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Financial Engineering Course: Lecture 3/14, part 1/2, (The HJM Framework)
Financial Engineering Course: Lecture 3/14, part 1/2, (The HJM Framework)
The speaker dives into the topic of arbitrage-free conditions in interest rate models, specifically focusing on the Heat, Jarrow, and Morton (HJM) framework. They set the agenda for the lecture and clarify the distinction between equilibrium models and term structure models. While emphasizing the power and significance of term structure models, which generate yield curves without requiring calibration, the speaker explains the derivation of arbitrage-free conditions within the HJM framework. The upcoming block will involve Monte Carlo simulations for two models, Julie and Hull-White, along with a provided homework assignment. It's worth noting that the HJM framework serves as a generic and arbitrage-free framework for all interest rate models.
Moving forward, the concept of short rates and interest rates is introduced, emphasizing that short rates are associated with infinitesimal periods of time. The first short rate model, the Ornstein-Uhlenbeck (OU) process, is discussed as an example of an endogenous model that requires calibration to the yield curve, potentially resulting in limited degrees of freedom and poor calibration. On the other hand, exogenous models take the yield curve as an input, avoiding the calibration issue. The lecture also provides insights into developing modeling skills and programming proficiency for interest rate modeling.
The HJM framework is explored, focusing on the transformation of endogenous models into exogenous models. This transformation ensures that regardless of the chosen model parameters, the yield curve remains the same. The lecturer highlights the exceptional power of the AJM framework, which provides a clear path from equilibrium models to term structure models. The lecture mentions that numerous models exist in the literature, with two popular ones being discussed. One such model is the Vasicek short rate model, which has faced criticism for its limitation in accommodating negative interest rates.
The issue of negative interest rates is addressed, and the speaker explains how financial engineers tackle this problem by employing the Cox-Ingersoll-Ross (CIR) process, which disallows negative rates but allows rates to reach zero. To shift this process, a parameter is introduced, allowing the distribution to move from zero to negative values, typically around two or three percent. The importance of fitting to the yield curve and the challenges of calibration are also discussed. The lecturer emphasizes that if the yield curve cannot be fitted, there is no point in attempting to fit other aspects of the model. Simulation examples are provided to illustrate the impact of varying parameters, such as speed of mean reversion and volatility coefficient.
The impact of the volatility coefficient on the paths of different models, including the HJM and CIR models, is discussed. Larger volatility coefficients result in greater spikes in paths and increased uncertainty, while smaller coefficients lead to narrower distributions. The lecturer also explains how mean reversion and interest rates affect the behavior of these models. Python code is utilized to simulate paths using Euler discretization and standardization, while imposing conditions to prevent paths from becoming negative.
The presenter provides an in-depth discussion of the HJM (Heath-Jarrow-Morton) framework, which serves as a global framework encompassing all interest rate models. The dynamics of instantaneous forward rates, representing rates over future periods from today's perspective, are modeled within the HJM framework. The AJM framework is presented as a fundamental basis for interest rate models due to its explicit relationship between the volatility of instantaneous forward rates and arbitrage-free drift, ensuring the model is always free of arbitrage. The framework is explored in the context of both short rate and LIBOR market models, which are special cases of the AJM framework.
The relationship between arbitrage-freeness and the drift is discussed, particularly in relation to the volatility of instantaneous forward rates. Adjusting the volatility allows for switching between different models. While the HJM framework accommodates different volatility structures, obtaining analytical expressions for short rates or LIBOR market models is challenging. However, for certain cases, the HJM framework does provide analytical expressions for zero coupon bonds based on the specified volatility. This framework plays a crucial role in transitioning from equilibrium models to term structure models, as it enables the use of observable yields as input for the model. A comparison is made with other models, such as short rate models under the HJM framework, which are likened to Ferraris in terms of fast calibration but lack flexibility in calibration and implementation for multiple market instruments. The primary objective of a short rate model for interest rates is to ensure the accuracy of the yield curve and zero coupon bonds.
The limitations of various term structure models employed in financial engineering are discussed by the lecturer. While the HJM framework offers more flexibility in calibrating to the yield curve, its simplicity with only two parameters makes it challenging to calibrate for complex exotic options evaluated over extended periods. The market model with stochastic volatility, despite its high maintenance costs and calibration challenges, is deemed ideal for pricing exotics and volatility. The lecturer proceeds to define instantaneous forward rates using zero coupon bonds and illustrates how to construct a forward rate over a specific period using a refinancing strategy, thus extracting an effective rate.
The speaker delves into the concept of an arbitrage-free refinancing strategy and explains how to imply rates from zero components. They introduce a functional form for the forward rate and impose a structure that ensures it takes an exponential form with an accrual times rate. By taking the logarithm of the expression and multiplying it by a negative sign, they identify the rate that satisfies the equation for both the short rate and the forward rate. The instantaneous forward rate is defined as f dt, and the speaker emphasizes that it is always with respect to maturity.
Next, the lecture introduces the notion of the instantaneous forward rate, defined as the derivative of the logarithm of the zero coupon bond with respect to maturity. This serves as a fundamental building block within the HJM framework, as all quantities are expressed in terms of instantaneous forward rates. The importance of differentiating between zero coupon bonds and money savings accounts is highlighted, with the former being a deterministic value and the latter a stochastic quantity. The dynamics of the instantaneous forward rate are a focal point within the HJM framework, aiming to understand and model the dynamics of interest rates.
The professor proceeds to describe the dynamics of the instantaneous forward rate under the p-measure and the objective of determining the dynamics when switching the measure from p to q. The HJM framework encompasses the dynamics of the instantaneous forward rate, the money savings account (integral of the short rate), and the relation of zero coupon bonds. To define the dynamics of the instantaneous forward rate under the q-measure, specific quantities must function as martingales. The relationship between the short rate and the instantaneous forward rate is explained, emphasizing the interdependence between different instantaneous rates and the connections among various parameters.
Continuing the lecture, the speaker emphasizes the importance of understanding the relationship between arbitrage-freeness and the drift in interest rate models, particularly in terms of the volatility of the instantaneous forward rate. By adjusting the volatility, one can switch between different models within the HJM framework. This framework allows for various volatility structures, although obtaining analytical expressions for short rates or a LIBOR market model can be challenging. However, in some cases, the HJM framework does provide analytical expressions for zero coupon bonds based on the specified volatility.
The lecturer highlights that the HJM framework is a generic and arbitrage-free framework for all interest rate models. It offers a clear path from equilibrium models towards term structure models, making it a powerful tool in the field. There are numerous models available in the literature, but two popular ones are discussed in detail.
First, the short rate model by Vasicek is examined. The lecturer acknowledges that this model has faced criticism for not allowing negative interest rates. To address this issue, some financial engineers adopt the Cox-Ingersoll-Ross (CIR) process, which does not permit negative rates but allows rates to reach a level of zero. However, the lecturer mentions that it is possible to introduce a shift parameter to the CIR process, effectively shifting the distribution from zero to a negative value, such as negative two or three percent. Fitting the model to the yield curve is emphasized as a critical aspect, and the issue of calibration is discussed. The lecturer states that if the yield curve cannot be accurately fit, there is no point in fitting any other parameters.
Next, the speaker introduces Monte Carlo simulations for two models: Julie and Hull-White. The simulations aim to provide practical examples and illustrate the impact of varying parameters, such as the speed of mean reversion and volatility coefficient, on the model's paths. Python code, utilizing Euler discretization and standardization, is used to simulate these paths. Conditions are imposed to restrict paths from becoming negative.
The lecture moves on to discuss the impact of the volatility coefficient on the paths of various models, including the HJM and CIR models. Larger volatility coefficients result in more significant spikes in paths and increased uncertainty, while smaller coefficients lead to narrower distributions. The influence of mean reversion and interest rates on the behavior of these models is also explained.
The lecturer concludes by summarizing the key points covered, reiterating the power and significance of term structure models within the HJM framework. The ability to self-generate yield curves without requiring calibration to the yield curve is emphasized. Finally, a homework assignment is provided, encouraging students to further explore and apply the concepts and techniques discussed in the lecture.
The lecture provides an in-depth exploration of arbitrage-free conditions in interest rate models, specifically within the HJM framework. It covers the differences between equilibrium models and term structure models, the derivation of arbitrage-free conditions, and practical examples through Monte Carlo simulations. The importance of fitting to the yield curve, calibration challenges, and the impact of varying parameters are thoroughly discussed, providing students with valuable insights into interest rate modeling and programming skills.
Financial Engineering Course: Lecture 3/14, part 2/2, (The HJM Framework)
Financial Engineering Course: Lecture 3/14, part 2/2, (The HJM Framework)
In the lecture, the focus is on the HJM framework and its assumptions for interest rate modeling. The lecturer begins by discussing the arbitrage-free conditions in the HJM framework, which are crucial for any interest rate model within this framework. These conditions ensure that every asset discounted with the money savings account behaves as a martingale. By applying Itō's formula to zero coupon bonds and the money savings account, the dynamics of the asset divided by the money savings account are obtained, leading to the famous HJM lemma regarding arbitrage-free conditions for instantaneous forward rates.
Next, the lecturer explores how the drift of instantaneous forward rates is determined within the HJM framework. The volatility of the instantaneous forward rate plays a key role in defining the drift if one wants to be in the risk-neutral and arbitrage-free world. The lecturer explains that to model short rates or instantaneous forward rates, it is essential to specify the volatility for the instantaneous forward rate. Once this is defined, the dynamics for the instantaneous forward rate are known, ensuring an arbitrage-free environment. The lecture also covers the computation of the dynamics of the short rate, which involves the maturity curve, a constant deterministic function, and an integral with respect to the partial derivative of volatility.
The lecture further delves into the practical aspects of the HJM framework. The lecturer discusses how different short rate models can be generated by specifying the volatility within the framework. A constant volatility is presented as the simplest form, allowing for the calculation of the alpha function under the HJM condition. The dynamics of the short rate can then be derived by substituting the specified sigma and alpha into the framework, using the zero-coupon bond curve as an input. The importance of the yield curve, which is estimated from market instruments, is emphasized as a key component in interest rate derivatives pricing.
Special attention is given to the Uli model, which belongs to the affine class of processes and offers time-dependent drift and sigma parameters. The lecturer explains how this model enables the calculation of zero coupon bonds in an exponential form without the need for nested Monte Carlo simulations, thus saving computational power. The relationship between short rates and the known deterministic functions in b is explicitly expressed, and the potential use of the Longstaff Schwarz algorithm for estimating expectations is mentioned.
The lecture also highlights the importance of representing models in a zero-compounded and elegant way. The HJM framework is recognized as a powerful tool for achieving this goal. A Python experiment is conducted to demonstrate how simulated paths can be used to calculate zero coupon bonds, comparing them to the input yields. It is emphasized that the HJM framework ensures that the simulated paths always yield the same zero coupon bonds as those incorporated in the yield input.
Monte Carlo simulation methods within the HJM framework are discussed as a means to generate yield curves. The lecturer presents an approach that involves specifying a yield curve, estimating the zero component curve, and calculating theta and sigma parameters. Monte Carlo simulations are then performed, and the resulting discount factors are used to plot the zero-coupon bond curves from the model and the market. The lecturer showcases the flexibility of the approach in handling changes in parameter values and highlights the perfect match between the input and output yields.
Calibration of models within the HJM framework is also addressed, focusing on the advantage of calibrating to relevant products without the need for separate calibration to the yield curve. The difficulties often encountered in yield curve calibration are discussed, highlighting the benefits of the HJM framework in this regard. The derivation of the constant volatility model in short rate models using the HJM assumptions is explained, showcasing a simplified form of the short rate dynamics that facilitates model evaluation.
The lecture concludes by summarizing the main points covered and providing three exercises for students to apply the concepts and calculations learned. The exercises involve Ito's dynamics calculation,
Financial Engineering Course: Lecture 4/14, part 1/2, (Yield Curve Dynamics under Short Rate)
Financial Engineering Course: Lecture 4/14, part 1/2, (Yield Curve Dynamics under Short Rate)
The presenter delivers an informative lecture on short rate models and their connection to yield curve dynamics. They begin by introducing the concept of short rate models and discussing their relevance. To enhance the understanding, they extend the discussion from a single-factor cool white model to a more comprehensive multi-factor model, conducting several simulations along the way.
A comprehensive introduction to yield curves follows, with an exploration of different yield curve shapes and their relation to short rate dynamics. The presenter establishes a connection between these concepts and real market experiments, shedding light on their practical applications. While exploring the single-factor model's limitations, the presenter also presents potential solutions, including the construction and simulation of a two-factor model.
In the subsequent segment, the instructor focuses on mean-reverting processes and demonstrates how to generate paths for these processes. They present a 3D plot showcasing the distribution of interest rates over time. Introducing a transformation called "yt," the instructor explains how this process extracts the mean-reverting part from the whole white model. By applying the Ito lemma to yt and substituting the dynamics for the whole white model, they derive the solution for the white model's distribution.
The dynamics of yt take the center stage as the lecturer highlights its stochastic component independence, effectively removing the dependence on rt and yt. They proceed to find the solution for the process rt through integration. The solution for the whole rate model encompasses a scaling constant, a time-dependent drift function, a volatility component with an exponent, and a decay coefficient. The deterministic nature of the expression makes integrating time-dependent functions easy, and the resulting integral is normally distributed. Consequently, rt follows a normal distribution with an expectation and variance, where the long-term expectation converges to the theta t function. The class of affine diffusion processes is also briefly discussed.
Moving on to jump diffusion processes, the lecturer delves into the characteristics specific to the Hull-White model and interest rate models. They emphasize that the Hull-White model belongs to the class of affine jump diffusion processes, enabling the derivation of the characteristic function for this process and analytical expressions for zero coupon bonds. The derivation of the characteristic function and the application of the Hull-White model's decomposition are explained in detail. Time-dependent parameters are identified as significant factors impacting the model's functions, with the possibility of taking them outside the expectation.
The professor proceeds to discuss the solution to the model and highlights the importance of the Dupey-Duffy-Singleton theorem. They explain that the solution takes the form of a Riccati-type equation, and the theorem facilitates the derivation of functions A and B. This theorem's significance lies in expressing the conditional expectation solely in terms of dependence at the specific point of Rt paths, thus improving simulation. This feature proves particularly valuable for portfolio evaluations requiring multiple nested Monte Carlo simulations. Furthermore, the closed-form nature and ease of implementation of functions A and B make them highly adopted models in the industry, avoiding the need for costly recalibration while effectively calibrating to yield curve dynamics.
The instructor emphasizes a powerful expression that allows for the evaluation of zero coupon bonds without resorting to nested Monte Carlo simulations. This expression eliminates the need for additional simulations, significantly enhancing the efficiency of pricing swaps with long-term maturities. Functions A and B, which depend on maturity, play a pivotal role in this process and can be directly evaluated. The lecturer provides closed-form relations between zero coupon bonds and functions A and B, involving a theta function, volatility, and a minimum speedometer version. Moreover, they demonstrate two approaches to evaluating zero coupon bonds from the model: using the analytical expression or avoiding integrations.
Continuing the lecture, the instructor explains how to calculate zero coupon bonds within the full white model, employing a faster and more efficient method than nested Monte Carlo simulation. They present the expression for the zero coupon bond as a function of variables a and b, as well as the shortest instantaneously forward rate, r0. This method proves advantageous in terms of speed and efficiency compared to the previous nested Monte Carlo simulation approach. The importance of the yield curve in determining present values of future cash flows is also emphasized. The yield curve serves as a crucial tool for mapping quotes of liquid instruments to a unified curve, with different maturities of zero coupon bonds being utilized to construct forward rates. The primary objective of the yield curve is to provide an expectation of future rates under various scenarios.
The lecture further explores the significance of selecting the most liquid instruments when constructing a yield curve. These instruments are chosen due to their frequent usage in hedging and pricing exotic derivatives. The interpolation of points on the yield curve is discussed, as it can have a substantial impact on the overall discount curve used in calculations. Additionally, the yield curve is viewed as a leading indicator of a country's economic direction and can be influenced by the monetary policies of central banks. The mapping of zero coupon bonds to yield is explained, with yields typically expressed as effective rates in units of years. It is noted that the yield curve reflects not only interest rate expectations but also investors' risk attitudes and their preference for bonds with different maturities.
Continuing with the lecture, the lecturer explains the mechanics of yield curves and their dependence on the demand for short-term bonds. Yield curves are represented by a set of nodes, each associated with a corresponding pair. These pairs are used to define spine points on the curve, and the curve itself is a function that maps a set of zero rates to real numbers. The determination of spine points involves calibration instruments, and the interpolation method between these points can vary based on market conventions or individual trader preferences. This interpolation is necessary for obtaining bond values between spine points. The mapping of zero coupon bonds to the yield curve and the construction of the yield curve are also discussed in detail.
The speaker highlights the crucial role of interpolation in calculating bond values and emphasizes its impact on hedging performance. The choice of interpolation method significantly influences sensitivities and risks associated with the yield curves. Furthermore, the construction of the yield curve has a profound impact on hedging strategies. The lecture delves into the conventions regarding the naming of yield curves and yields, with specific examples, such as a five percent yield over five years being related to zero coupon bonds and spine points on the yield curve. The session concludes by foreshadowing the next segment, which will explore yield curve construction in more depth, addressing the sensitivity of instruments, the impact of different interpolation techniques, and the influence of interpolation on hedging performance.
In the subsequent part of the lecture, the speaker emphasizes the importance of accurately calculating yields and stresses the need to employ the full expression instead of solely relying on the expectation of a single term. This is due to the fact that integral and exponent functions do not possess equivalent expectations. Yield curve dynamics are introduced and various shapes of yield curves are explored, including the normal yield curve, which indicates a healthy economy. The speaker further explains how central banks utilize quantitative easing to drive short-term rates low, consequently impacting the shape of the yield curve.
The instructor discusses different shapes of yield curves, including the flat curve and the inverted yield curve. The latter is typically associated with market crises or impending crises. It represents a transition from a normal curve to an inverted curve and may result in banks being hesitant to issue more loans, leading to limited stimulation of the overall economy. The lecture showcases a graph from the US Treasury displaying yield curve dynamics over time, providing insights into future economic trends. The parallel shift of yield curves and its impact on positions in the interest rate realm are also covered.
Shifting the focus to yield curve dynamics under short rates, the lecturer presents a video demonstration that showcases the dynamics of the yield curve. In the video, the blue line represents the effective fed funds rate, which can be considered a short rate since it reflects overnight rates. The green line corresponds to the yield implied by the market, representing market expectations. The video illustrates various crises, such as the 2008 financial crisis, where the yield curve flattened and inverted, leading investors to move from the stock market to Treasury bonds.
The lecturer provides a link to the video, encouraging viewers to explore the dynamics of the yield curve themselves. Understanding the relationship between short rates and yield curve movements is essential for effective risk management. By simulating short rates and constructing yield curves for each path using formulas that incorporate zero coupon bonds, one can gain insights into the dynamics and behavior of yield curves.
Building upon this understanding, the subsequent part of the lecture will delve into more realistic yield curve dynamics derived from short rates. This exploration aims to provide a comprehensive understanding of the interplay between short rates and yield curves, enabling better risk assessment and management in financial markets.
Financial Engineering Course: Lecture 4/14, part 2/2, (Yield Curve Dynamics under Short Rate)
Financial Engineering Course: Lecture 4/14, part 2/2, (Yield Curve Dynamics under Short Rate)
The instructor dives into the topic of simulating short rate models and their application in measuring the dynamics of yield curves. Yield curves represent the market's expectations of future yields and are influenced by market perceptions and expectations. To analyze these dynamics, the instructor presents an experiment that involves observing the continuously compounded rate for each realization of the short rate and generating yield curves for each scenario. This simulation helps assess the realism of the short rate model and the driving function theta t. Real market data is utilized in this experiment to enhance accuracy.
The lecturer highlights the utility of short rate simulations for risk analysis. By generating yield curves for different scenarios, it becomes possible to evaluate the present value of a portfolio comprising interest rate products. To demonstrate this, the lecturer simulates multiple paths for short rates and computes the zero-coupon bonds for each path. Interestingly, the lecture points out that yield curves generated using the full white model exhibit a parallel shift, which is unrealistic in practice. The lecture concludes by showcasing Python code used to generate the yield curves.
Continuing the discussion, the importance of having a continuum in zero coupon bonds for calculating the function theta is emphasized. The lecture stresses the significance of interpolation, particularly interpolating on the rate itself instead of the exponent, to ensure numerical stability. Various choices for interpolation and the number of points for bond calculation are explored. Additionally, the lecture delves into simulating and generating zero coupon bonds and yields, underscoring the importance of implementing these processes consistently and robustly. Finally, the lecture presents the yield curve generated from market data and the simulated Monte Carlo paths of the worldwide model, revealing a healthy yet remarkably low rate.
The lecture proceeds to address the limitations of the full white model. While the model allows for calibrating the entire yield curve, it falls short in calibrating the entire forward curve, which is a common limitation in most short rate models. To overcome this limitation, the lecturer introduces the Labor Market Model, which is well-suited to addressing the forward curve and yield curve calibration. Additionally, the full white model encounters issues with perfectly correlated zero components, further reducing its effectiveness.
Moving on, the limitations of the single-factor Hull-White model are discussed. These limitations include high correlation between bonds with close maturities but lower correlation for bonds with distant maturities, rendering it impossible to calibrate the model to the entire term structure of different interest rates. The model is also deemed unsuitable for risk management purposes as it assumes a correlation of one between zero coupon bonds and short rate dynamics. To address these issues, an extension to the two-factor Hull-White model is introduced. However, this extension is primarily used for risk management and scenario analysis rather than pricing. The dynamics of the two-factor model are explained, with the first factor representing the level of the yield curve and the second factor representing the skewness of the yield curve.
The lecturer proceeds to discuss the Gaussian two-factor Hull-White model, which is a variation of the single-factor model. A comparison between the two models is presented, emphasizing that the parameters' meanings may differ when switching between them. The lecture highlights the advantages of the Gaussian two-factor Hull-White model in terms of simulating processes and its efficient implementation in Monte Carlo simulations. The lecture explores the integral function of the model and its application in zero coupon bond pricing.
Simulating yield curves for given realizations using the full white two-factor model is then explained. The zero coupon bond for this model has a closed analytical form and involves a Gaussian process system. Simulating the Gaussian two-factor model entails simulating two mean-reverting processes that correspond to the term structure, employing expressions for volatilities and correlation coefficients. The lecture differentiates between the processes X and Y, where X represents the level of the yield curve and Y represents the steepness or skewness of the curve. The correlation between the two Brownian motions associated with these processes is negative, indicating a stiffening effect on the curve.
The lecture also delves into the correlation between bonds when applying the same technique to the two-factor model. Unlike the single-factor model, the correlation between corresponding yields is not equal to one in the two-factor model. This finding confirms that adding an additional factor to the model leads to a more realistic implied volatility shape, particularly when pricing caps. However, it's important to note that increasing the number of factors in the model adds complexity and calibration difficulties. Despite this, the two-factor model consistently generates the same yield curve, making it an AJM (arbitrage-free joint model) framework.
The lecture further discusses the limitations of incorporating more factors into the Gaussian model. It is explained that even with a large number of parameters, the flexibility in terms of implied volatilities remains limited due to the absence of stochastic volatility. The lecture then proceeds to simulate paths for the two-factor model, examining the yield curve yields implied by the whole white two-factor model with additional correlation coefficients. The resulting yields exhibit not only a parallel shift but also reflect the impact of correlations and dynamics. This feature proves valuable for risk management purposes. The lecturer concludes this section by sharing the Python code used for the simulation.
Emphasizing the significance of choosing appropriate interpolation techniques when modeling yield curves, the lecturer highlights that the selection of interpolation method can significantly influence the results. The next lectures will cover topics such as yield reconstruction, the impact of different interpolations, common pitfalls to avoid, and methods to ensure realistic interpolation. Additionally, the lecture introduces the concept of a grid for zero coupon bonds. A comparison is made between zero coupon bonds generated from the market and those calculated using the Hull-White model. A Monte Carlo simulation is performed, generating yield curves for both the single-factor and two-factor models over a ten-year period. The lecture concludes with a comparison of the yield calculations obtained from these two models.
Next, the lecture focuses on presenting the simulation results for the two-factor model of yield curve dynamics. These results are compared with those from the one-factor model as well as the analytical results derived from the market. It becomes evident that the two-factor model provides a more realistic and comprehensive representation of the yield curve dynamics. While the overall volatility in the two-factor model is higher due to the additional volatility factor, it does not significantly alter the overall picture. The key takeaway is that incorporating an extra factor in the Gaussian two-factor model leads to a much more realistic depiction of yield dynamics in the Monte Carlo simulation. Finally, the lecturer summarizes the main learnings from the lecture, including solving the Hull-White model and relating zero coupon bonds to the characteristic function, and briefly introduces the construction of the yield curve and its limitations.
Concluding the lecture, the limitations of the Cool White model are discussed. These limitations primarily revolve around the correlations between bonds with different maturities and the model's inability to calibrate to a wide range of instruments in the market due to its limited parameter set. To address these issues, the lecture suggests extending the model to a two-factor framework, allowing for the relaxation of the perfect correlation assumption between zero coupon bonds. The lecture concludes by assigning two exercises for homework: one involving expectations under the t forward measure and the other utilizing Laplace transforms to demonstrate certain expectations.
Throughout the lecture, the importance of understanding and selecting appropriate models for risk analysis and yield curve dynamics becomes evident. While the Hull-White model and its variations offer valuable insights and tools, it's essential to acknowledge their limitations and explore alternative models to address specific challenges.
One such alternative model introduced in the lecture is the Labor Market Model, which provides a solution to the limitation of the Hull-White model in calibrating the entire forward curve. The Labor Market Model allows for more comprehensive calibration of both the forward curve and the yield curve, making it a suitable choice for certain risk management applications.
Additionally, the lecture highlights the significance of interpolation techniques in yield curve modeling. Choosing the right interpolation method is crucial for accurately capturing the behavior and shape of the yield curve. The lecturer emphasizes that interpolation is not just a technical detail but an art that requires careful consideration and understanding of the underlying dynamics. To illustrate the impact of interpolation, the lecture offers a comparison between yield curves generated from market data and those calculated using the Hull-White model. The lecturer demonstrates how different interpolation choices can result in varying yield curve shapes and values. This analysis underscores the importance of selecting an interpolation method that aligns with the desired characteristics and realism of the yield curve.
As the lecture progresses, the topic of simulating yield curves for different scenarios emerges. Monte Carlo simulations prove to be a valuable tool for generating yield curves and assessing the potential risks associated with interest rate products. By simulating multiple paths for short rates and computing the zero-coupon bonds for each path, analysts can evaluate the present value of a portfolio of interest rate products under different market scenarios.
The lecture concludes with a demonstration of Python code used to generate yield curves. The code showcases the practical implementation of the concepts discussed throughout the lecture, offering learners a hands-on experience and reinforcing their understanding of the subject matter.
In summary, the lecture provides an in-depth exploration of short rate models, yield curve dynamics, and their implications for risk analysis. It discusses the limitations of the Hull-White model and introduces alternative models such as the Labor Market Model and the Gaussian two-factor Hull-White model. The importance of selecting appropriate interpolation techniques and conducting Monte Carlo simulations is emphasized. Through examples and practical demonstrations, the lecture equips learners with the knowledge and tools necessary to model and analyze yield curves effectively in various financial contexts.
Financial Engineering Course: Lecture 5/14, part 1/2, (Interest Rate Products)
Financial Engineering Course: Lecture 5/14, part 1/2, (Interest Rate Products)
The lecture begins by introducing various interest rate products, such as interest rate swaps, forward rate agreements, and floating rate notes. These products rely on volatilities like floorlets and couplets for pricing. The lecturer emphasizes that the LIBOR forward rate serves as a fundamental component in all interest rate contracts.
Linear and non-linear products are discussed, and the lecture delves into the concept of the simple compounded forward LIBOR rate, which is extensively used in different interest rate products, including swaps and derivatives. This forward rate helps in establishing expectations regarding the interest rate period. It is important to note that until the reset date, the interest rate remains a stochastic random variable, but after the reset date, it becomes fixed without any uncertainty.
The lecturer explores the exchange of forward rates between two counterparties, leading to forward rate agreements. The cash flows in these agreements are divided by one plus tau times the LIBOR rate for discounting purposes. The forward LIBOR rate is defined over a specific period, and its definition can be related to zero coupon bonds. Pricing the agreement involves using a risk-neutral measure and discounting, with a fixed rate and accruing period playing key roles.
The concept of tradable assets under the risk-neutral measure, including the money savings account, being martingales is explained. The lecturer demonstrates that the value of a forward can be represented as the difference between two bonds and emphasizes that forwards are traded at zero value, implying that the fixed rate should be equal to that amount. The lecture also covers floating rate notes, which are heavily traded interest rate products. Initially, payments for such contracts are set at zero and later adjusted to account for the convenience of not needing to pay anything at the contract's inception.
The lecture focuses on floating rate notes (FRNs), which are defined based on LIBOR rates and involve coupons as fractions of the notional multiplied by accruing periods. Since the LIBOR rate is stochastic, the FRN receives a floating rate. The value of the contract is determined by summing all the payments, which are individually discounted to present value using expectations in the risk-neutral measure. The measure for FRNs changes to the TK forward measure, and determining the expectations requires finding the joint distribution between the empty and LIBOR rate, which is crucial for payment calculations.
The lecture begins by introducing various interest rate products, such as interest rate swaps, forward rate agreements, and floating rate notes. These products rely on volatilities like floorlets and couplets for pricing. The lecturer emphasizes that the LIBOR forward rate serves as a fundamental component in all interest rate contracts.
Linear and non-linear products are discussed, and the lecture delves into the concept of the simple compounded forward LIBOR rate, which is extensively used in different interest rate products, including swaps and derivatives. This forward rate helps in establishing expectations regarding the interest rate period. It is important to note that until the reset date, the interest rate remains a stochastic random variable, but after the reset date, it becomes fixed without any uncertainty.
The lecturer explores the exchange of forward rates between two counterparties, leading to forward rate agreements. The cash flows in these agreements are divided by one plus tau times the LIBOR rate for discounting purposes. The forward LIBOR rate is defined over a specific period, and its definition can be related to zero coupon bonds. Pricing the agreement involves using a risk-neutral measure and discounting, with a fixed rate and accruing period playing key roles.
The concept of tradable assets under the risk-neutral measure, including the money savings account, being martingales is explained. The lecturer demonstrates that the value of a forward can be represented as the difference between two bonds and emphasizes that forwards are traded at zero value, implying that the fixed rate should be equal to that amount. The lecture also covers floating rate notes, which are heavily traded interest rate products. Initially, payments for such contracts are set at zero and later adjusted to account for the convenience of not needing to pay anything at the contract's inception.
The lecture focuses on floating rate notes (FRNs), which are defined based on LIBOR rates and involve coupons as fractions of the notional multiplied by accruing periods. Since the LIBOR rate is stochastic, the FRN receives a floating rate. The value of the contract is determined by summing all the payments, which are individually discounted to present value using expectations in the risk-neutral measure. The measure for FRNs changes to the TK forward measure, and determining the expectations requires finding the joint distribution between the empty and LIBOR rate, which is crucial for payment calculations.
The lecture addresses the misalignment between payment dates and measuring dates and highlights the need for correct evaluation. The measure corresponds to the numerator in a payment schedule, and any corrections or adjustments are necessary if it doesn't align correctly. The Libor with a payment at time tk under the tk forward measure is a martingale, enabling pricing of floating rate notes. The pricing equation involves taking the expectation of the Libor rate over a given period, and the contract is referred to as a swap, where one party receives payment while the other pays based on fixed rates.
Swap contracts are discussed in detail, involving the exchange of cash flows over a specified period. Swaps are commonly used to hedge risks in the mortgage market. There are two options: swap payer, where an individual pays a fixed rate and receives a floating rate, and swap receiver, where an individual receives a fixed rate and pays a floating rate. The notional amount can be deterministic, stochastic, or time-decaying, and the payment frequency can vary. The fixed part remains constant, while the floating part carries uncertainty related to LIBOR rate dynamics.
The lecture emphasizes the importance of hedging in financial engineering, particularly in contracts with stochastic payments. Hedging is crucial to offset potential losses due to fluctuations in underlying assets when a financial institution has obligations to receive fixed or floating rate payments.
The lecturer continues to explain how the value of a swap contract can be calculated by utilizing the summation of accruing periods over zero coupon bonds and establishing a linear relationship between the Libor rate and strike. This calculation provides insight into the value of a swap and highlights the role of zero coupon bonds in hedging.
The lecture further emphasizes that the value of a swap depends on the first and last payments of the bond and can be effectively hedged with the first and last zero coupon bonds. The annuity factor is a crucial component when dealing with swaps as it acts as a tradable asset. Interest rate swaps are considered perfect instruments that allow two parties to hedge their specific exposures, and banks can utilize them to hedge loans from individuals, resulting in significantly large value notions.
The lecture shifts its focus specifically to interest rate swaps, noting that they are often considered at a portfolio level, and the value at inception is typically set to zero, enabling a free deal. The swap rate, which is the strike that makes the swap value equal to zero, can be expressed as a weighted sum of the Libor rates. Basic interest rate swaps can be priced without making underlying model assumptions by utilizing interest rate instruments available in the market and mapping them to a yield curve. The construction of a yield curve based on market instruments will be further discussed in an upcoming lecture.
The lecturer delves into the different types of notionals in a swap, which can be time-dependent, determined by market instruments, or random. Additionally, the conditions necessary for a martingale are explained, which include using traded assets or linear combinations of them. It is highlighted that if a non-linear formula, such as an asset's square, is employed, the relationship between the measure and the asset cannot be considered a martingale. The application of Ito's lemma to the squared Libor demonstrates that L squared is not a martingale under the D forward measure due to the presence of a drift effect.
The lecture progresses to explain how to evaluate a swap using a yield curve and the Hulument model. A yield curve specification is provided, and swaps for different strikes are generated using this model. The value of a swap changes linearly with the strike, and the swap rate is determined using the Newton-Raphson algorithm. The lecture concludes by noting that if the par swap equals 0.03808, then the value of the swap is close to zero, indicating that the strike for which the value of the swap is zero has been found.
This section of the lecture provides a comprehensive overview of interest rate products, with a focus on interest rate swaps. It covers various topics, including the pricing of swaps, hedging strategies, the role of zero coupon bonds, and the evaluation of swaps using yield curves. By understanding these concepts, students gain valuable insights into financial engineering and the calculation of swap contract values.
Financial Engineering Course: Lecture 5/14, part 2/2, (Interest Rate Products)
Financial Engineering Course: Lecture 5/14, part 2/2, (Interest Rate Products)
In this lecture, the focus is on the pricing of derivatives that involve volatility. The speaker begins by covering the concept of measure changes for interest rates, particularly in the context of the Hull-White model. They derive the Rhodom/Nichodemus derivative and apply the Girsanov theorem to calculate the measure changes. This understanding of measure changes is crucial in pricing options on interest rate products.
Next, the lecture explores the dynamics of zero coupon bonds under different measures using the AJM framework. The speaker discusses how these dynamics relate to the pricing of options on these bonds. They highlight the substitution of the instantaneous forward rate for the integral and dz in the expression for the dynamics of a zero coupon bond, providing a derived final expression. The lecture also delves into the dynamics of zero coupon bonds under the Hull-White model and the T-forward measure. The importance of changing the measure, particularly in stochastic discounting, is emphasized to avoid complex calculations.
The speaker introduces the Kirizanov, Loefler, and Radon-Nikodym derivative as tools to switch between different measures. They explain how to find the dynamics of the bond and the money savings account by applying Ito's lemma to the Radon-Nikodym derivative. This leads to the Girsanov theorem, which establishes the relationship between the T-forward measure and the risk-neutral measure and highlights the additional drift when switching between measures. By substituting the Brownian motion under the risk-neutral measure with the T-forward measure, the dynamics of the Hull-White model are derived.
The lecture then introduces a measure short rate model represented by lambda and a theta function dependent on maturity. They define mu theta with two arguments, small t and capital m t, and apply the Girsanov theorem to change the measure from the risk-neutral measure to the T-forward measure. The focus shifts to pricing options on zero coupon bonds, requiring a measure change from the risk-neutral measure to the zero-forward measure. The speaker discusses the dynamics of the zero coupon bond and its distribution under the T-forward measure, providing an expression for the bond and adjusting the strike to a constant time-dependent function. They also discuss the distribution of the process r under this measure.
Moving forward, the lecture explains how the distribution of r under the T-forward measure can be solved using the Black-Scholes model with adjusted parameters. Changing the measure allows for analytical pricing of zero coupon bonds using normal cumulative distribution functions and closed-form solutions. The speaker conducts an experiment to price a zero coupon bond and compares the analytical expression with a Monte Carlo simulation using standard Euler discretization. Code for the simulation is provided, and the calculation of option prices for different strikes is discussed.
The lecture emphasizes the pricing of European-type options on zero coupon bonds, highlighting their importance as they are closely linked to the pricing of options on a forward LIBOR rate. Two approaches for pricing these options are explained: one based on the full light model and the other by directly imposing a distribution or stochastic process on the LIBOR rate. The formula for pricing European call options or couplets is provided, and the method for changing the measure from the risk-neutral measure to the T-forward measure is explained. The focus remains on call options, with a mention that a put option or floor on it will be given as a homework exercise.
Additionally, the dynamics and pricing of LIBOR rates are discussed. The lecture acknowledges that the LIBOR rate is a martingale under the given measure, allowing for the assumption of driftless dynamics. However, using a log-normal distribution to represent LIBOR rates poses challenges, such as the possibility of negative rates, especially in pricing exotic derivatives. Calibration to market data, particularly using cap and floor rates, is deemed necessary, and the interest rate cap is described as a means of providing insurance for a holder with a loan on a floating rate.
The lecture proceeds by discussing the pricing of caplets, which can be decomposed into basic contracts known as couplets. The speaker notes that pricing caplets using a log-normal distribution poses problems due to the potential for negative interest rates. To address this, a shift parameter is introduced to impose on the distribution. The pricing of a caplet using an underlying model is then explained, which is closely related to the pricing of an option on a zero coupon bond. By substituting the definition of a LIBOR rate in terms of zero components, the pricing equation is simplified, resulting in the pricing of a call option on a zero coupon bond with a slightly different strike. The lecture concludes with a brief presentation of the pricing code, which involves a simplified yield curve.
Furthermore, the speaker delves into the pricing of put options on zero coupon bonds, also known as "couplets," and emphasizes the importance of adjusting not only the strike but also the notional when pricing. They acknowledge the close match between Monte Carlo simulation and theoretical pricing for options on zero coupon bonds and yield curves. However, they highlight the significance of market model parameters such as mean reversion and volatility in shaping implied volatility surfaces. They note that while these parameters may have a limited impact on the Hull-White model, it cannot generate an implied volatility smile, only skew. Finally, the speaker summarizes the two main blocks covered in the lecture, which include simple interest rate products and the pricing of simple options in the context of the Hull-White model.
Towards the end of the lecture, the instructor informs the students that the course will solely focus on European-type payoffs, while more exotic derivatives will be addressed in a subsequent course. Homework is assigned, including pricing a floored option and deriving Black's formula for a new variant of the shifted log-normal distribution. Students are instructed to compare the results obtained from Black's formula with their numerical results and introduce a shift to the log-normal stochastic differential equation to reflect the necessary adjustments.
The lecture provides an in-depth exploration of pricing derivatives involving volatility, specifically focusing on the dynamics and pricing of zero coupon bonds, options on these bonds, and LIBOR rates. The concept of measure changes, the use of Radon-Nikodym derivatives, and the application of the Girsanov theorem are covered to facilitate these pricing calculations. The lecture emphasizes the importance of adjusting measures, strike prices, and notional values while highlighting the impact of market model parameters on implied volatility surfaces.
Financial Engineering Course: Lecture 6/14, part 1/3, (Construction of Yield Curve and Multi-Curves)
Financial Engineering Course: Lecture 6/14, part 1/3, (Construction of Yield Curve and Multi-Curves)
Continuing with the topic of yield curves, the lecture emphasizes the significance of constructing an accurate yield curve, which serves as a vital component in valuing interest rate derivatives and financial analyses. The instructor explains that yield curves are essential for discounting future cash flows, determining present values of payments, and valuing companies, among other applications. The construction of a yield curve typically relies on liquid instruments, which introduce less uncertainty into the valuation process. From a mathematical perspective, yield curves map market quotes of these liquid instruments.
Moving on, the instructor provides further insights into the nature of yield curves. They explain that yield curves connect various market instruments in the interest rate world and represent expectations of future rates. While the yield curve may appear stochastic when observed from day to day, its price is deterministic from today's perspective based on expectations. The construction of a yield curve involves selecting a discrete set of liquid instruments and interpolating to connect the spine points. The instructor emphasizes the importance of choosing instruments of similar quality and notes that the number of instruments may change over time. They highlight that the yield curve serves not only as a mathematical tool but also offers valuable economic insights, acting as a barometer of current market conditions.
The lecture delves deeper into the construction and interpretation of yield curves. The instructor discusses how yield curves reflect the allocation of money in the market, whether it is invested in stocks or bonds, and if bonds are preferred, whether they are long-term or short-term. Yield curves provide insights into investors' expectations about future interest rates and their attitudes toward risk. However, the instructor cautions that yield curves have limitations in accurately forecasting the future due to factors such as interventions from central banks and external investments. Therefore, constructing a yield curve meticulously and considering changes that occur over many years is crucial to ensure its accuracy.
The term structure of interest rates is also explained in relation to yield curves. The instructor highlights that yield curves represent the time relation between yields of different maturities and are dependent on the local economy. They mention that the US Treasury bond curve holds significant importance as a global economic indicator due to the US's position as one of the largest economies and the use of the dollar as a reserve currency. Government bonds, such as US Treasury bonds, are typically considered default-free when issued in the local currency, while bonds issued in foreign currencies carry a higher risk of default. The concept of risk premium is also discussed as a factor influencing yield or interest rates.
The lecture explores various shapes of yield curves and their implications for the economy. A standard normal shape indicates that longer-term yields are significantly higher than shorter-term yields, reflecting a normal economic situation. In contrast, an inverted yield curve, where long-term yields decrease while short-term yields remain stable, can signify an unhealthy scenario that may create challenges for banks and pensions. The instructor provides examples of different yield curve shapes and explains how they can affect the market.
The impact of inflation on yields is discussed, highlighting that an increase in inflation expectations leads to higher yields as investors require compensation for the negative real return on their investments. The lecture also covers the concepts of steepening and flattening of the yield curve due to changes in the economy. The spread between a 10-year constant maturity swap and a 2-year swap can indicate the direction of a steepening curve, while the inversion of the yield curve signifies a flattening curve. Graphical examples are used to demonstrate how these different curves and spreads have influenced the economy in the past.
The lecture introduces the concept of yield control and its influence on interest rates. Yield control refers to the central bank's ability to influence the yield curve by adjusting interest rates to achieve targets related to inflation and employment. Central banks can buy or sell bonds to influence demand and stimulate the economy. However, these actions also carry risks and limitations, especially if inflationary pressures increase. The instructor explains that the yield curve is mathematically defined by spline points and corresponding discount factors, which represent expectations of short rates.
Moving on, the instructor delves into the construction of the yield curve and multi-curves in financial engineering. They explain that the curve is constructed by combining spine points obtained from the market with an interpolation routine. Several requirements must be met for a well-constructed yield curve, including pricing the curve using the selected instruments, ensuring continuous forward rates, and employing a local interpolation method for accurate hedging. Constructing the curve also involves defining an optimization problem and determining the vector of zero coupon bonds as spine points at different maturities.
The professor provides a step-by-step explanation of how to construct a yield curve and multi-curves. The process involves finding a vector of Present Value of a contract (PVI) that depends on all the spine points of the curve. The goal is to ensure that the market quote matches the curve price for all the instruments used in constructing the curve. To solve this problem, an optimization technique using the L norm is employed. The professor illustrates how to solve the problem in single-dimensional cases using the Newton-Raphson algorithm, which minimizes the absolute difference to arrive at an optimal solution. Next, the speaker discusses the iteration process used to find the optimal sigma for a Black-Scholes model. He explains the stopping criteria for the model and the requirements for achieving convergence. The speaker emphasizes the interdependence of spine points on the curve and highlights the need to iterate for multiple strikes to build an implied volatility smile or skew. The construction of the interpolation and optimization techniques required for this process, including the building of a Jacobian, are also explained.
The importance of interpolation in constructing various curves, particularly the yield curve and the implied volatility smile, is emphasized by the speaker. They note that while interpolation in yield curves is relatively straightforward due to continuity and differentiability conditions, selecting the appropriate interpolation method is even more critical for implied volatility smile, as an incorrect choice can introduce significant pricing arbitrage. The speaker highlights that interpolation plays a crucial role in all cases, and careful attention to detail is necessary when choosing the appropriate interpolation routine.
The lecture provides comprehensive coverage of the construction and interpretation of yield curves. It highlights their importance in valuing interest rate derivatives and understanding market dynamics. The lecture also explores the mathematical formulation, the impact of different curve shapes on the economy, and the role of yield control. Additionally, it delves into the construction of yield curves and multi-curves, discussing optimization techniques, interpolation choices, and their implications in financial engineering.
Financial Engineering Course: Lecture 6/14, part 2/3, (Construction of Yield Curve and Multi-Curves)
Financial Engineering Course: Lecture 6/14, part 2/3, (Construction of Yield Curve and Multi-Curves)
In the lecture, the speaker delves into the practical aspects of building an algorithm for yield curve construction. They emphasize the importance of curve calibration and analyze Python code used to construct the yield curve using market instruments like swaps. The impact of different interpolation methods on hedging is also explored. The lecturer explains the iteration routine for constructing a yield curve, which involves algebraic calculations with vectors and matrices. They demonstrate how to optimize the curve by setting the next iteration to zero.
Moving on, the instructor explains the process of finding optimal spine points to build a matrix. This process entails iteratively adjusting vector discount factors (dfs) until convergence is achieved. The adjustments are based on a Jacobian matrix, and the inverse of the Jacobian determines the adjustment for the delta of the dfs. The lecture emphasizes the importance of specifying grids (pairs of ti and discount factors) for building the curve before finding optimal zero bonds. A practical example of building a yield curve for a two-year and a five-year interest rate swap is provided, highlighting the challenge of solving a system with more unknowns than equations.
The challenges of constructing a yield curve using swap payments for spine points are discussed due to an underdetermined system. The solution is to consider only the final payment as the spine point and interpolate the points in between. It is emphasized that the number of instruments should equal the number of spine points to avoid confusion. The process of constructing a yield curve using a forward rate agreement and a swap is explained, with an emphasis on numerical implementation.
The lecture emphasizes the importance of building a yield curve and the impact of market quotes, which are typically zero. The definition of the LIDOR rate is discussed, along with expressing the Present Value of a contract (PV1) in terms of the LIDOR rate. The PV1 depends only on the discount factor (df1), which can be calculated using the first set of equations. The second set of equations involves the swap with two payment dates. The lecture explains the use of a lower triangular matrix and efficient inversion for curve building when only swaps are used.
The process of building a yield curve using market data from the U.S. Department of Treasury is explored. Quotes for LIBOR rates and swaps with varying maturities are used to build the yield curve. The lecture introduces the multi-dimensional Newton-Raphson function used to calibrate the curve and emphasizes the importance of selecting the right interpolation method. The function for evaluating a swap instrument on a vector of spine points is also introduced.
The lecture focuses on the construction of yield curves and multi-curves. The process begins with defining a swap and then moves on to constructing a yield curve using an array of instruments and maturities. A multivariate Newton's method is employed to optimize the yield curve during the construction process. The importance of choosing a tolerance value is stressed, and the challenge of optimization with a tolerance of 10 to the power of 10 is highlighted. The lecture concludes by emphasizing the fast convergence achieved with this optimization method.
The evaluation of instruments using spine points and interpolation methods is explained. The yield curve is constructed using spine points and an interpolation method, followed by the evaluation of each swap as a function of the zero coupon bonds based on the current spine point state. A Jacobian, representing the sensitivity of each individual Present Value (PV) to all the spine points, is calculated numerically by performing a shock on each individual spine point and evaluating all the swaps. The lecture highlights the compact and efficient function for calculating the Jacobian.
The lecture discusses the process of building the yield curve and multi-curves using the Newton-Raphson iteration method, the Jacobian matrix, and the numpy linear algebra toolset. After building the yield curve, the swaps are evaluated before building the curve. The lecture emphasizes the need to set a limit on the number of evaluations to avoid overwhelming the Python code and suggests incorporating protections to prevent this issue. Furthermore, the lecture demonstrates how to calculate the present value (PV) of the swaps using both the initial yield curve and the calibrated yield curve obtained from the iteration process involving the spine points.
The professor further explores the optimization routine and yield curve calibration for interest rate swaps. It is noted that the yield curve calibration using swaps yields highly accurate results, even when encountering values below zero. The lecture also highlights areas for improvement, such as employing analytical calculations for derivative sensitivities to enhance computational efficiency and accuracy.
The concept of "hedging" is introduced as a focus for the subsequent section. The impact of different interpolation routines on hedging outcomes is discussed, and various interpolation methods are explored. The professor recommends consulting the existing literature to explore additional options for interpolation. The lecture concludes by emphasizing the significance of conducting tests under small conditions and considering the effects of interpolation routines on the yield curve.
In the lecture, the speaker examines different interpolation routines employed in yield curve construction and their influence on the results. The drawbacks of straightforward interpolation, such as simple linear interpolation, are highlighted, particularly when using a model-based yield curve. It is explained that the behavior of the short rate term structure can become erratic if small details are overlooked in the interpolation, as the instantaneous forward rate depends on the logarithm of a zero-coupon bond. To overcome these limitations, one method suggested is differentiating on log discount factors.
The lecture also explores local and global interpolation, emphasizing the importance of localizing the impact of a shock or change to a spine point to avoid affecting a large number of points on the curve. Additionally, the lecturer stresses the significance of selecting an interpolation method that considers the characteristics of the instruments on the curve and their impact on its performance.
The construction of yield curves and multi-curves is discussed from a financial engineering perspective. A Python experiment is presented, demonstrating a function developed to calibrate a yield curve through small adjustments. The experiment includes the construction of an instrument set as a function and the incorporation of quadratic and cubic interpolation. Furthermore, the pricing of an off-market swap and the sensitivity analysis of the swap to all the market instruments used in constructing the curve are demonstrated through differentiation and curve recalibration for each shocked instrument in the portfolio set.
The speaker explains how to construct a yield curve and multi-curves using shock and delta. The process involves repeating the entire procedure for each instrument with a shocked fixed rate and redefining delta, which represents the derivative of the swap with respect to each market instrument. Delta values are approximated by dividing the shock size, rebuilding the curve, and assessing the resulting impact. With these delta values, it becomes possible to determine the required usage of each market instrument for curve construction, enabling effective hedging of futures. Linear interpolation is employed to illustrate the hedging of a four-year swap using instruments with three and five-year maturities, aligning with expected outcomes. Finally, a comparison between linear and cubic interpolation reveals that cubic interpolation is more computationally expensive but leads to substantial differences in results.
The speaker discusses the construction of yield curves and multi-curves within a financial engineering context. A comparison between cubic interpolation and linear interpolation is made, highlighting that cubic interpolation is more advanced but also slower. The impact of interpolation on hedging is addressed, noting that while cubic interpolation may result in a smoother curve, it can lead to larger hedging expenses due to sensitivities to products with maturities far beyond that of the swaps. The speaker suggests exploring quadratic interpolation as an alternative and emphasizes that the impact of interpolation on hedging should not be overlooked.
Continuing the lecture, the speaker elaborates on the construction of yield curves and multi-curves using shock and delta. This method involves recalibrating the entire process for each instrument with a shocked fixed rate. The delta, which represents the derivative of the swap with respect to each market instrument, is redefined by dividing the size of the shock and approximating the resulting impact on the curve. By analyzing the delta values, it becomes possible to determine the appropriate allocation of each market instrument for curve construction, enabling effective hedging of futures. The speaker demonstrates the use of linear interpolation to illustrate hedging a four-year swap using instruments with three and five-year maturities, aligning with expected outcomes.
The lecture emphasizes the importance of choosing the right interpolation method, as it significantly impacts the shape and behavior of the yield curve. While cubic interpolation may offer a smoother curve, it often incurs larger hedging expenses due to its sensitivity to products with maturities well beyond that of the swaps. Therefore, the speaker suggests exploring quadratic interpolation as an alternative that strikes a balance between accuracy and computational efficiency.
Furthermore, the lecture stresses the need to consider the characteristics of the instruments used in constructing the curve and their impact on its performance. Different instruments may require different interpolation methods or adjustments to ensure accurate pricing and risk management. It is essential to carefully analyze and understand the instruments' behavior within the context of the yield curve construction process.
The lecture concludes by encouraging further research and exploration of interpolation options. While cubic interpolation is more advanced and offers a smoother curve, it may not always be the optimal choice. Financial professionals and researchers are encouraged to delve into the existing literature and study various interpolation routines to identify the most suitable approach for their specific needs.
The construction of yield curves and multi-curves involves a combination of mathematical techniques, calibration methods, and interpolation routines. It is a complex process that requires careful consideration of various factors, such as instrument characteristics, computational efficiency, and hedging implications. By employing the right methods and understanding the underlying principles, financial practitioners can construct robust yield curves that accurately reflect market conditions and support effective risk management strategies.Financial Engineering Course: Lecture 6/14, part 3/3, (Construction of Yield Curve and Multi-Curves)
Financial Engineering Course: Lecture 6/14, part 3/3, (Construction of Yield Curve and Multi-Curves)
In the lecture, the concept of multi-curves is introduced, which incorporates default probabilities of counterparties when constructing yield curves. This additional information accounts for the frequency of payments and the associated risks of default. The speaker highlights that lending money for a longer duration to a counterparty increases the risk compared to shorter-term lending. Multi-curves emerged as a development in financial mathematics after the 2008-2009 financial crisis and remain prevalent in today's market.
The lecture includes a Python implementation of multi-curves and assigns a homework task to students, challenging them to enhance the existing code by incorporating additional instruments for curve calibration and hedging aspects.
The construction of yield curves and multi-curves in financial engineering is discussed, emphasizing the impact of payment frequency on the curve type and risk management. Higher payment frequency reduces the potential loss in case of counterparty default, making it a safer choice. The motivation behind multi-curves stems from the crisis of 2007-2009 when basis spreads between different tenors became significant, leading to multiple basis points of difference between varying frequency curves.
The speaker explains that different instruments exhibit varying liquidity and credit risk premiums, influencing their yield curves. Before the financial crisis, pricing was based on a single curve. However, post-crisis, additional risk premia need to be considered for different tenor structures. The speaker illustrates the risk premium spread between different tenors using an illustration of instantaneous forward rates. The market consensus is to discount future cash flows based on the highest frequency tenure, and the optimal choice for discounting is a curve with the least credit risk, typically associated with a 10 of one day.
The lecture delves into the inclusion of default probabilities in pricing and the development of a framework for pricing derivatives within the multi-curve context. Curves such as the Euro Overnight Index Average and the US Federal Reserve Overnight Rate are discussed. Practitioners first observed the market, and later the theory was developed, necessitating the inclusion of default probabilities in the multi-curve framework. The library definition needs to be modified to incorporate the risk-free curve and the counterparty's default probabilities. The speaker highlights the need for extended versions of the LIBOR rate and measure changes to accommodate this modification. By incorporating default probabilities and verifying the existence of the counterparty before executing transactions, practitioners gain a better understanding of derivative pricing within the multi-curve framework.
The concept of probability of default is explained in the context of pricing derivatives with credit risk. The probability of default represents the risk of default occurring over a specific period and is typically derived from market instruments like credit default swaps. When market instruments are unavailable, banks and financial institutions assign a probability of default based on industry risk association. Pricing derivatives with credit risk involves discounting all future cash flows and assuming independence between interest rates and the probability of default. The expected payoff is then calculated using an indicator function of the probability of default.
The lecture discusses how probabilities of default and enhancement rates relate to survival probabilities and hazard rates. Credit default swaps (CDXs) are introduced as traded derivatives used to estimate the probability of default. By examining market quotes of CDXs, the risk premium can be calculated, providing insights into the likelihood of default. The risky yield curve incorporates the probability of default and adjusts zero coupon bonds using risk adjustments. In practice, D(t0, ti) is typically interpreted as a discount factor, enabling the construction of a yield curve as a collection of zero coupon bond discount factors.
The video explains the process of determining a fair price for an unsecured liability that considers probabilities of default by constructing a curve corresponding to a specific term on top of a discount curve. It demonstrates computing risk-free zero coupon bonds and a zero coupon bond with an additional risk premium, representing the adjustment factor for the curve. The video also covers how the pricing of an interest rate swap can be calculated in a multi-curve setting. It combines the concepts of a risky liability and the rate of the overnight index swap, approximating the pricing by computing the expectation of the forward LIBOR under the corresponding martingale measure.
The lecturer emphasizes the circular dependency between different curves and the construction of yield curves in practice. The discount curve is typically built first, followed by the construction of three-month and six-month curves based on the discount curve and additional market quotes. However, complications arise when there are spreads involved, necessitating calibration of all curves simultaneously rather than individually. While it may be more complex, maintaining consistency in hedging other risks allows using the wrong interest rate in the Black-Scholes model to match the market quote.
The video provides guidance on implementing multi-curves in Python for pricing and constructing multiple yield curves. It builds upon previously developed codes for single yield curves and extends them to handle multi-curves. An extension of the swap definition is introduced to facilitate pricing in the multi-curve context. The video also emphasizes the importance of performing a sanity check to ensure consistency between the new interest rate swap and a single curve setting. This is achieved by using two instances of the same curve to verify that they yield the same value.
The speaker discusses the calibration of the yield curve and introduces four swaps corresponding to the new curve with initial guesses separate from the previous case. The objective remains to match market prices with model prices. The discount curve is based on the bootstrap curve, and the swaps are defined as lambda expressions of the forward curve. The speaker explains the search for zero coupon bonds or yield curves for the swaps and the optimization of values that make the swap zero for the specific yield goal. The calibration of the curve is double-checked, and the values of the swaps are plotted. The sanity check confirms the consistency of the new swap implementation, and finally, the new curve is bootstrapped.
The speaker discusses the results of the calibration and bootstrapping process, noting that the pricing returns to par. The discount curve and the forecasting curve are plotted, illustrating the spread curve between them. The speaker highlights that the forward curve is lower due to the limited number of instruments, resulting in a lack of smooth transition between different maturities. The calibration process is relatively fast, requiring optimization iterations compared to the server for the discount curve. In conclusion, the speaker summarizes the key concepts covered in the lecture, including the yield curve's dynamic nature, mathematical formulation, problem formulation, spine points, optimization routine, and analytical examples.
Lastly, the speaker discusses the extension of the existing code for the beginning of a curve and the inclusion of additional instruments. The practical importance of developing a hedging framework to understand the impacts of different interpretations is emphasized. The video explains the significance of multi-curves and their relationship with default probabilities and forecasting. It concludes by demonstrating Python code to implement and extend the existing framework to handle multi-curves. As a homework assignment, the audience is tasked with extending the existing code for a new curve and incorporating an additional layer of a forward curve based on six months, three months, and available market instruments.
The video explains how to calculate the fair price of an unsecured liability that considers the probabilities of default. This involves building a curve corresponding to a specific term on top of a discount curve. The video demonstrates the computation of risk-free zero coupon bonds and an additional risk premium-based zero coupon bond, representing the adjustment factor for the curve. Furthermore, the pricing of an interest rate swap is discussed, combining the concepts of a risky liability and the rate of the overnight index swap. The pricing approximation involves computing the expectation of the forward LIBOR under the corresponding martingale measure.
To conclude, the lecturer reiterates the importance of yield curve construction, multi-curves, and their practical implications in financial engineering. The lecture covers various aspects such as curve calibration, hedging, probability of default, pricing derivatives with credit risk, and the implementation of multi-curves in Python. By extending the existing code and incorporating additional instruments, students are challenged to deepen their understanding of multi-curves and gain hands-on experience in curve calibration and pricing aspects within a multi-curve framework.
Financial Engineering Course: Lecture 7/14, part 1/2, (Swaptions and Negative Interest Rates)
Financial Engineering Course: Lecture 7/14, part 1/2, (Swaptions and Negative Interest Rates)
The lecture begins with a review of previous topics, including swaps, interest rates, yield curve construction, and basic product pricing. It then progresses to more advanced subjects: swaption pricing and pricing under negative interest rates. Swaptions, which depend on volatility, are explored, along with options in interest rates such as couplets and flow rates.
The concept of a caplet is introduced as a European option that plays a role in calibrating the Hull-White model. Caplets are used in path-dependent models and require calibration to market instruments. The lecturer discusses the Black-76 model for pricing caplets and distinguishes between Black-Scholes equations and Black's equations for interest rate forwards. The implied volatility surface for interest rates and exotic derivatives pricing is briefly mentioned as a topic for a future course.
The lecture delves into parameter calibration for the full white model using market prices for couplers. Implied volatilities using Black's model are introduced and used in the calibration process. The distinction between Black's implied volatility and implied volatility from the model is emphasized. The lecture covers the formula for a library dependent on two zero bonds and its substitution in pricing. A new strike is defined to remove constant or time-dependent components outside the expectation, enabling the exploration of dynamics or distributions under the TK measure.
The pricing of swaptions is discussed in relation to zero coupon bond pricing in a zero coupon model. The difference lies in the timing of payments, with zero coupon bonds paying at the beginning and swaptions at the end. The lecture introduces the concept of conditioning on a signal field and using the definition of a money service account to solve this issue. It leads to an expression for the swaption's price as the expectation of the ratio of two money service accounts under the forward measure.
The lecture further explores the relationship between caplets, bonds, and options on zero coupon bonds. The Black-Scholes model is utilized for calculating implied volatilities, with periodic calibration of the model's parameters. The lecture emphasizes the importance of correctly choosing simulation dates and matching measures and expectations in option pricing.
The generation of implied volatility smiles using interest rate products and pricing options on zero coupon bonds is discussed. The code is inspected to ensure accurate evaluations, and a comparison is made between market and model-derived yield curve zero coupon bonds. The pricing of options on zero coupon bonds, including put options, is covered, and experiments are performed to analyze the impact of volatility and model versions on pricing.
The lecture introduces an iteration process to find implied volatility that satisfies the constraint of equal market value and Black '76 price for an option. Grids of different volatility levels are defined and interpolated as a starting point for Newton-Raphson. The mean reversion parameter's impact on implied volatilities is discussed, with a recommendation for fixing it while calibrating the volatility parameter. Time-dependent parameters are emphasized for XVA considerations.
The limitations of adding stochastic volatility to the HJM model in derivative pricing are addressed, including the impact on the implied volatility skew and calibration challenges. The lecture highlights the significance of the annuity component in swaps and the need to account for it when changing the measure. Understanding interest rate swaps and improving models while maintaining computational efficiency is crucial due to their prevalence in financial institutions.
The pricing of swaps is focused on, assuming a single curve. The value of a swap depends on two payments, initially and at the end, and can be represented as the difference of two zero components with the strike multiplied by the annuity. Par pricing is explained, where the strike is chosen to make the value zero, resulting in no cash payments. Volatility is necessary for pricing exotic derivatives, requiring calibration to market instruments.
The use of swaptions in financial engineering to gauge market volatility is discussed. Swaptions are European derivatives that provide the holder with the right, but not the obligation, to enter into a swap at a predetermined future date. The strike price of the swaption determines whether the holder will be a payer or receiver of the swap. By substituting the swap's definition, the valuation equation for swaptions is derived, and the numerator of the equation is identified as a suitable candidate for a measure change. This allows for the cancellation of the annuity component and simplification of the equation.
The speaker explains the use of annuity measures and geometric Brownian motion to calculate swaption prices, assuming that swap rates cannot be negative. The annuity measure is considered an appropriate choice for measurement, and under this measure, the swap must be a martingale. The Black-Scholes equation is introduced as a pricing model for swaptions. However, the speaker acknowledges that in practice, swaps can have negative values, which can pose challenges for the pricing equation. They mention that a solution to this issue will be presented later in the lecture. The ultimate goal is to determine the price under the BlueWise model, which will be used for simulation in future lectures.
The lecturer discusses the formulation of a swap in terms of zero coupon bonds and how it can be redefined as a single summation of zero coupon bonds with different weights. This formulation proves convenient when seeking a solution for pricing options under full white dynamics. The lecture covers the process of changing the measure from a risk-neutral measure to a measure associated with a zero coupon bond, which helps address the challenge of pricing a swap. The Jambchidian Flick is introduced as a technique to exchange the expectation of the maximum of a sum with a sum of expectations, a crucial step in finding a closed-form solution for pricing swaptions. This method aids in simplifying the pricing process and obtaining accurate results.
The instructor's discussion highlights the importance of understanding and effectively pricing swaptions, as they provide valuable information about market volatility. The ability to accurately assess and price these derivatives contributes to informed decision-making and risk management in financial markets.
The lecture covers various advanced topics related to pricing in the context of swaptions and negative interest rates. It explores the intricacies of calibrating models, determining implied volatilities, and understanding the nuances of different pricing approaches. The lecturer emphasizes the significance of carefully selecting parameters, matching measures and expectations, and considering the limitations and challenges associated with pricing derivatives in complex financial environments.