Quantitative trading - page 17
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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.