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How is the money savings account related to a zero-coupon bond?
How is the money savings account related to a zero-coupon bond?
Welcome to today's Q&A session on computational finance. In this session, we will discuss question number two, which is based on the material covered in lecture number one. For a detailed understanding, I recommend revisiting lecture number one. Today's question focuses on the relationship between a money savings account and a zero-coupon bond, particularly in the context of interest rates.
To begin, let's define a money savings account. The time value of money states that if we have one euro today and we are interested in its future value, considering a simple interest rate, the amount we will receive in one year will be one euro times (1 + interest rate). This interest rate is expressed as a percentage. This is a straightforward calculation in the case of deterministic interest rates.
However, when we introduce stochastic interest rates, the relationship becomes more complex and interesting. In such cases, the difference between managing a savings account and a zero-coupon bond becomes crucial. Let's define the money savings account and the zero-coupon bond to understand the difference more clearly.
The money savings account (MSA) at time T is defined as the initial value (which can be considered as one for simplicity) multiplied by e^(RT), where R represents the interest rate. You can find detailed derivations of the MSA in lecture number one. In the case of stochastic interest rates, the MSA can be expressed as M(T) = M(0) * e^(∫[0 to T] R(s) ds), where R(s) represents the stochastic interest rate and the integral accounts for the integration of the stochastic quantity.
Now let's discuss the definition of a zero-coupon bond. A zero-coupon bond is a contract that pays one euro at a future time T. The pricing problem associated with a zero-coupon bond is to determine its value today. In other words, we want to find the present value of the future payment. This is a fundamental problem in computational finance, as we always focus on determining the value of contracts today to establish their fair value.
In the case of stochastic interest rates, the fundamental pricing theorem states that the value of a contract with a future payment at time T, discounted to today under the risk-neutral measure, can be expressed as an expectation. Specifically, it is the expectation of the integral of interest rates. This can be seen as an extension of the concept of the MSA, where the expectation and the negative sign differentiate it from the MSA. So, the zero-coupon bond can be expressed as an expectation of -∫[0 to T] R(s) ds.
To summarize, the relationship between the money savings account and the zero-coupon bond can be described as follows: M(T) = initial value * e^(∫[0 to T] R(s) ds) for the MSA, while the zero-coupon bond is defined as the expectation of -∫[0 to T] R(s) ds. In deterministic cases, the relationship is more straightforward, with the zero-coupon bond being equal to 1 / M(T), where M(T) is the MSA value at time T.
Understanding this relationship is essential in computational finance, especially when dealing with stochastic interest rates. It plays a crucial role in financial engineering and pricing problems. The concept of change of measure, as explained in this course, is a powerful tool that simplifies complex payoffs and often allows us to find analytical pricing equations. If you're interested in this topic, I recommend exploring the course on financial engineering available on this channel.
I hope this explanation clarifies the differences between the money savings account and the zero-coupon bond. The main distinction lies in the expectation term, which becomes significant when dealing with stochastic interest rates. In the absence of stochastic interest rates, the relation between the money savings account and the zero-coupon bond is more straightforward. In such cases, if we have a constant interest rate, the expression for the zero-coupon bond would simply be 1 / M(T), where M(T) represents the value of the money savings account at time T.
However, when stochastic interest rates are introduced, the expectation term becomes crucial. The integration of the stochastic interest rates in the zero-coupon bond calculation accounts for the uncertainty and variability of the interest rates over time. This adds complexity to the relationship between the two financial instruments.
Understanding the dynamics and relation between the money savings account and the zero-coupon bond is essential in the field of computational finance. It enables us to analyze and evaluate the values of various financial contracts and determine their fair prices. The concept of change of measure, covered in this course, provides a powerful framework for simplifying complex payoffs and deriving pricing equations.
In conclusion, the money savings account and the zero-coupon bond are closely related, but they differ in terms of their mathematical formulation. The money savings account represents a compounded value of a principal amount over time, whereas the zero-coupon bond calculates the present value of a future payment through an expectation of integrated interest rates. This distinction becomes more significant and intriguing when dealing with stochastic interest rates. By understanding this relationship, financial professionals can make informed decisions and effectively navigate the world of computational finance.
What are the challenges in the calculation of implied volatilities?
What are the challenges in the calculation of implied volatilities?
Welcome to the questions and answers based on the course of computational finance. Today, we will delve into question number three, which relates to the challenges in calculating implied volatilities, specifically in the context of the Heston model.
When discussing implied volatilities, we typically refer to Black-Scholes implied volatilities, unless stated otherwise. Therefore, for the Heston model, if we are asked how to derive implied volatility, we cannot simply invert the Heston formula solely for the long-term mean or initial variance. Implied volatility in the Heston model requires a two-step process: calculating prices based on the Heston model and then utilizing these prices in the Black-Scholes formula for inversion to find the corresponding sigma.
The Heston model introduces multiple parameters for the variance, which adds complexity to the calculation. Unlike the Black-Scholes model, where we have a single parameter, the Heston model's multiple parameters prevent us from re-inverting to obtain a unique set of parameters.
Implied volatilities are valuable tools for comparing the behavior and performance of different stocks, as they allow for relative comparisons that consider the current value of the stock. Implied volatility incorporates uncertainty, which helps evaluate the risk and uncertainty associated with option valuations.
The concept of implied volatility has been around for many years, and it became apparent that the Black-Scholes model was not suitable for pricing options due to its single parameter. In practice, different options with varying strikes and expirations often exhibit different implied volatilities. This discrepancy suggests that a constant volatility assumption is not appropriate for pricing all options simultaneously. Thus, the challenge lies in finding the implied volatilities that align the prices from the model with the prices observed in the market.
The calculation of implied volatilities involves inverting the Black-Scholes formula, which is a non-trivial task. Several numerical routines, such as Newton's method or Brent's method, are commonly used for this purpose. These methods aim to find the unknown implied volatility by solving an equation that equates the Black-Scholes price from the model with the market price of the option.
Efficient calculation of implied volatilities is crucial, especially in high-frequency trading or when calibrating models to market data. The speed of calculation can significantly impact trading strategies or the effectiveness of model calibration. Therefore, developing fast and accurate numerical routines for implied volatility calculations is of great importance.
The challenge intensifies when dealing with out-of-the-money options, where the call option surface becomes extremely flat. In such cases, iterative search algorithms can struggle to converge or may require a large number of iterations to find the optimal point due to the lack of accurate gradients. Thus, determining a suitable initial guess becomes crucial to ensure the efficiency and effectiveness of the calculation.
It's worth noting that implied volatilities are primarily associated with Black-Scholes implied volatility. However, it is possible to have implied volatilities based on other models, such as arithmetic Brownian motion or shifted log-normal distributions. In such cases, it is essential to explicitly state the model used for the calculations.
In conclusion, calculating implied volatilities poses challenges related to speed, especially when dealing with out-of-the-money options. Efficient numerical routines and careful consideration of initial guesses are necessary for accurate and fast computations. Implied volatilities play a vital role in options pricing, risk assessment, and model calibration, making their calculation and understanding crucial in computational finance.
Can you price options using Arithmetic Brownian motion?
Can you price options using Arithmetic Brownian motion?
Welcome to the Computational Finance course Q&A session!
Today's question is number four, which focuses on pricing options using arithmetic Brownian motion. This question is based on the materials discussed in Lecture Number Two.
Arithmetic Brownian motion is a slightly different process compared to geometric Brownian motion, which we have seen before. When it comes to pricing options, such as using the Black-Scholes model, the main difference lies in the volatility and drift. In this simplified version of the model, the volatility term and derivative are adjusted.
In a market scenario, let's consider a specific strike price (K) and expiry (T). We observe an option price (C1). Based on our knowledge, we can easily find the implied volatility for geometric Brownian motion. Similarly, in this case, we can find an implied volatility (Sigma tilde) that matches the observed option price perfectly in the market. However, it's important to note that the two models are not equivalent. The difference between them becomes apparent when we examine the sensitivities, also known as the Greeks.
Arithmetic Brownian motion assumes that stock realizations can become negative, which is unrealistic. In contrast, geometric Brownian motion assumes only positive stock paths. This difference necessitates adjusting our hedging strategy to account for negative stock realizations, making the assumption of arithmetic Brownian motion less realistic.
While comparing option prices may provide some insights, it is not always the best criterion to determine whether a model is good enough. Additionally, both geometric and arithmetic Brownian motion models are unable to calibrate to implied volatility smile or skew. However, in this specific case, where we consider a market with only one particular option, we can easily compare the two models and determine which one is more suitable.
Similar considerations can be made for the OU process, where the volatility parameter (Sigma) is fixed. However, the OU process faces additional issues, such as the drift, which is not well defined under the risk-neutral measure in terms of a stock divided by money savings accounts. Hence, it is not a viable process for pricing options.
To provide visual examples, I have prepared a few realization paths for the three stochastic differential equations: geometric Brownian motion, arithmetic Brownian motion, and the OU process. In the simulations, the same Brownian motion is used, resulting in similar shapes and patterns among the paths.
In summary, while it is possible to price options using arithmetic Brownian motion, it may not always be the most sensible approach. The adequacy of a model depends on whether the underlying assumptions and dynamics of the asset reflect the physical properties of the market. That is the key element to consider.
What is the difference between a stochastic process and a random variable?
What is the difference between a stochastic process and a random variable?
Welcome to the Computational Finance course Q&A session!
Today's question is number five, which focuses on the difference between a stochastic process and a random variable. This question is based on the materials discussed in Lecture Number Two.
A stochastic process is essentially a collection of random variables that are parametrized with respect to time. Formally, we can represent a stochastic process as X(t), where we have two arguments: time (t) and Omega (Ω), which corresponds to the probability space. In contrast, a random variable is a simpler concept that does not have this time dependence. For example, if we are tossing a coin and considering the outcomes of "tails" or "heads," it is a random variable. However, if we introduce time into the equation and consider the occurrences of "tails" or "heads" over time, it becomes a stochastic process.
In both industry and academia, we often neglect the second argument (Omega) when discussing stochastic processes. Instead, we refer to the process as X(t) rather than dX(t, Ω), which would provide a complete definition of a stochastic process.
It's also important to understand how to interpret simulated Monte Carlo paths and their connection to time and Omega. If we plot the values of the process X(t) over time, we can observe multiple Monte Carlo paths. Each path represents a possible realization of the process. If we fix a specific time, let's say t*, and look at the distribution of all realizations at that point, we are considering different outcomes (Omegas) at a given time. On the other hand, we can fix a specific realization (Omega) and observe how the process evolves over time, resulting in a single path. Therefore, we have two dimensions to consider: fixing time to analyze distributions of outcomes or fixing a realization to observe the process's behavior over time.
In summary, a stochastic process is a collection of random variables that are parameterized with respect to time. It represents the evolution of a system over time and can be observed through Monte Carlo paths. A random variable, on the other hand, is a simpler concept that does not depend on time. Understanding this distinction is crucial when studying computational finance.
What are the advantages and disadvantages of using ABM/GBM for modelling a stock process?
What are the advantages and disadvantages of using ABM/GBM for modelling a stock process?
Welcome to the Questions and Answers session on Computational Finance!
Today's question is number six, which explores the advantages and disadvantages of using arithmetic Brownian motion or geometric Brownian motion for modeling a stock process. This question is based on Question Number Two and is similar to the one discussed in a previous session where arithmetic Brownian motion was used for pricing options.
The difference between these two processes is relatively small, primarily concerning whether we consider an asset that allows for both positive and negative values or focus only on positive assets like stocks. Today, we will delve into the aspects that help us determine whether arithmetic Brownian motion or geometric Brownian motion is suitable for pricing a particular derivative in various scenarios.
Let's consider a case where we have an exotic derivative that we need to price. This derivative is complex, possibly involving callability features. To assess whether arithmetic or geometric Brownian motion is adequate for pricing, we need to examine certain factors.
The first question to ask is whether the market for exotic derivatives in this asset class is rich. If there are other exotic derivatives available, it suggests that we should consider a model that allows calibration to these market prices. We can then extrapolate the pricing to the derivative of interest. However, if the market is not rich, it means we can price the exotic derivative, but there are no additional exotic derivatives available for calibration.
In the latter case, we move to the next step and check if there are options available for this market. If there is an option market, we should first calibrate our model to these options, typically liquid instruments. This calibration helps determine the model parameters. Once we have the calibrated model parameters, we can use them to price the exotic derivative.
If there are no calls and puts available in the market, we encounter a scenario where there are no market instruments to utilize. In such cases, for example, a market with no implied volatilities for calls and puts, we may consider that the Black-Scholes model or geometric Brownian motion is suitable for pricing the exotic derivative. However, in this situation, it is essential to note that calibration of the Sigma parameter should be sufficient. One might argue that if we lack hedging instruments, such as underlying calls and put options, for a derivative with advanced features like callability, it may not be advisable to trade that derivative. Nevertheless, from a purely theoretical perspective, geometric Brownian motion can be used in such scenarios with limited market information.
It's crucial to understand that if there are more instruments available in the market, such as other exotic derivatives or calls and puts, pricing the exotic derivative using geometric Brownian motion is not suitable. The model cannot calibrate sufficiently well to the implied volatility smile and skew with only one free parameter.
In summary, the choice of a pricing model is always based on the type of derivative we aim to price. We need to consider the availability of market instruments to judge the adequacy of a model. If there are market instruments available, models like geometric Brownian motion or simple Black-Scholes models are not suitable. However, for pricing implied volatilities, geometric Brownian motion is still applicable. But for pricing exotic derivatives and more complex assets, it is not the preferred choice.
In terms of advantages and disadvantages, the advantages of these models are minimal. They allow for a physical representation that considers whether the market permits positive or negative assets. However, they have limited degrees of freedom for model calibration, which makes them unsuitable for pricing exotic derivatives.
I hope this explanation clarifies the advantages and disadvantages of using arithmetic Brownian motion or geometric Brownian motion for modeling stock processes and pricing derivatives. See you next time! Goodbye.
What sanity checks can you perform for a simulated stock process?
What sanity checks can you perform for a simulated stock process?
Welcome to the Questions and Answers session based on the Computational Finance course.
Today's question is number seven, which focuses on the sanity checks that can be performed for a simulated stochastic process. This question relates to practical exercises involving the simulation of a discretized stochastic differential equation for pricing purposes. It is essential to perform certain checks to ensure that the implementation is correct and to gain confidence in the validity of the results.
To address this question, let's examine several steps and checks that can be performed. Firstly, it is important to consider the particular asset class being simulated. For example, if we simulate a stock process, a simple check is to assess whether the discounted stock follows the Martingale property. The expectation of the stock at maturity, discounted to today, should be equal to the initial stock value. In reality, there may be a slight difference, which should decrease as the number of simulation paths increases or the grid size decreases. Monitoring and minimizing this difference can help improve the simulation accuracy.
Another aspect to check is whether the derivative being priced can be simplified. For instance, if a call option with a strike price of zero is chosen, it essentially reduces to the first check mentioned above. Verifying the proper implementation of the derivative's payoff is crucial.
Stability is another important consideration. It involves assessing the impact of increasing the number of Monte Carlo paths and the stability of results when changing the random seeds. If simulations with different seeds yield significantly different prices, it indicates potential instability in the model. Adjustments such as drift correction or Martingale correction terms may be necessary to ensure stability.
Additionally, it is valuable to observe how the results vary when changing the discretization step size of time intervals. This helps assess the sensitivity of the simulation to different time resolutions.
One critical check is whether the simulated process can price back the market instruments. If the model parameters are calibrated to market instruments such as options, comparing the model's prices to the market prices is essential. If the prices significantly differ, it suggests that the model is not performing well and may require adjustments or additional calibration.
These are some of the basic sanity checks that can be performed for simulated stochastic processes. It is worth noting that the specific checks may vary depending on the type of pricing contract being considered. For instance, for options with exercise dates, it is important to ensure that they collapse to European-type payoffs as a base case scenario.
Performing these checks helps validate the simulation and identify any potential issues or bugs in the implementation.
What is the Feynman-Kac formula?
What is the Feynman-Kac formula?
Welcome to the Serious Questions and Answers session on Computational Finance.
Today's question is number eight from Lecture number three, which focuses on the Feynman-Katz formula and its application. The Feynman-Katz formula establishes a crucial connection between partial differential equations (PDEs) and stochastic processes, providing a method for solving specific PDEs through the simulation of random paths. This powerful machinery enables us to solve complex problems by combining PDEs with stochastic processes.
The formula itself relates to a particular form of a partial differential equation. Consider a PDE with a time derivative term (dt), a drift term (μ), a first-order derivative term (dX), a volatility term (σ²/2), and a second-order derivative term (d²X). The PDE also includes a terminal condition, where the value V takes on a deterministic function ETA(X) at time T. Here, X represents a state variable.
The Feynman-Katz theorem states that the solution to this PDE can be expressed as the expectation of the deterministic function ETA evaluated at time T, considering it as a function of a stochastic process. The stochastic process, denoted by X(t), can be defined as follows: dX(t) = μ dt + σ dW(t), where dW(t) represents a Wiener process (Brownian motion). The drift term μ and the volatility term σ² are determined by the coefficients of the PDE.
If we have a PDE in the form of dt + μ dX + (σ²/2) d²X = 0, along with a terminal condition, we can express the solution as the expectation of the terminal condition evaluated at X(t), the stochastic process at time T.
Let's consider a simple example where the PDE only includes the second-order derivative term and a terminal condition. By applying the Feynman-Katz theorem, we know that the solution will be the expectation of the function ETA, which in this case is x². Thus, the solution can be written as the expectation of X(t)², where X(t) is a scaled Brownian motion with some initial state. Calculating the expectation yields Sigma²(T-t) + X².
The Feynman-Katz formula is a powerful tool in finance, particularly in pricing options. For instance, in the Black-Scholes equation, we start with a replicating portfolio, which leads to a pricing PDE. By following the same strategy, the pricing PDE can be elegantly related to the simulation of the expectation of the terminal payoff based on the stochastic process. This connection between expectation and the PDE provides a comprehensive framework for option pricing, where we can replicate the portfolio, derive the pricing PDE, and then simulate the expectation through Monte Carlo paths or simulated stochastic processes.
Understanding and utilizing the Feynman-Katz formula is essential in various financial applications. It offers a powerful method for solving PDEs and provides a clear link between stochastic processes and partial differential equations.
Thank you, and see you next time!
What is the implied volatility term structure?
What is the implied volatility term structure?
Welcome to the Questions and Answers session based on lectures in Computational Finance.
Today's question is number nine, which is related to the material covered in Lecture number four. The question is, "What is the implied volatility term structure?" This question often arises when discussing the impact of time-dependent volatility on the Black-Scholes model and whether it can generate implied volatility smile or skew. Unfortunately, the common answer stating that a time-dependent volatility can produce smile or skew is incorrect. Let's explore the implied volatility term structure and its connection to the Black-Scholes model.
To understand implied volatility, we need to know how it is calculated and its meaning in the context of the Black-Scholes model. In the standard Black-Scholes framework, given the market price of a call option, we aim to find the implied volatility (Sigma_imp) that makes the difference between the market price and the Black-Scholes price zero. This implied volatility is derived by inverting the Black-Scholes pricing equation.
When comparing option prices obtained from the model with those observed in the market, it is challenging to determine the presence of implied volatility smile or skew solely based on prices. Instead, we should focus on implied volatilities. Looking at the implied volatilities, we observe that market option prices decrease for increasing strike prices (k), which is expected. However, the behavior of implied volatilities can vary significantly. In some cases, they may be flat, while in others, they may exhibit skew. It is crucial to examine implied volatilities rather than prices to assess the presence of volatility smile or skew accurately.
Implied volatilities can take various shapes, including smile, skew, or even a hockey stick shape, depending on the market conditions. Different types of markets exhibit different implied volatility patterns, and accordingly, different models and calibration procedures are required to match those patterns.
Now, let's discuss the term structure of implied volatility. In the term structure, we focus on varying the option expiration while keeping the strike price fixed. If we introduce time-dependent volatility in the Black-Scholes model (replacing a constant Sigma with sigma(T)), we find that the implied volatility term structure does not generate smile or skew. Instead, it demonstrates how implied volatilities change for at-the-money options over time. The term structure describes the evolution of implied volatilities as the expirations of options change. In a 3D plot, we observe that for at-the-money options, the implied volatility remains constant as long as the expiration is the same (flat surface). However, as we vary the option expiration, the implied volatilities change, illustrating the implied volatility term structure.
It is essential to note that introducing time-dependent volatility in the Black-Scholes model does not generate implied volatility smile or skew. The model still lacks smile or skew, but it allows calibration to at-the-money options in terms of their implied volatilities over time. In my book and Lecture number four, you will find materials on how to represent option prices (both calls and puts) using time-dependent volatilities by compressing the time dependence into a constant Sigma, known as Sigma star. This allows you to reuse the Black-Scholes pricing framework while considering the term structure associated with at-the-money options.
In conclusion, time-dependent volatility in the Black-Scholes model does not generate implied volatility smile or skew. It solely affects the implied volatilities associated with the term structure of at-the-money options. To assess the presence of smile or skew, always examine implied volatilities rather than option prices.
I hope this explanation clarifies the concept. See you next time. Bye-bye, and thank you!
What are the deficiencies of the Black-Scholes model? Why is the Black-Scholes model still used?
What are the deficiencies of the Black-Scholes model? Why is the BS model still used?
Welcome to the Questions and Answers session based on the Computational Finance course.
Today's question is number 10, which is related to Lecture number four. The question is, "What are the deficiencies of the Black-Scholes model, and why is it still used?"
The Black-Scholes model, as discussed in this course, is a fundamental model for pricing derivatives. It assumes a single stochastic differential equation (SDE) with geometric Brownian motion to represent the stock price. This simple process is then used to price options. However, we have learned that the assumptions of the model are not adequate for the current market conditions.
One major drawback of the Black-Scholes model is its reliance on a single parameter, Sigma, which represents volatility. This single parameter is insufficient to capture the complexity of implied volatility smiles and skews observed in the market. The model's assumption of constant interest rates is also unrealistic, although interest rates have a minimal impact on option pricing compared to volatility.
Another disadvantage of the Black-Scholes model is that the returns generated by geometric Brownian motion are not heavily tailed enough. This means that extreme events with very low probabilities are not adequately accounted for, making the model unrealistic.
Now, why is the Black-Scholes model still used despite these deficiencies? The answer is multifaceted. While the Black-Scholes model is not suitable for pricing exotic derivatives, it can still be used for pricing European options. European options are simpler and have more liquid markets, allowing for easier hedging using vanilla European options. Therefore, if there are no other market instruments available, the Black-Scholes model may be used to price exotic derivatives. However, it is important to note that this approach is risky as it lacks the ability to hedge the exotic derivatives effectively.
Additionally, the Black-Scholes model is widely used in the calculation of implied volatilities. Implied volatilities are an essential tool for option traders and are derived using the Black-Scholes formula. Even when employing more complex models like the Heston model or models with jumps, the implied volatilities associated with those models are still calculated using the Black-Scholes formula. Implied volatilities are preferred because they provide a measure of volatility independent of the asset's level, allowing for meaningful risk comparison across different assets.
In this course, we have explored various alternatives to the Black-Scholes model, such as stochastic volatility models and local volatility models, which offer improvements over the Black-Scholes framework. I encourage you to revisit the lectures if you need a more in-depth understanding of these alternatives.
Thank you very much, and I look forward to our next session.
How does Ito’s table look like if we include the Poisson jump process?
How does Ito’s table look like if we include the Poisson jump process?
Welcome to the Questions and Answers session on Computational Finance. Today we will be discussing question number 11, which is based on the materials covered in lecture five. The question is: How does the Ethos table look like when we include the Poisson jump process?
To begin, let's recall the application of the Ethos lemma to processes involving Brownian motion. We know that in order to find the dynamics of a function of a process, we need to apply the Ethos lemma, which involves a Taylor expansion. The Ethos table for Brownian motion includes terms with dt, dw, dtdw, and dwdw. If we have cross terms with dt multiplied by dw or dtdw, they are considered zero due to symmetry. And dwdw is simply dt.
Now, let's consider the case where we not only have Brownian motion but also a Poisson process included in the dynamics of the process. The Poisson jump process can be represented as a series of jumps that occur at each point in time. If we discretize the process, we can have multiple jumps in a finite interval. However, when considering infinitesimally small intervals, only a single jump occurs. We introduce the notation xt- and xt to represent the left-hand limit and the value of the process just before the jump, respectively.
Now, let's focus on the function G(xt). If we apply the Ethos lemma to a function of a process with a Poisson jump, we obtain an expression that includes a drift term, a jump term, and the increment of G due to the jump. The drift term is similar to the one in the Ethos lemma for Brownian motion, but without the diffusive part. The jump term depends on the Poisson process and consists of the product of the jump size and the indicator function for the occurrence of a jump.
To summarize, the Ethos table for a Poisson jump process includes the terms from the Ethos table for Brownian motion, as well as an additional term that arises from the product of two increments of the Poisson process. This additional term is crucial in the application of the Ethos lemma to jump processes.
It is important to understand the Ethos lemma and its application to jump processes, as it is a powerful tool in finance for analyzing the dynamics of functions of stochastic processes. Further details on this topic can be found in lecture five and relevant literature. Feel free to ask any further questions. Goodbye!