients becomes an extensive study unique to each institution.

In the lecture, the speaker also discusses the estimation of coefficients used in mortgage prepayment models, emphasizing that they are not market-driven but solely based on historical behavior estimations. Moreover, the concept of an index amortizing swap is defined, highlighting its utilization of refinancing incentives and prepayment rates, which are determined based on historical data, to establish mortgage notional values. By evaluating these expectations, one can ascertain the overall value of a mortgage portfolio and make necessary adjustments according to market conditions.

The instructor further elaborates on the complexities involved in the decomposition of notionals, explaining that they cannot be further divided as they depend on the swap rate, which, in turn, is not independent of the Libor swap rate. While assuming independence is possible, it is not recommended without careful study of the correlation's impact. Instead, employing Monte Carlo simulation is advisable. This entire process entails several steps, including pricing a swap rate, estimating the refinancing function, constructing a function based on the mortgage type, and adjusting notionals. The upcoming block of the lecture will focus on simulating the north node, which provides insights into how notionals behave over time based on the type of mortgage. It is crucial to approach this process with meticulous attention to detail and careful consideration of each step involved.

In summary, today's lecture has emphasized the interplay between refinancing incentives, prepayments, and different types of mortgages. We have explored the concept of amortizing swaps, both with and without uncertainty, and introduced the index amortizing swap, which incorporates market-driven prepayment behavior. By linking refinancing incentives, benchmark swap rates, and derivative pricing, we can effectively manage a mortgage portfolio's amortization over time.

Stochastic factors such as the swap rate and volatility play a significant role in pricing and assessing prepayment risk. The use of Ito's lemma becomes essential to evaluate observed quantities' martingale properties accurately. It is also important to differentiate between fixed-rate and floating-rate mortgages when considering prepayment risk.

We have delved into the intricacies of the index amortizing swap, an exotic derivative that combines a plain vanilla swap with partial absorption. Although typically designed for sophisticated investors, it offers valuable insights into mortgage pricing, prepayment behavior, and market observations. Deterministic amortization schemes align well with this type of swap, simplifying its processing and incorporating embedded optionality.

The lecture has emphasized the modeling of the notional of an index amortizing swap, considering stochastic amortization and the intricate function tied to the mortgage type. The estimation of coefficients for prepayment models requires historical data and detailed analysis, varying among banks based on their unique client portfolios.

Furthermore, we have discussed the challenges associated with decomposing notionals and the importance of understanding the correlation between swap rates and Libor rates. Employing Monte Carlo simulation is recommended for pricing derivatives with stochastic notions, offering a comprehensive approach to handle the complexity of the process.

This lecture has shed light on the connection between refinancing incentives, prepayments, and various mortgage types. By incorporating market observations, historical data, and advanced modeling techniques, we can effectively manage prepayment risk and navigate the complexities of pricing mortgage portfolios.

• 00:00:00 In this section of the lecture, the objective is to connect the concepts of refinancing incentives, prepayments, and different types of mortgages. The first step involves looking at a constant payment rate and relating mortgages to an amortizing swap without uncertainty. Then, the concept of an index amortizing swap is introduced, which includes the willingness of clients to prepay or refinance depending on market circumstances. Next, we link refinancing incentives and the benchmark swap rate into derivative pricing, which is applied to a mortgage portfolio that is amortizing over time. The amortization can be a deterministic function, but in the more advanced case, it becomes a stochastic function of the swap rate, which is the ultimate goal of today's lecture. Finally, different mortgage types are defined in terms of their payment schedules and amortization schedules, which allow us to construct a function connecting bullet and annuity mortgages.
  
  
• 00:05:00 In this section of the lecture, the correlation between notional and payment date is discussed for both bullet and annuity mortgages. The concept of a prepayment right and a multiplier function called psi are introduced, and it is shown that the notional can be represented by a generic formulation for both mortgage types. The lecture then shifts to the perspective of a bank owning a portfolio of mortgages and how it can be hedged with an amortizing swap. The importance of accrued periods and prepayment rates is emphasized, and it is noted that stochasticity can complicate the matter. The lecture concludes with the concept of a constant prepayment rate and how it can simplify the calculation of expectations.
  
  
• 00:10:00 In this section of the lecture, the instructor discusses how to establish the repayment rate for mortgages and the difficulty of creating a time-dependent function for prepayments. It is easier to estimate the prepayment rate as a constant using historical data of clients but developing an accurate procedure for a time-dependent function requires richer data. The pricing of amortizing swaps is discussed, and it is explained that there will be a decay of the notion due to the prepayment rates and it may not be linear. The pricing expression for an amortizing swap is demonstrated using the continued expectation and changing measures from summation elements to the ti forward measure. It is important to note that cancellation of terms is not as elegant as in the case of a regular interest rate swap because the elements will be multiplied by different coefficients. Finally, the instructor explains how to incorporate the prepayment rate and swap rate into the lambda function and the sigmoid function, respectively.
  
  
• 00:15:00 In this section of the lecture, the focus is on establishing a clear link between market simulation and the impact of prepayments on a mortgage portfolio. By introducing stochasticity, the speaker points out that clients may behave irrationally, and this element is incorporated by observing certain rates in the market, which are included in the pricing of an amortizing swap. To mitigate the problem of pricing a stochastic notion, the speaker shows how an index amortizing swap can be represented as a function of swaptions and explains that this can be achieved by mapping the optionality into the pricing of European options. However, the problem of pricing with stochasticity is highlighted, and it is shown that the standard approach cannot be used, leading to the need for advanced counterparties to create these kinds of derivatives.
  
  
• 00:20:00 In this section, the speaker discusses how stochastic factors in mortgages such as the swap rate and volatility can affect pricing and prepayment risk. It is important to use Ito's lemma to check whether the observed quantity is a martingale or not, particularly in cases where the factor being observed is a function of a Libor, otherwise, the drift term might be missed. It is also noted that prepayment risk only exists in fixed-rate mortgages as there is no incentive with a floating rate mortgage. The speaker concludes by emphasizing the importance of the principles behind index amortizing swaps to manage prepayment risk and reduce interest rate risk.
  
  
• 00:25:00 In this section, the lecturer discusses the index amortizing swap, which is an over-the-counter interest rate swap that combines a plain vanilla swap with partial absorption. This exotic derivative involves large notionals and requires a financial institution to create, making it typically designed for sophisticated investors. Mortgages are usually not included in XVA evaluations, but the concept of pricing mortgages and connecting prepayment with refinancing incentives and market observations is worth exploring. Deterministic amortization schemes for amortizing swaps are commonly traded instruments, making them easier to process. The notional of a mortgage portfolio is amortizing, making it a natural fit for index amortizing swap, which shares the same embedded optionality.
  
  
• 00:30:00 In this section of the lecture, the concept of an index amortizing swap is introduced, with the notional based on a prepayment rate that is a function of the swap rate and refinancing incentive that is historically determined based on client behavior. The prepayment rate will be itself defined as a function of the swap rate, and the objective is to evaluate a variation of index amortization. The challenge is in the modeling of the notional of the index amortizing swap, which embodies the possibility of stochastic amortization via an involved function of the type of a mortgage, with a historical estimation of refinancing incentive that is a function of various factors.
  
  
• 00:35:00 In this section, the focus is on the main factor of refinancing incentives and how to define an index amortizing swap. The prepayment rate is assumed to be a function only of the refinancing incentive, which is dependent on the observable quantity that is the swap rate. The prepayment rate depends on the client's willingness to prepay, influenced by factors such as age, income, wealth, and taxes. The refinancing incentive is assumed to be either fully rational or more realistically, a sigmoid function with estimated coefficients. The estimation of these coefficients would vary among banks based on their portfolio of clients, making it an extensive study.
  
  
• 00:40:00 In this section, the speaker discusses the coefficients used in mortgage prepayment models and how they are estimated using historical data. He emphasizes that these coefficients are not market-driven and are only based on historical estimations of behavior. Additionally, the speaker explains refinancing incentives and how they impact prepayment rates. He defines the index amortizing swap and how it uses refinancing incentives and prepayment rates based on historical data to determine mortgage notional values. The speaker concludes that by evaluating these expectations, one can determine the overall value of the mortgage portfolio and adjust it according to market conditions.
  
  
• 00:45:00 In this section, the instructor explains that although the expectation can be divided, the notionals cannot be decomposed any further because they depend on the swap rate, which is not independent of the library swap rate. While we could assume independence, this is not recommended unless careful studies have been performed to understand the impact of the correlation. Monte Carlo simulation is recommended instead. This whole process requires several steps, including pricing a swap rate, estimating the refinancing function, constructing a function depending on the mortgage type, and adjusting notionals. In the next block, the instructor will simulate the north node, which will show how the notionals behave over time depending on the type of mortgage. Overall, this is a rather complex process that requires careful consideration and attention to detail.

[MetaQuotes]

Financial Engineering Course: Lecture 8/14, part 4/4, (Mortgages and Prepayments)

Financial Engineering Course: Lecture 8/14, part 4/4, (Mortgages and Prepayments)

In the lecture, the pricing of mortgages takes center stage, and the instructor demonstrates a Python experiment that combines the knowledge of pricing annuities and mortgages, including refinancing incentives, to simulate the stochasticity in notional values. The lecture covers various aspects such as Swaps, pricing models, and the associated risks, including pipeline options, that banks face.

A significant part of the lecture focuses on the behavior of the notional profile for bullet and annuity mortgages and how they can be simulated. It is highlighted that the randomness of simulated paths has a substantial influence on the notional profile. Prepayments are shown to have a significant impact on the notional value, especially for bullet mortgages, while annuity mortgages are comparatively less affected. The lecturer presents Python codes that are extended to make the constant prepayment rate time-dependent, requiring inputs such as the zero coupon bond curve, swap rate, and stochastic paths at each time step.

The speaker delves into the prepayment rate for mortgages and its influence on the outstanding notional and incentive function, which is dependent on market factors like the swap rate. Two mortgage payment profiles, bullet and annuity, are presented, and their indexing for time and prepayment behavior is explained. The lecture introduces two incentive functions, sigmoid and logistic, and emphasizes that the yield curve used for market simulation is fixed at five percent. The Monte Carlo paths generated for interest rate parts serve as the basis for evaluating the incentive functions.

The instructor further discusses the simulation of swap rates, considering the client's perspective and their outstanding mortgage. They define the incentive function based on the client's mortgage and iterate over time steps to create notional schedules. The incentive function is evaluated for the mortgage profile at each time step, and this information is stored in metrics, resulting in a stochastic notional that depends on the incentive function, stochastic interest rates, and the type of mortgage. The lecture includes plotted results, showcasing the paths with and without prepayment options.

The lecturer emphasizes the significance of incentive functions and stochasticity in the context of mortgages and prepayments. Various examples of notional profiles are shown, illustrating their behavior under different scenarios, including rational and irrational behavior using the sigmoid function. The impact of increasing uncertainty and volatility is discussed, emphasizing the role of the incentive function in risk exposure and the need for buying or selling index amortizing swaps or swoptions. The number of steps in the simulation is shown to impact the notion profile, and practical adjustments are highlighted.

An in-depth discussion is held on annuity mortgages in the rational setting, with a graph depicting how prepayment incentives work and how clients determine their maximum prepayment. Limitations such as legal restrictions or penalties may exist, influencing the client's choices. A comparison between bullet mortgages and annuity mortgages reveals that uncertainty strongly depends on the schedule, with a reduction in notional leading to lower uncertainty. Decomposing a complex order portfolio into linear and non-linear parts is explained, with financial engineering offering a possibility for financing without necessarily resorting to index amortizing swaps.

The calculation of payments and the notional value of a mortgage are explained using a simplified case of a two-period mortgage. The notional value is split into two parts: n-up and the difference between n-up and n-low. The latter part handles mortgage prepayment and is only positive if the strike is greater than L-K, similar to a call option's nonlinear effect. The calculation for the second payment involves a summation of two payments, with the first payment being deterministic and the second payment being discounted based on possible outcomes of n-up and n-low.

The lecture redefines the index amortizing swap as a combination of a deterministic amortizing swap and a nonlinear floorlet. The lecturer highlights that purchasing a mortgage can be seen as entering into a long position in a swap, with prepayments reducing the mortgage's notion, which is akin to an option to enter a swap. The composition of an index amortizing swap can be optimized to replicate its risk profile, and advanced exotic derivatives like this can be hedged or replicated using simplified liquid instruments available in the market. The lecture consistently emphasizes the prepayment risks and their impact on the notion of the mortgage portfolio.

Another topic discussed in the video is the additional risk associated with European mortgages or Dutch mortgages, specifically related to the client's ability to choose the fixing rate of the mortgage. The lecture highlights two critical dates: t0, the quotation day, and t1, the time when the client signs a contract with the bank. The risk for the bank is that the client may choose the lower rate, leading to substantial losses. This risk is referred to as pipeline risk, and it is crucial to manage it effectively to protect the bank's profits.

The discussion revolves around pricing pipeline risk for mortgages and prepayments. Hedging pipeline risk poses challenges as it requires the use of swaptions, necessitating continuous recalculation of values and associated profiles. This process is not a one-time occurrence for a single client; it applies to each individual client. Furthermore, risks are accumulated in a portfolio, necessitating bundling of mortgages into a larger portfolio that needs to be aged. The lecture concludes by focusing on pricing pipeline risk, incorporating optionality for clients to choose the rate at the quotation date or settlement date, depending on which rate is smaller.

The lecturer explains the decomposition of the index amortizing swap into a linear product and the remaining swaption part. This decomposition strategy is common in finance when dealing with structures involving optionality. To handle the associated risk, Black's formula is introduced as a straightforward approach, requiring only volatility for the swaption of those configurations. The lecture emphasizes the importance of considering client behavior and incentives, along with pricing in the risk-neutral world when working with mortgages.

In addition, the speaker compares bullet mortgages and annuity mortgages, highlighting that annuity mortgages involve regular repayments over time instead of a lump sum payment at the contract's end. The lecture explores the factors that lead to client prepayments, such as refinancing incentives, and presents numerical experiments on notional simulation based on market and incentive functions of mortgages. The discussion also covers the risks associated with transitioning from an index amortizing swap to stochastic prepayment and options.

Towards the end of the lecture, exercises are provided for students to simulate notionals and price mortgage contracts. The focus shifts to the concept of convexity and its impact on expectations in finance. Students are tasked with determining the side of a function that yields equality when compared to a library with a martingale payment measure, using analytical or numerical methods. The lecture introduces the concept of convexity collection and explores its effects on expectations. Students are also encouraged to modify code to ensure that prepayments occur only a few times during the lifetime of the mortgage contract, further developing their programming skills in Python.

Overall, the lecture provides a comprehensive understanding of mortgage pricing, covering various complexities such as prepayment risks, incentive functions, stochasticity, pipeline risk, and the decomposition of index amortizing swaps. It equips students with the necessary knowledge and practical skills to analyze and simulate mortgage portfolios while considering market factors and client behavior.

• 00:00:00 In this section of the financial engineering lecture, the focus is on the pricing of mortgages. The lecture features a Python experiment that combines the knowledge of pricing annuities and mortgages, including refinancing incentive, to simulate the stochasticity in notional values. Using a short-rate process, the simulation of Swaps and pricing models is demonstrated. The lecture also delves into the risk associated with pipeline options, which is another source of risk for banks. The importance of index amortizing swaps to Swaptions is also discussed in this section, particularly in hedging mortgage portfolios. Overall, the lecture provides a comprehensive view of mortgage pricing and its various complexities.
  
  
• 00:05:00 In this section, the lecturer discusses the behavior of the notional profile for bullet and annuity mortgages along with how those paths can be simulated. It is observed that the randomness of simulated paths highly influences the notional profile. The stochasticity of the mortgages comes into play as prepayments impact the notional value significantly more for the bullet option, and this impact would be much smaller in the case of an annuity type of mortgage. The lecturer also presents Python codes that are extended to make the constant prepayment rate time-dependent. The inputs required are the zero coupon bond curve, swap rate, and stochastic paths at every time t.
  
  
• 00:10:00 In this section of the lecture, the speaker discusses the prepayment rate for mortgages and its impact on the outstanding notional and incentive function, which depends on market factors such as swap rate. The speaker presents two mortgage payment profiles: bullet and annuity, both of which have an additional indexing for time and prepayment behavior. The code used for simulation is introduced with two incentive functions: a sigmoid and a logistic function. The speaker explains that the yield curve used for market simulation is fixed at five percent and that the Monte Carlo paths generated for interest rate parts serve as the basis for evaluating the incentive functions.
  
  
• 00:15:00 In this section of the Financial Engineering Course, the instructor discusses how they simulate the value of a swap rate in order to assume that their client will look at the constant maturity swap which will always be based on the third-year swap rate. They also adjust the function to ensure that it doesn't have any zeros. The client would usually look at the swap corresponding to their outstanding mortgage, which could be for a shorter period and the instructor goes on to define the incentive function based on the client's outstanding mortgage. The instructor then goes on to iterate over not time steps in order to create the notional schedules and evaluate the incentive function for the mortgage profile at every time step. They store this information in their metrics, which creates a stochastic notional depending on incentive function, stochastic interest rates, and the type of mortgage being used. They plot the results, showing the paths with and without prepayment options.
  
  
• 00:20:00 In this section of the lecture, the instructor discusses incentive functions and stochasticity in the context of mortgages and prepayments. They show examples of notional profiles and how they behave under different scenarios, such as rational behavior and irrational behavior using the sigmoid function. The impact of increasing uncertainty and volatility is also discussed, and the importance of the incentive function is highlighted as it affects risk exposure and the need for buying or selling index amortizing swaps or swoptions. The instructor also explains how the number of steps in the simulation impacts the profile of the notion, and adjustments that need to be made for practical applications.
  
  
• 00:25:00 In this section, the lecturer discusses annuity in the rational setting with a graph showing how prepayment incentives work and how clients determine their maximum prepayment, which may be limited by law or penalties. The comparison between bullet mortgage and annuity shows that the uncertainty strongly depends on the schedule, with a reduction in notional leading to lower uncertainty. Decomposing a complicated order portfolio into a linear part and a non-linear part is discussed, with the possibility of financing through financial engineering, indicating that there is no need to necessarily go to the index amortizing swap and buy over the counter.
  
  
• 00:30:00 In this section of the lecture, the speaker discusses the calculation of payments and the notional value of a mortgage in a simplified case of a two-period mortgage. The notional value of the mortgage is split into two parts - n-up and the difference between n-up and n low. The latter part handles prepayment of the mortgage and is only positive if the strike is greater than L-K. This nonlinear effect is similar to that of a call option. The calculation for the second payment requires a summation of two payments, with the first payment being deterministic and the second payment being discounted based on the possible outcomes of n-up and n low.
  
  
• 00:35:00 In this section of the lecture, the speaker redefines the index amortizing swap as a combination of a deterministic amortizing swap and a nonlinear floorlet. They explain that purchasing a mortgage can be seen as entering into a long position in a swap, and that prepayments reduce the notion of the mortgage, which is equivalent to an option to enter a swap. The speaker notes that the composition of an index amortizing swap can be done through optimization to replicate its risk profile and that advanced exotic derivatives like this can be hedged or replicated by simplified liquid instruments available in the market. Overall, the lecture focuses on the prepayment risks and their impact on the notion of the mortgage portfolio.
  
  
• 00:40:00 In this section, the video discusses the additional type of risk associated with mortgages, particularly European mortgages or Dutch mortgages, that is related to the possibility of the client to choose the rate, or the fixing rate, of the mortgage. Here, there are two important dates: t0, which is the quotation day, and t1, which is the time when the client will sign a contract with the bank. The client must choose between the rates at these two dates, and the risk for the bank is that the client may choose the lower rate, which could lead to significant losses for the bank. This is called pipeline risk and is a substantial risk that must be managed properly. If not, the profits of the bank will be consumed.
  
  
• 00:45:00 In this section of the financial engineering lecture, the discussion centers on how to price pipeline risk for mortgages and prepayments. The main challenge in hedging pipeline risk is that it's hedged using swaptions, meaning the process is a continuous one that requires continuous recalculation of values and associated profiles. This process is not something that occurs just once for one client; instead, it occurs per client. Furthermore, the risks are accumulated in a portfolio, meaning mortgages must be bundled together in a big portfolio, and the portfolio must be aged. The discussion concludes with a focus on how to price pipeline risk, incorporating optionality for the client to choose the rate at the quotation date or the settlement date, depending on which rate is smaller.
  
  
• 00:50:00 In this section, the lecturer discusses the decomposition of the index amortizing swap into a linear product and the remaining swaption part. This is a common strategy in finance for structures that involve optionality. The easiest way to handle the associated risk is by using Black's formula, which requires only volatility for the swaption of those configurations. The lecturer explains that it is necessary to consider client behavior and their incentives, in addition to pricing in the risk neutral world, when dealing with mortgages. With this understanding, the lecture on mortgages concludes.
  
  
• 00:55:00 In this section of the lecture on mortgages and prepayments, the speaker discusses the difference between bullet mortgage and annuity mortgage, with annuity mortgage involving regular repayments over time as opposed to a lump sum payment at the end of the contract. The speaker also covers determined factors that lead to client prepayments, including refinancing incentives, as well as numeric experiments on notional simulation depending on a mortgage's market and incentive functions. Additionally, the section covers the pipeline risk linked to the transition from index amortizing swap to the rising of stochastic prepayment and options. Finally, the lecture includes exercises for students related to simulating notional and pricing the contract.
  
  
• 01:00:00 In this section of the Financial Engineering Course, the focus is on the concept of convexity and the impact it has on expectations in finance. The task given is to determine what side of a function yields equality when compared to a library with a martingale payment measure, using either analytical or numerical methods. The concept of convexity collection is introduced and its impact on expectations is explored. The final task involves modifying code to ensure that prepayments only occur a few times in the lifetime of the mortgage contract. These exercises are designed to provide insight into the pricing of mortgages, an introduction to follow-up lectures on convexity, and to further develop programming skills in Python.

[MetaQuotes]

Financial Engineering Course: Lecture 9/14, part 1/2, (Hybrid Models and Stochastic Interest Rates)

Financial Engineering Course: Lecture 9/14, part 1/2, (Hybrid Models and Stochastic Interest Rates)

In the lecture, the focus is on hybrid models and their significance within the portfolios of financial institutions. These models are utilized to simulate future scenarios for various asset classes, including interest rate swaps, foreign exchange contracts, and stocks. The lecturer begins by discussing the importance of employing hybrid models for xVA (valuation adjustments) and VaR (value at risk) calculations. They introduce the Black-Scholes hybrid model, which establishes a connection between stocks and interest rates and can be easily extended to forex pricing. This model serves as a foundation for further discussions on stochastic volatility models.

The lecture is divided into blocks, with the second block centering on stochastic volatility models. The Heston-Hull-White model is discussed, which involves incorporating stochastic volatility into the hybrid model framework. The lecturer provides an overview of the model's dynamics and highlights its application in simulating potential future values of portfolios. The aim is to evaluate risks and assess the value of portfolios that encompass multiple asset classes, such as interest rates, stocks, foreign exchange, commodities, credit, and inflation. The speaker emphasizes the correlation between different asset classes and the need to account for their interdependencies.

The lecture also emphasizes the calibration of multi-dimensional stochastic differential equations (SDEs) to market quotes, particularly for simulating correlated processes from different asset classes. Hybrid models are particularly useful for hybrid payoffs and were initially popular for pricing exotic derivatives. However, due to cost considerations and regulatory restrictions, they have found more efficiency in the xVA and hVAR (hybrid value at risk) framework. The concept of netting effect, which considers the offset values of different asset classes due to their correlations, is highlighted as an important factor in portfolio evaluation and exposure calculation.

While hybrid models offer benefits in evaluating call options and potential future exposures, the lecture acknowledges the challenges associated with these models. The instructor suggests keeping the models as simple as possible to facilitate fast evaluations, as speed is crucial in pricing derivative products. Calibration to market data and considering correlations between different stochastic differential equations are essential. Some approximations may be necessary when dealing with non-zero correlations. The lecture suggests Monte Carlo simulations or partial differential equations (PDEs) as methods to evaluate hybrid models.

The limitations of using PDEs for valuing portfolios with assets from different classes are discussed due to the high dimensionality involved. The lecture advocates for the use of Monte Carlo simulations, which provide a more practical approach. Efficient valuation and calibration are highlighted as crucial for portfolio evaluation, as thousands of evaluations are typically required. The lecturer mentions the extension of the Black-Scholes model with Hull-White for interest rates, emphasizing the role of stochasticity and time dependence in hybrid models. The remaining mechanics of the model remain similar to the standard Black-Scholes model.

The lecturer also delves into the concept of changing the measure from risk-neutral to the T forward measure to leverage the advantages of hybrid models in dealing with stochastic discounting. They discuss the calculation of expectations for European payoff types based on time and underlying variables, using integral forms and the Radon-Nikodym derivatives from measure transformations. The dynamics of stock and discounted stock are explained, emphasizing the need for them to be martingale processes. The concept of forward stock price is introduced to simplify the process.

Further explanations are provided on the derivation of the forward stock price stochastic differential equation (SDE) and the importance of performing log transformations to make it linear in state variables. The lecturer applies Ito's lemma to the forward stock price SDE and addresses the measure transformation required for the process. The resulting driftless SDE features two separate Brownian motions, corresponding to the stock and interest rates, with correlation between them. The factorization of the two Brownian motions is discussed in terms of their distributional properties.

The dynamics of the forward stock are explored in the lecture using a hybrid model with two stochastic differential equations. It is emphasized that the volatility of the forward stock is no longer constant but influenced by the volatility of interest rates. The speaker discusses the calculation of implied volatilities within the context of stochastic interest rates. They suggest using prices to determine implied volatilities and highlight the importance of switching between risk-neutral and T-forward measures to exclude stochastic discounting from payoffs. This section underscores the complexities involved in working with stochastic interest rates in financial engineering.

The lecture introduces a stochastic interest rate model with a one-dimensional process and a time-dependent volatility function reminiscent of the Black-Scholes equation without interest rates. The discounting component is factored outside of the expectation, and the pricing process for European options involves only the constant value of the integral of the time-dependent function. The speaker also presents the cost method for pricing, leveraging the affinity of the Black-Scholes model, and provides insights into how stochastic discounting is handled within this approach.

In the subsequent segment, the speaker discusses the integration process required to obtain the expression for the constant "c" and its relevance in pricing with a stochastic interest rate. They explain that the Black-Scholes model with a stochastic interest rate can represent European option prices as a modified Black-Scholes equation with adjusted volatility. However, it is noted that even with a two-dimensional stochastic differential equation for the interest rate, there is no impact on implied volatility for stock options. The inclusion of interest rates only results in a time-dependent volatility for stocks, without additional stochasticity, leading to a flat volatility across different strike prices. The speaker conducts an experiment to illustrate the influence of different parameters on the term structure of implied volatility.

The lecture further delves into the utilization of forward values in option price implied volatility calibration using actual data. The impact of the speed of mean reversion (lambda) on the implied volatility term structure of stocks is discussed, along with the volatility of interest rates. The speaker highlights that fixing one of these parameters can result in a similar shape of implied volatilities, simplifying the calibration process. Moreover, the effect of correlation on implied volatilities is addressed, where the positivity or negativity of the overall variance of sigma_f impacts the implied volatilities accordingly.

The lecture emphasizes the importance of hybrid models in financial institutions' portfolios, particularly for xVA and VaR calculations. It explores the dynamics and complexities of stochastic volatility models, discusses the calibration of multi-dimensional stochastic differential equations, and highlights the correlations between different asset classes. The lecture also covers the application of measure transformations, the derivation of forward stock price SDEs, and the challenges and considerations related to stochastic interest rates. The calibration of implied volatilities and the impact of various parameters on the term structure of implied volatility are also addressed.

• 00:00:00 In this section of the Financial Engineering Course lecture, the focus is on hybrid models and their importance in financial institutions' portfolios. Hybrid models are used to simulate future scenarios for different asset classes, such as interest rate swaps, foreign exchange contracts, and stocks. The first block of the lecture discusses the necessity of using hybrid models for xva and var calculations and introduces the Black-Scholes hybrid model, which connects stocks and interest rates and can easily be extended to forex pricing. The second block covers stochastic volatility models, with a discussion on the Heston-Hull-White model, and concludes with a summary and homework assignments. The ultimate goal of the course is to be able to simulate xva and hvar.
  
  
• 00:05:00 In this section, the speaker discusses two approaches for simulating the value of a portfolio: the Monte Carlo simulation and the historical simulation. These methods are used to determine the potential future values of portfolios and are important when dealing with multiple asset classes such as interest rate, stocks, foreign exchange, commodities, credit, and inflation. The speaker emphasizes that different asset classes are correlated, and changes in one can have an impact on the other. Thus, it is important to be able to simulate potential future realizations of these asset classes to evaluate the portfolio's risks and value.
  
  
• 00:10:00 In this section, the focus is on simulating correlated processes from different asset classes and calibrating multi-dimensional stochastic differential equations (SDEs) to market quotes. Hybrid models, which involve multiple asset classes, can be used for hybrid payoffs, and they were initially popular for pricing exotic derivatives. However, due to the high costs and regulatory restrictions, it is more efficient to use hybrids in xVA and hVAR framework. Netting effect is considered important in portfolio evaluation and exposure calculation since correlations between different asset classes can have an impact on the portfolio and offset values of each asset.
  
  
• 00:15:00 In this section of the financial engineering lecture, the focus is on hybrid models and how they relate to different asset classes. Hybrid models can be used for evaluating call options and potential future exposures, but market practice is to keep them as simple as possible to facilitate fast evaluations. These models can be difficult to deal with because they require calibration to market data and strong dependence on the availability of fast pricing for European type options. Correlations between different stochastic differential equations need to be considered when using hybrid models, and some approximations may need to be made if the correlation is non-zero. The models can be evaluated using Monte Carlo simulations or PDEs.
  
  
• 00:20:00 In this section, the instructor discusses the limitations of using PDEs for the valuation of portfolios with assets from different classes, due to high dimensionality, and advises using Monte Carlo simulations instead. He emphasizes the importance of speed in pricing derivative products and recommends calibrating the hybrid European instruments due to their liquidity. The instructor mentions that highly efficient valuation and calibration are crucial for the portfolio evaluation, which requires thousands of evaluations. Further, he talks about the extension of the Black-Scholes model with whole white for interest rates and emphasizes that the stochasticity and time dependence play an important role in hybrid models. The rest of the model's mechanics stay the same as the standard Black-Scholes model.
  
  
• 00:25:00 In this section of the lecture, the professor discusses the Black-Scholes case, which is exponential and normally distributed, and introduces hybrid models and stochastic interest rates. They explain that for XVA or VAR calculations, volatility and eta are typically considered time-dependent, and it is important to calibrate interest rates accurately, which will be discussed in a follow-up course. They then explain the dynamics of the model and how performing a log transformation can make it linear in state variables. They conclude by discussing the currency function and how the same technology and methodologies can be used for hybrid models with affinity.
  
  
• 00:30:00 In this section, the instructor explains how characteristic functions can be used to price European options, and how fast Fourier transformation can be used for this. Affine models have currency functions with accuracy functions that are not in closed form. These can be solved using a recup-type of equation with special matrices. For example, the black Scholes model for European options can be solved analytically. However, some hybrid models cannot be solved analytically and require a numerical solution. Stochastic discounting should be handled using the Radon-Nikodym derivative.
  
  
• 00:35:00 In this section, the lecturer explains the concept of changing the measure from the risk-neutral to the T forward measure in order to benefit from the ending of hybrid models when dealing with stochastic discounting. They define the expectation of a European type of payoff based on the time t and the underlying s, which can be exchanged and written in integral form with the random Nicodem derivatives from the measure transformation. They also discuss the dynamics of the stock and the discounted stock, which must be martingale and introduce the definition of a forward stock price to simplify the process.
  
  
• 00:40:00 In this section, the lecturer discusses the derivation of the forward stock price stochastic differential equation (SDE). He defines the forward stock price as this quantity that is driftless and shows how it is equal to the stock value under certain conditions. The lecturer also performs an Ito's lemma on the forward stock price SDE and performs the required measure transformation for the process. Ultimately, he ends up with a driftless SDE but with two separate Brownian motions corresponding to the stock and interest rates, which are correlated. The lecturer then performs a factorization of the two Brownian motions and explains that this can only be done in a distribution sense.
  
  
• 00:45:00 In this section of the lecture, the speaker explains the dynamics of a forward stock using a hybrid model with two stochastic differential equations. They note that the volatility for the forward stock is no longer constant but is impacted by the interest rate volatility. The speaker then discusses the calculation of implied volatilities in the context of stochastic interest rates. They suggest using prices to find implied volatilities and switching between measures from a risk-neutral to a t-forward measure to neglect the stochastic discounting from payoffs. Overall, this section highlights the complexities of working with stochastic interest rates in financial engineering.
  
  
• 00:50:00 In this section, the speaker explains a stochastic interest rate model with a one-dimensional process and a time-dependent volatility function that resembles the Black-Scholes equation with no interest rates. The discounting part is taken outside of the expectation, and the pricing process for European options involves only the constant value of the integral of the time-dependent function. The speaker also introduces the cost method for pricing using the affinity of the Black-Scholes model and the details of how stochastic discounting is handled in the cost method.
  
  
• 00:55:00 In this section of the lecture, the speaker discusses the integration involved in obtaining the expression for constant c and how it can be used in pricing with a stochastic interest rate. The model in particular, Black-Scholes with a stochastic interest rate, can represent European option prices as just a Black-Scholes equation with a proper adjusted volatility. However, the speaker notes that even with a two-dimensional stochastic differential equation for the interest rate, there is no impact on implied volatility for options on the stock. This is because the mapping from the inclusion of interest rates only results in a time dependent volatility for stock, without any extra stochasticity, leading to a flat volatility for every strike. The speaker then presents an experiment on different parameters and their impact on implied volatility on the term structure volatility.
  
  
• 01:00:00 In this section of the lecture, the speaker explains the use of a forward value for option price in the implied volatility calibration process using actual data. The impact of the speed of mean reversion (lambda) on stock implied volatility term structure is discussed, along with the volatility of interest rates. The speaker notes that fixing one of these parameters can lead to obtaining a similar shape of implied volatilities, thus simplifying the calibration process. The impact of correlation on implant utilities is also addressed, with the positivity or negativity of the overall variance of sigma f affecting the implant utilities accordingly.

[MetaQuotes]

Financial Engineering Course: Lecture 9/14, part 2/2, (Hybrid Models and Stochastic Interest Rates)

Financial Engineering Course: Lecture 9/14, part 2/2, (Hybrid Models and Stochastic Interest Rates)

In this lecture, the focus is on advanced hybrid models, particularly stochastic volatility hybrid models like the Scholes-Black, Heston, and Shobel-Zoo full white models. The lecturer demonstrates the impact of different correlation coefficients on the hybrid payoff of a basket consisting of a stock and a bond. Efficient simulation techniques for these hybrid models using Monte Carlo simulation are also discussed.

The lecture delves into the Shobel-Zoo full white model, which extends the Black-Scholes model by introducing a normally distributed process for volatility. However, it has limitations due to its structural model. The lecturer discusses the constraints and limitations of the Schobel-Zhu model compared to the Heston model. The volatility structure of the Schobel-Zhu model is less flexible, resulting in a more limited range of implied volatility skew and smiles compared to the Heston model.

Another model discussed is the Shwartz-Zhao model, which introduces an additional process for sigma squared and extends the set of state variables. However, solving the characteristic function analytically becomes computationally expensive due to the complex set of Riccati equations involved. The lecturer shows the shapes of implied volatilities and skews for different parameters and compares them to the Heston model.

The impact of correlations on the pricing of hybrid payoffs is explored. An experiment is conducted to evaluate the derivative's value for different correlations between stock and interest rate motions. The importance of calibrating correlations to market data before calibrating other model parameters is emphasized. The lecture briefly mentions more advanced discretization methods for hybrid models that will be discussed later.

The lecture focuses on extending the flexibility and calibration of the Heston model with stochastic interest rates. Introducing an extra dimension for interest rates creates challenges with instantaneous covariance metrics. Approximations are used to find the connector function and solve the correlation problem. The importance of maintaining the correlation between stock and interest rates for evaluating the characteristic function and calibrating the model to market data is highlighted.

Approximation methods, such as the delta method and Taylor series expansion, are discussed to simplify the evaluation of variance and characteristic functions. The lecturer provides formulas and techniques for approximating variances and discusses the limitations of these approximations.

The time-dependent function of stock volatility and the mapping of the function over time are explained, along with the Euler discretization method of simulation. The lecturer mentions that later on, they will compare the estimates of the simulation against Monte Carlo brute force and Fourier transformation. The iterative step of the Euler discretization method for approximating the integral is also covered.

The lecture addresses the issue of zero attainability by the volatility paths in the CIR model and provides fixes for Euler discretization. The importance of keeping the variances of hybrid models as independent as possible for better simulation results is emphasized. The process for x(t) is discussed, including its correlation matrix and Cholesky decomposition, highlighting the need to maintain independence from the variance.

The challenges of dealing with non-positive definite matrices in financial engineering are discussed, and the importance of adjusting correlations to satisfy the condition for positive terms under the square root is emphasized. The lecture also covers the generic form of discretization and important steps for modeling stochastic interest rates.

The lecturer introduces the trick and representation for almost exact simulation of the Heston model, applicable to the Heston-Hull-White model as well. The simplification achieved through special cases for the variance process and the evaluation of integrals using Euler discretization and non-central chi-squared distributions is explained. The concept of almost exact simulation is discussed, emphasizing the importance of the variance process in determining accuracy. The lecturer highlights the need to use a whole vector of samples for v life and establishes the order of simulation as first sampling the variance process, followed by the short rate.

The lecturer provides an overview of a simulation performed on the Heston for White model and compares it with other methods. Euler discretization, almost exact simulation, and the COS (Characteristic Function-Based Option Pricing Method) method are compared. The results demonstrate that all methods yield good results. The lecturer shares the code for the simulation, including the configuration for the Heston for White model and the three-dimensional discretization of the hybrid model using the Euler method. Adjustments are made to ensure that the realizations for the variance are capped and floored from zero. The COS method for the Heston for White model is also discussed, and the approximation for the characteristic function is derived and coded.

The focus shifts to comparing different methods for hybrid models and stochastic interest rates. The Monte Carlo simulation results show good accuracy with 10,000 samples, but a larger number of Monte Carlo paths is recommended for improved accuracy. Various hybrid models such as Black-Scholes, Heston, and Schulz-Zucchi models are covered. The lecture also touches upon the application of hybrid models in pricing different asset classes within a single evaluation and their use in xVA calculations. Two exercises are assigned to students, one on advanced models like Heston CIR and the other on developing a Monte Carlo simulation.

In the final part of the lecture, the speaker discusses the development of a Monte Carlo simulation using a white model for stochastic interest rates. It is suggested to derive the corresponding ordinary differential equations to achieve faster Monte Carlo simulations that allow for larger steps. This approach will be compared to the Euler discretization method. The speaker concludes the lecture and expresses anticipation for the students' presence in the next session.

This lecture covers various advanced hybrid models, their limitations, calibration techniques, impact of correlations on pricing, approximation methods, simulation techniques, and comparisons between different methods. The focus is on understanding the intricacies of these models and their practical applications in financial engineering.

• 00:00:00 In this section of the Financial Engineering Course, the focus is on advanced hybrid models, particularly the stochastic volatility hybrid models such as the Scholes-Black, Heston, and Shobel-Zoo full white models. The lecturer shows the impact of different correlation coefficients on the hybrid payoff of a basket consisting of a stock and a bond, and how to perform efficient simulation of these hybrid models using Monte Carlo simulation. The lecture also discusses the Shobel-Zoo full white model, which extends the Black-Scholes model by introducing normally distributed process for volatility but has limitations due to its structural model. The lecture concludes with a summary of the discussed models and homework assignments.
  
  
• 00:05:00 In this section of the lecture, the limitations and constraints of the Schobel-Zhu model are discussed in comparison to the Heston model. The volatility structure of the Schobel-Zhu model is less flexible, which means that it cannot achieve all the shapes of implied volatility skew and smiles that can be achieved using the Heston model. This is due to the fact that the square and product of the volatility parts of the Brownian motions are quadratic, which does not belong directly to the fine processes. However, the problem can be solved by introducing an additional process for dvt, which handles the sigma squared t, making the system of stochastic differential equations extended. This introduces a constraint on flexibility for getting implied volatility smiles and skews, making the range of the smiles and skews much more limited than the Heston model.
  
  
• 00:10:00 In this section, the lecturer discusses the Shwartz-Zhao model, which introduces an additional process for sigma squared and extends the set of state variables of this quadratic class of processes. However, due to the complicated set of Riccati equations involved, the characteristic function cannot be solved analytically and must be calculated numerically, which can be costly. The lecturer also shows the shapes of implied volatilities and skews for different parameters and compares them to the Heston model. The extension of the model does not significantly impact the dynamics of smiles and skews, and some parameters can be fixed during calibration to save time. The lecturer also provides Python codes for implementing the Shwartz-Zhao model and performing numerical integration.
  
  
• 00:15:00 In this section, the speaker discusses an experiment where a set of parameters is chosen and one by one they are changed to observe the impact on implied volatilities. The cost method is evaluated, which is adjusted for stochastic interest rates, and the implied volatility for the Black76 is looked at. The performance of a zero coupon bond is also examined, and a hybrid payoff is discussed, which depends on two asset classes. The speaker emphasizes that although the payoff is hybrid, the nature of it is still European and straightforward, and its variance is mainly driven by the correlation between the performance of the two asset classes.
  
  
• 00:20:00 In this section of the lecture, the speaker discusses the impact of correlations on the pricing of hybrid payoffs. The speaker shows an experiment where the value of the derivative is evaluated for three different correlations between the stock and interest rate motions. The results of this experiment show that depending on the weighting factor, the impact on the price can be significant. The speaker explains that correlations play an important role in the pricing of hybrid payoffs and that it is crucial to calibrate the correlations to the market data before calibrating the remaining model parameters. The speaker also briefly mentions more advanced discretizations for hybrid models that will be discussed later in the lecture.
  
  
• 00:25:00 In this section of the lecture, the focus is on extending the flexibility and calibration of the Heston model with stochastic interest rates. The Heston model is a stochastic volatility model with a variance process defined by a square root process and can be extended with a full-wide short rate model for interest rates. However, introducing an extra dimension creates a problem with instantaneous covariance metrics,
  and an attempt to extend the model using a new variable is not successful. Instead, the approach is to use approximations to find the connector C function to solve the problem of correlation between stock and interest rates. Historically, the correlation between short-term interest rates and the stock market is not strong, but it varies depending on economic circumstances and the market overall.
  
  
• 00:30:00 In this section, the lecture discusses the limitations of hybrid models, which are not truly hybrid when they do not have any correlations. This simplifies the model into one that is essentially a Heston model with non-correlated stochastic interest rates. The lecture emphasizes the importance of keeping this correlation to evaluate the characteristic function and calibrate the model to market data. The lecture also mentions the approximation of the quantity that drives the value of European options, allowing for the introduction of approximations with less importance. The lecture then presents a straightforward approach to mapping the square root of the variance process onto its expectations and a limitation that this expectation can be computationally expensive to calculate at every point in time.
  
  
• 00:35:00 In this section, the lecturer discusses an approach to approximating a function using the delta method, which involves expanding the function around its expectation using the Taylor series. This method is useful when calculating the variance of a function, which can be approximated by the variance of an equivalent expression. The lecturer provides a formula for approximating the variance of the square root of variance in a continuous time stochastic process and demonstrates how this can be simplified further using the closed-form solutions for the expectation and variance of a CIR process. By substituting this approximation into the instantaneous covariance matrix, it is possible to evaluate the characteristic function analytically. The limitations of this approximation are discussed, however, as the term under the square root may sometimes become negative.
  
  
• 00:40:00 In this section of the video, the speaker discusses the time-dependent function of the stock volatility and the mapping of the function over time, along with the Euler discretization method of simulation. The speaker also mentions that later on, they will compare the estimates of the simulation against Monte Carlo brute force and Fourier transformation. The objective is to concentrate on the hybrid models of the Black-Scholes, Shaw, Zou and Heston-Holloway models, and compare their approximations and quantifying the error using those approximations. The video also covers the iterative step of the Euler discretization method for approximating the integral of the entire interval between times t_i and t_i+1.
  
  
• 00:45:00 In this section of the lecture on hybrid models and stochastic interest rates, the lecturer discusses the issue of zero attainability by the volatility paths in the CIR model if the feather condition is not satisfied. This leads to problems with Euler discretization, but there are fixes for this issue which will be covered in the next part of the lecture on almost exact simulation. The lecturer then recommends keeping the variances of hybrid models as independent as possible to simplify the model and obtain better results in simulation. Lastly, the process for x(t) is discussed, with its correlation matrix and Cholesky decomposition. It is advised to keep x as the last process to maintain independence from the variance and to ensure that the square root of one minus the correlation terms is not negative.
  
  
• 00:50:00 In this section, the lecturer discusses the challenges of dealing with a non-positive definite matrix in financial engineering. If a matrix is not positive definite, numerical techniques can be used to make it positive definite, but this means that estimated correlations are not properly estimated. Therefore, it is important to adjust the correlations to satisfy the condition that the term under the squared has to be positive. The lecture goes on to discuss the generic form of discretization and the important steps that need to be taken care of. The approach towards modeling stochastic interest rates is not difficult since it only involves the integral over a normal process, and the difficult part is the one that evolves. The lecture concludes with a discussion on how the calibration of the model is critical, and if there is no fast approximation for pricing, the model will not be used.
  
  
• 00:55:00 In this section, the lecturer discusses a trick and a representation for almost exact simulation of the Heston model, which can also be applied to the Heston-Hull-White model. By choosing special cases for the variance process and using the representation, it is possible to take all of the elements on the left-hand side and obtain a nice expression for a complicated integral in terms of known values. This allows for the evaluation of two integrals corresponding to brownian motion and the approximation of two integrals by evaluating Euler discretization. The remaining terms consist of constant coefficients expressed in model parameters and the sampling of non-central high squared distributions.
  
  
• 01:00:00 In this section of the lecture, the focus is on the concept of almost exact simulation, which concentrates on the variance process as a key process for determining accuracy. The goal is to achieve satisfactory results with a few time steps simulations that are still beneficial in terms of accuracy compared to other discretizations. A sampling from two independent standard normals is used to simplify the representation, and the process of short rate is followed by Euler discretization. The need to use a whole vector of samples for v life is emphasized, and the order of simulation is established to first sample the variance process, followed by the short rate.
  
  
• 01:05:00 In this section, the lecturer gives an overview of a simulation performed on the Heston for White model and compares it with other methods. The simulation involves comparing Euler discretization, almost exact simulation, and the COS (Characteristic Function-Based Option Pricing Method) method. The results show that all methods produce good results. The lecturer then provides the code for the simulation, including the configuration for the Heston for White model and the three-dimensional discretization of the hybrid model using the Euler method, with adjustments to make sure the realizations for the variance are capped and floored from zero. Lastly, the COS method for the Heston for White model is discussed, and the approximation for the characteristic function is derived and coded.
  
  
• 01:10:00 In this section of the lecture, the focus is on comparing different methods for hybrid models and stochastic interest rates. The results from a Monte Carlo simulation show good accuracy, with 10,000 samples used, although a larger number of Monte Carlo paths is recommended for better accuracy. The lecture covers various hybrid models, including Black-Scholes, Heston, and Schulz-Zucchi models. The lecture also discusses the use of hybrid models for pricing different asset classes in one pay evaluation and the application of the models in xVA calculations. Two exercises are given for students, one on advanced models like Heston cir and the other on developing a Monte Carlo simulation.
  
  
• 01:15:00 In this section, the speaker discusses the development of a Monte Carlo simulation using a white model for stochastic interest rates. He recommends deriving the corresponding ordinary differential equations to achieve faster Monte Carlo simulations that allow for larger steps. This will be compared to the Euler discretization. The speaker concludes the lecture and looks forward to seeing his students next week.

[MetaQuotes]

Financial Engineering Course: Lecture 10/14, part 1/3, (Foreign Exchange (FX) and Inflation)

Financial Engineering Course: Lecture 10/14, part 1/3, (Foreign Exchange (FX) and Inflation)

The instructor delves into the realm of financial engineering, focusing on two crucial asset classes: foreign exchange and inflation. He provides a comprehensive understanding of the modeling process for each asset class and demonstrates how options can be priced accordingly. Additionally, the instructor delves into the inclusion of stochastic volatility and stochastic interest rates in the evaluation of these assets.

The lecture begins by exploring the history of foreign exchange, highlighting its significant growth in recent years attributed to globalization. The instructor discusses the impact of the gold standard, which limited private ownership of currency, and how the Bretton Woods system established the current framework of multiple currencies backed by gold. The lecture concludes with the assignment of homework tasks to reinforce the covered material.

Furthermore, the video delves into the historical aspect of currencies and the role of gold within them. Specifically, it outlines the transition that occurred in 1971 when the United States ceased using gold as the standard for determining the value of its currency. This pivotal shift led to the current worldwide system where currencies are exchanged based on their relative strength rather than being backed by gold.

Risk assessment is another significant topic addressed in the video. It explores the various risks investors may encounter when engaging with bonds, foreign exchange, and inflation. The lecture elucidates the intricate relationships and complexities associated with these risk factors. The determination of foreign exchange rates through supply and demand dynamics is also thoroughly discussed. The video emphasizes how central banks manipulate these rates through the utilization of reserves. Moreover, it dispels the notion that gold is an investment and clarifies that owning gold is not a necessity for investors.

Financial engineering concepts take the spotlight, with the video showcasing the replication of a forward FX contract. An example is provided to illustrate the initiation of a forward FX contract and how the exchange rate between the original currencies and the new currency is determined. The application of financial engineering in pricing forward foreign exchange contracts is also examined. The video demonstrates the calculation of the forward rate, which is derived by multiplying the spot rate by the effect rate.

The lecture further delves into the concept of financial engineering, exploring its application in pricing assets and liabilities. The equivalence of two pricing approaches is demonstrated, enabling the calculation of a forward rate using these approaches.

Managing exposure to foreign currencies and inflation through derivatives is a significant aspect of financial engineering. The lecture highlights the determination of a forward rate, which depends on the exchange rate at which a country will trade its currency for another. Additionally, the basis spread adjusts for the difference in demand and supply of various currencies.

The intricacies of foreign exchange (FX) and inflation are explained, with the lecture emphasizing that different rules apply depending on the specific type of FX swap contract being executed.

Valuing a foreign exchange contract while considering the effects of foreign exchange rates and discounting is thoroughly discussed. The instructor demonstrates the calculation process, including the utilization of a forward FX contract for the same purpose.

Finally, the lecture explores how foreign exchange (FX) and inflation impact swaps. It delves into the calculation of the swap's value in domestic and foreign currencies while accounting for exchange rate fluctuations.

• 00:00:00 In this lecture, the instructor discusses the two important asset classes financial engineering, foreign exchange and inflation. He explains the modeling process for each, and demonstrates how to price options for both. Finally, he discusses how to include stochastic volatility and stochastic interest rates in the evaluations.
  
  
• 00:05:00 In this lecture, Professor covers the history of foreign exchange, discussing how it has grown significantly in recent years due to globalization. He goes on to discuss how the gold standard limited private ownership of currency, and how Bretton Woods established the current system of multiple currencies being backed by gold. He wraps up the lecture by discussing some homework assignments.
  
  
• 00:10:00 This video discusses the history of currencies and the role of gold in them. It explains how, beginning in 1971, the United States stopped using gold as a standard for measuring the value of its currency. This has led to a worldwide system in which currencies are exchanged based on their relative strength, rather than being backed by gold.
  
  
• 00:15:00 The video discusses the various risks that an investor may face when investing in bonds, foreign exchange, and inflation. It also discusses the complexities of these relationships.
  
  
• 00:20:00 The video discusses how foreign exchange rates are determined by supply and demand and how central banks use reserves to manipulate these rates. The lecture also discusses how gold is not an investment and how it is not necessary for investors to have gold in their portfolio.
  
  
• 00:25:00 The video discusses financial engineering concepts and demonstrates how a forward FX contract can be replicated. The video provides an example of how the forward FX contract would be initiated and how the exchange rate between the original currencies and the new currency would be determined.
  
  
• 00:30:00 The video discusses how financial engineering can be used to price forward foreign exchange contracts. The example shown demonstrates how to calculate the forward rate, which is equal to the spot rate multiplied by the effect rate.
  
  
• 00:35:00 In this lecture, Professor discusses the concept of financial engineering and how it can be used to price assets and liabilities. He demonstrates how two approaches to pricing these assets and liabilities are equivalent, and shows how these approaches can be used to calculate a forward rate.
  
  
• 00:40:00 Financial engineering involves the use of derivatives to manage exposure to foreign currencies and inflation. A forward rate is determined by the rate at which a country will exchange its currency for another currency, while the basis spread adjusts for the difference in demand and supply for different currencies.
  
  
• 00:45:00 This video explains how foreign exchange (FX) and inflation work. Different rules apply depending on the type of FX swap contract being executed.
  
  
• 00:50:00 In this lecture, Professor discusses how to valuation a foreign exchange contract, including taking into account the effects of foreign exchange rates and discounting. He also shows how to do the same calculation using a forward FX contract.
  
  
• 00:55:00 In this lecture, the author explains how foreign exchange (FX) and inflation affect swaps. He explains how to calculate the value of a swap in domestic and foreign currencies, and how to account for exchange rate changes.

[MetaQuotes]

Financial Engineering Course: Lecture 10/14, part 2/3, (Foreign Exchange (FX) and Inflation)

Financial Engineering Course: Lecture 10/14, part 2/3, (Foreign Exchange (FX) and Inflation)

The instructor's focus is on pricing options related to foreign exchange or off options, utilizing the Black-Scholes framework as a starting point. The lecture extensively covers the derivation of differential equations for domestic risk-neutral measures and their impact on the dynamics of stochastic differential equations. To illustrate these concepts, Python experiments are conducted, comparing the Western Corridor model in two currencies using both Monte Carlo simulation and Fourier transformation with the COS method. The section also delves into the dynamics of the foreign exchange process and the establishment of martingales as market quantities and their corresponding value.

Moving forward, the lecture addresses the dynamics of foreign exchange (FX) and inflation. It begins by defining a generic effects process and then focuses on pricing, transitioning to the risk-neutral domestic measure for FX. The lecture explains the utilization of the high function to manage foreign money savings accounts, which are subsequently exchanged in domestic amounts and discounted using the domestic money savings account. By applying the Ethos lemma and simplifying the equation, the lecture concludes that the dynamics of FX and inflation do not represent a marked yield under this measure. However, valuable insights are provided that can be applied effectively.

A significant topic covered by the speaker is the process of measure transformation from E to Q, creating a new process used for option pricing evaluation. The derived process represents the FX process under the risk-neutral measure of domestic risk information, ensuring that when foreign money savings accounts are exchanged for local currency, the quantity is marked. This enables the pricing of European options using Black-Scholes equations, with the only differences being the discounting of options under the risk-neutral measure and the inclusion of the drift term rd-rf. The FX market model is an extension of a standard log-normal model, and European options can be priced using the same methodology of changing measures and identifying martingales.

Expanding on the foreign exchange market, the lecture focuses on augmenting the Black-Scholes model with stochastic volatility and stochastic interest rates. While previous lectures discussed deterministic interest rates, introducing stochasticity becomes essential for XVA calculations and VAR simulations. Additionally, the correlation between different stochastic factors is emphasized, highlighting the potential pitfalls of relying solely on deterministic interest rates. The foreign exchange market's complexity arises from its non-tradable nature and the necessity to exchange assets across different columns to enforce martingale conditions. Furthermore, the effects world introduces an additional term in the stochastic differential equations that requires careful analysis and calibration to the market.

The speaker delves into the calibration of various asset classes, including stocks from small companies and interest rate products, one of the largest asset classes globally. It is noted that attempting to calibrate all parameters simultaneously can be challenging, leading to the recommendation of calibrating individual parameters and incorporating them into the stock dynamics. The lecture also explores the evaluation of European options through Fourier transformation, discussing the approximations employed. Furthermore, the importance of defining measures for interest rates in the foreign market and transforming them into the risk-neutral measure under domestic markets is addressed.

Affine models for zero coupon bonds and binary savings accounts are discussed, with a focus on their dynamics and the calibration of options, caps, and tablets. The use of stochastic differential equations to derive models for effects and leverage calibrated parameters for each individual process is proposed. The lecture delves into the complexities of pricing derivatives with intricate drift terms, emphasizing the accurate handling of this additional term. The primary driver of option pricing is the volatility corresponding to the FX process, with higher-order returns influencing interest rate volatility.

Volatility's significance in foreign exchange is emphasized by the speaker, particularly due to the non-linear nature of the process, including the presence of the square root of a term. The challenges associated with drift handling and the necessity of employing a stochastic interest rate are discussed. Two stochastic differential equations corresponding to the foreign zero coupon and couple with domestic measures are explained, emphasizing the requirement for them to be martingales under specific conditions. The importance of correlation between foreign markets and FX is highlighted, emphasizing that it cannot be assumed to be zero. Finally, the speaker derives the pricing equation for European options for FX, incorporating all the discussed concepts.

The professor introduces the payoff of a European call option with a maximum value of yt minus k, involving a discounting process with the domestic money savings account. To address stochastic interest rates, the first step is to transition from a measure flow to the t-forward measure associated with the bond maturity capital t. As the dynamics of FX exhibit no drift, the professor only needs to incorporate volatilities into the diameter. Applying the Ethos lemma to this quantity, the professor includes three different elements in the dynamics, including the previously discussed zero components and the dynamics of yt in the FX process.

Moving forward, the speaker delves into the dynamics of the FX forward and variance processes in the short-rate model, where the volatility parameter remains constant. However, the volatility contribution from FX is time-dependent and not constant, resulting in a reduction of dimensionality from four to two. The speaker also mentions the additional quantum correction that arises when switching measures from risk-neutral to domestic t-forward measure, which poses challenges when using small time steps. The section concludes with a discussion on numerical experiments and approximations employed for the characteristic function.

The speaker emphasizes the importance of carefully selecting model parameters as they significantly impact pricing and hedging decisions. The Heston model is discussed, and the characteristic function is defined, enabling the pricing and calculation of FX impact volatilities. A comparison is made between Monte Carlo simulation and Fourier approximation, involving 20 different Monte Carlo runs with 1000 paths per run. The results demonstrate alignment between Monte Carlo option pricing and the Fourier approximation, with satisfactory differences for calibration to implied volatility market data. However, it is noted that the quality of results can vary depending on the specified model parameters.

The professor proceeds to discuss the Python code for the COS method and analyzes its accuracy. The code encompasses specifications for 500 expansion terms and incorporates different model parameters and configurations for domestic and foreign markets, as well as comprehensive metric collections. The professor emphasizes the significance of random samples in Monte Carlo simulations and suggests changing the random seeds to improve results. A Monte Carlo simulation with multiple runs is performed, evaluating option prices using the payoff evaluation method. The average of all runs is calculated, along with the expectation and standard deviation, allowing for error monitoring arising from changes in the random seed.

Lastly, the lecturer highlights the importance of accurate model parameter selection, as it greatly influences pricing and hedging decisions. The characteristic function for the Heston model is defined, enabling the pricing and calculation of FX impact volatilities. A comparison between Monte Carlo simulation and Fourier approximation is conducted, involving 20 Monte Carlo runs with 1000 paths per run. The results demonstrate satisfactory alignment between Monte Carlo option pricing and the Fourier approximation, providing calibration to implied volatility market data. However, the speaker emphasizes the influence of specified model parameters on result quality.

• 00:00:00 In this section of the financial engineering course, the focus is on pricing options on foreign exchange or off options, starting with a Black-Scholes framework. The lecture also covers the importance of deriving differential equations for domestic risk-neutral measures and the impact on the dynamics of stochastic differential equations. The lecture includes Python experiments where the Western Corridor model in two currencies is compared using Monte Carlo and Fourier transformation using the COS method. The section also covers the dynamics of the foreign exchange process and how to establish martingales as quantities in the market and their value.
  
  
• 00:05:00 In this section of the lecture, the instructor discusses the dynamics of foreign exchange (FX) and inflation. Starting with defining a generic effects process, the lecture focuses on pricing and moving to the risk-neutral domestic measure for FX. The lecture explains that the high function is used to manage foreign money savings accounts, which are then exchanged in domestic amounts and discounted with the domestic money savings account. Applying the Ethos lemma and simplifying the equation, the lecture concludes that the dynamics of FX and inflation are not a marked yield under this measure, but it provides insights that can be enforced.
  
  
• 00:10:00 In this section of the lecture, the speaker discusses the process of switching measures from E to Q using a measure transformation to make a new process, which is used to evaluate option pricing. The process derived is the FX process under the risk-neutral measure of domestic risk information, which guarantees that if one exchanges foreign money savings accounts to local currency, the quantity will be marked. This allows for the pricing of European options using Black-Scholes equations, with the only differences being the discounting of the options under the risk-neutral measure and the addition of the drift term rd-rf. The FX market model becomes an extension of a standard log normal, and European options can be priced using the same machinery of changing measures and finding martingales.
  
  
• 00:15:00 In this section, the focus is on extending the foreign exchange market driven by the Black-Scholes model with inclusion of stochastic volatility, as well as stochastic interest rates. While the previous lectures covered deterministic interest rates, it is necessary to make them stochastic for XVA calculations and VAR simulations. Moreover, the correlation between different stochastic factors is crucial, and the reliance on deterministic interest rates can be a trap. The foreign exchange market is more complicated since it is not tradable and requires the exchange of assets from different columns to impose martingale conditions. Additionally, the effects world has an additional term in the stochastic differential equations that is non-trivial to solve but can be operated and handled with proper analysis and calibration to the market.
  
  
• 00:20:00 In this section, the speaker discusses the calibration of different asset classes, such as a stock from a small company, to interest rate products, which is one of the world's largest asset classes. They explain how we can't qualify the parameters together and that calibration can become very difficult when trying to calibrate all the parameters at the same time. The speaker discusses the need to calibrate individually and then include those parameters in the stock dynamics. They also discuss the evaluation of European type of options through the Fourier transformation and how this framework is approximated. Finally, the speaker touches on the need to define the measures for interest rate in the foreign market and how to change them to the risk-neutral measure under domestic markets.
  
  
• 00:25:00 In this section, the lecturer discusses the affine models used for zero coupon bonds and binary savings accounts, with an emphasis on the dynamics and processes used for calibration of options, caps, and tablets. The lecturer also proposes the use of stochastic differential equations to derive models for effects and to benefit from the calibrated parameters for each individual process. The lecture further explores the FX model and the difficulty of pricing derivatives with complicated drift terms, highlighting the significance of handling this additional term accurately. The main driver of option pricing is volatility corresponding to the FX process, with higher order returns driving volatility for interest rates.
  
  
• 00:30:00 In this section, the speaker talks about the importance of volatility in foreign exchange and how the process is non-linear, especially due to the presence of the square root of a term. They also discuss the difficulty in handling the drift and how it must be corrected using a stochastic interest rate. They explain how the two stochastic differential equations correspond to the foreign zero coupon and couple with domestic measures and how they must be martingales in specific conditions. They discuss the importance of the correlation between foreign markets and FX and why it cannot be set to zero. Finally, the speaker goes on to derive the pricing equation for European options for FX.
  
  
• 00:35:00 In this section, the professor defines the payoff of a European call option with maximum y t minus k, which involves a discounting process with the domestic money savings account. To deal with stochastic interest rates, the first step is to move from a measure flow to the t forward measure associated with the bond maturity capital t. The dynamics of fx has no drift, so the professor only needs to include the volatilities in the diameter. The professor applies Ethos lemma to this quantity, which has three different elements to include in the dynamics, including the zero components from before and the dynamics of y in the fx process.
  
  
• 00:40:00 In this section of the lecture, the speaker discusses the dynamics of the fx forward and variance process in the short rate model, which has a constant volatility parameter. However, the contribution of the volatility of fx is time-dependent and not constant, leading to a reduction in dimensionality from four to two. The speaker also mentions the additional quantum correction that occurs when switching measures from risk-neutral to domestic t-forward measure, which is not ideal and cannot be handled with small time steps. The section concludes with a discussion of numerical experiments and approximations used for the characteristic function.
  
  
• 00:45:00 In this section, the speaker discusses the configuration choices made for the experiment, including choosing the yield curve for zero coupon bonds in domestic and foreign markets. They also talk about the importance of choosing the right parameters for volatilities and mean reversion speed for short rate models. The speaker emphasizes that choosing the right parameters is crucial to the simulation's accuracy, and if they are too big, the results may not make sense. Additionally, the speaker discusses the configuration options for the FX part of the simulation, including choosing the right correlation matrix, which is usually based on historical data except for correlations between effects and volatility, which is calibrated through the model. Finally, the speaker talks about the importance of evaluating stripes for FX and other markets and how varying the parameters can help find the most optimal options.
  
  
• 00:50:00 In this section, the lecturer discusses how to handle strikes in the options market. He explains that it is more convenient to describe strikes not just as a percentage of the spot value, but also in a formulaic way. A popular method for handling strikes is to use a log-fone formula that evaluates the strikes based on a stochastic process for effects. The lecturer demonstrates how industry prefers to handle those strikes as a function of time. He also explains that different conventions exist for how implied volatilities are reported in the industry and discusses the calibration of the model to the volatility service. Finally, he demonstrates a figure describing the movement of the distribution along the forward curve of effects.
  
  
• 00:55:00 In this section of the lecture, the professor discusses the Python code for the cos method and analyzes its accuracy. The code includes specifications for 500 expansion terms and uses different model parameters and configurations for domestic and foreign markets, as well as full metrics of collections. The professor also discusses the importance of random samples in Monte Carlo simulation and recommends changing the random seeds for better results. They perform a Monte Carlo simulation with multiple runs and evaluate the option prices using the payoff evaluation method. They also take the average of all runs and calculate the expectation and standard deviation to monitor the error that comes from changes in the random seed.
  
  
• 01:00:00 In this section, the speaker emphasizes the importance of carefully choosing model parameters, as they can significantly affect pricing and hedging decisions. The Heston model is discussed, and the characteristic function is defined, which allows for pricing and calculating FX impact volatilities. The speaker then compares the Monte Carlo simulation to the Fourier approximation, with 20 different Monte Carlo runs and 1000 paths per run. Results show that the Monte Carlo option pricing is aligned with the Fourier approximation, and the difference is satisfactory for calibration to implied volatility market data. Quality, however, can vary depending on the specified model parameters.

[MetaQuotes]

Financial Engineering Course: Lecture 10/14, part 3/3, (Foreign Exchange (FX) and Inflation)

Financial Engineering Course: Lecture 10/14, part 3/3, (Foreign Exchange (FX) and Inflation)

The lecturer delves into the topic of inflation, tracing its development over the past century. Initially, inflation was associated with monetary policy and the increase in money supply, but its definition has now shifted to encompass changes in price levels. The importance of inflation derivatives for hedging inflation risks, particularly for banks and pension funds, is highlighted. The pricing of these derivatives is closely linked to foreign exchange pricing, adding to their significance in the financial market. The section provides a concise overview of inflation and its relevance in the financial sector.

Moving forward, the lecturer examines the variations in inflation measures used across countries, with a specific focus on the European Harmonized Consumer Price Index (HICP) and the US Consumer Price Index (CPI). Comparing these measures is not always straightforward, as they may not accurately reflect actual price increases. However, they are still used to price derivatives contracts, with derivatives often linked to CPI index values. To illustrate historical inflation trends in the US, the lecturer presents a graph showcasing the fluctuation of CPI figures over time, using a reference date from 2000-2015.

In the subsequent part of the lecture, the instructor explores the non-linear nature of inflation and its evolution over different periods. A graph is presented, highlighting the impact of market crashes on deflation and the potential deflationary effects of globalization. The lecturer also delves into the concepts of sticky and transitory inflation, explaining their implications for prices and the economy. It is emphasized that due to its dynamic nature, inflation cannot be easily described by simple economic models. Various factors, such as demographics and the global economy, influence inflation, making it a complex phenomenon to analyze. Furthermore, changes in the composition of price measurement baskets over time can affect inflation figures significantly.

Continuing the discussion, the lecturer explains that comparing inflation over time is challenging due to the changing definitions associated with different goods and services. The lecture also sheds light on the composition of elements used in calculating the CPI index and the techniques employed to adjust and smooth out results. These techniques include the hedonic effect, which factors in the utility of a product when considering price increases, and substitution, where consumers switch to cheaper goods to mitigate rising prices.

Housing's impact on inflation and inflation measures is then examined. In the US, housing prices are not included in CPI or inflation measures because housing is seen as a capital investment. However, CPI measures do incorporate a "shelter impact," which estimates the cost of living in a rented house. The lecture emphasizes that the basket of products used for inflation calculations changes over time, leading to potentially unreliable inflation figures. While the CPI index is considered a lagging indicator of inflation, it serves as an underlying observable quantity for derivative pricing. Pension funds, insurance companies, and banks dealing with inflation-dependent derivatives are the primary users of inflation products, as inflation can significantly affect their payments. The break-even inflation rate is determined by the spread between legal and inflation-linked bonds.

Shifting the focus, the lecturer explains the distinction between nominal and real instruments in relation to inflation. Nominal instruments do not account for inflation and are considered nominal prices that do not protect against inflationary forces. Inflation swaps and inflation forwards are products that expose individuals to the difference between the real and nominal economies. The basic contract discussed is an inflation swap, where the performance is based on the CPI index at a given time, exchanging the floating and fixed parts. The lecturer highlights the importance of considering delays when modeling inflation, as inflation data is released with a lag and is based on past months.

The lecture goes on to discuss how commodities can provide a better representation of inflation compared to inflation figures, as commodity prices are immediately observable in daily markets, while inflation figures have a few months of lag. Forward inflation is defined as inflation observed at a particular time, and if forward inflation is available in the market and the yield curve for nominal zero coupon bonds is known, the real zero coupon bond can be calculated. The lecture also covers the pricing of inflation swaps using similar methodologies as foreign exchange and interest rate swaps. Additionally, the lecturer touches on pricing options using inflation processes and the possibility of defining and extending hybrid models for inflation with stochastic interest rates.

Expanding on the similarities and differences between foreign exchange and inflation, the professor explains the relationship between nominal and real rates. The transfer of funds between nominal and real economies creates a connection term that influences the risk-neutral measure. The lecture also delves into derivative options such as call options and explores year-on-year inflation, which measures the performance of inflation over a specific period of time. Furthermore, the professor examines the distribution of inflation in the log-normal case and how this ratio is affected in the Black-Scholes framework. The lecture encompasses various processes related to foreign exchange and inflation, including risk-neutral measures, derivative options, and inflation performance over time.

The professor further elaborates on the connection between inflation and foreign exchange in pricing inflation products and cross-currency swaps. The derivation of the characteristic function for the distribution of the log of forward inflation rates is explained using Fourier transformations and pricing techniques. The importance of pricing options is emphasized as it aids in calibrating volatility parameters to market instruments, enabling the evaluation of future portfolio exposures and the application of risk measures such as VAR calculations.

Shifting the focus to the foreign exchange (FX) market and inflation, the lecture covers the evaluation of FX rates, determining the fair value of FX contracts, and deriving the fair value of cross-currency. Pricing FX options is discussed, extending the pricing methodology to incorporate stochastic volatility and interest rates. Additionally, the lecture explores the definition of inflation forwards and the pricing of inflation swaps. The lecture concludes by presenting three exercises for students to apply their knowledge, including deriving the question function for year-on-year inflation within the Black-Scholes framework and using simulations to find the expectations of a function.

Lastly, the instructor presents an exercise centered around the Stochastic Differential Equation for Foreign Exchange. The objective of the exercise is to simplify the equation, factorize the Brownian motions to obtain Sigma hat, and subsequently determine the Sigma and Sigma Sigma hat terms. The instructor concludes the lecture by bidding farewell to the students and expressing hopes that they have enjoyed the course and the exercises.

• 00:00:00 In this section of the lecture, the speaker discusses inflation and its development over the last 100 years. The definition of inflation has changed over time; it was initially related to monetary policy and the increase of money supply, whereas now it is linked to price levels. The speaker talks about inflation derivatives and their importance for hedging inflation risks, especially for banks and pension funds. The pricing of inflation derivatives is closely aligned with foreign exchange pricing. Overall, the section provides a brief overview of inflation and its significance in the financial market.
  
  
• 00:05:00 In this section, the lecturer discusses the differences in inflation measures used across countries, focusing on the European Harmonized Consumer Price Index (HICP) and the US Consumer Price Index (CPI). These measures are not always straightforward to compare, meaning that official inflation figures may not necessarily provide an accurate picture of price increases. Nevertheless, they can still be used to price derivatives contracts, with derivatives anchored to the CPI index values. The lecturer then presents a graph of historical inflation development in the US, showing how CPI figures have fluctuated over time against a reference date from 2000-2015.
  
  
• 00:10:00 In this section, the lecturer discusses inflation and how it is not linear in growth, but rather changes over time. He presents a graph that shows the inflation figures in different periods, highlighting the deflationary impact that a market crash can have on inflation and how globalization can lead to deflation as well. He also explains the difference between sticky and transitory inflation and how it can affect prices and the economy. The lecturer notes that inflation is complex and difficult to describe by simple economic models due to its changing nature and that it is influenced by various factors such as demographics and the global economy. He also cautions that the baskets used to measure prices could be completely different than ones used in the past, which could affect inflation figures as well.
  
  
• 00:15:00 In this section, the instructor discusses how inflation is difficult to compare over time due to the changing definition of inflation depending on goods and services, which makes the policy of keeping inflation at a certain percentage a little bit of a floating policy. The lecture also explains the composition of elements used in the CPI index calculation and how inflation is measured, which involves techniques to adjust and smooth out results, such as the hedonic effect and substitution. The hedonic effect subtracts the utility of a product from the price increase, while substitution involves consumers switching to cheaper goods to avoid higher prices.
  
  
• 00:20:00 In this section, the speaker discusses the impact of housing on inflation and measures of inflation. In the US, the price of housing is not included in CPI or inflation measures because housing is seen as a capital investment. A "shelter impact" is, however, included in CPI measures, which estimates the cost of living in a house if it was rented. The basket of products used for inflation calculations varies over time, leading to unreliable inflation numbers. While many consider the CPI index to be a lagging indicator of inflation, it is used as an underlying observable quantity for derivative pricing. The modeling of inflation is very different from that of foreign exchange, but both have a strong correlation. Pension funds, insurance companies, and banks dealing with inflation-dependent derivatives are the main clients of inflation products because inflation can significantly impact their payments. The break-even inflation rate is determined by the spread between legal and inflation-linked bonds.
  
  
• 00:25:00 In this section, the lecturer explains the difference between nominal and real instruments and how they relate to inflation. Nominal instruments do not compensate for inflation and therefore, prices in the market are considered nominal and do not protect against inflationary forces. Inflation swaps and inflation forwards are products that will expose someone to the difference between the real and nominal economies. The basic contract for this is an inflation swap where the performance is exchanged based on the CPI index at a given time and we exchange the floating part and fixed part. The lecturer cautions that it's important to consider the delays when modeling inflation because inflation is always delayed in release and always coming from a past month.
  
  
• 00:30:00 In this section, the video explains that looking at commodities is considered a better representative of inflation because commodity prices can be seen immediately every day in the markets while inflation figures have a few months of delays. Forward inflation is defined as inflation seen at a particular time, and if forward inflation is available in the market and nominal zero coupon bond from a yield curve is known, then the real zero coupon bond can be calculated. The video also discusses how inflation swaps can be priced using the same methodology as foreign exchange and interest rate swaps. Finally, the lecture touches on pricing options using inflation processes and how hybrid models for inflation can be defined and extended with stochastic interest rates.
  
  
• 00:35:00 In this section, the professor discusses the similarities and differences between foreign exchange and inflation regarding nominal and real rates. The professor explains how the transfer of funds between economies from a nominal to a real economy creates a connection term that affects the risk-neutral measure. The lecture also covers derivative options such as call options, and year on year inflation where the performance of inflation is over a period of time. Additionally, the professor looks at finding the distribution of inflation in the log-normal case and how this ratio is affected in the black scholes case. Overall, the lecture discusses different processes regarding foreign exchange and inflation, including risk-neutral measures, derivative options, and inflation over a period of time.
  
  
• 00:40:00 In this section of the financial engineering course, the professor discusses the relationship between inflation and foreign exchange in pricing inflation products and cross-currency swaps. He explains how to derive the characteristic function for the distribution of the log of forward inflation rates with the help of the pricing machinery using Fourier transformations. The pricing of options is crucial in this process as it helps in calibrating volatility parameters to market instruments, ultimately leading to the evaluation of future exposures for portfolios and the application of measures like VAR calculations.
  
  
• 00:45:00 In this section of the video, the focus is on the foreign exchange (FX) market and inflation. The lecturer covers how to evaluate FX rates, find the fair value of FX contracts, and derive the fair value of cross-currency. They also discuss pricing FX options, extending the pricing with stochastic volatility and interest rates, defining inflation forwards, and pricing inflation swaps. The lecture concludes with three exercises for students to complete, including deriving the question function for year-on-year inflation in the Black-Scholes White case and finding the expectations of a function using simulations.
  
  
• 00:50:00 In this section of the video, the instructor presents an exercise on Stochastic Differential Equation for Foreign Exchange. The objective of the exercise is to simplify the equation and factorize the Brownian motions to obtain Sigma hat, then find the Sigma and Sigma Sigma hat terms. The instructor concludes the lecture by saying goodbye and hoping the students have enjoyed the course and the exercises.