Quantitative trading - page 17

 

Financial Engineering Course: Lecture 11/14, part 1/2, (Market Models and Convexity Adjustments)



Financial Engineering Course: Lecture 11/14, part 1/2, (Market Models and Convexity Adjustments)

In this lecture, the focus is primarily on the library market model and its extensions, specifically stochastic volatility. The library market model aims to consolidate individual measures of Libor rates into a unified and consistent measure for evaluating derivative prices. After providing an overview of the model's history and specifications, the speaker delves into the derivation of the model, exploring popular choices such as log-normal and stochastic volatility.

The second subject covered is convexity correction, which entails defining and modeling these adjustments. The lecture addresses when convexity corrections occur, how to identify them, and their relevance in evaluating derivatives that involve convexity adjustments.

The lecturer emphasizes the significance of market models and convexity adjustments in the realm of financial engineering. Market models offer powerful solutions to various complex problems, particularly in pricing exotic derivatives with intricate payoff structures. However, these models can be cumbersome and expensive. Nevertheless, the Libor market model, or market models in general, have been designed to handle such complications, especially in pricing exotic derivatives dependent on multiple Libor rates.

Furthermore, the lecture explores the development of a unified measure to incorporate multiple Libor rates, a crucial prerequisite for accurate pricing. The machinery employed relies on major change techniques and the forward measure associated with zero coupon bonds. Although closed-form solutions are possible in some cases, the machinery itself is complex and multidimensional.

The speaker discusses the framework for defining interest rate models, highlighting the importance of specifying drift and volatility conditions to ensure the model is well-defined and free of arbitrage opportunities. Valuing complex fixed income products, including exotic derivatives, necessitates advanced models due to their dependency on multiple libraries, making it impossible to decompose them into independent payments. To address this, the Libor Market Model is introduced, developed with a practical approach to maintain consistency with market practices and existing pricing methods for swaptions or options on libraries. This model enables advanced valuation and is arbitrage-free, making it indispensable for pricing complex fixed income products.

The lecture emphasizes the significance of the BGM (Brace Gatarek Musiela) model, which revolutionized the pricing of exotic derivatives. Built upon existing market foundations, the BGM model introduced additional elements that allowed it to be widely accepted as the market practice for pricing derivatives tied to multiple libraries and complex volatility structures. Monte Carlo simulations are often used to separate the processes involved in the BGM model due to the challenges posed by dealing with multiple Libor rates under different measures. The model aims to provide arbitrage-free dynamics for Libor rates, enabling the pricing of caplets and florets in a manner similar to the market convention set by the Black-Scholes formula. While the BGM model simplifies to this fundamental block, it offers additional features to facilitate the pricing of exotic derivatives.

The speaker proceeds to explain the process of obtaining library rates by defining a forward zero bond as a refinancing strategy between time t1 and time d2. Various considerations, such as reset dates, reset delay, and pay delay, need to be taken into account, as mismatches between product payment and discounting require convexity adjustments. Moving forward, the lecture delves into the specification of a multi-dimensional Libor market model, starting with the determination of the required number of Libor rates.

The lecture explores the structure of stochastic differential equations for a system of Libor rates over time. As time progresses, the dimensionality of the system decreases as certain Libor rates become fixed at specific points. The speaker emphasizes the importance of the correlation structure between the Libor rates and its parameterization to ensure a positive definite correlation matrix. The lecture also mentions the role of the forward measure and zero coupon bonds in defining martingales.

Tradable assets and zero-coupon bonds are introduced as martingales. The Libor rate, L(T), and TI-1 are considered martingales under certain conditions. Functions σ(i) and σ(j) are introduced as drivers of the Brownian motion, which must be defined under a consistent measure. The lecture highlights the need for consistency between the expectation measure and the Brownian motion measure used to evaluate expressions. The Libor market model, also known as the BGM model, combines individual sets according to market practices derived from Black-Scholes models, serving as a key point in the model's framework.

The lecture delves into the concept of the Libor Market Model, which utilizes multiple stochastic differential equations to unify different processes under a consistent forward measure. Each Libor rate, under its own measure, acts as a martingale. However, when measures are changed for each Libor rate, it affects the dynamics and drift term. The crucial element of the Libor Market Model lies in determining the transition of the drift and how it behaves when measures change for each Libor rate. This drift term can be complex, and the lecture discusses two common possibilities for choosing the terminal measure or spot measure for pricing derivatives. Additionally, the lecture explores the relationship between the Libor Market Model and other models like AJM (Andersen-Jessup-Merton), Brace Gatarek Musiela Model, and HJM (Heath-Jarrow-Morton), providing insights into their interconnections. The use of full wide volatility for the instantaneous forward rate within the Libor Market Model is also examined.

The lecture addresses the relationship between the instantaneous forward rate and the Libor rate, emphasizing their strong correlation, particularly when the two times approach each other and a running index is present. The process of changing the measure from i to j and finding the drift term through measure transformations is thoroughly explained. The lecture underscores the importance of grasping the concepts covered in previous lectures to comprehend the array of tools and simulations required in the final two lectures.

The instructor delves into measure transformations and the dynamics of the Libor rate under different measures. By employing Girsanov's theorem and making appropriate substitutions, an equation is derived to represent the measure transformation from i-1 to i or vice versa. This equation serves as a basis for representing the LIBOR rate under different measures. The lecture highlights the significance of selecting the appropriate spot or terminal measure for accurate derivative pricing.

The lecture further explains the process of adjusting the drift for different Libor rates within the market model to ensure consistency with the terminal measure. The adjustment involves accumulating all the necessary adjustments for the Libor rates between the first and last rates until reaching the terminal measure. The transition from one measure to another can be derived iteratively, and the process of adjusting the drift is central to the Libor Market Model. However, a challenge arises with the terminal measure, where the shortest period, closest to the present, becomes more stochastic as it involves all the subsequent processes, which may seem counterintuitive. Nevertheless, the Libor Market Model primarily operates under the spot measure as a consensus default, unless a specific payoff is designated to be in the terminal measure.

The speaker addresses certain issues with the library market model, particularly the lack of continuity concerning the times in between the specified tenor grid. To address this, the speaker introduces the strategy of using a discrete three-discretely rebalanced money savings account to define the spot measure for the library market model. This strategy involves observing how one unit of currency invested today can accumulate given the existing tender structure of zero coupon bonds. The strategy is defined not at t0, but at t1, involving the purchase of a bond at t1, receiving the accrued amount at maturity, and reinvesting it for the second bond at t2.

The lecture explains the concept of compounding within a discrete interval structure, which allows investment in zero coupon bonds while reinvesting the received amounts in new bonds. The product of all the zero coupon bond components defines the amount the investor would receive at a specified time. The accumulated amount can be continuously defined by discounting from the last point on the grid to the present point. The lecture introduces the concept of the spot-Libor measure, which allows the running numerator to switch from a ti measure to a tm measure. Additionally, the concept of mt is introduced as the minimum i such that ti is the biggest than t, establishing a link between t and the next bond.

Moving forward, the speaker explains the process of defining the measure transformation from the M_t measure to the M_t+1 measure. This is achieved by employing the Radon-Nikodym derivative. The lecture delves into the dynamics for lambda and psi, which determine the measure transformation and the relationship between Brownian motions under t and n. Finally, the speaker presents the final representation of the library market model, which closely resembles the previously discussed measure transformations in models like the market mode.

Next, the lecture focuses on the dynamics of the Libor market model, particularly its application in pricing advanced and complex exotic products in the interest rate domain. The model poses a high-dimensional problem with a complex drift that encompasses multiple Libor rates, making its implementation challenging. However, the model serves as a valuable problem-solving tool. The lecture explores extensions of the model to incorporate volatility smiles and discusses the selection of the stochastic volatility process while keeping the model's dynamics as simplified as possible. It is noted that the log-normality of the model exists only under the marginal measure and involves a summation of different independent processes, indicating that it is not log-normal in the general case.

The lecture series on the Libor Market Model and its extensions, particularly stochastic volatility, delves into various aspects of the model's framework. It covers the unification of individual Libor rates into a consistent measure, the derivation of the model using popular choices like log-normal and stochastic volatility, and the concept of convexity corrections for pricing derivatives. The lecture emphasizes the importance of understanding measure transformations, dynamics under different measures, and choosing appropriate spot or terminal measures. The model's ability to handle complex fixed income products, its relationship to other market models, and its dynamics and challenges are thoroughly explored. By comprehending these concepts and tools, financial engineers can effectively price exotic derivatives and navigate the intricacies of the interest rate world.

  • 00:00:00 In this section of the Financial Engineering Course lecture, the focus is on the first subject of the library market model and its extensions, specifically stochastic volatility. The library market model aims to bring all individual measures of Libor rates into one consistent measure and evaluate the price derivative. After discussing the history and model specifications, the lecture covers the derivation of the model, including the popular choices of log-normal and stochastic volatility. The second subject is convexity correction, which includes defining what convexity corrections are, identifying when they occur, modeling them, and evaluating derivatives that involve convexity adjustments.

  • 00:05:00 In this section, the lecturer discusses market models and convexity adjustments in relation to financial engineering. Market models are extremely powerful and can be used to solve a variety of serious problems but can also be problematic due to their clumsy and expensive nature. However, the Libor market model or market models are designed to handle very complicated advanced payoff structures in pricing exotic derivatives. The development of a unified measure to incorporate multiple Libor rates into one framework is also discussed, which is necessary for pricing purposes. The machinery relies on major change techniques and the forward measure associated with zero coupon bonds. While closed form solutions are possible in some cases, the machinery itself is complex and multi-dimensional.

  • 00:10:00 In this section, the speaker discusses the framework for defining interest rate models, which involve specifying conditions for drift and volatility to ensure the model is well-defined and arbitrage-free. Complex fixed income products, such as exotic derivatives, require advanced models for valuation, as their payoff depends on multiple libraries and cannot be decomposed into independent payments. The speaker introduces the Libor Market Model, which is developed with a practical approach to ensure it is consistent with market practices and doesn't disturb current pricing practices for swaptions or options on libraries. This model allows for advanced valuation and is arbitrage-free, making it useful in the pricing of complex fixed income products.

  • 00:15:00 In this section of the lecture, the importance of the BGM model and how it allowed for the pricing of exotic derivatives is discussed. The BGM model was based on existing building blocks in the market and added something extra to the framework, allowing it to be accepted as market practice for pricing of exotic derivatives that depend on multiple libraries and complex volatility structures. The separation of the processes involved in the BGM model is mostly done using Monte Carlo, due to the dimensionality issue when dealing with multiple libor rates under different measures. The concept of developing a new model is to provide arbitrage-free dynamics for libor rates and facilitate the pricing of caplets and florets in a similar fashion as the market convention, Black-Scholes formula. The BGM model collapses to this basic fundamental block while still providing something extra to the framework to allow for pricing of exotic derivatives.

  • 00:20:00 In this section of the lecture, the speaker discusses how to arrive at the library rates by defining forward zero bond as a refinancing strategy between time t1 and time d2. The reset dates can be slightly changed, and there are additional dates to take into account, such as the reset delay and pay delay. When there is a mismatch between the payment of the product and the discounting, the speaker explains that convexity adjustments must be made to account for the impact. The speaker then moves on to discuss the specification of a multi-dimensional libor market model, starting with defining the number of libor rates needed.

  • 00:25:00 In this section of the Financial Engineering lecture, the speaker discusses the structure of stochastic differential equations for a system of Libor rates over time. As time progresses, the dimensionality of the system decreases as some of the Libor rates become fixed at certain points in time. The speaker explains that the correlation structure between the Libor rates is also important and can be parameterized to ensure that the matrix of correlations is positive definite. Forward measure and zero coupon bonds are also mentioned in relation to defining martingales.

  • 00:30:00 In this section of the lecture, the concept of the tradable assets and zero-coupon bonds as a martingale are discussed. If we know the Libor, L(T), and TI-1 to be a martingale, then we can define functions σ(i) and σ(j), which become the drivers of the Brownian motion. However, these drivers have to be defined under one consistent measure, and there has to be consistency between the expectation measure and the Brownian motion measure used to evaluate some expression. This is the key point of the Libor market model or BGM model, which combines individual sets according to market practices of pricing using Black-Scholes models.

  • 00:35:00 In this section of the lecture, the concept of the Libor Market Model is explored. This model uses multiple stochastic differential equations to bring different processes together under one consistent forward measure. Each Libor under its own measure is a martingale, but changing measures introduces consequences for the corresponding Libor rate's dynamics and drift term. The Libor Market Model's key element is to determine the transition of the drift and how it behaves when the corresponding measures are changed for each Libor rate. This drift term can be quite complicated, and the lecture will discuss two common possibilities for choosing the terminal measure or spot measure to price derivatives. Additionally, the relation of the Libor Market Model to the AJM, Brace Gatarek Musiela Model, and HJM is discussed, and the lecture explores the use of a full wide volatility for the instantaneous forward rate in the Libor Market model.

  • 00:40:00 In this section of the lecture, the speaker discusses the relationship between the instantaneous forward rate and the Libor rate, which are highly related, especially in cases where the two times are approaching each other, and a running index is present. The lecture also goes into detail about changing the measure from i to j, and finding the drift term relying on measure transformations, which is a key element in transforming Brownian motions under different measures. The lecture emphasizes the importance of understanding previous lectures' concepts to understand the variety of tools and simulations required in the course's last two lectures.

  • 00:45:00 In this section of the lecture, the professor discusses measure transformations and the dynamics of the library under different measures. By using Girsanov's theorem and some substitutions, the professor arrives at an equation that shows the measure transformation from i-1 to i or vice versa. The professor then explains how this equation can be used to represent the LIBOR rate under different measures. The lecture also highlights the importance of choosing the appropriate spot or terminal measure for pricing derivatives.

  • 00:50:00 In this section of the Financial Engineering Course lecture, the instructor explains how to adjust the drift for different libraries in the market model to be consistent with the terminal measure. He explains that all the libraries in between the first and last one need to be adjusted, being accumulated until that terminal measure. The transition from one measure to another can be derived iteratively, and the process of adjusting the drift is the essence of the Libor Market Model. However, the problem associated with the terminal measure is that the process for the shortest period, which is closest to today, ends up being more stochastic because it involves all the processes that are after that point, which is counterintuitive. Nevertheless, the Libor Market Model works under the spot measure as a consensus default unless a payoff is specified to be in the terminal measure.

  • 00:55:00 In this section of the lecture, the speaker discusses the issues with the library market model, particularly the lack of continuity in terms of times in between the specified tenant grid. As such, the speaker explains the strategy of using a discrete three discretely rebalanced money savings account to define the spot measure for the library market model. This involves observing how the investment of one unit of currency today can be accumulated given the existing tender structure of zero coupon bonds. The strategy is defined not at t0 but at t1 and involves buying a bond at t1, receiving the accrued amount at maturity, and reinvesting it for the second bond at t2.

  • 01:00:00 In this section, the concept of compounding in a discrete interval structure is explained as a way to invest in zero coupon bonds while reinvesting the received amounts in new bonds. The product of all the zero components defines the amount the investor would receive at a specified time, and the accumulated amount can be defined continuously by discounting from the last point on the grid to the present point. The concept of the spot-libor measure is also introduced, which allows the running numerator to switch from a ti measure to a tm measure. Additionally, the concept of mt is introduced as the minimum i such that ti is the biggest than t to link t to the next bond.

  • 01:05:00 In this section of the lecture, the speaker goes through the process of defining the measure transformation from the M_t measure to the M_t+1 measure. This is accomplished by using the Radon-Nikodym derivative. The speaker also explains the dynamics for lambda and psi, which will determine the measure transformation and the link between Brownian motions under t and n. Finally, the speaker presents the final representation of the library market model, which is similar to what was seen before in the changing measure of measures like the market mode.

  • 01:10:00 In this section, the speaker discusses the dynamics of the Libor market model, which is used for advanced and complicated exotical products in the interest rate world. The model involves a high dimensional problem, with a complicated drift that includes multiple Libors, making it difficult to implement. However, the model is a problem solver, and the speaker goes on to discuss extensions of the model to include volatility smiles and how to choose the stochastic volatility process while keeping the model's dynamics as simplified as possible. The speaker notes that the log-normality of the model only exists under the marginal measure and that it involves a summation of different autonomous processes, making it not log-normal in the general case.
Financial Engineering Course: Lecture 11/14, part 1/2, (Market Models and Convexity Adjustments)
Financial Engineering Course: Lecture 11/14, part 1/2, (Market Models and Convexity Adjustments)
  • 2022.03.10
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Financial Engineering: Interest Rates and xVALecture 11- part 1/2, Market Models and Convexity Adjustments▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This course ...
 

Financial Engineering Course: Lecture 11/14, part 2/2, (Market Models and Convexity Adjustments)



Financial Engineering Course: Lecture 11/14, part 2/2, (Market Models and Convexity Adjustments)

The lecture series on the Libor Market Model and its extensions with stochastic volatility provides a comprehensive understanding of the model's framework and its applications in financial engineering. The speaker emphasizes the importance of considering measure transformations, dynamics under different measures, and choosing appropriate spot or terminal measures. The log-normal assumption in the model is discussed, along with its limitations and the challenges of handling stochastic volatility.

One of the key topics covered is the concept of convexity adjustments, which are necessary to account for payment delays or mismatches in financial instruments. The lecturer explains the challenges that arise when including Libor dynamics into the variance dynamics and discusses potential solutions, such as imposing correlations between Libor and volatility. However, the lecturer cautions that these solutions may not be realistic or well-calibrated to the market's implied volatility data.

To address these challenges, the lecturer introduces the concept of displaced diffusion stochastic volatility model, which offers a better approach for modeling stochastic volatility in the Libor Market Model. By using a stochastic volatility process and a displacement method, the model can change the distribution of process values while preserving smile and skew characteristics. The lecturer explains how the displacement factor, controlled by the beta function, determines the interpolation between initial and process values. The independence of the variance process is achieved by assuming zero correlation between the variance and Libor dynamics.

The lecture further explores the implementation and calibration of the displaced diffusion stochastic volatility model. The lecturer demonstrates how to link the dynamics of the model to the representation of the master model, which is a special case of the Hassle model. The benefits of using this model for calibration are discussed, emphasizing the ease of calibrating each Libor under its own measure without additional drift corrections. The lecturer also highlights the impact of beta and sigma on the implied volatility shape and explains how to pass the model to the Hassle model for pricing.

In addition, the lecture addresses the issue of convexity adjustments in the Libor Market Model. The lecturer explains how to adjust the initial value and volatility of a displaced diffusion stochastic volatility process to account for market convexity. A new variable is introduced, and constant corrections and adjustments are applied to the displacement and Libor terms. The resulting process is a displaced diffusion stochastic volatility process that incorporates market convexity.

The lecture series also touches upon the freezing technique, which is used to fix the stochasticity of variables and simplify models. However, the lecturer cautions about the potential pitfalls of using this technique and emphasizes the importance of accurately calibrating the model to market data.

To reinforce the concepts discussed, the lecture series concludes with several homework assignments. These assignments include exercises on calculating convexity adjustments, determining correlation matrices, and exploring different model specifications.

The lecture series provides a thorough exploration of the Libor Market Model, its extensions with stochastic volatility, and the challenges and techniques involved in implementing and calibrating the model for pricing and risk management in the interest rate domain.

  • 00:00:00 In this section of the lecture, the focus is on the library market model and its extensions with stochastic volatility. The log-normal assumption in the model is discussed, and it is shown that a simple and naive approach to handling stochastic volatility may lead to a complex system of sds. The freezing technique for approximating the model is introduced, along with its pitfalls and what problems can be encountered when trying to apply it. Lastly, convexity corrections and adjustments are addressed, along with the inclusion of impact volatility smile and skew in the calculations. Three assignments are given for further insight into the library market model and convexity adjustments.

  • 00:05:00 In this section, the speaker discusses the dynamics of the process and applies the necessary measured changes to the model. They assume that the libraries are correlated, which they state is a key element, and that the Libor is correlated with the variance. By assuming zero correlation, the model only has a smile. They then redefine the model in terms of independent Brownian motions since it's more convenient when performing measured transformations. Finally, they substitute the definition of the dynamics into the model and demonstrate the dynamics of the Libor and variance process after substitution.

  • 00:10:00 In this section, the speaker explains the complications of using market models and convexity adjustments in financial engineering. Specifically, they discuss the challenges that arise when including libor dynamics into the variance dynamics. While there are potential solutions, such as imposing correlations between libor and volatility, these solutions may not be realistic or well-calibrated to the market's implied volatility data. As a result, the speaker suggests using displaced diffusion as an alternative option for modeling stochastic volatility in the libor market model.

  • 00:15:00 In this section, the speaker discusses the displaced diffusion stochastic volatility model, which is a better candidate for labor market models due to its ability to satisfy important model conditions. The model involves using a stochastic volatility process and a displacement method to change the distribution of process values while preserving smile and screw. The speaker explains that the displacement factor is controlled by the beta function, which determines the interpolation between initial and process values. The independence of the variance process is achieved by assuming zero correlation between the variance and light bulb dynamics. The model can be used to introduce skew and compensate for the skew lost due to the assumed zero correlation.

  • 00:20:00 In this section, the speaker discusses how to link the displays diffusion dynamics to the representation of the master model, which is a special case of the Hassle model. They explain the benefits of using this model for calibration and how each libor can be calibrated under its own measure, without any additional drift corrections, making it easier to evaluate derivatives. The speaker then shows the impact of beta and sigma on the implied volatility shape and how introducing the process for smile can give enough flexibility to the model to be calibrated to market instruments. They also briefly discuss the python implementation and how to link displays diffusion with stochastic volatility to pass the model to the Hassle model for pricing.

  • 00:25:00 In this section of the lecture, the speaker explains how to adjust the initial value and volatility of a displaced diffusion stochastic volatility process to account for convexity in the market. To do this, they introduce a new variable and perform constant adjustments to the displacement and library terms. After applying the constant corrections and adjustments, the form of the new process for v is defined, with a new correction or adjustment for the variances that is handled with the variable eta-hat. The resulting process is a displaced diffusion stochastic volatility process that accounts for market convexity.

  • 00:30:00 lecture delves into market models and convexity adjustments, specifically the Heston model, which allows for both smile skew and the handling of issues with measures in the Libor market model. The lecture also covers the freezing technique, a method used to fix the stochasticity of variables and simplify models. While this technique can be useful in some scenarios, the lecturer highlights that it is often abused and can lead to inaccurate results, ultimately rendering the model useless.

  • 00:35:00 In this section of the video, the lecturer discusses the concept of convexity adjustment and its importance in interest rate markets. Convexity adjustment is required when payment delays or mismatches occur between the payment date of a contract and the corresponding numerator. The lecturer explains that this can cause problems in pricing when the payment does not coincide with the payment date of the observable asset being priced. However, this issue can be avoided by using full Monte Carlo models and simulating Libor dynamics. The lecturer explains that it is important to consider the structure of the contract and the market scenarios before using convexity adjustment techniques, which should only be used when absolutely necessary.

  • 00:40:00 In this section, the lecturer explains the challenges of relying on the yield curve when payments are not aligned in a financial instrument. If the instrument is slightly different from what is available in the market, the expectation has to be estimated, which is often related to convexity. He illustrates with an example where the payment in the contract is different from what is seen in the market, so the expectation cannot be calculated from the yield curve. The lecturer demonstrates how to express the expectation in terms of observables in the market and switch the measure to the forward measure. The remaining expectation is not something that has been seen before, and the convexity adjustment functions or convexity correction would come into play here. He emphasizes that market instruments, such as swaps, are presented in their natural measures, which is not always the same as the measure used to calculate the expectation.

  • 00:45:00 The section focuses on dealing with terms and expectations under different measures and how to handle convexity corrections. The excerpt explains that switching from ti minus one to ti measure would correspond to the payment date of the slide bar. However, this leads to an interesting combination where the product of libor and the zero coupon bond is not martingale. To reformulate the problem, the section suggests adding and subtracting libor to determine the convexity correction term and ultimately finding the adjustment needed to have equality in the expression for the trade value at time t zero.

  • 00:50:00 In this section, the instructor discusses the challenges of avoiding simulations in financial modeling and instead utilizing the simplest possible blocks, specifically yield graphs, in calculating trade value. The issue with calculating the libor divided by zero coupon bonds is that it is not a martingale, making it problematic because of the squares regarding zero-coupon bonds. What is necessary is to find an expectation under t-forward measure to get the value valuation of a trade. The instructor then defines dynamics for the libor and discusses solutions for the expression's expectations, which will depend on a single libor, making it simple to do.

  • 00:55:00 In this section of the lecture, the concept of convexity correction is discussed in relation to the dynamics of the LIBOR market and the unknown variable c. It is noted that the choice of sigma is problematic as there is no clear indication of the volatility from the given expectation. The simplest choice would be to take the volatility at the money level, but this would overlook the impact of the volatility smile. A Python experiment is presented to illustrate the impact of changing sigma on the convexity adjustment, highlighting that the optimal sigma to match the market is around 0.22. To extract the correct sigma from the market, financial engineers would need to look at market instruments and use methods such as the Newton-Raphson algorithm to calibrate it.

  • 01:00:00 In this section, the speaker explains the implementation of the Hull-White model for generation of paths and calculating the convexity correction. The model calculates the zero coupon bonds for a specific period and calculates the expectation with one over, discounting that library. The Monte Carlo paths are generated until time t1, after which point the bonds can be calculated from t1 to any future point. The speaker emphasizes the importance of checking the match between yield curves from the market and the model simulation as well as being fluent in measure changes when dealing with convexity corrections. The speaker also mentions an alternative approach where the implied volatility smile and skew can be taken into account for evaluating expectations, eliminating the need to specify a particular sigma parameter.

  • 01:05:00 In this section of the lecture, the Brandon Litzenberger approach is discussed as a technique for expressing the expectation of a variable in terms of the in the money value and then calculating a correction term involving the integration of poles inputs of implied volatility smile based on the implied volatility smile. This approach is powerful because it allows for the calculation of all sorts of expectations and doesn't rely on the market availability of a sigma of any product. However, it does rely on the availability of an implied volatility surface, so assuming some log-normal distribution or another type of distribution for libero dynamics may be more efficient and straightforward if an implied volatility surface is not available. The lecture also discussed the two main subjects of the day, which were the liberal market model and possible extensions with stochastic volatility and convexity corrections.

  • 01:10:00 In this section of the lecture, the focus is on the contribution of the model's outers to leakage between different Libors defined under different measures and creating one uniform measure that can be used to evaluate derivatives that depend on multiple libraries. The lecture dives into the dynamics for the Libor under the P measure, the t-forward measure, and the differences between terminal and spot measures. The discussion also covers stochastic volatility, including the naive way of approaching the problem, adding correlated relativity to the Libor dynamics, and the problem of a complex volatility structure. The lecture ended with a focus on convexity corrections and how to solve and specify a model for evaluating nonlinear expectations. The homework assignments include an algebraic exercise and an extension of the Heston model, where instead of having one volatility driver, we have two.

  • 01:15:00 In this section of the video, the instructor assigns three homework problems related to market models and convexity adjustments. The first problem involves finding the values of psi bar and the initial process for two given equations. The second problem is the same as the convexity adjustment calculation, but with a shift parameter introduced to handle negative rates in the market. The third problem is to determine the correlation matrix for a given set of processes.
Financial Engineering Course: Lecture 11/14, part 2/2, (Market Models and Convexity Adjustments)
Financial Engineering Course: Lecture 11/14, part 2/2, (Market Models and Convexity Adjustments)
  • 2022.03.18
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Financial Engineering: Interest Rates and xVALecture 11- part 2/2, Market Models and Convexity Adjustments▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This course ...
 

Financial Engineering Course: Lecture 12/14, part 1/3, (Valuation Adjustments- xVA)



Financial Engineering Course: Lecture 12/14, part 1/3, (Valuation Adjustments- xVA)

In the lecture, the concept of xVA is introduced as a valuation adjustment that holds significant importance for banks, particularly in the context of pricing exotic derivatives. The lecturer delves into the intricacies of exposure calculations and potential future exposure, emphasizing their crucial role in effective risk management. Moreover, the lecture explores expected exposure, which serves as a connection between the measures employed for exposure calculations and simplified cases for computing xVA. Practical examples involving interest rate swaps, FX products, and stocks are provided, and a Python implementation is offered for generating multiple realizations samples from stochastic differential equations.

The video delves into the realm of counterparty credit risk and its relationship with xVA. It elucidates how the inclusion of counterparty default probability impacts derivative pricing and valuation. While the concept of risk-neutral measure was previously discussed in earlier lectures, the scope now widens to encompass a broader framework that incorporates risks like counterparty credit. To illustrate the concept of counterparty credit risk and its influence on pricing, a simple example of an interest rate swap is presented.

A scenario involving a swap transaction is discussed in the video, wherein the market has experienced a shift resulting in a positive value for the contract due to an increase in float rates. However, the counterparty's default probability has also risen, introducing a wrong-way risk as both exposure and default probability have amplified. The video emphasizes the necessity of incorporating this additional risk in valuation adjustments, which will be further explored in subsequent sections.

The lecturer elucidates the risks associated with default situations and highlights the regulatory requirements that financial institutions must consider. Counterparty credit risk (CCR) arises when a counterparty fails to fulfill its obligations and is directly linked to default risk. If the counterparty defaults before the contract's expiration and fails to make the necessary payments, it is referred to as Issuer Risk (ISR). Such payment failures can lead to the loss of potential future profits, forcing the financial institution to re-enter the swap and consequently exposing itself to further risks. Overall, financial institutions must account for these risks as they significantly impact derivative valuation.

The video delves into the impact of default probabilities on the valuation of derivative contracts. The speaker explains that a derivative contract involving a defaultable counterparty holds a lower value compared to a contract with a risk-free counterparty due to the additional risk that needs to be factored into the derivative price. The 2007 financial crisis is cited as a catalyst for changes in risk perception, including alterations in default probabilities and counterparty credit risk. The collapse of major financial institutions triggered a widespread propagation of default risk, resulting in systemic risk within the financial sector. As a response, regulators intervened to establish new methodologies and regulations aimed at minimizing risk and ensuring transparency in derivative positions.

The professor discusses the impact of regulations on exotic derivatives and elucidates how these derivatives have become more expensive due to increased capital requirements and maintenance costs. The professor explains that selling exotic derivatives in the market is not as straightforward and necessitates finding interested counterparties for such trades. Furthermore, the prolonged low-rate environment has diminished the attractiveness of exotic derivatives. However, with higher interest rates, the costs associated with maintaining exotic models can be offset. The professor emphasizes the importance of incorporating counterparty default probability in the pricing of financial derivatives, which has transformed simple products into exotic derivatives. This necessitates the use of hybrid models for pricing exotic products and extending risk measures beyond exotic derivatives.

The video discusses the inclusion of default probability risk in the pricing of financial derivatives. The probability of defaults on exotic derivatives needs to be factored in to account for risk, and counterparties are charged an additional premium that is integrated into risk-neutral pricing. Default probabilities are incorporated into the fair price of derivatives to compensate for counterparty risk. Due to the lack of confidence in the financial system, there has been a reduction in complexity, leading to a greater focus on estimating and maintaining simple financial products. The video also delves into various types of valuation adjustments, including counterparty valuation adjustment (CVA), funding valuation adjustment (FVA), and capital valuation adjustment (KVA), all aimed at achieving the ultimate goal of accurately pricing financial derivatives.

The professor proceeds to explain how financial institutions employ a technique called mapping to approximate the probabilities of default for a company, even in the absence of specific contracts like credit default swaps (CDSs) to reference. This section also covers the concept of exposures, emphasizing the significance of positive and negative exposures in the context of xVA. The professor clarifies that the value of a derivative at a given time, denoted as vt, is defined by the exposures at a later time, denoted as g, which is the maximum of vt and zero. The value of vt undergoes stochastic changes based on the filtration for the subsequent day, and the exposure represents the maximum amount of money that can be lost if the counterparty defaults.

The instructor shifts the focus to valuation adjustments or xVAs. The first aspect explored is exposure, which denotes the disparity between the amount one party owes and what the counterparty owes in a transaction. This exposure can lead to either losses or gains, with a maximum positive amount defined. The instructor explains that in the event of a counterparty default, the obligation to pay the full amount remains, and any recovery of funds is contingent upon the quality of the underlying assets. Furthermore, potential future exposure is introduced as a measure of the maximum potential loss, calculated based on the worst-case scenario exposure, considering the distribution of potential outcomes.

The concept of potential future exposures (PFE) is then discussed as a means to estimate the tail risk of a portfolio. PFE represents a quantile of exposures based on the valuation of a portfolio in future realizations. The lecture also covers the aggregation of trades within a portfolio, either at the contract level or at the counterparty level, highlighting the benefits of netting to offset risks. Netting, akin to hedging, involves acquiring offsetting contracts to reduce risks or cash flows.

The instructor proceeds to explain the advantages and limitations of netting, delving into credit valuation adjustments (CVA) in detail. It is clarified that only homogeneous trades that can be legally netted as per ISDA master agreements can be utilized for netting, and not every trade is eligible. The recovery rate is established once the legal process commences and is associated with the value of assets held by the bankrupt firm. A simple example involving a default scenario is presented to illustrate the benefits of netting, whereby the cost incurred due to a defaulting counterparty can be significantly reduced, benefiting the counterparty involved.

The professor further elaborates on the impact of netting on portfolios and its legal justifications. After calculating exposures, potential future exposures can be computed based on the distribution or realization of the portfolio. The professor emphasizes that exposure stands as the most crucial component when it comes to xVA and other adjustments. Additionally, an interesting approach to calculating potential future exposures is introduced, involving the utilization of expected loss as an interpretation of expected exposure.

The instructor delves into potential future exposures (PFE) once again, highlighting its role as a measure of tail risk. PFE indicates the point at which the probability of losses exceeds the potential future exposure, focusing solely on the remaining segment of tail risk. A debate surrounding the calculation of PFE is mentioned, questioning whether it should be based on the q-measure or calibrated using historical data under the p-measure. Risk managers may prefer incorporating scenarios that have occurred in the past, in addition to market expectations of the future, to effectively account for tail risk.

The speaker concludes the lecture by discussing various approaches to evaluating and managing risk in financial engineering. Different methods, such as adjusting exposures based on market data or specifying extreme scenarios manually, are employed depending on the discretion of risk managers. The choice of risk management approach is crucial, as the measures used play a significant role in managing risk. These measures help determine limitations for traders and the types and quantities of risks permitted when trading derivatives.

The lecture provides a comprehensive overview of xVA and its importance in the banking sector, particularly in the pricing of exotic derivatives. It covers exposure calculations, potential future exposure, and expected exposure, highlighting their significance in risk management. The inclusion of default probabilities and counterparty credit risk is emphasized, given their impact on derivative valuation. The lecture also explores the regulatory landscape, the increasing costs associated with exotic derivatives, and the use of hybrid models for pricing. Netting and various valuation adjustments, such as CVA, are discussed as means to mitigate risk. The role of potential future exposures (PFE) in estimating tail risk and the debate surrounding its calculation methodology are also addressed. Ultimately, the lecture emphasizes the importance of effective risk management in financial engineering and the role of valuation adjustments in pricing financial derivatives.

  • 00:00:00 In this section of the financial engineering course, the lecturer introduces the concept of xVA, a valuation adjustment that is important for banks, especially in relation to pricing exotic derivatives. The lecture will cover exposure calculations and potential future exposure, which are both crucial in risk management. The expected exposure will also be discussed, providing a link between the measures used for exposure calculations and simplified cases for calculating xVA. The lecture will also include examples of interest rate swaps, FX products, and stocks and will provide a python implementation for generating multiple realizations samples from stochastic differential equations.

  • 00:05:00 In this section of the lecture, the concept of counterparty credit risk and xVA are introduced. The lecture covers how to include the probability that the counterparty will not fulfill their obligations in derivative pricing, and how this affects valuation. The concept of risk-neutral measure was discussed in previous lectures, but now the lecture moves into a wider framework that includes risks such as counterparty credit. The lecture starts with a simple example of an interest rate swap to explain the concept of counterparty credit risk and how it affects pricing.

  • 00:10:00 In this section, the video discusses the scenario of a swap transaction where the market has moved and the value of the contract has become positive due to an increase in float rates. However, the counterparty's risks of default probability has also increased, which creates a wrong-way risk as our exposure has increased along with the probability of a default. The video suggests the need to include this additional risk in our valuation adjustments, which will be discussed further in the following sections.

  • 00:15:00 In this section, the lecturer explains the risks associated with default situations and how financial institutions need to account for them due to regulatory requirements. Counterparty credit risk (CCR) is the situation when a counterparty doesn't meet its obligations, and it is associated with a default risk. If the counterparty defaults before the expiration of the contract and doesn't make all required payments, it's called Issuer Risk (ISR). The failure to make these payments could result in a loss of potential future profits, and the financial institution would need to re-enter the swap, leading to further risks. Overall, financial institutions need to account for these risks as they impact the valuation of derivatives.

  • 00:20:00 In this section, the video discusses the impact of default probabilities on the valuation of derivative contracts. The speaker explains that a derivative contract with a defaultable counterparty is worth less than a contract with a risk-free counterparty due to the additional risk that needs to be included in a derivative price. The 2007 financial crisis is mentioned as a catalyst for changes in risk perception, including changes in default probabilities and counterparty credit risk. The collapse of large financial institutions triggered a widespread propagation of default risk, creating systemic risk in the financial world. Regulators stepped in to create new methodologies and regulations aimed at minimizing risk and ensuring transparency in derivative positions.

  • 00:25:00 In this section of the lecture, the professor discusses the impact of regulations on exotic derivatives and how they have become more expensive due to the increased capital requirements and maintenance costs. He explains that exotic derivatives cannot be easily sold in the market and require finding a counterparty interested in that kind of trade. Furthermore, the low rate environment over the years has made exotics less attractive, but with higher interest rates, the costs associated with maintaining exotic models can be offset. The professor also highlights the importance of incorporating the probability of default of a counterparty in the pricing of financial derivatives, which has turned simple products into exotic derivatives. This requires the use of hybrid models for pricing exotic products and pricing of risk measures beyond exotic derivatives.

  • 00:30:00 In this section, the video discusses the inclusion of the risk of default probabilities in the pricing of financial derivatives. The probability of defaults on exotics needs to be included to account for risk, and counter parties are charged with an additional premium, which is put on risk-neutral pricing. Default probabilities are added to the fair price of derivatives to compensate for counterparty risk. Due to the lack of confidence in the financial system, there has been a reduction in complexity, and simple financial products are easier to estimate and maintain. The video also discusses the various types of valuation adjustments, such as counterparty valuation adjustment, funding valuation adjustment, and capital valuation adjustment, which are used to achieve the ultimate goal of pricing financial derivatives.

  • 00:35:00 In this section of the lecture, the professor explains how financial institutions use a technique of mapping to approximate the probabilities of default for a company even if it doesn't have contracts like credit default swaps (CDSs) to map to a particular probability of default. This section also covers the concept of exposures, where the positive and negative exposures are important for xVA. The professor explains that the value of derivative at time t is defined as exposures at time g, which is the maximum vt and then zero. The value of vt changes stochastically based on the filtration for tomorrow, and the exposure is the maximum amount of money that can be lost if the counterparty defaults.

  • 00:40:00 In this section of the Financial Engineering course, the instructor discusses valuation adjustments or xVAs. The first aspect is exposure, which is the difference between what a party owes and what the counterparty owes in a transaction. The exposure amount could lead to losses or gains, and there is a maximum positive amount. The instructor explains that if a party defaults, the obligation to pay the full amount remains, and any recovery of the funds is based on the asset quality. The potential future exposure measures the maximum potential loss, which is calculated based on the worst-case scenario exposure, considering the distribution of outcomes.

  • 00:45:00 In this section of the lecture, the concept of potential future exposures (PFE) is discussed as a way to estimate the tail risk of a portfolio. PFE is a quantile of exposures based on the value of a portfolio evaluated in future realizations. The lecture also covers how to aggregate trades together in a portfolio, such as at the contract level or at the counterparty level, and the benefits of netting to offset risks. Netting is a concept similar to hedging, where offsetting contracts are bought to reduce risks or cash flows.

  • 00:50:00 In this section of the Financial Engineering course, the instructor explains the benefits and limitations of netting and discusses CVA (credit valuation adjustments) in detail. Only homogeneous trades that can be legally netted as per ISDA master agreements can be used and not every trade can be netted. The recovery rate is established once the legal process starts and is associated with the value of assets of the bankrupt firm. A simple example with a default scenario is given to explain the benefits of netting where the cost of a counterparty that defaulted can be reduced significantly which would be beneficial to the counterparty.

  • 00:55:00 In this section of the lecture, the professor discusses netting effects on portfolios and how they are legally justified. After calculating exposures, potential future exposures can be calculated based on the distribution or realization of the portfolio. The professor emphasizes that exposure is the most important ingredient when it comes to xVA and other adjustments. Additionally, there is an interesting approach to calculating potential future exposures which involves leveraging expected loss as an interpretation of expected exposure.

  • 01:00:00 In this section of the lecture, the instructor discusses potential future exposures (PFE) as a measure of tail risk. PFE represents where the probability of losses can exceed the potential future exposure, which only accounts for the remaining part of the tail risk. The instructor also mentions a debate on how PFE should be calculated, whether it should be based on the q-measure or calibration using historical data under the p-measure. Risk managers may prefer taking into account scenarios that happened in the past, in addition to what the market expects about the future, to account for tail risk.

  • 01:05:00 In this section, the speaker discusses different ways to evaluate and manage risk in financial engineering, such as adjusting exposures based on market data or specifying extreme scenarios by hand. The choice of risk management approach depends on the risk manager's discretion and the measures used are important for managing risk, such as determining limitations for traders and the types and amounts of risks allowed when trading derivatives.
Financial Engineering Course: Lecture 12/14, part 1/3, (Valuation Adjustments- xVA)
Financial Engineering Course: Lecture 12/14, part 1/3, (Valuation Adjustments- xVA)
  • 2022.03.24
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Financial Engineering: Interest Rates and xVALecture 12- part 1/3, Valuation Adjustments- xVA (CVA, BCVA and FVA)▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This ...
 

Financial Engineering Course: Lecture 12/14, part 2/3, (Valuation Adjustments- xVA)



Financial Engineering Course: Lecture 12/14, part 2/3, (Valuation Adjustments- xVA)

The lecturer continues to delve into the topic of valuation adjustments (xVA) in financial engineering, providing additional examples and insights. They discuss cases where expected exposures can be calculated analytically, such as for portfolios consisting of a single stock, and highlight the increased complexity and option-like characteristics that arise when calculating exposure in expected exposure. The importance of martingales, measures, and filtrations in financial engineering is also emphasized.

In one example, the lecturer explains how filtrations and conditional expectations are used to derive a simplified expression for expected exposure, which is then discounted. In another example, they apply principles from previous lectures to determine the discounted value of a swap at a specific time, considering the available cash flows and excluding the former ones. These examples underscore the significance of understanding and correctly applying concepts in financial engineering.

The lecturer revisits previous topics and demonstrates their connection to valuation adjustments. Using the example of an FX swap, they illustrate the process of changing the measure to the t-forward measure, resulting in the elimination of the domestic money savings account and leaving only the foreign currency's zero coupon bond multiplied by the notional. By utilizing the forward FX rate, the expectation can be simplified to a forward transaction.

The calculation of expected exposure in the domestic currency for a swap is also discussed. The stochastic nature of the zero coupon bond poses a challenge, which is addressed by using its definition as a ratio of the money savings account. The measurement is then changed from the domestic neutral measure to the t-forward domestic measure, enabling the pricing of an option using the European option price. Through the use of a stochastic differential equation, the expected exposure under the domestic measure can be determined by pricing the option. This process incorporates concepts such as interest rate capitalization and foreign exchange discussed in previous lectures. The section concludes with a numerical experiment in a one-dimensional case.

The speaker further explores the valuation of interest rate swaps using the Hull-White model and expresses swap valuation in terms of zero coupon bonds. They emphasize the importance of monitoring future cash flows for xVA evaluation, as they are exposed to counterparty default risk. The speaker highlights the balancing effect of increasing uncertainty and reducing risk associated with future cash flows in swaps. Additionally, the significance of the root in the Hull-White model for integrating multicolored paths to evaluate zero coupon bonds is discussed.

The computational challenges of determining the price of zero coupon bonds are addressed. Integrating pathways can be computationally expensive, but the Hull-White model's time-dependent function representation offers efficiency by evaluating functions instead of integrating paths. This makes it more efficient for xVA simulations of exposures and VAR calculations. Numerical results for an interest rate swap are provided, showing the increasing exposure profile due to volatility and the eventual reduction of exposure as cash flows are paid back. The value of swaps over time is also illustrated for a 20-year ex-swap.

The concept of expected exposures and potential future exposures in financial engineering is discussed. Negative expected exposures are defined as volumes and become significant when the exposure approaches zero. The speaker presents a graph of positive and negative exposures, specifying confidence intervals. A Monte Carlo simulation is performed, considering the number of paths, steps, and parameters for the Hull-White model. The calculation of swap value and money savings account value is explained. The section concludes by emphasizing the significance of confidence levels in potential future exposures.

Calculation of expected exposure and discounted expected exposure for single swaps and portfolios with netting is explained. The value of the swap is already expressed at a specific time, eliminating the need for discounting to the present. Numerical results from Monte Carlo simulations illustrate the potential values of swaps under different market scenarios, highlighting the importance of hedging to reduce exposures. Positive exposures and discounted expected exposures from the swap are depicted with varying levels of potential future exposure. Understanding the methodology in terms of filtration is emphasized, as it allows for a cohesive framework to simulate xVA of exposures.

The speaker further discusses the impact of netting on reducing potential future exposures. Adding swaps to a portfolio can be beneficial in minimizing exposures and potential future exposure. They emphasize the need to use hybrid models and construct multi-dimensional systems of stochastic differential equations when simulating multi-currency swaps in different economies. However, they caution that evaluating portfolios across multiple scenarios, although cheaper from a computational perspective, can still be time-consuming in practice.

The lecture addresses the challenges involved in evaluating xVA, particularly the computational cost associated with calculating exposures' sensitivity to specific risk factors or market changes. However, they highlight techniques to reduce the number of evaluations required to approximate the desired profile. The lecture emphasizes the importance of model selection and multiple evaluations, especially when dealing with multiple currencies and assessing exposures between the trade's inception and maturity. Finally, the lecture introduces the credit value adjustment (CVA) series as a means to account for the possibility of counterparty default in risk-free pricing.

The lecture further delves into the concept of credit value adjustment (CVA) in derivative pricing when considering default risk. It begins with a simple scenario where default occurs after the contract's last payment, providing a formula for valuing the derivative. The lecture then explores more complex cases where the possibility of default impacts derivative valuation. The notation for discounted payoff and the objective of linking the prices of derivatives with and without default risk are introduced. Various default scenarios and the corresponding amounts that can be received in each scenario are examined to determine the necessary adjustment in risk evaluation for the contract.

Different scenarios regarding the timing of default and recovery rates when dealing with a counterparty are discussed. If default occurs before a certain time, all payments are received until that point. If it happens after the contract's maturity, the outstanding balance may be recovered. However, if default occurs between these two points, there may be future obligations and a recovery rate to consider. The speaker demonstrates how to calculate the expectation of discounted future cash flows for four different cases and how to connect them using an equation.

The lecture moves on to the next step after calculating expected exposure, which involves utilizing the linearity of expectation and dividing it into two components. The first component involves indicator functions dependent on different maturities, representing the contract's value from time tau until maturity time t. The second component considers cases where tau is greater than time t or less than t. As the contract's value is measurable with respect to filtration, the first three terms under the expectation term represent the risk-free value of the derivative. The second part introduces an adjustment to include the convex part with a maximum and recovery rate, resulting in the credit value adjustment (CVA). In summary, a risky derivative can be expressed as a risk-free derivative minus the CVA adjustment, which corresponds to the counterparty's default probability—an essential element in the relationship.

Lastly, the speaker explains the concept of calculating exposure for each time period until the contract's maturity, adjusting for default, and discounting all cash flows accordingly. The recovery rate is defined as the loss given default and is included in the credit value adjustment formula.

The lecture provides a comprehensive exploration of valuation adjustments (xVA) in financial engineering. It covers various examples, computational challenges, and methodologies for calculating exposures, expected exposures, and credit value adjustments. Understanding these concepts and applying them correctly is crucial for accurate risk assessment and pricing in financial markets.

  • 00:00:00 In this section of the lecture, the speaker continues with the topic of valuation adjustments (xVA) in financial engineering. They discuss special cases for which expected exposures can be calculated analytically, such as for a portfolio consisting only of a single stock. They emphasize that exposure calculation in expected exposure brings the complexity of one level higher and explain that the value of a simple contract, such as a single cash payment, can become an option due to this increased complexity. The section concludes with a reminder of the importance of martingales, measures, and filtrations in financial engineering.

  • 00:05:00 In this section, the lecturer discusses the valuation adjustments (xVA) in financial engineering using two examples. In the first example, he explains how to deal with filtrations and conditional expectations to arrive at a simple expression for the expected exposure, which is discounted. In the second example, he uses the principles from previous lectures to determine the discounted value of a swap at time t, taking into account the available cash flows and excluding the former ones. Both examples highlight the importance of understanding the concepts and applying them correctly in financial engineering.

  • 00:10:00 In this section of the lecture, the professor revisits previously discussed topics and shows how they connect with the current topic of valuation adjustments. He uses the example of an FX swap and goes through the process of changing the measure to the t forward measure. This allows for the cancellation of the domestic money savings account and the zero coupon bond, leaving only the zero coupon bond times the notional in the foreign currency. Through the use of the forward FX rate, the expectation can be simplified to just a forward transaction.

  • 00:15:00 In this section, the speaker discusses the process of calculating expected exposure in the domestic currency for a swap. The zero coupon bond becomes non-measurable due to its stochastic nature. This issue can be resolved by using the zero coupon bond definition as a ratio of the money savings account. The next step is to change the measurement from the domestic neutral measure to the t-forward domestic measure, which would allow the pricing of an option using the European option price. Through the use of a stochastic differential equation, the expected exposure under the domestic measure can be determined by pricing the option. The process involves several of the concepts discussed in previous lectures, such as interest rate capitalization and foreign exchange. The section concludes with a numerical experiment in a one-dimensional case.

  • 00:20:00 In this section, the speaker discusses the valuing of interest rate swaps using the Hull-White model and how to express this swap valuation in terms of the zero coupon bonds. They also mention the importance of monitoring future cash flows for evaluation in xVA, since they are exposed to counterparty default risk, and how the offsetting effects of increasing uncertainty and reducing risk associated with future cash flows balance out in swaps. Lastly, the root in the Hull-White model is highlighted as a crucial feature for integrating multicolored paths to evaluate zero coupon bonds.

  • 00:25:00 In this section, the speaker discusses the computational challenges of determining the look upon bond price, which requires integrating pathways, making it very computationally expensive. However, because the full white model belongs to a kind of fine class of processes, it can be represented as a time-dependent function, making it extremely powerful. This means that we can determine the value of zero coupon bonds without actually integrating paths, and we only need to evaluate functions, making it more efficient for XVA simulations of exposures and VAR calculations. The speaker provides numerical results for an interest rate swap, showing that the exposure profile increases because of the volatility, the impact of paying back the flows becomes significant, and eventually goes to zero. Additionally, a profile for an ex-swap of 20 years shows the value of swaps in time.

  • 00:30:00 In this section of the lecture, the speaker discusses the concept of expected exposures and potential future exposures in financial engineering. The negative expected exposures are defined as volumes, and it becomes significant when the exposure is almost at zero. The speaker shows a graph of positive and negative exposures, specifying the level of confidence intervals. The experiment involves a specification for the Monte Carlo simulation, including the number of paths and steps and parameters for the whole White model. The speaker also explains the process of calculating the value of a swap and the money savings account. The section concludes by discussing the significance of confidence levels in potential future exposures.

  • 00:35:00 In this section, the speaker discusses the calculation of expected exposure and discounted expected exposure for single swaps and portfolios with netting. The value of the swap is already expressed at time ti, so there is no need for discounting to today. They also demonstrate the numerical results of Monte Carlo simulations, showing the potential value of swaps depending on different market scenarios, and the importance of hedging to reduce exposures. They illustrate positive exposures and the discounted expected exposures from the swap, with different levels of potential future exposure. The speaker emphasizes the importance of understanding the methodology in terms of filtration to put all the blocks learned so far into one framework for easier simulation of xVA of exposures.

  • 00:40:00 In this section, the speaker discusses how netting can impact reducing potential future exposures and how adding swaps into a portfolio can be useful in reducing exposures and potential future exposure. It is essential to use hybrid models and build multi-dimensional systems of stochastic differential equations while simulating multi-currency swaps in different economies. The speaker also warns that although the Monte Carlo simulation is relatively cheaper from the computational perspective, it can still be time-consuming while evaluating portfolios on all those scenarios.

  • 00:45:00 In this section, the speaker discusses the challenges involved in evaluating xVA, specifically the computational cost which can be very big, especially when calculating the sensitivity of exposures to particular risk factors or market changes. However, there are techniques to reduce the number of evaluations needed to resemble the required profile. The lecture then delves into the idea of xVA and the different measures and techniques that can be applied to compute discounted expectations of exposures for a counterparty or portfolio. The importance of model choice and multiple evaluations is emphasized, especially when dealing with multiple currencies and evaluating exposures between the beginning and maturity of a trade. Finally, the credit value adjustment series is introduced as a way to account for the possibility of counterparty default in the risk-free pricing.

  • 00:50:00 In this section, the lecture discusses the credit value adjustment (CVA) in derivative pricing when default risk is taken into account. The lecture starts with a simple case where the time of default happens after the last payment of the contract, and a formula is given for the value of the derivative. The lecture then delves into more complex cases where the possibility of default of the institution being considered has an impact on the valuation of derivatives. The lecture also introduces the notation for discounted payoff and the objective of linking the derivative price that may default and the price that may not. From this configuration, the lecture goes on to explore the different scenarios of possible default and the amount of money that can be received in each scenario, which will be used to determine the adjustment needed in the risk evaluation of the contract.

  • 00:55:00 In this section, the speaker discusses different scenarios regarding the timing of default and recovery rates when dealing with a counterparty. If the default happens before a certain time, all payments are received up until that point, whereas if it happens after the maturity of the contract, the outstanding balance may be recovered. However, if the default happens in between, there may be future obligations and a recovery rate involved. The speaker then shows how to calculate the expectation of the discounted future tables for four different cases and how to link them using an equation.

  • 01:00:00 In this section, the lecture discusses the next step after calculating the expected exposure, which involves using the linearity of expectation and dividing the expectation into two pieces. The first piece involves indicator functions that depend on different maturities, representing the value of a contract until time tau and from tau until maturity time t. The second piece involves cases where tau is bigger than capital time t or smaller than t. The value of the contract is measurable with respect to filtration, so the first three terms under the expectation term are just the risk-free value of the derivative. The second part involves the adjustment to include the convex part with a maximum and recovery rate, resulting in a credit value adjustment or CVA. The bottom line is that a risky derivative equals a risk-free derivative minus the CVA adjustment, which corresponds to the default probability of a counterparty, a crucial element in the relationship.

  • 01:05:00 In this section of the video, the speaker explains the concept of calculating exposure for every time period until the maturity of a contract and then adjusting for default and discounting everything. The recovery rate is discussed as the loss given default and is represented in the formula for the credit value adjustment.
Financial Engineering Course: Lecture 12/14, part 2/3, (Valuation Adjustments- xVA)
Financial Engineering Course: Lecture 12/14, part 2/3, (Valuation Adjustments- xVA)
  • 2022.03.31
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Financial Engineering: Interest Rates and xVALecture 12- part 2/3, Valuation Adjustments- xVA (CVA, BCVA and FVA)▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This ...
 

Financial Engineering Course: Lecture 12/14, part 3/3, (Valuation Adjustments- xVA)



Financial Engineering Course: Lecture 12/14, part 3/3, (Valuation Adjustments- xVA)

During the lecture, the speaker delves into the market standard approximations used for estimating Credit Value Adjustment (CVA) and addresses the issue of symmetry with regards to Pseudo CVA (PCVA) and Volume CVA (VCVA). They explain that client charges based on default probabilities can differ, creating a hurdle for transactions to occur without adjustments. To tackle this problem, the concept of Depth Value Adjustment (DVA) is introduced, and the application of Heavy Rays for calculating expected exposures is explained.

Trade attributions for CVA are also discussed, along with the importance of weighting CVA in a portfolio to avoid additivity problems. In conclusion, the speaker provides a summary of the lecture and presents two exercises for the students.

Moving on, the speaker emphasizes the incorporation of risk in pricing and considers the recovery rate or loss given default as a constant. They explain that obtaining an approximation for CVA correction requires a joint distribution, which is a stochastic quantity correlated with the time of default. Furthermore, the terms "wrong way risk" and "right way risk" are explored, highlighting their relation to the correlation between exposures and default probabilities of counterparties. The speaker also mentions the availability of classical articles online that provide an introduction to techniques used to impose correlations when assuming independence between two variables.

Shifting focus, the professor discusses the market approach to approximating conditional expectation through expected exposure, emphasizing its significance in the course. They break down the three main elements comprising CVA and emphasize that the expected exposure part is the most costly. The lecture highlights the symmetry problem associated with CVA, where counterparties' prices differ due to conflicting views on default probabilities, hindering agreement. To address this issue, the lecturer concludes that bilateral Credit Value Adjustment (bCVA) needs to be explored.

Bilateral CVA takes into account the risk associated with both parties' default, ensuring symmetry in derivative pricing. This means that one party may not agree with the adjusted price calculated by the other party. Bilateral CVA ensures the inclusion of both parties' creditworthiness, ultimately determining the fair value price of a derivative by incorporating their respective probabilities of default.

The discussion then transitions to the valuation adjustments, collectively referred to as xVA, and stresses the importance of incorporating adjustments in the pricing of risk-free or default-free derivatives. The lecturer explains that Bilateral Credit Value Adjustment (BCVA) is the difference between CVA and Debit Value Adjustment (DVA). They touch on how Volume CVA (VCVA) can increase, leading to a decreased CVA portion due to a firm's increased default risk and the challenges associated with rising evaluations. The computation formula for Funding Value Adjustment (FVA) is explored, consisting of the cost of funding adjustment (FCA) and the funding benefit adjustment (FBA). The funding spread (SBE) represents the cost of funding for derivatives, typically tied to market funding costs. The formula assumes independence between the exposure value of the portfolio, probabilities of default, and the funding part. FVA incorporates two types of funding: funding generated from the business and funding required to support existing positions, both included in the Liquidity Value Adjustment (LVA).

Understanding the risk profiles of trades within a portfolio or net set is emphasized by the speaker. Knowledge of individual Credit Default Adjustments (CDAs) per trade facilitates the assessment of trades' contributions to risk profiles, allowing for risk mitigation through position selling or associated risk establishment. The objective is to decompose CVA into individual CVAs to express it as a summation of individual CVAs, providing insights into their role in CVA evaluation. While incremental CVA can be performed, it is computationally expensive. Thus, the objective is to find a decomposition method that ensures agreement between portfolio-level CVA and the sum of individual CV VAs.

To achieve the desired decomposition of xVA or expected exposures into individual contributors while preserving the total sum equal to the portfolio exposure, the instructor introduces the Euler allocation process and a homogeneity function. The function f is considered homogeneous of degree k if k times f of x equals the summation of all the elements of the derivative of this function with respect to each individual element of the vector times x i. This enables the decomposition of CVA or expected exposures into the sum of individual contributions, expressed as a discounting part and a smooth alpha component. By employing this approach, expected exposures can be evaluated and computed at each individual time and weighted with alpha coefficients to achieve a smooth product.

The lecturer highlights the benefits of calculating sensitivity with respect to alpha i, as it allows for reduced computations when evaluating expected exposures for a portfolio. By reformulating the CVAs, the individual CVAs for each trade can be expressed as a ratio, and the derivative can be calculated from the expected exposure without the need to repeat the Monte Carlo simulation. This approach is advantageous from a numerical perspective, but it relies on the homogeneity assumption, and the portfolio combination must satisfy the condition.

The lecture further discusses extending code for multiple dimensions and swaps, as well as calculating expected exposures for multiple risk factors such as inflation and stocks. The calculation of CVA encompasses the consideration of both the counterparty's and our own probability of default, while the concept of Funding Value Adjustments (FVA) is introduced. The section concludes with a discussion on decomposing XVA into individual risk contributors and attributions.

For the homework assignment, students are tasked with simulating a portfolio consisting of 10 stocks, 10 interest rate swaps, and 5 call options. They are required to calculate expected exposures, potential future exposures, and perform CVA evaluation. Additionally, students are asked to discuss the knitting effect and suggest derivatives that could reduce expected exposures.

The speaker concludes by presenting exercises aimed at evaluating the risk profiles of a portfolio and exploring methods to reduce them. The first exercise involves simulating expected exposures of a swap and implementing swaption pricing using a full white model to validate its equivalence to swaption pricing. The second exercise serves as a sanity check to ensure the correctness of the implementation. The upcoming lecture will focus on value at risk and utilize the knowledge acquired in this lecture.

Overall, the lecture covered the fundamentals of credit value adjustments, the simulation of expected exposures, potential future exposures, and the utilization of Monte Carlo simulations and Python coding in the process.

  • 00:00:00 In this section of the financial engineering course, the speaker discusses the market standard approximations for estimating CVA and addresses the problem of symmetry with PCVA and VCVA. They explain how client charges based on default probabilities may differ and why this can prevent transactions from taking place without adjustments. The concept of DVA or depth value adjustment is introduced and the heavy rays application of calculating expected exposures is explained. The speaker also discusses the issue of trade attributions for CVA and how to weight CVA in a portfolio to avoid additivity problems. Finally, they summarize the lecture and provide two exercises for students.

  • 00:05:00 In this section of the lecture, the speaker discusses how to incorporate the risk of pricing and considers the recovery rate or the loss given default as a constant. The speaker then explains that to get an approximation for CVA correction, they need joint distribution, which is a stochastic quantity that is correlated with the time of default. Additionally, the speaker discusses the terms "wrong way risk" and "right way risk" and their relation to the correlation between exposures and default probabilities of counterparties. Finally, the speaker explains that there are classical articles available online that provide an introduction to the techniques used to impose correlations when assuming independence between two variables.

  • 00:10:00 In this section of the lecture, the professor discusses the market approach to approximating conditional expectation by expected exposure, emphasizing that this is the main interest of this course. He then breaks down the three main elements that make up CVa and explains that the expected exposure part is the most expensive one. The main issue with CVa is the so-called symmetry problem where the prices of counterparties differ due to their conflicting views of the default probabilities, making it difficult to reach an agreement. To address this, he concludes the lecture, we need to go into the bilateral credit value adjustment (bCVA).

  • 00:15:00 counterparty's perspective will be the same when we incorporate bilateral CVA (credit value adjustment). Bilateral CVA takes into account the risk associated with our own default as well as the counterparty's default. This means that one party may not agree with the adjusted price as calculated by the other party. The adjusted value is calculated by the investor and is not the opposite of the adjusted value calculated by the counterparty. Bilateral CVA steps in to ensure symmetry in the pricing of derivatives, taking into account both parties' credit worthiness. The fair value price of a derivative is ultimately determined by the bilateral CVA, which includes both parties' respective probabilities of default.

  • 00:20:00 In this section, the lecturer discusses the valuation adjustments, or xVA, and the importance of incorporating adjustments in the pricing of the risk-free or default-free derivative. They explain how the bilateral credit value adjustment, or BCVA, is the difference between the credit value adjustment, or CVA, and the debit value adjustment, or DVA. The lecturer also touches on how VCVA can increase resulting in a decreased CVA part due to an increase in a firm's default and problems with increasing evaluations. They emphasize the importance of computing expected exposures as it is a crucial element in calculating xVA and the adjustments such as the funding value adjustment or FVA.

  • 00:25:00 section explores the computation formula for fva, which consists of two parts - the cost of funding adjustment (fca) and the funding benefit adjustment (fba). The funding spread is the cost of funding for derivatives, expressed by sbe, and is usually linked to the cost of obtaining funding in the market. The formula assumes that the exposure value of the portfolio is independent of the funding part, and the exposure and probabilities of default are also independent. The fva is calculated based on the expected exposures and estimated probabilities of default from the market. Additionally, fva includes two types of funding - the funding generated from the business and the funding that needs to be paid to support existing positions. Both these types of funding are included in the lva.

  • 00:30:00 In this section, the speaker explains the importance of understanding the risk profiles of trades when dealing with the value adjustments in a portfolio or net set. Having knowledge of the individual cdas per trade in a portfolio can help assess which trades contribute the most to the risk profiles, allowing for the establishment of associated risks or even the selling of positions for overall risk reduction. The objective is to decompose cva into individual cvas to find a way to express it as a summation with individual cvas to understand their role in cv evaluation. Incremental cva can also be performed, but it is computationally expensive, and the objective is to find a way to decompose cva into individual cvas such that they agree both at the portfolio level and the sum of individual ones.

  • 00:35:00 In this section, the instructor discusses the methodology of using the euler allocation process and a homogeneity function to achieve the desired decomposition of xVA, or expected exposures, into individual contributors that preserve the total sum equal to the portfolio exposure. The function f is said to be homogeneous of degree k if k times f of x is equal to the summation of all the elements of the derivative of this function to each individual element of the vector times x i. This enables the decomposition of CVA or expected exposures into the sum of individual contributions, which are then expressed as a discounting part and a smooth alpha component. By doing so, expected exposures can be evaluated and computed at each individual time and weighted with alpha coefficients to attain a smooth product.

  • 00:40:00 In this section, the lecturer discusses the benefits of calculating sensitivity with respect to alpha i and how it allows for the reduction of computations when evaluating the expected exposures for a portfolio. By using a reformulation of the CVAs, the individual CVAs for each trade can be expressed as a ratio and the derivative can be calculated from the expected exposure without the need for redoing the Monte Carlo simulation. This approach is beneficial from a numerical perspective, but it still relies on the homogeneity assumption and the portfolio combination must satisfy the condition. Overall, the lecture covered the basics of credit value adjustments and the simulation of expected exposures and potential future exposures, utilizing Monte Carlo simulations and Python coding.

  • 00:45:00 In this section of the Financial Engineering Course, the lecturer discusses how to extend code for multiple dimensions and swaps, as well as calculating expected exposures for multiple risk factors, including inflation and stocks. The lecture also covers the calculation of CVA, including the inclusion of both the counterparty's and our own probability of default, and introduces funding value adjustments (FVA). The section ends with a discussion of how to decompose XVA into individual risk contributors and attributions. The homework includes simulating a portfolio with 10 stocks, 10 interest rate swaps, and 5 call options, calculating expected exposures, potential future exposures, and performing CVA evaluation. Additionally, students are asked to discuss the knitting effect and suggest derivatives to reduce expected exposures.

  • 00:50:00 In this section of the video, the speaker discusses exercises that help to evaluate the risk profiles of a portfolio and how to reduce them, as well as the importance of hedging a position using as few additional positions as possible. The first exercise involves simulating expected exposures of a swap and implementing swaption pricing using a full white model to confirm that it is equivalent to the pricing of a swaption. The second exercise is a sanity check to ensure the correctness of the implementation. The next lecture will focus on value at risk and reuse the knowledge learned in this lecture.
Financial Engineering Course: Lecture 12/14, part 3/3, (Valuation Adjustments- xVA)
Financial Engineering Course: Lecture 12/14, part 3/3, (Valuation Adjustments- xVA)
  • 2022.04.07
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Financial Engineering: Interest Rates and xVALecture 12- part 3/3, Valuation Adjustments- xVA (CVA, BCVA and FVA)▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This ...
 

Financial Engineering Course: Lecture 13/14, part 1/2, (Value-at-Risk and Expected Shortfall)



Financial Engineering Course: Lecture 13/14, part 1/2, (Value-at-Risk and Expected Shortfall)

The lecturer begins by explaining the motivations behind value-at-risk (VaR) calculations and their relevance to risk management in a portfolio's profit and loss (P&L). VaR is introduced as a measure of potential losses associated with market fluctuations, aiming to provide a single number for the worst-case scenario over a specified time period. However, it is emphasized that VaR is not the only answer and that financial institutions must have sufficient capital to cover estimated losses based on various environmental factors.

The lecture covers the calculation and interpretation of VaR, including stressed VaR and expected shortfall. Stressed VaR involves considering historical data and worst-case events to prepare institutions for extreme market moves. Expected shortfall, on the other hand, calculates the average loss beyond the VaR level, providing a more conservative approach to risk management. The importance of incorporating multiple VaR calculations and diversification effects when making investment decisions is highlighted.

In the next segment, students learn about programming a VaR portfolio simulation using Python. The lecture focuses on simulating a portfolio with multiple interest rate products, downloading market data for yield curves, and calculating shocks. The significance of diversification and considering different VaR calculations is reiterated. The segment concludes with a summary and an assignment that tasks students with extending Python code to calculate VaR for a specific portfolio comprising stocks and interest rates.

The lecture also touches upon the acceptance and utilization of VaR by financial institutions for risk monitoring and capital adequacy purposes. The regulatory aspect is emphasized, with VaR being imposed to ensure institutions can withstand recessions or market sell-offs. An example of a portfolio's VaR is provided, indicating a 95% confidence level that the portfolio won't lose more than a million dollars within a single day.

Furthermore, the lecture explains the calculation of VaR using the distribution of portfolio values and possible market scenarios, drawing parallels to previous calculations of exposures and potential future exposures. The lecturer emphasizes the simplicity of VaR compared to expected exposures, which only consider the absolute value of the risk factor. Different approaches to VaR calculations are mentioned, such as parametric VaR, historical VaR, Monte Carlo simulation, and extreme value theory, with a focus on understanding their characteristics and limitations.

The concept of coherent risk measures is introduced, outlining the academic requirements for a risk measure to be considered good. The lecture acknowledges the criticism surrounding these requirements and highlights the practitioners' perspective on practicality and back-testing. The sub-additivity requirement is explained, emphasizing that the risk measure of a diversified portfolio should be less than or equal to the sum of the individual risk measures of its assets. While VaR is not a coherent measure, it is commonly used for risk management purposes. Nevertheless, risk managers are encouraged to consider multiple risk measures to gain a comprehensive understanding of their portfolio's risk profile and risk appetite.

The limitations of VaR as a risk management tool are discussed, leading to the introduction of expected shortfall as a more conservative alternative. Expected shortfall is presented as a coherent risk measure that considers the average loss exceeding the VaR level. By relying on multiple measures, such as VaR and expected shortfall, financial institutions can enhance their risk mitigation strategies and protect their portfolios effectively.

The lecture concludes by addressing the limitations of VaR calculations, such as their dependence on data quality and quantity. It emphasizes the importance of pragmatic risk management, avoiding excessive conservatism while choosing measures that are realistic and reliable.

  • 00:00:00 In this section of the course, the instructor covers the motivations for value-at-risk (VaR) calculations and how they relate to risks in a portfolio's profit and loss (P&L). The lecture also includes an explanation of stressed VaR, expected shortfall, and how these measures fit into a coherent risk management plan. In the second block of the lecture, students learn about programming a simulation of a VaR portfolio with multiple interest rate products, downloading market data for yield curves, and calculating shocks. The lecture emphasizes the importance of a diversified portfolio and the need to consider multiple VaR calculations when making investment decisions. The segment concludes with a summary and an assignment requiring students to extend Python code to calculate VaR for a specific portfolio consisting of stocks and interest rates.

  • 00:05:00 In this section of the lecture on financial engineering, the focus is on value-at-risk (VaR) and expected shortfall, which are used to measure potential losses associated with market fluctuations. VaR attempts to provide a single number for the worst-case scenario of potential losses over a given time period, but it's important to note that it's not the only answer. Banks are required to have enough capital to cover estimated potential losses, based on environmental factors. The lecture explains how VaR is calculated using the distribution of portfolio values and possible market scenarios, demonstrating its similarity to previous calculations of exposures and potential future exposures.

  • 00:10:00 In this section of the lecture, the importance of Value-at-Risk (VaR) and how it is used by financial institutions is discussed. VaR is used to help prepare financial institutions for worst-case scenarios by looking at historical data and worst-case events so that they have enough capital to sustain their business during periods of market dramatic moves. VaR is imposed by regulators to keep a better eye on monitoring positions and risks to ensure that financial institutions survive during recessions or market sell-offs. The lecture also explains how VaR numbers are calculated and interpreted, with a specific example of a portfolio's VaR indicating there is a 95% confidence level that the portfolio won't lose more than a million dollars within one day.

  • 00:15:00 In this section, the lecturer explains the idea behind the Value-at-Risk (VaR) method of measuring risk. VaR involves observing the daily fluctuations in the historical movements of an underlying asset, applying those to today's value and re-evaluating the portfolio to determine the distribution of profits and losses. The method is much simpler than the calculations performed in Expected Exposures, which only looked at the absolute value of the risk factor. The lecturer explains that VaR has been accepted in the industry for over 40 years and that there are different approaches to how the calculations can be conducted. Although VaR provides an estimate of the amount of risk involved with market movements, it does not guarantee a company's survival in the event of something catastrophic happening.

  • 00:20:00 In this section, the concept of Value-at-Risk (VaR) as a measure of risk is introduced. VaR calculates the amount of capital needed to support a specific level of risk, and adding capital will move the distribution to the right-hand side, which reduces the risk. The confidence level of VaR is set by regulators, and a common requirement is a one-sided confidence interval of 99%. While VaR enables the incorporation of diversification effects, it can be problematic. Improvements such as Expected Shortfalls are suggested to address VaR's limitations. Additionally, regulators require a holding period of 10 days for VaR calculation, but it is necessary to consider additional measures as well.

  • 00:25:00 In this section of the lecture on financial engineering, the professor explains that the wider the window of observation, the wider the distribution would be for the P&L distribution. Regulators require a 10-day holding period for value-at-risk and at least one year of historical data for market risk factors. The stress scenario, known as svar, involves looking at the market data from a violent and volatile period of time in the past. Although the model parameters are standardized, banks do not have to follow exactly the same particular approach for estimating value at risk. The four main methodologies for calculating value at risk include parametric var, historical var, Monte Carlo simulation, and extreme value theory. The professor notes that they will not be focusing on the parametric var method.

  • 00:30:00 In this section of the course, the lecturer discusses different approaches to calculating Value-at-Risk (VaR) for a portfolio. The first approach mentioned is the parametric form, where a distribution is imposed on the portfolio's returns and samples are taken from the distribution to evaluate the portfolio. However, this method is heavily biased, and if the distribution is not properly calibrated or suited to a particular risk factor, it can expose the portfolio to a significant amount of risk. The lecture then goes on to explain Monte Carlo simulation, where risk factors such as interest rates are simulated using a stochastic differential equation and then evaluated using the portfolio. Monte Carlo simulation can be done in two ways - by calibrating the model to the market's implied volatilities, or by calibrating it to historical data using a moving window of observable market shocks.

  • 00:35:00 In this section of the lecture, the concept of coherent risk measures is discussed, which refers to the academic requirements proposed for any risk measure to be considered a good measure. However, there is a lot of criticism surrounding these requirements as practitioners argue that some measures may not be practical and satisfy the best back-testing requirements. The sub-additivity requirement is also explained, which ensures that the risk measure of a diversified portfolio should be less than or equal to the sum of the individual risk measures of those assets. Although Value-at-Risk (VaR) is not a coherent measure, it is often still used by practitioners for risk management purposes, but risk managers are encouraged to consider multiple risk measures to better understand their risk profile and risk appetite.

  • 00:40:00 In this section, the requirements for a coherent measure are discussed, with the first requirement being that the measure should respond monotonically to risk. This means that if the Value-at-Risk (VaR) increases but the Expected Shortfall (ES) decreases while diversifying or hedging, then there is something happening in the portfolio that needs to be analyzed. The second requirement is that if one asset is worth less than or equal to another asset, then the risk measure of the former should be greater than or equal to the latter. Additionally, the section explains the limitations of VaR, including how it does not satisfy sub additivity, which may lead to wrong interpretations in financial solutions that use VaR but do not realize it violates some additivity.

  • 00:45:00 In this section, the lecture covers the limitations of using value-at-risk (VAR) as a risk management tool, and introduces the concept of expected shortfall as a more conservative alternative. While VAR is popular in the industry, it has the potential risk of misrepresenting the actual level of risk of a portfolio, leading to the assumption of too much risk or a failure to hedge when necessary. Expected shortfall is a coherent risk measure that takes VAR as an input and calculates the average loss that exceeds the VAR level, resulting in a more conservative approach to risk management. By relying on multiple measures, such as VAR and expected shortfall, financial institutions can better mitigate risks and protect their portfolios.

  • 00:50:00 In this section, the speaker discusses the limitations of Value-at-Risk (VaR) and suggests some potential improvements. The speaker notes that VaR calculations are highly dependent on data quality and quantity, so it's important to carefully consider the data being used. Additionally, the speaker cautions against being too conservative in risk management, as this can lead to unrealistic measures. Instead, it's necessary to be pragmatic and choose measures that are realistic and reliable.
Financial Engineering Course: Lecture 13/14, part 1/2, (Value-at-Risk and Expected Shortfall)
Financial Engineering Course: Lecture 13/14, part 1/2, (Value-at-Risk and Expected Shortfall)
  • 2022.04.14
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Financial Engineering: Interest Rates and xVALecture 13- part 1/2, Value-at-Risk and Expected Shortfall▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This course is ...
 

Financial Engineering Course: Lecture 13/14, part 2/2, (Value-at-Risk and Expected Shortfall)



Financial Engineering Course: Lecture 13/14, part 2/2, (Value-at-Risk and Expected Shortfall)

The instructor delivers a comprehensive lecture on performing a Python simulation and evaluating historical Value-at-Risk (VaR) using real market data for a portfolio of interest rate swaps. The lecture covers various crucial topics, including handling missing data, arbitrage, and the concept of re-reading yield curves to incorporate market data changes for generating VaR scenarios. The Monte Carlo method for VaR calculations is also explained, along with the use of backtesting to assess the performance of the VaR model. To conclude the lecture, an assignment is given to the students, challenging them to implement or enhance the historical VaR implementation by introducing an additional risk factor and contemplating risk diversification in their portfolio.

The concept of Value-at-Risk (VaR) is thoroughly elucidated by the instructor. VaR is employed to forecast or derive a distribution for potential profits and losses (P&L) in a portfolio, based on historical movements of risk factors. To ensure stable results, the portfolio remains constant, and historical evaluations of risk factors serve as input for VaR calculations. The instructor highlights the significance of including all relevant risk factors in the calculations and mentions that the time window length and confidence level can be specified. Furthermore, the instructor intends to analyze the impact of varying time window lengths on the distribution of the P&L profile in a Python experiment.

In the subsequent segment, the lecturer delves into estimating potential losses that a portfolio may encounter within a day. Emphasizing the importance of realistic risk factors and utilizing historical data, the lecturer describes how daily changes in risk factors can be applied to today's level to determine the range of possible outcomes and the distribution of probable losses over a period. It is stressed that effective risk control and management are essential for safeguarding the institution, going beyond mere compliance with regulatory conditions. Moreover, the lecturer explains that calculating VaR and managing a portfolio of simple derivatives is comparatively easier than dealing with interest rate products that necessitate the construction of yield curves for each scenario.

The lecturer proceeds to discuss the steps involved in pricing an interest rate portfolio and calculating Value-at-Risk (VaR) and Expected Shortfall. The construction of a yield curve for every scenario is an essential computational task in this process. An experiment is outlined, where a portfolio of swaps is evaluated over a 160-day period using historical data on daily treasury yield curves. By calculating daily shocks and subsequently reconstructing yield curves, the portfolio's value, VaR, and Expected Shortfall can be determined. The lecturer mentions that this procedure relies on the prior coverage of yield curve construction in a previous lecture. The objective of the experiment is to observe the distribution of potential profile losses with 95% confidence intervals.

The lecture covers the calculation of quantile for VaR and the expected value of the left-hand side from this quantile, which corresponds to the expected shortfall. Building a portfolio using zero coupon bonds and evaluating swaps with different configurations, rates, notionals, and settings is also discussed. Additionally, the lecture addresses the calculation of the yield curve based on historical data and the iterative process of obtaining the shocks required for yield curve adjustments in all scenarios.

The speaker proceeds to explain the utilization of historical data to estimate potential yield curve movements. This estimation of possible scenarios is valuable for risk management when other information is unavailable. Scenarios can also be specified manually, such as by a regulator. The speaker also delves into examining risk profiles based on historical data and handling special cases when dealing with changing instruments. The process of shocking market values and reconstructing yield curves for each scenario is explained, followed by the evaluation of the portfolio for each constructed curve. Lastly, the speaker outlines the methodology behind estimating the expected shortfall based on observations of the tail end of the distribution.

The speaker provides insights into the results obtained from running code to calculate the distribution of profits and losses (P&Ls), as well as the value-at-risk (VaR) and expected shortfall. The distribution of P&Ls exhibits a familiar shape with tails on both ends and the majority of values centered around ten thousand. The VaR is computed at minus seven thousand, indicating a five percent probability that tomorrow's losses will exceed that amount. On the other hand, the expected shortfall is determined to be minus sixteen thousand, nearly twice the impact of the VaR calculation. The speaker emphasizes the importance of consistent and high-quality market data in conducting accurate historical VaR computations. The homework assignment entails extending the function to incorporate additional risk factors like stocks and replicating the same experiment.

Also the lecturer explains how to handle missing market data in financial calculations, particularly when dealing with instruments that lack active trading or market-implied values. The process involves constructing a curve to interpolate missing data based on available instruments, while also considering delta constraints and volatilities. The lecturer underscores the significance of utilizing market-available instruments in risk management and establishing data quality standards for VaR and expected shortfall calculations. Additionally, the issue of negative volatilities is addressed, along with insights into methodologies to handle such occurrences.

Two types of arbitrage, namely calendar arbitrage and butterfly arbitrage, are discussed by the speaker. Calendar arbitrage occurs in the time dimension, while butterfly arbitrage is concerned with strikes. The speaker explains how the butterfly strategy approximates the second-order derivative of a call option with respect to strike, which corresponds to the density of a stock. However, applying inconsistent shocks to the volatility surface of the present day can introduce arbitrage opportunities and negative volatility, posing risks. Interpolating volatilities also presents challenges, especially in the context of VaR calculations. The speaker introduces VaR calculations based on Monte Carlo simulation, which can be calibrated to historical data or market instruments. The simulation is performed using Monte Carlo, and the model is associated with either the P or Q measure, depending on whether it is calibrated to historical data or market instruments.

The speaker further explains how Monte Carlo simulation can be employed to evaluate a portfolio. By simulating scenarios for a short-rate model and applying shocks or differences on a daily or 10-day basis, the portfolio can be assessed across various scenarios. Monte Carlo simulation provides more degrees of freedom and a broader range of scenarios compared to relying solely on historical data. Generating a large number of possible scenarios is crucial for improving risk management. The speaker acknowledges that certain choices within the methodology still require further exploration, but overall, the approach serves as a straightforward means to illustrate Monte Carlo simulation.

The speaker highlights that revaluing a portfolio in each scenario can be computationally demanding, particularly for large portfolios consisting of complex derivative securities. This process becomes the determining factor in the number of scenarios that can be generated, resulting in fewer scenarios for larger portfolios. To illustrate the evaluation of daily value-at-risk (VaR), the speaker demonstrates taking a 10-day difference between interest rates, calculating the portfolio, storing the results in a matrix, and estimating the quantile and expected shortfall for a given alpha of 0.05. The results indicate that the expected shortfall is twice as large as the VaR, underscoring the importance of effective risk management in mitigating substantial losses.

The lecture delves into the topic of backtesting for value-at-risk (VaR). Backtesting involves comparing the predicted losses from VaR to the realized profit and loss (P&L) derived from real market data. By conducting this analysis on a daily basis over a specific period, typically one year or 250 business days, the quality of the VaR model can be assessed, and potential issues such as missing risk factors or poorly calibrated models can be identified. However, it should be noted that backtesting is a backward-looking measure and may not accurately predict volatile events in forward-looking situations. To enhance the quality of backtesting, the use of Monte Carlo simulations and calibration with market data can be considered.

The video emphasizes the significance of balancing multiple models when estimating Value at Risk (VaR) and discusses the choice between using historical data versus stochastic processes. Calibrating the model to the market can provide additional information beyond what is derived solely from historical data. The speaker also explains how backtesting results play a crucial role in assessing the performance of a model. By comparing the model's predictions to a certain significance level, one can determine whether the model is performing well or poorly. The lecture concludes by summarizing the main points of the VaR discussion and underlining the importance of considering the expected shortfall in relation to VaR.

Further the speaker provides a summary of the second part of the lecture, which focused on practical issues such as handling missing data, arbitrage, and using Monte Carlo simulation for VaR computation. The speaker highlights the significance of gaining a comprehensive understanding of different VaR measures to effectively monitor the health and status of a portfolio. The homework assignment given requires students to extend a portfolio using historical value interest calculations, incorporate an additional risk factor such as a stock or foreign exchange, and consider diversifying derivatives to reduce variance. The speaker concludes the lecture by summarizing the key takeaways, including the calculation of VaR and the various VaR measures used to estimate the risks associated with potential market movements.

The lecture provides valuable insights into performing Python simulations and evaluating historical Value-at-Risk (VaR) based on real market data for a portfolio. It covers important topics such as handling missing data, arbitrage, re-reading yield curves, and employing Monte Carlo simulation for VaR calculations. The lecture also emphasizes the significance of backtesting to validate VaR models and the importance of considering the expected shortfall in addition to VaR. By exploring these concepts and completing the assigned tasks, students can develop a comprehensive understanding of risk management and portfolio evaluation in financial contexts.

  • 00:00:00 In this section of the Financial Engineering Course, the instructor discusses how to perform a Python simulation and evaluate historical value-at-risk (VaR) based on real market data for a portfolio of interest rate swaps. The lecture covers how to handle missing data, arbitrage, and the concepts of re-reading yield curves in the context of market data changes for generating scenarios for VaR calculations. The Monte Carlo method for VaR calculations is also discussed, along with backtesting for performance checking of the VaR model. The lecture concludes with an assignment that requires students to implement or extend the historical VaR implementation with an additional risk factor and think about diversifying risks in their portfolio.

  • 00:05:00 In this section, the instructor explains the concept of Value-at-Risk (VaR) and how it is used to forecast or give a distribution for the potential profits and losses (P&L) in a portfolio, based on possible historical movements of risk factors. The portfolio is kept constant in order to obtain stable results, and the historical evaluations of risk factors are used as input for VaR. The instructor emphasizes the importance of including all relevant risk factors in the VaR calculations. The length of the time window and the confidence level can also be specified. The instructor plans to analyze the impact of the length of the time window on the distribution of the P&L profile in the Python experiment.

  • 00:10:00 In this section, the lecturer discusses the process of estimating potential losses that a portfolio may encounter in a day. The lecturer emphasizes the importance of having realistic risk factors in the portfolio and using historical data to impose daily changes on today's level of risk factors. By applying these changes, it becomes possible to determine what is possible, and the distribution of offered probable and loss distribution over a period. The lecturer highlights that it is essential to control and manage risks and protect the institution, rather than just meeting the regulatory conditions. Finally, the lecture explains how a portfolio consisting of simple derivatives is much easier to calculate compared to interest rate products that require the construction of whole yield curves for every scenario.

  • 00:15:00 In this section of the video, the lecturer discusses the steps required to price an interest rate portfolio and calculate Value-at-Risk (VaR) and Expected Shortfall. To do this, a yield curve must be built for every scenario, which can be computationally intensive. The lecturer then outlines an experiment where they evaluate a portfolio of swaps over a period of 160 days using historical data on daily treasury yield curves. By calculating daily shocks and then rebuilding yield curves, they can value the portfolio and calculate VaR and Expected Shortfall. The lecturer notes that this process relies on the construction of a yield curve, which was covered in a previous lecture. The goal of the experiment is to see the distribution of possible profile losses with 95% confidence intervals.

  • 00:20:00 In this section of the Financial Engineering course, the topic of Value-at-Risk and Expected Shortfall is discussed. The lecture covers the calculation of the quantile for VAR and then the expected value of the left-hand side from this quantile, which will be the loss of the expected shortfall. The lecture also covers building a portfolio using zero coupon bonds and evaluating swaps with different configurations, rates, notionals, and settings. Additionally, the lecture discusses calculating the yield curve based on historical data and iterating over all the scenarios to obtain the shocks that need to be applied to the yield curve.

  • 00:25:00 In this section, the speaker explains how to use historical data to estimate the possible moves of a yield curve. This estimation of possible scenarios is of added value to manage risks without any other information available. Scenarios can also be specified by hand, for example by a regulator. The speaker also explains how to deal with different measures on looking at risk profiles based on historical data, and how to handle special cases when dealing with changing instruments. The process of shocking the market values and reconstructing yield curves for each scenario is explained, followed by evaluating the portfolio for every curve previously constructed. Finally, the speaker explains the methodology behind estimating expected shortfall based on observations of the tail end of the distribution.

  • 00:30:00 In this section of the Financial Engineering course, the speaker discusses the results of running a code to calculate the distribution of P&Ls and the value-at-risk (VaR) and expected shortfall. The distribution of P&Ls shows a familiar shape with tails on either end and the middle at ten thousand. The VaR is calculated at minus seven thousand with a five percent probability that tomorrow's losses will be bigger than that. The expected shortfall is minus sixteen thousand, which is almost twice the impact of the VaR calculation. The speaker also stresses the importance of having consistent and quality market data when conducting historical VaR computation. The homework involves extending the function to add extra risk factors such as stocks and performing the same experiment.

  • 00:35:00 In this section, the lecturer explains how to deal with missing market data in financial calculations, particularly in the case of instruments that are not actively traded or implied by the market. The process of creating a curve can be used to interpolate missing data based on available instruments, but additional criteria such as delta constraints and volatilities must be taken into account. The lecturer also notes the importance of using market-available instruments in risk management and establishing data quality standards for var and expected shortfall calculations. Additionally, he discusses the issue of negative volatilities and provides insights on methodologies to handle such events.

  • 00:40:00 In this section, the speaker discusses two types of arbitrage – one in the direction of time called calendar arbitrage and the other in the direction of strikes called butterfly arbitrage. They explain how the butterfly strategy approximates the second-order derivative of a call option with respect to strike, which is equivalent to the density of a stock. However, applying inconsistent shocks to today's volatility surface can introduce arbitrage and negative volatility, which can be risky. Interpolating volatilities is also challenging and requires attention, especially in the case of VAR calculations. The speaker then introduces VAR calculations based on Monte Carlo simulation, which can be calibrated to either historical data or market instruments. The simulation is performed with Monte Carlo, and the model is associated with either the P or Q measure, depending on whether it's calibrated to historical data or market instruments.

  • 00:45:00 In this section of the lecture on financial engineering, the speaker discusses using Monte Carlo simulation to evaluate a portfolio. By simulating scenarios for a short rate model and applying shocks or differences on a daily or 10-day basis, the portfolio can be evaluated based on various scenarios. By using Monte Carlo simulation, there are more degrees of freedom and more scenarios available than with historical data. It is important to generate as many possible scenarios as possible to improve risk management. The speaker explains that there are still many question marks regarding specific choices, but overall, the methodology is a straightforward approach to illustrate Monte Carlo simulation.

  • 00:50:00 In this section, the speaker explains that revaluing a portfolio in each scenario is computationally expensive, especially for large portfolios that consist of complex derivative securities. This process becomes the limiting factor in determining the number of scenarios that can be generated, and therefore, fewer scenarios can be generated for larger portfolios. The speaker also demonstrates the process of evaluating the daily value-at-risk (VaR) by taking a 10-day difference between the interest rates. They then calculate the portfolio, store it in a matrix, and estimate the quantile and expected shortfall for alpha 0.05. The results show that the expected shortfall is twice as large as the VaR, which demonstrates the importance of risk management in reducing large losses.

  • 00:55:00 In this section of the lecture, the topic of back testing for value-at-risk (VaR) is discussed. The main idea of back testing is to check if your VaR model accurately predicts losses by comparing the predicted losses from VaR to the realized profit and loss (P&L) from the real market data. This is done on a daily basis over a certain period of time, usually one year or 250 business days. Back testing can help assess the quality of the VaR model and identify potential issues such as missing risk factors or poorly calibrated models. However, back testing is a backward-looking measure and does not predict volatile events in forward-looking situations. The use of Monte Carlo simulations and calibration with market data can potentially improve the quality of back testing.

  • 01:00:00 In this section, the video discusses the importance of balancing multiple models when it comes to estimating Value at Risk (VaR) and using historical data versus stochastic processes. By calibrating to the market, one may receive more information than predictions based only on historical data. The video also explains back testing results and how they can help indicate whether a model is performing poorly or well by exceeding a certain significance level. Finally, the lecture summarizes the main points of the VaR discussion and mentions the importance of considering the expected shortfall in relation to VaR.

  • 01:05:00 In this section, the speaker summarizes the second part of the lecture which focused on practical issues such as missing data, arbitrage, and Monte Carlo simulation for VAR computation. The speaker also emphasizes the importance of having a good overview of different VAR measures to monitor the health and status of a portfolio. The homework assignment given requires the extension of a portfolio using historical value interest calculations and the addition of a risk factor such as a stock or foreign exchange. The assignment also requires the consideration of diversifying derivatives to reduce variance. The speaker concludes the lecture by summarizing the key takeaways, including how to calculate VAR and the different VAR measures used to estimate risks associated with possible market movements.
Financial Engineering Course: Lecture 13/14, part 2/2, (Value-at-Risk and Expected Shortfall)
Financial Engineering Course: Lecture 13/14, part 2/2, (Value-at-Risk and Expected Shortfall)
  • 2022.04.28
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Financial Engineering: Interest Rates and xVALecture 13- part 2/2, Value-at-Risk and Expected Shortfall▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This course is ...
 

Financial Engineering Course: Lecture 14/14, (The Summary of the Course)



Financial Engineering Course: Lecture 14/14, (The Summary of the Course)

The speaker concludes the Financial Engineering Course by recapping the 14 lectures that covered a wide range of topics. These topics included filtrations and measure changes, interest rate models, yield curve dynamics, pricing of swaptions, mortgages and prepayments, stochastic differential equations, market models, and evaluation and historical VAR adjustments. The course aimed to provide learners with a comprehensive understanding of financial engineering and equip them with the skills to implement their own derivative portfolios.

During the lecture, the speaker emphasizes the importance of understanding filtrations and measures, as well as performing simulations for portfolio evaluation and risk management. The benefits of conditional expectations in pricing options and reducing model complexity are discussed, along with the concept of changing measures and dimension reduction techniques. The lecture also covers the AJM framework of arbitrage-free short-rate models and two derived models, HJM and Hull-White, with simulations to compare yield curves used as input and output of the model. Additionally, the yield curve dynamics under short rate and the observation of the fed fund rate in experiments are explored.

In another segment, the speaker focuses on the relationship between yield curve dynamics and short rate models in Python simulations. He delves into the motivation behind developing a two-factor full-wide model as an extension of the single-factor model to capture yield curve dynamics. Interest rate products such as swaps, forward trade agreements, and volatility products are discussed, highlighting their importance for calibration to market data. The lecture also covers yield curve construction, including interpolation routines and multi-curves, and how these factors impact hedging and portfolio risk. Pricing swaptions and the challenges posed by negative interest rates are also addressed.

The final lectures of the course are summarized, covering topics such as the pricing of options using Jamshidian's trick, negative interest rates, and shift-like normal shifted implied volatility. Discussions on mortgages, hybrid models, prepayment risks, large time-step simulations, foreign exchange, and inflation are included as well. The importance of linking risk-neutral and real-world measures, observed market quantities, and calibration for model parameters is highlighted.

Furthermore, the application of financial engineering to multiple asset classes is explored, including interest rates, stocks, foreign exchange, and inflation. The challenges associated with models like the Heston model, convexity corrections, and the labored market model for pricing exotic derivatives are discussed. The course also focuses on measures of change and extends the standard normal libel market model to incorporate stochastic volatility. The primary objective is to calculate xVA and value at risk, considering exposure calculation, portfolio construction, and Python coding for evaluating exposure profit in a swaps portfolio. The speaker also mentions the importance of credit valuation adjustment (CVA) based on counterparty default probability and practical applications of xVA.

In the final recap, the lecturer reviews the lecture dedicated to value at risk. Historical value at risk, stress value at risk, Monte Carlo-based value at risk, and expected shortfalls were discussed, both from a theoretical perspective and through practical experiments involving market data and Monte Carlo calculations. The lecture also touched on the concept of backtesting to assess the quality of value at risk calculations. The lecturer expresses satisfaction with the course and congratulates the viewers on completing it, recognizing the practical and rewarding nature of the material covered.

  • 00:00:00 In this section, the speaker recaps the entire Financial Engineering Course consisting of 14 lectures. The course covered various topics including filtrations and measure changes, interest rate models, yield curve dynamics, pricing of swaptions, mortgages and prepayments, stochastic differential equations, market models, and evaluation and historical VAR adjustments. The speaker highlights the importance of understanding filtrations and measures, performing simulations, and implementing risk management techniques for portfolio evaluation. Overall, the course enabled learners to implement their own derivative portfolios.

  • 00:05:00 In this section, the speaker discusses the importance of understanding the composition and risks of a portfolio at a given time through filtration measurements and simulations. The lecture covers the benefits of conditional expectations in pricing options and reducing model complexity, as well as changing measures and dimension reduction techniques. The course also covers the AJM framework of arbitrage-free short-rate models and two derived models, HJM and Hull-White, with simulations to compare yield curves used as an input and output of the model. Additionally, the lecture covers yield curve dynamics under short rate, and the observation of the fed fund rate in experiments.

  • 00:10:00 In this section, the speaker discusses the relationship between yield curve dynamics and short rate models in Python simulations. They explore the motivation behind developing a two-factor full-wide model as an extension of the single-factor model to capture yield curve dynamics. They also cover interest rate products such as swaps, forward trade agreements, and volatility products, which are crucial for calibration to market data. Additionally, they delve into yield curve construction, including interpolation routines and multi-curves, and how they impact hedging and portfolio risk. The speaker ends this lecture by discussing the concept of pricing swaptions and the problem of negative interest rates.

  • 00:15:00 In this section, the speaker summarizes the final lectures of the Financial Engineering course, which covered topics such as the pricing of options and the application of Jamshidian's trick, negative interest rates, and shift-like normal shifted implied volatility. Lectures 8 and 9 were about mortgages and hybrid models, respectively, and included discussions on prepayment risks and large time-step simulations. The 10th and final lecture covered foreign exchange and inflation and included concepts such as cross currency swaps and pricing of FX options. The lectures provided insights into linking risk-neutral and real-world measures, observed market quantities, and the importance of calibration for model parameters.

  • 00:20:00 In this section, the speaker discusses the application of financial engineering to cover multiple asset classes, including interest rates, stocks, foreign exchange, and inflation. They also examine the challenges that arise from the Heston model, the inclusion of convexity corrections, and the labored market model, which is useful for pricing exotic derivatives. Through the course, students explored measures of change, such as the difference between Tamriel and spot measures, and extended the standard normal libel market model to incorporate stochastic volatility. The primary objective of the course is the calculation of xVA and the value at risk. The speaker examines the calculation of exposure, the construction of portfolios, and coding in Python to evaluate exposure profit for a profile of swaps. The ultimate goal is to derive the credit valuation adjustment (CVA) based on the probability of a counterparty's default and explore the practical applications of xVA.

  • 00:25:00 In this section of the transcript, the lecturer recaps the final lecture of the financial engineering course, which focused on value at risk. The lecture covered historical value at risk, stress value at risk, Monte Carlo-based value at risk, and expected shortfalls. The theoretical aspects and motivation behind these techniques were discussed in the first block of the lecture, while the second part involved a number of experiments, including historical var calculations for market data and Monte Carlo var calculations. The lecture also touched on back testing, which is used to measure the quality of avar calculations. Overall, the lecturer concludes that the course has been rewarding and practical, and congratulates viewers on completing it.
Financial Engineering Course: Lecture 14/14, (The Summary of the Course)
Financial Engineering Course: Lecture 14/14, (The Summary of the Course)
  • 2022.05.20
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Financial Engineering: Interest Rates and xVALecture 14, The Summary of the Course▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬This course is based on the book:"Ma...
 

Computational Finance Q&A, Volume 1, Introduction



Computational Finance Q&A, Volume 1, Introduction

Welcome to this channel! In this series of videos, I am offering a set of 30 questions and answers based on the course of Computational Finance. The questions in this course are not only useful as exam questions but also as potential interview questions for Quant-type jobs. The slides and lecture materials for this course can be found in the links provided in the description of these videos. The course consists of 14 lectures, covering topics such as stocks, stochastic, pricing of options, implied volatilities, jumps, fine diffusion models, stochastic volatility, and pricing of exotic derivatives.

For every lecture, I have prepared two to four questions, and for each question, I will provide you with a detailed answer. These answers can range from two to 15 minutes depending on the complexity of the question. The questions I have prepared cover a variety of topics, from global questions about different asset classes to more specific questions about the Heston model and time-dependent parameters.

In Lecture 1, we begin with simple questions about pricing models for different asset classes and the relationship between money savings accounts and zero-coupon bonds. Lecture 2 covers implied volatility, pricing of options using arithmetic Brownian motion, and the difference between stochastic processes and random variables. Lecture 3 focuses on the Feynman-Kac formula, a famous formula in computational finance, and how to perform sanity checks on simulated stocks. Lecture 4 delves into implied volatility term structures, deficiencies of the Black-Scholes model, and potential solutions to those deficiencies.

Lecture 5 covers jump processes, including the Eto's table and its relation to Poisson processes, implied volatility and jumps, and characteristic functions for models with jumps. Finally, Lecture 6 covers stochastic volatility models, including the Heston model and time-dependent parameters.

If you're interested in learning more about these topics, check out the playlist of lectures available on this channel.

Computational Finance Q&A, Volume 1, Introduction
Computational Finance Q&A, Volume 1, Introduction
  • 2023.01.03
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Computational Finance Q&A, Volume 1, Question 0/30▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬Materials discussed in this video are based on:1) FREE online course...
 

Can we use the same pricing models for different asset classes?



Can we use the same pricing models for different asset classes?

Today's computational finance course discussed the question of whether the same pricing models can be used for different asset classes. The question essentially asks whether a stochastic differential equation that has been successfully applied to one asset class, such as equities, can be used for modeling other asset classes as well. In the course, we explored various asset classes, including stocks, options, interest rates, exchange-traded commodities, over-the-counter electricity markets, and more. The aim was to determine whether models developed for one asset class can be effectively applied to others.

The short answer to this question is that it is generally possible to use the same pricing model across different asset classes, but it is not always the case. There are several criteria to consider when deciding whether a model can be applied to a different asset class. The first and most important criterion is whether the dynamics of the model align with the physical properties of the asset of interest. For instance, if a model assumes positive values, it may not be suitable for assets like interest rates that can be negative.

Another criterion is how the model parameters can be estimated. Are there option markets or historical data available for calibration? It's important to note that even if a model has an option market, such as the Black-Scholes model, it may not always fit well with the market's implied volatility smile or skew. Thus, it's crucial to assess whether the model aligns with the asset class and the specific pricing requirements. For example, if pricing a European option with a single strike and maturity, a simpler model like Black-Scholes may suffice, whereas more complex models with stochastic volatility may be necessary for other scenarios.

The existence of an option market, particularly the presence of implied volatility smiles or surfaces, is another factor to consider. If implied volatility patterns are observed in the market, models with stochastic volatility might be more suitable. However, if such patterns are absent, simpler models with less complex dynamics may be preferable.

Furthermore, understanding the market practice for modeling is essential. Is there an established consensus in the market? Are there documentation and guidelines available from exchanges or other sources? It is crucial to review existing literature and gain a comprehensive understanding of the asset class before selecting a stochastic process. Trying to fit a stochastic differential equation to an asset class without proper knowledge of its properties often leads to suboptimal results.

In the course, we covered various models, including those involving jumps and multiple differential equations. Two specific examples were discussed to illustrate the difference in dynamics: geometric Brownian motion and mean-reverting Ornstein-Uhlenbeck processes. The paths and realizations of these processes differ significantly, and it is important to choose a model that aligns with the specific characteristics of the asset class. Geometric Brownian motion is always positive, making it unsuitable for modeling interest rates, which can be negative. Similarly, an Ornstein-Uhlenbeck process may not be appropriate for modeling stocks, which can also exhibit negative behavior.

While there are numerous models available, such as the Heston model, local volatility models, or hybrid models, it is crucial to start with a good understanding of the asset class and its objectives. Different models have different strengths and weaknesses, and their applicability depends on the specific requirements and constraints of the market.

In conclusion, it is generally possible to use the same pricing models across different asset classes, but it is not guaranteed to be successful in all cases. The decision to apply a particular model should be based on a thorough understanding of the asset class, its dynamics, and the specific pricing requirements. By considering the criteria mentioned earlier and conducting a literature study, one can make informed decisions regarding model selection and application.

Can we use the same pricing models for different asset classes?
Can we use the same pricing models for different asset classes?
  • 2023.01.05
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Computational Finance Q&A, Volume 1, Question 1/30▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬Materials discussed in this video are based on:1) FREE online course...