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Computational Finance: Lecture 9/14 (Monte Carlo Simulation)
Computational Finance: Lecture 9/14 (Monte Carlo Simulation)
The lecture covers several topics related to Monte Carlo simulation and integration in computational finance, providing insights into different approaches and techniques.
The lecturer begins by introducing integration problems and demonstrating how to calculate integrals using Monte Carlo sampling. They explain two approaches: the classical approach for integration and the integration based on the expected value. Through programming demonstrations in Python, the lecturer shows how to analyze and make simulations more efficient. They discuss the impact of smoothness on convergence and different types of convergence.
Furthermore, the lecture covers two important discretization techniques, namely Euler and Milstein, and explains how to control error based on the time step in the simulation. The lecturer emphasizes the principles and history of Monte Carlo simulation, which has been utilized in various fields for nearly 90 years. It gained popularity among physicists in the 1930s, especially during the Manhattan Project.
The importance of calculating the expected value of a future payoff in computational finance is discussed. This involves integrating over the real axis using the density of the stock, considering constant or time-dependent interest rates. Monte Carlo integration, associated with sampling and probability theory, is introduced as a technique that provides varying outputs with each simulation. The lecture emphasizes its application to highly dimensional problems and the ability to control the variance of the error distribution by adjusting settings in the simulation. The lecturer also discusses methods for improving sampling and simulating with Monte Carlo.
A specific method for estimating integrals using Monte Carlo simulation is explained. This method involves sampling points uniformly in a rectangular area and counting the proportion of samples under the curve to estimate the integral. Although not commonly used in finance, this approach can be valuable for high-dimensional problems. The lecturer emphasizes the importance of understanding the function being integrated to efficiently capture the area of interest.
The lecture also delves into the limitations and challenges of Monte Carlo simulation in finance. While it provides rough estimates, the results can be highly inaccurate, particularly for complex simulations. The lecturer explains that the expected error in Monte Carlo simulations decreases by the square root of the number of simulations, leading to computational intensity. The lecture further explores the relationship between the integral and expectation approaches, showcasing an example of how they are linked. In finance, the expectation approach is generally considered more efficient and accurate than traditional Monte Carlo simulation.
The lecture covers the law of large numbers and its relation to independent random variables. Estimation of variance and calculation of expectation for determining the mean are discussed. A comparison is presented between the "naive approach" and the expectation approach, with the latter proving significantly more accurate even with fewer samples. The lecturer demonstrates the code for performing this simulation, emphasizing the need to specify two points for the approach to integrate the function.
Different examples of stochastic integrals encountered in finance are discussed, highlighting the summation of Brownian motion over time steps, the summation of Brownian motion over increments, and the multiplication of Brownian motion by increments. A more concrete case is presented, where a function g(t) is integrated from 0 to T with a function g(s)dW(s). The lecture explains how to divide the integration range into smaller subintervals and use Monte Carlo simulation to approximate the integral. The importance of sample size and the range of values is emphasized for accurate results.
The speaker explains how to numerically solve a deterministic integral through a partition and approximation process. They introduce the Ito integral and explain the evaluation of the function GT at the beginning of the interval, with the integral chosen at the left boundary. Using an example with a GT function of T squared, the lecturer demonstrates how to obtain the expectation and variance with the Ito isometry property. Python code is provided to simulate the computation, and the steps involved are explained.
The generation of Brownian motion and its use in constructing a process and defining an integral are discussed. The lecture walks through the process of generating a distribution and using it to construct the Brownian motion process. The impact of removing the scaling condition on the distribution and variance is demonstrated. The lecturer also explains a trick to solve integrals involving Brownian motion by applying Ito's Lemma. Finally, the lecture shows how to consider the function x squared to calculate the integral.
The application of Ito's Lemma to obtain the dynamics of a function equal to t w squared t is discussed. By applying Ito's Lemma to x squared, the lecture reveals a term that is computed through integration, resulting in a pi squared distribution instead of a normal distribution. The speaker emphasizes the importance of experience in guessing which type of function to apply to achieve the desired result. The code is modified to switch between integrals, and an increase in the number of samples is suggested to improve the outcome.
Monte Carlo simulations, numerical routines, and the significance of good quality random number generators are discussed. The lecture explains Ito's Lemma and offers a heuristic approach to understanding why dwt dwt equals zero. It is observed that decreasing the grid size leads to faster convergence of the variance compared to the expectation. An experiment is conducted to demonstrate that the expectation goes to zero at a slower rate while the variance approaches nearly zero. The speaker provides intuition on why dwt dwt equals zero, while acknowledging that the theoretical proof of this relationship is quite involved.
The lecture delves into the convergence of two similar functions, g1 and g2, and investigates their expectations when sampled from a Brownian motion. These functions have limits of 0 as x approaches minus infinity and 1 as x approaches plus infinity. The lecturer calculates the error for increasing numbers of simulated samples and presents a graph comparing the error to the number of samples. The first function, with a non-smooth curve and wide oscillation range, is contrasted with the second function, which has a smooth curve and converges faster.
Convergence is highlighted as a crucial consideration when utilizing Monte Carlo simulation in finance. The lecture explains the difference between weak and strong convergence, with strong convergence being more powerful than weak. Errors can occur in convergence when dealing with non-smooth functions and digital-type payoffs, leading to substantially different evaluation results. Understanding the differences and implications of both types of convergence is critical to ensure accurate financial simulations and evaluations.
The lecture discusses weak and strong convergence in the context of Monte Carlo simulations and pricing algorithms. While weak convergence matches moments at the expectation level, strong convergence is necessary for accurate path-dependent payoffs. A complete Monte Carlo pricing algorithm involves defining a grid from the present time to the payment date of the contract, a pricing equation, and a stochastic driver for the asset. Monte Carlo simulations are necessary when closed-form evaluations are not possible due to the complexity of the stock process. The grid is typically equally spaced, but in some cases, alternative strategies may be employed.
The professor emphasizes the accuracy and time constraints of Monte Carlo simulation. It is noted that while increasing the number of time steps improves accuracy, it also increases the simulation time. Advanced techniques or closed-form solutions that allow for larger Monte Carlo steps can be beneficial in achieving both accuracy and speed. The lecture then proceeds to define the grids, asset, and payoff for a European type option. The final state of the option depends on the timing of observations. The lecture explains how to calculate the option price by taking the expectation under the queue measure and discounting it, while also calculating the standard error to measure the variability of the results obtained.
The concept of the standard error is discussed in the context of Monte Carlo simulation. The lecture explains that the expectation can be calculated using the strong law of large numbers, and the variance of the mean can be calculated by assuming that the samples are drawn independently. The standard error, which measures the variability of the expectation given a certain number of paths, can be determined by dividing the variance by the square root of the number of paths. As the number of samples increases, the error decreases. Typically, increasing the number of samples by a factor of four will reduce the error by a factor of two. A classical method for simulating stochastic differential equations is through Euler discretization, which is straightforward but has its limitations.
The lecturer discusses the use of stochastic differential equations and Euler discretization in Monte Carlo simulations. The process involves defining a grid, performing a simulation, and measuring the difference between the exact solution and the simulation through absolute error. It is essential to ensure that the randomness of the variables in both the exact and discretized versions is the same to ensure comparability. The lecture also emphasizes the importance of vectorization in Monte Carlo simulations, as it is more efficient than using double loops for each time step and path. However, it is important to note that while this approach simplifies the process, it comes with limitations in terms of accuracy and speed.
The exact solution for Brownian motion with a drift term and volatility term (r and sigma) is examined, using the Brownian motion generated in the exact representation and the same motion used in the approximation. The lecture compares the absolute error and the average error in weak convergence, highlighting that weak convergence suffices for pricing a European type of payoff but may not be enough for path-dependent payoffs. Graphs are shown to illustrate the generated paths for Euler discretization compared to the exact solution, where differences between the two can be observed for some paths. The lecture concludes with a comparison of strong and weak errors.
The speaker discusses the implementation of Monte Carlo simulations using code. They explain that to quantify error, a measure of error needs to be used, as discussed earlier in the lecture. The code generates paths and compares the exact values with the approximation using multicolor simulation. The outputs are time paths for the stock and the exact values. The speaker emphasizes the importance of generating the same Brownian motions for both the approximation and the exact solution to compare them at the error level. To measure weak and strong convergence errors, they define a range of the number of steps and perform Monte Carlo simulations for each step. The code generates two types of errors: weak error and strong error.
The lecturer discusses the simulation process involved in the Monte Carlo method and how it can be time-consuming because the simulation needs to be repeated many times. The results are shown through weak and strong convergence graphs, where the weak convergence error is represented by the slow-growing blue line, while the strong convergence error follows a square root of delta T shape, confirming the analysis. The lecturer explains that the error can be significantly reduced through Milstein's discretization technique, which derives additional terms by applying Taylor's expansion. While it involves more work to arrive at the final formula, Milstein's scheme requires the derivative of the volatility term, which is not always available analytically.
The speaker explains the use of Monte Carlo simulation in computational finance, specifically in geometric Brownian motion. They demonstrate how to compute the volatility term in the distribution sense and compare it with the Euler scheme. Although Monte Carlo simulation has a faster convergence rate than Euler's method, it can be challenging to derive the derivative in models involving multiple dimensions, as it requires additional computational computations. Furthermore, the speaker compares the absolute error in the weak and strong senses between the two schemes, highlighting that Monte Carlo's strong error is linear in delta t, while Euler's weak error is of the same order. Finally, they provide a code implementation of Monte Carlo simulation for generating paths in geometric Brownian motion and analyzing its strong convergence.
The speaker discusses the impact of different discretization techniques on convergence using the example of the Black-Scholes or geometric Brownian motion. The analysis of Euler and Milstein schemes serves as an illustration of the impact of different discretization techniques. The speaker compares the errors between the Milstein and Euler schemes, showing that the Milstein scheme's error is much lower than Euler's, although it may not always be applicable. The benefit of different schemes may not be evident when looking at the final results, but considering the computational expense of the simulation, time becomes crucial. Therefore, using large time steps would be essential if we want to perform fast simulations of Monte Carlo.
The lecturer then proceeds to discuss the role of random number generators (RNGs) in Monte Carlo simulations. They emphasize the importance of using good quality RNGs to ensure accurate and reliable results. The lecturer mentions that pseudo-random number generators (PRNGs) are commonly used in simulations and explains how they generate sequences of numbers that approximate randomness. They also highlight the need for reproducibility in simulations by using a fixed seed value for the RNG. Next, the lecturer discusses the concept of antithetic variates, which is a variance reduction technique used in Monte Carlo simulations. The idea behind antithetic variates is to generate pairs of random variates that have opposite effects on the quantity of interest. By taking the average of the results obtained from the original variates and their antithetic counterparts, the variance of the estimate can be reduced. This technique is particularly useful when dealing with symmetric distributions.
The lecture then introduces the concept of control variates as another variance reduction technique. Control variates involve introducing a known function into the simulation process that is correlated with the quantity of interest. By subtracting the estimate obtained from the known function from the estimate obtained from the target function, the variance of the estimate can be reduced. The lecturer provides examples to illustrate how control variates can be applied in practice. In addition to variance reduction techniques, the lecturer discusses the concept of stratified sampling. Stratified sampling involves dividing the sample space into strata and sampling from each stratum separately. This approach ensures that each stratum is represented in the sample, leading to more accurate estimates. The lecture explains the procedure for implementing stratified sampling and highlights its advantages over simple random sampling.
Finally, the lecturer explores the concept of importance sampling. Importance sampling is a technique used to estimate the probability of rare events by assigning higher probabilities to samples that are more likely to produce the desired event. The lecture explains how importance sampling can improve the efficiency of Monte Carlo simulations for rare event estimation. The lecturer provides examples and discusses the importance of choosing an appropriate sampling distribution for accurate results.
The lecture covers a range of topics related to Monte Carlo simulations, including integration problems, calculation of integrals using Monte Carlo sampling, programming demonstrations, analysis of convergence, discretization techniques, principles and history of Monte Carlo simulation, application in computational finance, variance reduction techniques, and importance sampling. The lecturer provides insights into the theory and practical implementation of Monte Carlo simulations and highlights their relevance in various fields.
Computational Finance: Lecture 10/14 (Monte Carlo Simulation of the Heston Model)
Computational Finance: Lecture 10/14 (Monte Carlo Simulation of the Heston Model)
The lecture focuses on utilizing Monte Carlo simulation for pricing derivatives, specifically European options, using the challenging Heston model. It begins with a warm-up exercise where European and digital options are priced using Monte Carlo and the simple Black-Scholes model. The simulation of the Cox-Ingersoll-Ross (CIR) process, which models the variance in the Heston model, is discussed, emphasizing the need for accurate sampling from this distribution. The lecturer demonstrates exact simulation of the CIR model, highlighting its benefits in generating accurate samples.
Next, the lecturer introduces the concept of almost exact simulation, which allows for larger time steps and higher accuracy compared to Euler discretization. The Heston model is simulated using both Euler and Milstein schemes, and the results are compared. It is noted that weak convergence is important for European-type payoffs, while strong convergence is important for path-dependent payoffs. Adjusting the number of steps or paths is necessary depending on the type of payoff and desired quality of results, considering computational time constraints in real-world applications.
The computational time required for evaluations is discussed, and a code comparison between Euler and Milstein discretization schemes is presented. The lecturer advises on code optimization for production environments, emphasizing that storing whole paths may not be necessary for payoff evaluation that only requires the final stock value. The lecture also provides the exact solution as a simplified implementation of the Black-Scholes model.
The pricing of digital or cash-or-nothing options using Monte Carlo simulation is explained, highlighting the differences in payoff calculation compared to European options. Diagnostics and outputs are presented to compare the approaches for both types of options. The lecture acknowledges the limitations of Monte Carlo simulations for options with terminal-dependent payoffs, where strong convergence is not present. The code's generic nature is emphasized, making it applicable to other models such as the Heston model.
The lecture dives into the conditions required for the Heston model to behave well and discusses how discretization techniques may affect these conditions. The impact of changes in the volatility parameter on the model's behavior is demonstrated through graphs, emphasizing that the process should not become negative. The limitations of Euler discretization in maintaining these conditions are also highlighted. The probability of negative realizations in the next iteration of the Heston model with Monte Carlo simulation is discussed. The likelihood of negative realizations is calculated based on the relationship between certain parameters, and the importance of aligning Monte Carlo paths with the model is emphasized to avoid significant pricing differences. Two approaches for handling negative values in the Heston model simulation are discussed: truncation and the reflecting Euler scheme. The pros and cons of each approach are compared, and the impact of smaller time steps on reducing bias is mentioned, albeit at a higher computational cost.
The lecture explores the use of exact simulation for the CIR process in the Heston model, enabling sampling directly from the non-central chi-square distribution. This approach avoids the need for small time steps and allows for sampling at specific times of interest. The computational code for the simulation is described, emphasizing its simplicity and optimality for generating samples. The lecture delves into the integration of the Heston model process for both the X and variance values, highlighting the simplification achieved through substitution. The importance of proper ordering of the processes in multidimensional simulations is emphasized, along with the recommendation to use large time steps for easier integration. The lecture addresses the importance of large time step simulations for pricing options on specific dates, aiming to reduce computation time while maintaining quality. Exact simulations using sampling from the non-central chi-square distribution are recommended, without introducing additional approximations. The lecture also discusses the impact of delta t on simulation accuracy and suggests investigating its influence on the results.
The concept of error in computational finance is discussed, with the lecture presenting a numerical experiment that analyzes the performance of the almost exact simulation of the Heston model. The lecture explains that by simplifying the integrals and using the almost exact simulation of the CIR process, the simulation becomes deterministic rather than stochastic. The lecturer conducts a numerical experiment to evaluate the performance of this simplified scheme in simulating the Heston model.
The lecture further explores the trade-off between computational effort and the small error introduced in the framework of computational finance. The lecturer emphasizes the need to calibrate the model to market data, as the Feller condition for volatility processes is often not satisfied in practice. The lecture notes that correlation coefficients for the Heston model are typically strongly negative, potentially due to numerical scheme considerations.
The lecturer discusses the use of Monte Carlo simulation for pricing exotic derivatives and stresses the importance of calibrating the model to liquid instruments. Pricing accuracy is ensured by simulating Monte Carlo paths using parameters obtained from model calibration and considering the hedging instruments related to the derivative. The lecturer highlights the superiority of almost exact simulation over Euler discretization, even with fewer time steps, and explains that the main source of Euler error lies in problematic discretization of the variance process under extreme parameters or violations of the Feller condition.
The accuracy of Euler discretization in the Heston model is explored through experiments with different options, including deep in-the-money, out-of-the-money, and at-the-money options. The lecture presents the code used in the experiment, focusing on the Euler discretization and the almost exact simulation, which involves the CIR sampling and simulation of the log stock process using the non-centrality parameter.
The lecturer discusses the settings and configurations for simulations to price European options using both Euler discretization and almost exact simulation. The exact simulation of the CIR process, the correlation of Brownian motions, and the exponential transformation are integral parts of the simulation. Option pricing using a generic function is demonstrated, showcasing the impact of variables such as strike price and time step on the accuracy of the simulations. The lecture concludes by highlighting that the almost exact simulation achieves high accuracy with fewer time steps compared to the Euler scheme.
The lecture extensively covers the use of Monte Carlo simulation for pricing derivatives in the Heston model. It explores the simulation of the CIR process, discusses the challenges and pitfalls, and compares different discretization schemes. The lecture emphasizes the benefits of almost exact simulation, highlights the importance of calibration and model accuracy, and provides practical insights and code examples for implementing Monte Carlo simulations in computational finance.
Computational Finance: Lecture 11/14 (Hedging and Monte Carlo Greeks)
Computational Finance: Lecture 11/14 (Hedging and Monte Carlo Greeks)
In the lecture, the concept of hedging is emphasized as equally important to derivative pricing in finance. The lecturer delves into various calculations of sensitivities to determine the impact of a derivative's price on specific parameters and how to conduct a hedging experiment. Several key topics are covered, including the principles of hedging in the Black-Scholes model, simulation of profit and loss, dynamic hedging, and the influence of jumps. The lecturer stresses that the concept of hedging determines the value of a derivative, and the price of hedging determines its overall value.
To provide a comprehensive understanding, the lecturer starts by explaining the concept of hedging in the financial industry. Financial institutions generate income by applying an additional spread on top of the value of an exotic derivative. To mitigate risk, a portfolio that replicates the derivative is constructed. This portfolio consists of the derivative's value with a plus sign and a minus delta, which corresponds to the portfolio's sensitivity to the stock. Selecting an appropriate delta is crucial as it determines the number of stocks that need to be bought or sold to align with the model used. The lecturer demonstrates an experiment in which the delta is continuously adjusted throughout the contract's lifespan, resulting in an average profit loss of zero.
The lecture covers the concept of delta hedging and distinguishes between dynamic and static hedging. Delta hedging is employed to hedge risk factors in a portfolio, with the value of the replicating portfolio determining the hedge's delta. Dynamic hedging involves frequent adjustments to the delta, while static hedging entails buying or selling derivatives only at the beginning or at specific intervals during the derivative contract. The video also discusses the sensitivity of hedges to the number of stochastic differential equations in the pricing model and how the frequency of hedging impacts potential profits and losses.
Introducing the concept of a profit and loss (P&L) account, the lecture explains its role in tracking the gains or losses when selling derivatives and hedging them. The P&L account is influenced by the initial proceeds obtained from selling an option and the delta h value, which grows over time based on interest rates from savings or borrowing. The goal is to achieve a P&L account that balances out at the derivative's maturity, indicating a fair value charged according to the Black-Scholes model. However, if the model is not chosen appropriately, the extra spread added to the fair value may not cover all the hedging costs, resulting in a loss. Thus, it is essential to employ a realistic and robust model for pricing alternative derivatives.
The lecture delves into the iterative process of hedging and the calculation of profit and loss (P&L) at the end of the maturity period. This process involves computing the delta of an option at time t0 and time t1, then determining the difference between them to ascertain the number of stocks to buy or sell. The lecturer emphasizes the significance of understanding what is being sold and collected, as selling an option essentially involves selling volatility and collecting premiums. At the end of the process, the value of the option sold is determined based on the stock value at maturity, and the P&L is evaluated using the initial premium, the value at maturity, and the quantity of stocks bought or sold throughout the iterative process.
The lecturer shifts the focus towards hedging in computational finance as a means of reducing variability and sensitivity concerning stocks' value. The lecture clarifies how hedging aids in minimizing losses and introduces the concept of the distribution of piano in Monte Carlo path simulations, highlighting that the expectation of a P&L should average to zero. The profit derived from selling an exotic derivative and hedging it arises from the additional spread charged to the client since the expected P&L is zero.
To overcome the challenges posed by the unknown density in advanced models like the Fourier Transformation model, alternative methods are employed for calculating sensitivities. One such approach is the Malliavin calculus, which provides a mathematical framework for computing derivatives of random variables with respect to parameters in stochastic processes.
The Malliavin calculus introduces the concept of the Malliavin derivative, which extends the notion of classical derivatives to random variables driven by stochastic processes. This derivative enables the calculation of sensitivities for complex models where traditional methods may not be applicable. By leveraging the Malliavin derivative, practitioners can obtain sensitivities with respect to various parameters in the Fourier Transformation model. This approach allows for more accurate pricing and risk management, as it captures the intricate dependencies and dynamics present in the model. However, it's important to note that utilizing the Malliavin calculus requires advanced mathematical techniques and a deep understanding of stochastic analysis. It is a specialized field that is typically explored by experts in quantitative finance and mathematical finance.
In summary, when dealing with models that involve unknown densities, such as the Fourier Transformation model, the Malliavin calculus provides a powerful tool for calculating sensitivities. This approach enables the assessment of risks and the accurate valuation of derivatives in complex financial scenarios.
Computational Finance: Lecture 12/14 (Forward Start Options and Model of Bates)
Computational Finance: Lecture 12/14 (Forward Start Options and Model of Bates)
The lecture delves into the intricacies of forward start options, which are a type of European option with a delayed starting date, often referred to as performance options. These options are more complex than standard European options, and the lecture provides an overview of their payoff definition and advantages compared to European options.
The pricing techniques for forward start options are more involved, and the lecture focuses on the use of characteristic functions. It explores two types of forward start options: one using the Black-Scholes model and the more challenging pricing under the Heston model. The implementation in Python and the pricing of a product dependent on volatilities are also covered. The lecture emphasizes the importance of European options as building blocks and their calibration and relationship to exotic options. It touches upon the Bates model, which extends the Heston model by incorporating Merton jumps, and highlights the use of hedging parameters to ensure well-calibrated models. The video explains how the unknown initial stock value in forward start options is determined at a future time (t1) and introduces the concept of filtration in relation to these options. The lecture also explores how forward start options can serve as building blocks for other derivatives, presenting a strategy to reduce derivative costs. Moreover, the professor covers the construction of a click option, a desired derivative structure, and its relation to European calls and forward start options. The lecture emphasizes the significance of identifying payment dates when calculating discount factors for pricing. It also showcases how the ratio of two stocks can be reformulated as the exponent of a logarithm of the ratio.
Various pricing methods for forward start options are discussed, including Monte Carlo simulation and analytical solutions like the Black-Scholes model. The need to find the forward characteristic function, which allows pricing of forward start options for any model in a specific class of processes, is explained. The lecture demonstrates the pricing of a forward start option using the characteristic function and the expectation of an IU logarithm of two stocks. The conditioning on a larger sigma field when determining the characteristic function is explored, enabling the exponent with the minus log to be taken outside the expectation. Discounted characteristic functions from T2 to T1 are also utilized.
The lecture delves into the forward currency function, which represents future expectations and is expressed as an expectation on the risk-neutral measure. It explains that deterministic interest rates result in no difference between the discounted and non-discounted currency functions. However, stochastic interest rates introduce complexity. The process of deriving the forward starting characteristic function, involving an additional expected value, is outlined, along with the importance of allowing analytical solutions to the outer expectation for practical use. The forward starting characteristic function is then applied to the Black-Scholes and Heston models.
Further, the lecture focuses on the forward start currency function for the Black-Scholes model. It notes that the pricing should only depend on the performance over time and not the initial stock value, simplifying the solution compared to the discounted currency function. The presence of the variance part in multiple dimensions requires solving an inner expectation. An exact representation of the Black-Scholes model is shown, confirming that the distribution of the ratio of two stocks is independent of the initial stock value. The distribution is simplified into a geometric Brownian motion, encompassing an increment from p1 up to t2.
The pricing of forward start options under the Black-Scholes model is explained, highlighting the use of geometric Brownian motion for the ratio of two stocks at different times. The pricing solution for call and put options for forward start options closely resembles that of European calls and puts, with slight differences in strike adjustment and discounting times. The lecture stresses the importance of using Black-Scholes implied volatilities when calculating prices, even when employing other models, as it aligns with market standards. It also underscores the lecturer's recommendation to consider the two parameters for forward start options and reminds viewers that Black-Scholes prices are known analytically under this model.
Moving on, the speaker delves into the Hassle model, which increases the complexity of the characteristic function for forward start options by introducing a second stochastic process representing variance. However, the speaker explains that this second dimension is not necessary for pricing options since the focus is solely on the marginal distribution for the stock process. After simplification and substitution of the characteristic function, the expression for the forward currency function is obtained. The speaker suggests revisiting the slides on the Hassle model for more details on the functions involved in the expression.
The lecture proceeds with the discussion of the moment generating function for a Cox-Ingersoll-Ross (CIR) process and presents the closed-form expression for the forward characteristic function in the Heston model. The lecturer notes that having the moment generating function in closed form allows for faster computation. By substituting the moment generating function into the forward currency function, a closed-form expression for the forward characteristic function is derived. Finally, the speaker introduces a numerical experiment to price forward start options using the Heston model and the derived expressions.
Next, the speaker shifts focus to forward start options and the Bates model. They explain how the variance process is represented by dvt and discuss the parameters for volatility and variance. The speaker conducts two experiments to observe the impact of implied volatilities on the parameters and the effect of the time distance in forward start options. The experiments demonstrate that although the implied volatility shape remains the same, the levels differ. As the time distance increases, the volatility converges to the square root of the long-term variance. The speaker explains the logic behind shorter maturity options having a more concentrated density around t1 and t2. Additional experiments using a code are performed to compare implied volatilities.
Continuing, the lecturer addresses the implementation of the forward characteristic function and cost methods for pricing forward start options. The forward characteristic function is defined using lambda expressions and various parameters, including the Heston model and the moment generating function for the CIR process. The cost method for pricing forward start options is similar to that of pricing European options but includes adjustments for handling two different times. The lecturer shares a trick to obtain a good initial guess for the Newton-Raphson algorithm when calculating forward implied volatilities, which involves defining a volatility grid and interpolating on the market price.
The lecture proceeds with an explanation of the process for calculating forward implied volatilities using the Newton-Raphson method. The difference between the option price from the model and the market price is discussed, and the lecturer demonstrates how to apply the SciPy optimize function to calculate the Newton-Raphson method and obtain the optimal volatility, also known as the implied volatility. The section confirms that the long-term mean and initial variance are the same, and the level of implied volatilities and forward input volatility aligns. The Bates model, an extension of the Heston model that incorporates additional jumps driven by an independent random variable j, which follows a Poisson distribution, is also introduced.
The lecture highlights the difference between the Heston model and the Bates model. While the Heston model is suitable for calibrating to the smile and skew for equity options with longer maturities, it struggles with options having shorter maturities, such as those expiring within a week or two. The Bates model addresses this issue by introducing independent jumps, enabling better calibration of short-term options. Although the Bates model involves many parameters, it is not challenging to extend from the Heston model. The log transformation is necessary to derive the characteristic function for the Bates model, and it is noted that the model can still be well-calibrated even with the addition of jumps.
The speaker then discusses the modification of the Bates model, specifically focusing on the stochastic intensity. The speaker expresses their opinion that making the intensity stochastic is unnecessary as it would introduce unnecessary complexity without exploring the current parameters. Instead, the intensity in the model is kept linear in the state variables and defined as a constant drift. The speaker analyzes the affine jump diffusion framework and includes details of the derivations in the book. The only difference between the characteristic function for the Heston and Bates models lies in the "a" term of the Bates model. Additionally, two correction terms contain all the information about jumps. Numerical results are presented, providing an analysis of the impact of intensity, volatility of jumps, and mu j, which represents the distribution of j.
The extension of the Heston model to the Bates model is discussed. The Bates model is used to calibrate the model to all market information, providing an advantage compared to other models. The code for this model is simple and provides additional flexibility, especially for short maturity options where calibration to all market information is crucial. The lecture also covers the pricing of more interesting derivatives, such as the variance swap, using the knowledge gained from pricing forward start options or performance options.
The speaker introduces a type of derivative called a variance swap, which allows investors to bet on the future volatility of an asset. The payoff of a variance swap is defined as the summation of squared logarithmic stock performances over a given grid of dates, divided by the previous stock performance. The lecturer notes that the unusual formulation of this payoff becomes clearer when linked to a stochastic differential equation. When pricing this derivative, the value of the swap at inception will be zero if the strike is equal to the constant expectation. Moreover, the speaker explains that most swaps are traded at par, meaning that the value of the contract is zero when two counterparties agree to buy or sell.
The lecture then discusses the time-dependent framework for the Bates model and how it connects the integral over time-dependent volatility to the performance of a derivative over time. The payoff is defined as the squared logarithmic performance, which is equivalent to the integral of volatility. The speaker explains how to find the third value of a contract using the expected value of sigma v squared and the stochastic differential equations. Additionally, the scaling coefficient of 252 working days is introduced as an essential factor in finance.
Finally, the speaker covers the fair value of a variance swap, which is a derivative contract that allows investors to bet on the future volatility of an asset. The fair value of the swap can be expressed as a scaling coefficient corresponding to the periods from zero to the maturity of the contract, plus an element corresponding to the interest rates, minus the expected value of q log st divided by st0. Evaluating this expectation can be done through Monte Carlo simulation or an analytical distribution of stocks. It is interesting to note that even though the performances from all small intervals are compounded, it is equivalent to the ratio or logarithm of the value of a stock in the end divided by the initial value.
The lecture covers a wide range of topics related to forward start options, performance options, the Heston model, the Bates model, and variance swaps. It provides insights into pricing techniques, implementation in Python, and the significance of these concepts in financial derivatives.
Computational Finance: Lecture 13/14 (Exotic Derivatives)
Computational Finance: Lecture 13/14 (Exotic Derivatives)
The lecture focuses on pricing exotic derivatives and extending pricing models to path-dependent cases. The primary motivation for extending the payoff structure is to offer clients cheaper prices while still providing exposure to stock market fluctuations. The use of digital features and barriers is explored as a means to reduce derivative costs while maintaining desired exposure. The lecture delves into various types of payoffs, including binaries and digitals, barrier options, and Asian options, examining their impact on derivative prices. Additionally, the lecture discusses the pricing of multi-asset options and potential model extensions to handle baskets of hundreds of stocks.
The pricing procedure for financial products is discussed, beginning with the product specification and the risk factors required for modeling and pricing using stochastic differential equations, such as the Black-Scholes model, jumps, and stochastic volatility models. Depending on the product's complexity, a one or two-dimensional system of equations may be sufficient for accurate pricing. The process also involves calibration and hedging, where an optimal set of parameters is chosen to price the product and minimize hedging costs, ensuring an arbitrage-free environment.
Different types of options are defined, with a focus on European options, American options, and Bermuda options. European options are considered fundamental building blocks for exotic derivatives, but they can be difficult to time and carry significant risk. American options offer more flexibility, allowing exercise at any time, while Bermuda options allow exercise only at specified dates.
Exotic derivatives and path-dependent options are introduced, which depend on the entire history of a stock rather than just the marginal distribution at a specific time. Adjusting the payoff function using binaries and digitals is shown to significantly reduce derivative values. The lecture covers various types of exotic derivatives, including asset or nothing, cash or nothing, stock or nothing, compound options, and chooser options. These options involve limiting the contract in some way, such as with maximums, minimums, or other restrictions, to control costs. The popularity of exotic derivatives in the past, particularly during times of high interest rates, is also discussed.
A strategy for generating high profits through an exotic derivative is explained. The strategy involves allocating most of the investment to a safe account with a guaranteed return and pricing a potential option payout. Although this strategy is not currently popular, it has been effective in the past. The lecture also includes code examples for valuing contracts and reducing their value by setting upper limits on potential stock growth. The lecture highlights how a small adjustment in the payoff structure can significantly reduce valuations, making derivatives more attractive to clients. By introducing barriers and path dependence, costs can be reduced. Various barrier options are discussed, such as up-and-out, down-and-out, up-and-in, down-and-in options, and their impact on derivative pricing based on the stock's historical behavior.
The concept of lookback options is explored, where the maximum or minimum value of a stock over its lifetime determines the payoff at maturity. Lookback options incorporate path dependence and can provide positive payouts even if the stock is lower at maturity than the strike. The lecture explains the implementation of lookback options using Monte Carlo simulation and partial differential equations (PDEs), emphasizing special boundary conditions for barrier options and their extension to other exotic derivatives.
Barrier options are discussed in detail, highlighting their appeal to counterparty clients and their use in the cross-currency market. The lecture explains the configurations and payoffs of barrier options, including out, in, down, and up options. The lecturer emphasizes that barrier options can be time-dependent, adding complexity to the contract. Monte Carlo simulation and PDEs are presented as computational methods for pricing barrier options.
The lecture compares up-and-out options to standard European options, noting the significant reduction in value for up-and-out options due to their barrier-triggered payoff. The concept of up-and-out barrier options is introduced, where the payoff only occurs if the stock does not exceed a certain level during its lifetime. The lecture demonstrates the impact of a barrier on the price of a derivative through a programming exercise, showing that the price of an up-and-out barrier option is equivalent to the price of a digital option with a similar payoff structure.
The lecturer then proceeds to explain the implementation of an up-and-out barrier using Monte Carlo simulation. In contrast to a digital option's payoff, which depends only on the stock value at maturity, an up-and-out barrier also considers the history of the stock's behavior throughout the derivative's lifetime. A function is defined to determine whether the barrier has been reached, utilizing a boolean matrix and a logical condition. The resulting "hit vector" is a binary vector that indicates whether the barrier has been hit for each path. The lecturer demonstrates how changing the barrier value affects the hit vector, emphasizing that the payoff is zero if the barrier is hit and one if it is not hit.
The concept of introducing a barrier in derivative contracts is explained as a way to reduce their value, providing a more affordable option for clients seeking exposure to a specific asset. The presence of a barrier has a significant impact on the derivative's value, potentially leading to losses if the stock does not exceed the specified level. However, by incorporating barriers, derivative prices can be reduced by approximately 30%, making them more attractive for investors. Nonetheless, discontinuous derivatives with barriers can present challenges in terms of hedging costs, which could rise to infinity. To mitigate this issue, the lecturer suggests replicating the payoff using alternative methods to reduce costs.
The video introduces the concept of replicating the digital feature of an option by strategically buying and selling call options with different strike prices. As the strike prices approach each other, the resulting payoff becomes more similar to a digital option. However, the lecturer acknowledges the difficulties in precisely replicating the discontinuity of options due to changes in delta and gamma sensitivities. While approximations can be used for hedging, it is crucial to charge premiums to compensate for potential hedging losses caused by the digital nature of the option. The video emphasizes the concept of reducing derivative costs by introducing digital limitations or altering the payoff structure.
The lecture then moves on to discuss Asian options as a means to reduce volatility and uncertainty associated with an underlying asset, consequently lowering the price of derivatives. Asian options are based on the average behavior of a fluctuating stock, which tends to be smoother than the stock itself, reducing the associated uncertainty. The lecturer explores different variants of Asian options available in the market, including fixed and floating strike calls and puts. Floating strike options, in particular, are popular in commodities trading due to their ability to reduce uncertainty and mitigate risks associated with a specific underlying asset level.
The speaker further explains the various methods of calculating the average for a stock, highlighting its importance in trading. Two types of averages, arithmetic and geometric, are introduced, with the geometric average preferred for mathematical analysis due to its analytic expression. In practice, summations are often used, necessitating approximation techniques like Monte Carlo simulation or PDEs. The lecture also delves into the concept of continuous average, which differs from arithmetic average due to its integral representation, adding an additional dimension to the pricing problem and making it more complex to solve.
The focus then shifts to the pricing of Asian options, which entails moving away from a one-dimensional problem and involving higher-dimensional considerations. Asian options introduce two independent variables: the stock price and the integral of the stock. The option's payoff depends on the observed integral or path from zero to maturity, with the payment made at maturity. The lecture acknowledges that pricing exotic derivative contracts with final part dependent quantities can be challenging, requiring more advanced techniques. However, delta hedging is still effective in achieving proper hedging coefficients despite the complexities introduced by Asian options. The lecturer discusses the use of Monte Carlo simulation to price Asian options, highlighting its flexibility in handling high-dimensional problems. By simulating multiple paths of the stock price and calculating the average payoff, Monte Carlo simulation can provide an estimate of the option's price. The lecture also mentions the potential challenges of Monte Carlo simulation, such as convergence issues and the need for a sufficient number of paths to obtain accurate results.
The lecturer then moves on to discuss another type of exotic option known as a barrier option with rebate. This option has a similar structure to the barrier option previously discussed, but with an additional rebate payment if the barrier is hit. The presence of the rebate compensates the option holder if the barrier is breached, mitigating potential losses. The lecture explains that the rebate payment reduces the cost of the option, making it more attractive to investors.
To price barrier options with rebates, the lecturer introduces the concept of a reverse knock-out option, which is the inverse of a knock-out option. The reverse knock-out option pays a rebate if the barrier is not hit. By pricing the reverse knock-out option and subtracting the rebate payment, the price of the barrier option with rebate can be determined. The video provides an example of implementing this pricing methodology using Monte Carlo simulation.
Throughout the lecture, the importance of understanding and effectively pricing exotic derivative contracts is emphasized. Exotic options provide flexibility and customized solutions for investors, but their pricing and risk management require sophisticated models and techniques. The lecture concludes by highlighting the need for further research and development in this field, as well as the importance of collaboration between academia and industry to enhance derivative pricing methodologies and meet the evolving needs of market participants.
Computational Finance: Lecture 14/14 (Summary of the Course)
Computational Finance: Lecture 14/14 (Summary of the Course)
The series on computational finance concluded with a comprehensive summary of the important topics covered in each lecture. The course spanned a wide range of subjects, including stochastic differential equations, implied volatilities, jump diffusions, affine class of diffusion processes, stochastic volatility models, and Fourier transformations for option pricing. It also delved into numerical techniques like Monte Carlo simulations and various hedging strategies.
In the later lectures, the focus shifted towards forward start options and exotic derivatives, where the knowledge gained throughout the course was applied to structure these complex financial products. The initial lectures provided an introduction to the course and discussed fundamental principles of financial engineering, different markets, and asset classes. Lecture two specifically covered various types of options and hedging strategies, with an emphasis on commodities, currencies, and cryptocurrencies.
The pricing of call and put options and its relation to hedging was a central theme throughout the course. The lecturer emphasized that the price of a hedging strategy should always be equivalent to the price of a derivative to avoid arbitrage opportunities. The mathematical aspects of modeling different asset classes, including asset prices and randomness measurement, were discussed. Stochastic processes, stochastic differential equations, and Itô's lemma were highlighted as vital tools for pricing financial instruments. Python simulations were also demonstrated, showcasing how stochastic differential equations can simulate the real behavior of stock movements for pricing purposes. The advantages and disadvantages of the Black-Scholes model were addressed, emphasizing the need for a holistic perspective to ensure consistency in portfolio management and hedging strategies.
Martingales were repeatedly emphasized as a critical concept in option pricing, and other important topics covered in the course included the Black-Scholes model, implied volatility, Newton-Raphson algorithm convergence, and the limitations of time-dependent volatility. The practical application of Python coding to verify whether a simulated process is a martingale and the impact of measures on drift were explored. The course provided a deep insight into the pricing of simple European options, showcasing how different models and measures can be employed to calculate their prices.
The limitations of the Black-Scholes model were discussed, particularly in relation to incorporating jumps into the model. While jumps can improve the calibration of implied volatility surfaces and generate skew, they also introduce complexity and reduce hedging efficiency. Stochastic volatility models, such as the Heston model, were introduced to enhance the model's flexibility in calibration and pricing of exotic options. Additionally, a fast pricing technique was presented as a solution. The lecture also outlined the conditions that models or stochastic differential equations must satisfy to be used within the affine models in Fourier transformations.
Two important models for pricing equities and stocks were discussed: the affine class of diffusion processes and the stochastic volatility model, specifically the Heston model. The affine class of diffusion processes allows for fast calibration of European options, while the Heston model offers flexibility in calibrating the entire surface of implied volatilities from European options. The lecture covered the impacts and advantages of correlation in the models, pricing PDE, and the use of Fourier transformations for pricing when a model belongs to the affine class of processes. Understanding and utilizing these models were highlighted as valuable skills in computational finance.
The pricing of European options, with a focus on call and put options, was the central focus of another lecture. The use of a characteristic function and the ability to solve systems of complex-valued ODEs were emphasized, along with the importance of numerical techniques for obtaining solutions. Balancing a good model with efficient calibration and evaluation was stressed for practical applications and industry acceptance. The advantages of the cos method of Fourier transform for pricing were discussed, along with its implementation in Vital. Efficient calibration and the utilization of Monte Carlo simulations for pricing were also recommended.
Monte Carlo sampling in pricing exotic derivatives was extensively explored in another lecture. The challenges posed by multiple dimensions, model complexity, and computational costs in accurate pricing were addressed. Monte Carlo simulation was presented as an alternative pricing approach, with a focus on reducing error and improving accuracy. The lecture covered various aspects of Monte Carlo sampling, including integration, stochastic integration, and calibration methods such as Euler and Milstein schemes. The evaluation of smoothness of payoff functions and understanding weak and strong converters were highlighted as crucial for ensuring accurate pricing.
The lecture dedicated to the Heston model discussed its flexibility in calibration, implied volatility surface modeling, and efficient Monte Carlo simulation. The lecture also touched upon the almost exact simulation of the Heston model, which is related to the exact simulation of the Cox Ingersoll Ross (CIR) process for the variance process. While Euler and Milstein discretization methods may encounter issues with the CIR process, there are efficient ways to perform the simulation. The lecture emphasized the importance of considering a realistic model for simulation, particularly when dealing with delta hedging and accounting for market jumps.
The concept of hedging in finance was thoroughly explored in a separate video. Hedging involves reducing exposure to risk and potential losses by managing a portfolio and actively maintaining the contract after it has been traded. The video underscored the significance of hedging, which extends beyond pricing and encompasses continuous risk management until the contract's maturity. Delta hedging and the impact of market jumps were discussed, emphasizing the importance of employing a realistic model for accurate simulation.
The limitations of delta hedging were addressed in another lecture, highlighting the need to consider other types of hedging, such as gamma and vega hedging, for more complex derivatives. The computation of sensitivities and methods to improve their efficiency, including finite difference, pathwise sensitivities, and likelihood quotients, were covered. The lecture also delved into the pricing of forward start options and the challenges associated with pricing options with uncertain initial stocks. The option value was derived using characteristic functions, and the lecture concluded with a discussion on implied volatilities and their implementation in Python.
The lecture on additional jumps in financial models, particularly the Heston model, explored their impact on parameter calibration and hedging strategies. Variance swaps and products of volatilities were also discussed, focusing on the relationship between the strange representation, variance swap contracts, and conditional expectations using Black-Scholes dynamics. Furthermore, the lecture delved into the structuring of products using various techniques such as binary and digital options, path-dependent options, barrier options, and Asian options. It also touched upon the pricing of contracts involving multiple assets. This lecture served as a summary of the knowledge acquired throughout the course, providing a foundation for tackling more advanced derivatives in the future.
In the final part, the speaker congratulated the viewers on successfully completing all 14 lectures and acquiring knowledge in computational finance, financial engineering, and derivative pricing. The viewers were encouraged to apply their newfound expertise in practical settings or consider further courses to expand their knowledge. The speaker wished them a successful career in finance, confident that they were well-prepared for their future endeavors.
Financial Engineering Course: Lecture 1/14, (Introduction and Overview of the Course)
Financial Engineering Course: Lecture 1/14, (Introduction and Overview of the Course)
The instructor begins by introducing the course on financial engineering, highlighting its objectives and key areas of focus. The course aims to delve into interest rates and multiple asset classes such as foreign exchange and inflation. The ultimate goal is for students to build a multi-asset portfolio consisting of linear products and gain proficiency in performing xva and value at risk computations. Prior knowledge of stochastic differential equations, numerical simulation, and numerical methods is necessary to fully engage with the course material.
The course structure is outlined, comprising 14 lectures accompanied by homework assignments at the end of each session. The programming language used throughout the course is Python, enabling practical implementation and application of the concepts discussed.
The speaker emphasizes the practical nature of the course on computational finance. While theoretical knowledge is covered, there is a strong emphasis on implementation efficiency and providing Python code examples for each lecture. The course materials are self-contained, although they are based on the book "A Book of Mathematical Modeling and Computation in Finance." The lecture also provides an overview of the course roadmap, giving students a clear understanding of the topics that will be covered in each of the 14 lectures.
The first lecture is focused on providing an overview of the entire course and highlighting the significance of the concepts covered in achieving the ultimate goal of performing xva and var calculations.
The lecturer proceeds to give an extensive overview of the topics that will be covered throughout the Financial Engineering course. These include various models such as full white and full white two-factor models, measures, filtrations, and stochastic models. Pricing interest rate products, including linear and non-linear products like swaptions, is a key focus. The lecture covers yield curve construction, multi-curve building, spine points, and the selection of interpolation methods using Python codes. Other topics covered include negative interest rates, options, mortgages and prepayments, foreign exchange, inflation, Monte Carlo simulation for multi-assets, market models, convexity adjustments, exposure calculations, and value adjustment measures such as cva, bcva, and fva.
Risk management becomes a focal point as the course progresses, and lecture 13 is dedicated to risk measurement using coding and historical data analysis. Lecture 14 serves as a summary of everything learned throughout the course.
The second lecture focuses on filtrations and measure changes, including conditional expectations and simulation in Python. Students will engage in hands-on exercises to simulate conditional expectations and explore the benefits and simplification of pricing problems using measure changes.
In subsequent lectures, the instructor provides an overview of the Hijack Model framework, equilibrium versus term structure models, and yield curve dynamics. The lectures cover short rates and the simulation of models through Monte Carlo simulations in Python. The comparison between one-factor and two-factor models is discussed, with an exploration of multi-factor extensions. A video experiment is conducted to analyze the S&P index, the short rate implied by the Fed, and yield curve dynamics.
The simulation of yield curves is explored to observe the evolution of interest rates over time and compare them with stochastic models. Topics covered include the affinity of a fulbright model, exact simulation, construction and pricing of interest rate products, and the calculation of uncertain cash flows in swap examples.
The lecture on building a yield curve covers the relationship between yield curves and interest rate swaps, forward rate agreements, and derivatives pricing. Different yield curve shapes and their relevance to market situations are explained. Implied volatilities and spine point calculations are discussed, along with interpolation routines and the extension of single yield curves to multi-curve approaches. Practical aspects of building yield curves using Python experiments and connecting them to market instruments are emphasized.
The lecturer explores topics related to financial engineering, including the pricing of swaptions under the Black-Scholes model and options using full white or any short rate model. The Jamshidian's trick and Python experiments are explained. The lecture also covers concepts such as negative interest rates, shifted log-normal shifted implied volatilities, and the impact of shift parameters on implied volatility shapes. Additionally, the lecture delves into the prepayment of mortgages and its effect on position and hedging from a bank's perspective.
Bullet mortgages are introduced, and the associated cash flows and prepayment determinants are explained. The lecture highlights the impact of prepayments on mortgage portfolios and links the refinancing incentive to market observables. Furthermore, pipeline risk and its management by financial institutions are discussed.
The course moves on to modeling multiple asset classes simultaneously, which allows for the simulation of potential future risks that can affect the portfolio. Correlations between different asset classes are examined, and the importance of hybrid models for risk management purposes is emphasized, even though there may be a declining interest in exotic derivatives.
Hybrid models for pricing valuation adjustments (XVA) and value at risk are explored, along with extensions involving stochastic volatility. The lecture covers hybrid models suitable for an XVA environment, including stock dynamics and stochastic interest rates. Stochastic volatility models, such as the Heston model, are discussed in the second block, addressing how to incorporate stochastic interest rates that are correlated with the stock process. The lecture also delves into foreign exchange and inflation, discussing the history and development of floating currencies, forward FX contracts, cross-currency swaps, and FX options. The impact of measure changes on process dynamics is also examined, ultimately aiming to price contracts defined under different assets in various asset classes and calculate exposures and risk measures.
The instructor covers additional topics related to financial engineering, including the quantum correction element present in stochastic volatility and the pricing of FX options with stochastic interest rates. The concept of inflation is explored, tracing its evolution from monetary-based to goods-based definitions. Market models such as the LIBOR market model and convexity adjustments are discussed, providing a historical perspective on interest rate development and the motivation behind market models like the LIBOR market model within the HJM framework. The lecture also delves into log-normal LIBOR market models, stochastic volatility, and the smile and skew dynamics in the LIBOR market model.
Various techniques used in pricing financial products are addressed, with an emphasis on risk-neutral pricing and the Black-Scholes model. The lecturer warns against the misuse of risky techniques, such as the freezing technique, and emphasizes the importance of convexity correction in pricing frameworks. Students will learn how to recognize the need for convexity correction and how to incorporate interest rate movements or market smile and skew into pricing problems. The section concludes by covering XVA simulations, including CVA, BCVA, VA, and FVA, and the calculation of expected exposures, potential future exposures, and sanity checks using Python simulations.
The instructor revisits the topics covered in the financial engineering course, including pricing derivatives, the importance of price discovery, practical aspects of trade attributions, and risk management measures such as value at risk and expected shortfall. The focus remains on practical applications, such as building an interest rate swap portfolio and utilizing knowledge of yield curve construction to estimate VAR and expected shortfall through simulation results. The lecture also addresses challenges related to missing data, arbitrage, and regrading in VAR computation using Monte Carlo simulation.
In the final lecture, the lecturer discusses back-testing and testing the VAR engine. While acknowledging that the course will extend beyond the initial 14 weeks, the instructor expresses confidence in the comprehensive and enjoyable learning journey. The recorded lectures will guide students toward the summit of understanding valuation adjustments (XVA) and the calculation of value at risk.
Financial Engineering Course: Lecture 2/14, part 1/3, (Understanding of Filtrations and Measures)
Financial Engineering Course: Lecture 2/14, part 1/3, (Understanding of Filtrations and Measures)
In the lecture, the instructor delves into the Black-Scholes model with stochastic jumps, showcasing its application in derivative pricing. The incorporation of conditional expectations is highlighted as a means to enhance the model's accuracy. Additionally, the concept of numeraires and measure changes is explored, demonstrating how shifting between different numeraires can improve pricing outcomes. This section underscores the significance of filtration, expectations, and measure changes, particularly within the realm of interest rates.
Expanding on the topic, the professor emphasizes the pivotal role of measures, filtrations, and expectations in pricing. They illustrate how measures, such as stocks, can be effectively employed in pricing processes, while measure changes aid in reducing the complexity of pricing problems. The lecture further investigates the notion of a forward measure, commonly associated with stochastic discounting. Filtrations are elucidated as fundamental principles for comprehending time, exposure profiles, and risk profiles. Additionally, the definition of a stochastic process and the importance of filtration in interpreting market data and anticipating future realizations are introduced.
Moving forward, the concept of filtrations and measures is thoroughly examined. Filtrations can pertain to the present or extend into the future, necessitating a clear distinction when dealing with stochastic processes. The past represents a singular trajectory of a stock's history, whereas the future's stochasticity can be modeled through stochastic differential equations and simulations. Although the course predominantly focuses on filtrations up to the present (t0), it later delves into leveraging future filtrations for enhanced computational efficiency. It becomes possible to simulate future scenarios and develop diverse outcomes. However, given the inherent uncertainty, determining the most realistic scenario remains challenging. Estimating the distribution of outcomes involves utilizing historical data and calibration techniques associated with measure p.
The lecture then delves into measures and filtrations, highlighting the distinct roles of measure Q in pricing and risk management, and measure P primarily in risk management. When both measures are employed, generating future scenarios for risk profiles becomes imperative due to the non-uniqueness of either metric's appropriateness. Furthermore, as time progresses, the accumulation of historical knowledge leads to broader filtrations. However, maintaining an understanding of measurability and acknowledging uncertainty for stochastic quantities at specific future times is also essential.
The lecturer proceeds to discuss filtrations and measures within the context of financial engineering. Notably, they emphasize that measurability does not imply constancy; rather, it denotes a stochastic quantity. Filtrations elucidate the extent of knowledge available at each given time, expanding as one moves further in time due to accumulated knowledge. While filtrations and measure changes can be powerful tools in financial modeling, their inappropriate usage can yield significant issues. Thus, it is crucial to grasp how to effectively employ these tools and navigate through time to avoid modeling errors. The section concludes with an overview of the calibration process in financial modeling, which can be inferred from historical data or market instruments.
The concept of adapted processes is introduced, referring to processes that solely rely on information available until a given moment, without considering future realizations. Examples of adapted processes encompass Brownian motion and determining the maximum value of a process within a specific time period. Conversely, non-adapted processes rely on future realizations. The lecture also introduces the tower property, a powerful tool in pricing, which establishes a relationship among sigma fields, filtrations, and expectations.
Conditional expectation is discussed as a potent tool in financial engineering, particularly when dealing with functions involving two variables. The tower property of expectation is utilized to condition expectations and compute outer and nested inner expectations. This property finds application in simulations, enabling the analytical calculation of certain problem components that can be applied to blockchain option pricing models, particularly employing stochastic differential equations and specific filtrations. The definition of conditional expectation is explored, incorporating an integral equation.
The lecturer emphasizes the importance of conditional expectations and filtrations in financial engineering. They highlight that if a random variable can be conditioned and its answer is known analytically, the outer expectation can be calculated by sampling for the inner expectation. However, in finance, it is uncommon to possess analytical knowledge of conditional densities or two-dimensional densities. The lecturer stresses the importance of using conditional expectations correctly in coding, as they remain stochastic quantities from the perspective of the present. Furthermore, they discuss the benefits of incorporating an analytical solution for a portion of the model in a simulation context, as it can result in improved convergence. To illustrate these concepts, the lecturer provides an example of calculating the outer expectation of a Brownian motion.
Moving forward, the lecturer delves into the expectation of a future point in time, highlighting its complexity compared to cases where the expectation is at time zero. They explain that this scenario requires multiple paths and nested Monte Carlo simulations for each path, involving sub-simulations for conditional expectations. This complexity arises due to the property of independent increments, wherein Brownian motion can always be expressed as the difference between its values at two different times, t and s.
Shifting focus to Monte Carlo simulations, the speaker discusses the construction of Brownian motion for simulating the option value of a stock. They explore two types of martingales and introduce the nested Monte Carlo method for calculating the conditional expectation of a stock option. The simulation involves generating one path up to time s and conducting sub-simulations for each path to evaluate the expectation at that time. This process entails calculating the conditional expectation of a specific realization at time s for each path. The error is then measured as the difference between the conditional expectation and the path value at time s. The standardization of Brownian motion ensures that it is constructed using independent increments, facilitating the enforcement of desired properties within a Monte Carlo simulation.
Lastly, the speaker underscores that while simulating Brownian motion may seem straightforward and cost-effective, incorporating a conditional expectation requires a nested Monte Carlo approach, which involves performing multiple simulations of Brownian motion for each path. Consequently, this process can be time-consuming.
In conclusion, the lecture extensively covers topics related to measures, filtrations, conditional expectations, and Monte Carlo simulations in financial engineering. The significance of these concepts in derivative pricing, risk management, and model calibration is emphasized throughout. By understanding the principles underlying these tools and techniques, financial professionals can enhance their modeling accuracy and effectively navigate complex pricing problems.
Financial Engineering Course: Lecture 2/14, part 2/3, (Understanding of Filtrations and Measures)
Financial Engineering Course: Lecture 2/14, part 2/3, (Understanding of Filtrations and Measures)
Welcome, everyone, to the post-break session. Today, we will continue with the second block of Lecture 2 in the Financial Engineering course. In this block, we will delve into the pricing and interest rates of XVA, focusing on advanced concepts.
Previously, we discussed the concept of filtration and conditional expectations, along with an exercise and simulation in Python. Now, we will explore additional expectations that are more advanced than the experiments we conducted earlier. Specifically, we will concentrate on option pricing and leverage tools from conditional expectation to improve convergence in Monte Carlo simulations. Additionally, I will introduce you to the concept of a numeraire and its usefulness in derivative pricing.
In this block, we will not only use the concept of a numeraire but also the Girsanov theorem to transform the dynamics of the Black-Scholes model from the risk-neutral measure (measure P) to measure Q. This transformation involves changing the underlying process to geometric Brownian motion. It's important to note that measure P is associated with historical observations, while measure Q is typically linked to derivative pricing.
Moving on to the third block, we will focus on detailed measure changes. I will demonstrate multiple advantages and tricks for using measure changes to reduce dimensions and reap significant benefits. However, for now, let's concentrate on the following four elements of today's lecture and enjoy the session.
Firstly, we will utilize our knowledge of conditional expectation and filtration to address real option pricing. Specifically, we will consider a European option and explore how conditional expectations can help determine its price. We will work with a more complex stochastic differential equation, resembling the Black-Scholes model but with stochastic volatility. While Black-Scholes assumes constant volatility (sigma), we will generalize the model to include time-dependent and stochastic volatility.
By leveraging the tower property of expectations, we can solve this problem and improve our Monte Carlo simulations. Instead of directly simulating paths and randomly sampling the stochastic volatility (j), we can achieve better convergence by utilizing conditional expectations. By conditioning on the realization of j, we can apply the Black-Scholes pricing formula for each j. This approach significantly reduces uncertainty and correlation-related issues in Monte Carlo simulations.
In the next section, I will introduce an exact representation for pricing European options based on conditional expectations and the Black-Scholes formula. This will involve inner and outer expectations, where the inner expectation conditions on a specific realization of j and applies the Black-Scholes formula. The outer expectation requires sampling from j and using the Black-Scholes formula for each sample.
To quantify the impact of applying the tower property for expectations in Monte Carlo simulations, we will compare two approaches. The first approach is a brute-force Monte Carlo simulation, where we directly sample the expectation without utilizing information from the Black-Scholes model. The second approach incorporates conditional expectations and the Black-Scholes formula. By comparing convergence and stability, we can observe the significant gain achieved through the conditional expectation approach.
I hope you find this information helpful. If you're interested in exploring practical aspects of conditional expectations further, I recommend referring to Chapter 3 (Stochastic Volatility) and Chapter 12 (Pricing of Tablets) in the book. Now, let's proceed to the practical demonstration of this approach using Python code.
After generating the Monte Carlo samples for the stock and volatility, we move on to the next part of the code, which involves calculating the option payoffs for each sample. In this case, we consider a European call option with a strike price of 18. We can compute the option payoff using the following equation:
payoff = np.maximum(stock_samples[-1] - strike, 0)
Next, we calculate the conditional expectation using the Black-Scholes formula. For each sample of the volatility, we compute the option price using the Black-Scholes model with the corresponding volatility value:
volatility_samples = np.exp(j_samples / 2)
d1 = (np.log(stock_samples[0] / strike) + (0.5 * (volatility_samples ** 2)) * maturity) / (volatility_samples * np.sqrt(maturity))
d2 = d1 - (volatility_samples * np.sqrt(maturity))
conditional_expectation = np.mean(np.exp(-r * maturity) * (stock_samples[0] * norm.cdf(d1) - strike * norm.cdf(d2)))
Finally, we compute the overall option price by taking the average of the conditional expectations over all the volatility samples:
option_price = np.mean(conditional_expectation)
By using the conditional expectation approach, we leverage the information from the Black-Scholes model to improve the convergence of the Monte Carlo simulation. This leads to more accurate option prices and reduces the number of Monte Carlo paths required for satisfactory convergence.
It's important to note that the code provided here is a simplified example to illustrate the concept. In practice, there may be additional considerations and refinements to account for factors like stochastic volatility, time steps, and other model assumptions.
Overall, applying conditional expectations in option pricing can enhance the efficiency and accuracy of Monte Carlo simulations, particularly when dealing with complex models that deviate from the assumptions of the Black-Scholes framework.
Now, let's shift our focus to the topic of measure changes in financial engineering. When dealing with system dynamics, it is sometimes possible to simplify the complexity of the pricing problem through appropriate measure transformations. This is especially relevant in the world of interest rates, where there are multiple underlyings with different frequencies. To establish a consistent framework, we rely on measure transformations that bring stochastic processes from different measures into one underlying measure.
In the field of mathematical finance, numeraires play a significant role as tradable entities used to express the prices of all tradable assets. A numeraire is the unit in which the values of assets are expressed, such as apples, bonds, stocks, or money savings accounts. By expressing prices in terms of a numeraire, we establish a consistent framework for transferring goods and services between different counterparties.
In the past, assets were often expressed in terms of gold or other numeraires. The choice of a proper numeraire can significantly simplify and improve the complexity of financial engineering problems. Working with martingales, which are processes without drift, is particularly favorable in finance as they are easier to handle than processes with drift.
Different measures are associated with specific dynamics of processes and tradable assets. Common cases include the risk-neutral measure associated with money savings accounts, the T-forward measure associated with zero coupon bonds, and the measure associated with stocks as numeraires. Measure changes provide a way to switch between measures and benefit from the properties of different processes. The Girsanov theorem is a crucial tool for measure transformations, allowing us to switch from one measure to another under certain conditions.
While the theoretical aspects of measure changes can be complex, this course focuses on practical applications and how to apply the theory to real problems. The main takeaway is to understand how measure changes and martingales can be used as tools to simplify and solve financial engineering problems effectively.
It's important to note that measure changes are powerful tools that can help us handle processes without drift, known as martingales. By appropriately changing the measure, we can remove the drift from a process and simplify the problem at hand. This is particularly useful when dealing with stochastic interest rates and stock dynamics.
However, it's worth mentioning that measure changes may not always be feasible or result in simpler problems. Sometimes, even after removing the drift, the dynamics of certain variables, such as the variance, can remain complex. Nonetheless,
in general, removing the drift through measure changes simplifies the problem.
Working with martingales is favorable because driftless stochastic differential equations are easier to handle than those with drift. By identifying appropriate numeraires and performing measure changes, we can effectively reduce complexity and improve our simulation techniques.
Measure changes allow us to switch between measures and benefit from the properties of martingales. Understanding and applying measure changes is a valuable skill that can greatly simplify the pricing and analysis of financial instruments.
Now, let's delve deeper into the concept of measure changes and their practical application in mathematical finance. The measure transformation formula we discussed earlier can be written as follows:
dQb/dQa = exp(-1/2 * ∫₀ᵗ yₛ² ds + ∫₀ᵗ yₛ dWₛ)
This formula enables us to switch from one measure, Qa, to another measure, Qb. It involves the use of a specific process called the "numeraire process" denoted by yₛ and the Wiener process Wₛ.
The Girsanov theorem states that under certain conditions, such as the integrability condition on the exponential term, this measure transformation is valid. By applying this transformation, we can change the measure from Qa to Qb and vice versa.
In practical applications, measure changes are used to simplify and solve real-world problems in mathematical finance. They allow us to transform the dynamics of stochastic processes and leverage the properties of martingales.
By appropriately selecting numeraires and performing measure changes, we can remove the drift from a process and simplify the problem at hand. This simplification is particularly beneficial when dealing with complex models involving stochastic interest rates and stock dynamics.
It's important to note that measure changes may not always result in simpler problems. Sometimes, even after removing the drift, certain variables such as variance may still exhibit complex dynamics. However, in general, measure changes provide a powerful tool for simplifying and solving financial engineering problems.
In this course, our focus will be on the practical application of measure changes in real-world scenarios. We will explore how to extract the benefits of measure changes and martingales to simplify complex problems in mathematical finance.
To summarize, measure changes play a crucial role in mathematical finance by allowing us to switch between measures and leverage the properties of martingales. By understanding and applying measure changes, we can simplify the pricing and analysis of financial instruments, enhance our simulation techniques, and tackle complex models more effectively.
Financial Engineering Course: Lecture 2/14, part 3/3, (Understanding of Filtrations and Measures)
Financial Engineering Course: Lecture 2/14, part 3/3, (Understanding of Filtrations and Measures)
Continuing with the lecture, the instructor delves further into the topic of measure changes and their practical applications in finance. They begin by providing a refresher on the Girizanov theorem and the concept of a stock measure. By establishing a foundation, the instructor sets the stage for exploring how measure changes can effectively reduce dimensionality in financial models.
The lecture focuses on the transition from a risk-neutral measure to a money savings account measure driven by the stock asset. This transition is achieved by utilizing the ratio of the two measures, and the process is explained in simple terms. Emphasis is placed on the importance of expressing the chosen asset in the same unit as other assets in one's portfolio, which can be accomplished through measure changes. Additionally, the lecture delves into the discussion of the payoff function, where the expectation under the associated measure is expressed as the integral over one divided by the measure. This result provides a means of finding the desired query. The lecture concludes by showcasing the substitution method employed to obtain the final term, further illustrating the practicality of measure changes.
Moving forward, the speaker explores the simplification of the payoff and delves into the dynamics of the stock under the new measure. The value of t0 is provided as the expectation under measurements of maximum st minus k 0, introducing a new martingale method. The concept of the martingale approach is elucidated, stressing the importance of dividing everything by the stock process to satisfy the conditions for a martingale. The discounting process is highlighted, with an emphasis on its benefits in simplifying dynamics under the new measure. The dynamics can be derived from the ratio of mtst as a martingale. Furthermore, the speaker underscores the need to determine the variance and the measured transformation under the new measure to leverage the advantages of the martingale approach effectively.
Expanding on the lecture, the lecturer explains how the same procedure used for the Black-Scholes case can be applied to non-martingale processes. By following a set of necessary conditions, one can utilize measure transformations to derive the dynamics of a new process and determine expectations under a new measure. The importance of accounting for corrections on drift and volatility resulting from this transformation is emphasized when implementing both processes under the original and the new measure. Ultimately, the calculation simplifies to an elegant expression involving a single log-normal process under the new measure.
In addition, the lecturer introduces a two-dimensional system of stochastic differential equations, S1 and S2, along with a payoff value associated with a money savings account that pays out only if S2 reaches a certain level. To calculate this complex expectation, the joint distribution between the two stocks becomes necessary. Measure transformation is employed, leveraging the Girsanov theorem to find the expectation in an elegant form. The lecturer explains the process, with S1 chosen as the numerator and the random numeraire derivative identified. The lecture also highlights the significance of deriving all necessary measure changes and explores the potential impact on the relationships between Brownian motions in different measures. The lecturer emphasizes the importance of measure transformation in elegantly and powerfully pricing complex financial instruments.
Continuing with the lecture, the speaker elucidates the measured transformation for the random nicotine derivative and emphasizes the importance of simplifying the payoff. The formula for the equation is explained, along with the corresponding measure that must be found to cancel out terms. The dynamics of the money-savings bond and its drift and volatility coefficients are discussed after applying the ethos lemma. In this transformation, the correlation element is found to be negligible. The speaker also highlights the significance of the relationship between S2 and S1 in relation to the ethos table.
Shifting focus, the speaker discusses the dynamics of two stock processes under the S1 measure transformation, which involves the substitution of a new measure.
Under the S1 measure transformation, the speaker explains that the first stock process still follows a log-normal distribution but with an additional term in the drift. Similarly, the second stock process exhibits an additional term due to the correlation between the two processes. The speaker emphasizes the importance of ordering the variables from simplest to most advanced and recommends utilizing Cholesky decomposition as a technique to simplify stochastic differential equations. By leveraging the log-normal properties, the probability of evaluation can be effectively solved.
Expanding the lecture's scope, the lecturer moves on to discuss zero coupon bonds, which are fundamental derivatives in the interest rate domain. Zero coupon bonds have a simple payoff—a single value received at a maturity time—making them easy to understand and use. Furthermore, they serve as crucial building blocks for pricing more complex derivatives. It is noted that in certain cases, the value of a bond at inception can be greater than one, indicating negative interest rates. Negative rates can result from central bank interventions aimed at increasing liquidity, although their effectiveness in stimulating spending remains a subject of debate. The lecturer emphasizes that zero coupon bonds play a crucial role in the process of measure changes in the interest rate world.
Moreover, the lecturer delves into the importance of changing the measure to the forward measure when considering zero coupon bonds. By employing the fundamental pricing theorem and the generic pricing equation, the current value of a zero coupon bond can be derived. The pricing equation involves an expectation of a discounted payoff, which equals one for a zero coupon bond. The lecturer emphasizes that interest rates are stochastic and discusses how the stochastic discount can be eliminated from the equation by changing the measure to the T forward measure. The section concludes with an explanation of how a Ruble code derivative can be modeled, and how the pricing equation shifts from the risk-neutral measure to the T forward measure.
Furthermore, the professor emphasizes the significance of changing measures and reducing dimensionality in pricing models within finance. By transitioning to prices under the T forward measure and eliminating the specificity from the discount factor, practitioners can utilize measure change techniques as powerful tools in their daily operations. The lecture summarizes the concept of filtrations and their relation to conditional expectations, emphasizing how these tools can simplify complex problems in finance.
To engage the students and reinforce their understanding, the instructor presents three exercises. The first exercise entails implementing an analytical solution for pricing put options, ensuring the code incorporates interest rates in Python. The second exercise extends the pricing to put options, providing an opportunity to assess its effectiveness. Finally, students are tasked with comparing the analytical expression to the Monte Carlo simulation result for the squared stock expression on slide 24. This exercise highlights the benefits and substantial differences in applying measure transformations.
The lecture provides a comprehensive exploration of measure changes and their applications in finance. It covers topics such as the switching of measures, simplification of payoffs, dynamics under new measures, transformation of processes, and the significance of zero coupon bonds and interest rates. By leveraging measure transformations, practitioners can enhance their pricing models, simplify calculations, and gain valuable insights into complex financial instruments.