Interest Rate Models  Theory and Practice: With Smile, Inflation and Credit Hardcover – 2 August 2007
by
Damiano Brigo
(Author),
Fabio Mercurio
(Author)
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Product details
 ISBN10 : 3540221492
 Hardcover : 982 pages
 ISBN13 : 9783540221494
 Language: : English

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4.1 out of 5 stars
13 reviews
Harold Bean
1.0 out of 5 stars
Good information, but terrible binding
5 March 2019 
Published on Amazon.comVerified Purchase
The content of the book is very good, but I would not purchase in hardcover format if you want to use as a reference. The book is very thick (980 page) and the binding broke as soon as I opened the first time. The second time I opened, pages are starting to come loose. I recommend against the hard cover version.
2 people found this helpful
Dr. Lee D. Carlson
5.0 out of 5 stars
Extremely detailed
7 July 2011 
Published on Amazon.comVerified Purchase
The modeling of interest rates is now a multimillion dollar business, and this is likely to grow in the years ahead as worries about quantitative easing, government budgets, housing markets, and corporate borrowing have shown no sign of abatement. The approach that the authors take in this book has been branded as too "theoretical" by some, particularly those on the trading floors, or those antithetic to modeling in the first place. The authors though are aware of such reactions to financial modeling, and actually devote the end of the book to a hypothetical conversation between traders and modelers (but omitting some of the vituperation that can occur between these groups). The book is written very well, with calculation steps for the most part included in detail. Since it is a monograph, there are no exercises, but readers will find ample opportunities to fill in some of the calculations or speculate on some of the many questions that the authors list in the beginning to motivate the book. These questions are invaluable for newcomers to the field, or those readers, such as this reviewer, who are not currently involved in financial modeling but are very curious as to the mathematical issues involved. There is also an excellent list of "theoretical" and "practical" questions in the preface that the authors use to motivate the book, along with a detailed summary of upcoming chapters.
The first part of the book sets the tone for the rest of the book, and can be considered as an elementary introduction to the theory of contingent claim valuation. In this discussion the authors focus on a portfolio consisting of riskless security (bond) and a risky security (stock) that pays no dividend. The object is to follow the time evolution of the price of these two securities. The time evolution of the riskless bond is merely exponential, as expected, but that of the risky security is random according to a geometric Brownian motion. The `trading strategy' consists of holding a number of units of each of these securities at each time. All changes in the value of the portfolio can be shown to be entirely due to capital gains, with none resulting from the withdrawal or infusion of cash. The authors refer to this as a `selffinancing' strategy, and the initial investment results in a pattern of cash flows that replicates that of a call option. This option is attainable by dealing only in a stock and a bond. This leads to the question as to what class of contingent claims a group of investors can actually attain, where a contingent claim is viewed as a nonnegative random variable which is measurable with respect to a filtration of a probability space. This filtration can be viewed as essentially a collection of events that occur or not depending on the history of the stock price. The bearer will obtain a payment at expiry, the size of which depends on the prior price history.
A contingent claim is said to be `attainable' at a particular price if there exists a selffinancing trading strategy, along with an associated market value process that equals the initial prices and equals the contingent claim at expiry. It is shown that every contingent claim is attainable in a complete market. The goal is then to find conditions under which arbitrage is impossible, i.e. conditions that prevent the occurrence of a zero investment and through some trading strategy is able to obtain a positive expected wealth at some time in the future. The authors show that a market is free of arbitrage if and only if there is a martingale measure, and that a market is complete if and only if the martingale measure is unique.
It was primarily the interest of this reviewer in analytical models rather than Monte Carlo simulations, even though there is a thorough discussion of the latter in this book, including the most important topic of the standard error estimation in simulation models. For analytical modeling, the Vasicek model is usually the first one discussed in the literature, and this book is no exception. But the Vasicek model allows negative interest rates and is mean reverting. The authors want to go beyond this model by searching for one that will reproduce any observed term structure of interest rates but that will preserve analytical tractability. One of these, the CoxIngersollRoss (CIR) model, is analytically tractable and preserves the positivity of the instantaneous short rate. Ample space in the book is devoted to a discussion of this model, which is essentially one where one adds a "square root" to the diffusion coefficient.
Physicists who aspire to become financial engineers may find the discussion on the change of numeraire to be similar to the "change in gauge" in quantum field theory. In the latter, a clever choice of gauge can make calculations a lot easier. The same goes for a choice of numeraire for pricing a contingent claim, and the authors give a detailed overview of what is involved in doing so. Of particular importance in this discussion is the role of the RadonNikodym derivative, a concept that arises in measure theory, and also the use of Bayes rule for conditional expectations. To fully appreciate this discussion, if not the entire book, readers will have to have a solid understanding of these concepts along with stochastic calculus and numerical solution of stochastic differential equations.
Interestingly, the authors devote a part of the book to the connection between interest rate models and credit derivatives, wherein they argue that credit derivatives are not only interesting in and of themselves, but that the tools used to model interest rate swaps can be applied to credit default swaps to a large degree. Of particular importance is the appearance of copulas in chapter 21, which have been criticized lately for their alleged role in the "financial crisis". The authors give an overview of these entities for the curious reader but do not use them in the book.
Some readers may find when first exposed to `reduced form models' that they might seem too extreme or judged to be inapplicable because default is viewed as being essentially independent of market observables. Instead default is modeled by an exogenous jump stochastic process. The authors spend a fair amount of time explaining why these models are suitable for credit spreads. In particular, they show that the probability to default after a given time, i.e. the `survival' probability, can be interpreted as a zero coupon bond and the intensities as instantaneous credit spreads. Positive interest shortrate models can therefore be used to do default modeling. The lack of an economic interpretation for the default event is to be contrasted with term structure models, and the authors discuss this in detail.
Structural models on the other hand are tied to economic factors, namely the value of the firm, i.e. its ability to pay back its debt. If this value drops below a certain level, the firm is taken to be insolvent. The authors give a brief overview of structural models, emphasizing their similarities to barrierfree option models, but do not treat them in detail in the book, since they do not have any analogues to interest rate models. For credit risk, the defaultable zero coupon bond is the analog of the zero coupon bond for interest rate curves. The forward rate for credit default swaps also has an analog with LIBOR and SWAP rates. Readers interested in counterparty risk will be exposed to an interesting assertion, namely that the value of a (generic) claim that has counterparty risk is always less than the value of a similar claim whose counterparty has a probability of default equal to zero. The authors give a rigorous formulation of this assertion by proving a general counterparty risk pricing formula.
Poisson processes, used heavily in network modeling and queuing theory, are discussed here in the authors' elaboration of intensity models, along with Cox processes where the intensity is stochastic. Detailed examples are given which illustrate how to use reduced form models and market quotes to estimate default probabilities.
Monte Carlo simulations, which are the bread and butter of financial modeling (along with many other fields of modeling) are used to simulate the default time. The authors address the problem of large variance and the consequent large number of simulations needed if the standard error is just one basis point. Techniques of variance reduction in Monte Carlo simulation are wellknown, and the authors discuss one of these, the control variate technique.
Also discussed is a hybrid model where both interest rates and stochastic intensities are involved, and the authors show how to calibrate survival probabilities and discount factors separately when there is no correlation between the interest rates and intensities. The calibration must then be done simultaneously when this is not the case. One is led to ask in this case, and in general, whether interest rate data can serve as a proxy of default calibration, and vice versa. Not really, but the authors do explain how the correlation can be ignored, since it has little impact on credit default swaps.
Ensuring that interest rates remain positive is thought of as an important side constraint by many modelers, who point to the large negative rates that may occur in Gaussian models of interest rates. One model that particularly stands out in this regard is due to B. Flesaker and L. Hughston, and which is discussed in one of the appendices in the book. Their strategy is to enforce positivity via the discount factor, and doing this in such a way so as to eliminate the possibility of "explosions", i.e. situations where the payoff can become infinite in an arbitrarily short time. Their model can essentially be characterized by an integral representation for discount bonds in terms of a family of kernel functions. The members of this family are positive martingales, and this ensures the required positivity. Their behavior under a change of measure involves a ratio called the `stateprice density' or `pricing kernel', and this shows that the FlesakerHughston model can be interpreted as a general model of interest rates. Arguments are given as to whether all choices of kernel can result in viable interest rate models. Examples are given illustrating that not all can be, but the FlesakerHughston model is interesting also in that it does not depend on possibly highly complex systems of stochastic differential equations for interest rate processes. The authors unfortunately do not include a discussion on how to calibrate this model to market data, but instead delegate it to the references.
Note: This book was read and studied between the dates of September 2007 and July 2011.
The first part of the book sets the tone for the rest of the book, and can be considered as an elementary introduction to the theory of contingent claim valuation. In this discussion the authors focus on a portfolio consisting of riskless security (bond) and a risky security (stock) that pays no dividend. The object is to follow the time evolution of the price of these two securities. The time evolution of the riskless bond is merely exponential, as expected, but that of the risky security is random according to a geometric Brownian motion. The `trading strategy' consists of holding a number of units of each of these securities at each time. All changes in the value of the portfolio can be shown to be entirely due to capital gains, with none resulting from the withdrawal or infusion of cash. The authors refer to this as a `selffinancing' strategy, and the initial investment results in a pattern of cash flows that replicates that of a call option. This option is attainable by dealing only in a stock and a bond. This leads to the question as to what class of contingent claims a group of investors can actually attain, where a contingent claim is viewed as a nonnegative random variable which is measurable with respect to a filtration of a probability space. This filtration can be viewed as essentially a collection of events that occur or not depending on the history of the stock price. The bearer will obtain a payment at expiry, the size of which depends on the prior price history.
A contingent claim is said to be `attainable' at a particular price if there exists a selffinancing trading strategy, along with an associated market value process that equals the initial prices and equals the contingent claim at expiry. It is shown that every contingent claim is attainable in a complete market. The goal is then to find conditions under which arbitrage is impossible, i.e. conditions that prevent the occurrence of a zero investment and through some trading strategy is able to obtain a positive expected wealth at some time in the future. The authors show that a market is free of arbitrage if and only if there is a martingale measure, and that a market is complete if and only if the martingale measure is unique.
It was primarily the interest of this reviewer in analytical models rather than Monte Carlo simulations, even though there is a thorough discussion of the latter in this book, including the most important topic of the standard error estimation in simulation models. For analytical modeling, the Vasicek model is usually the first one discussed in the literature, and this book is no exception. But the Vasicek model allows negative interest rates and is mean reverting. The authors want to go beyond this model by searching for one that will reproduce any observed term structure of interest rates but that will preserve analytical tractability. One of these, the CoxIngersollRoss (CIR) model, is analytically tractable and preserves the positivity of the instantaneous short rate. Ample space in the book is devoted to a discussion of this model, which is essentially one where one adds a "square root" to the diffusion coefficient.
Physicists who aspire to become financial engineers may find the discussion on the change of numeraire to be similar to the "change in gauge" in quantum field theory. In the latter, a clever choice of gauge can make calculations a lot easier. The same goes for a choice of numeraire for pricing a contingent claim, and the authors give a detailed overview of what is involved in doing so. Of particular importance in this discussion is the role of the RadonNikodym derivative, a concept that arises in measure theory, and also the use of Bayes rule for conditional expectations. To fully appreciate this discussion, if not the entire book, readers will have to have a solid understanding of these concepts along with stochastic calculus and numerical solution of stochastic differential equations.
Interestingly, the authors devote a part of the book to the connection between interest rate models and credit derivatives, wherein they argue that credit derivatives are not only interesting in and of themselves, but that the tools used to model interest rate swaps can be applied to credit default swaps to a large degree. Of particular importance is the appearance of copulas in chapter 21, which have been criticized lately for their alleged role in the "financial crisis". The authors give an overview of these entities for the curious reader but do not use them in the book.
Some readers may find when first exposed to `reduced form models' that they might seem too extreme or judged to be inapplicable because default is viewed as being essentially independent of market observables. Instead default is modeled by an exogenous jump stochastic process. The authors spend a fair amount of time explaining why these models are suitable for credit spreads. In particular, they show that the probability to default after a given time, i.e. the `survival' probability, can be interpreted as a zero coupon bond and the intensities as instantaneous credit spreads. Positive interest shortrate models can therefore be used to do default modeling. The lack of an economic interpretation for the default event is to be contrasted with term structure models, and the authors discuss this in detail.
Structural models on the other hand are tied to economic factors, namely the value of the firm, i.e. its ability to pay back its debt. If this value drops below a certain level, the firm is taken to be insolvent. The authors give a brief overview of structural models, emphasizing their similarities to barrierfree option models, but do not treat them in detail in the book, since they do not have any analogues to interest rate models. For credit risk, the defaultable zero coupon bond is the analog of the zero coupon bond for interest rate curves. The forward rate for credit default swaps also has an analog with LIBOR and SWAP rates. Readers interested in counterparty risk will be exposed to an interesting assertion, namely that the value of a (generic) claim that has counterparty risk is always less than the value of a similar claim whose counterparty has a probability of default equal to zero. The authors give a rigorous formulation of this assertion by proving a general counterparty risk pricing formula.
Poisson processes, used heavily in network modeling and queuing theory, are discussed here in the authors' elaboration of intensity models, along with Cox processes where the intensity is stochastic. Detailed examples are given which illustrate how to use reduced form models and market quotes to estimate default probabilities.
Monte Carlo simulations, which are the bread and butter of financial modeling (along with many other fields of modeling) are used to simulate the default time. The authors address the problem of large variance and the consequent large number of simulations needed if the standard error is just one basis point. Techniques of variance reduction in Monte Carlo simulation are wellknown, and the authors discuss one of these, the control variate technique.
Also discussed is a hybrid model where both interest rates and stochastic intensities are involved, and the authors show how to calibrate survival probabilities and discount factors separately when there is no correlation between the interest rates and intensities. The calibration must then be done simultaneously when this is not the case. One is led to ask in this case, and in general, whether interest rate data can serve as a proxy of default calibration, and vice versa. Not really, but the authors do explain how the correlation can be ignored, since it has little impact on credit default swaps.
Ensuring that interest rates remain positive is thought of as an important side constraint by many modelers, who point to the large negative rates that may occur in Gaussian models of interest rates. One model that particularly stands out in this regard is due to B. Flesaker and L. Hughston, and which is discussed in one of the appendices in the book. Their strategy is to enforce positivity via the discount factor, and doing this in such a way so as to eliminate the possibility of "explosions", i.e. situations where the payoff can become infinite in an arbitrarily short time. Their model can essentially be characterized by an integral representation for discount bonds in terms of a family of kernel functions. The members of this family are positive martingales, and this ensures the required positivity. Their behavior under a change of measure involves a ratio called the `stateprice density' or `pricing kernel', and this shows that the FlesakerHughston model can be interpreted as a general model of interest rates. Arguments are given as to whether all choices of kernel can result in viable interest rate models. Examples are given illustrating that not all can be, but the FlesakerHughston model is interesting also in that it does not depend on possibly highly complex systems of stochastic differential equations for interest rate processes. The authors unfortunately do not include a discussion on how to calibrate this model to market data, but instead delegate it to the references.
Note: This book was read and studied between the dates of September 2007 and July 2011.
9 people found this helpful
Ronnie
5.0 out of 5 stars
His class is really fantastic as well as the book he wrote
2 April 2018 
Published on Amazon.comVerified Purchase
Fabio is one of the instructors teaching Interest rate and FX models at NYU Courant and I am in his class right now. His class is really fantastic as well as the book he wrote. I really enjoyed the experience having him as my Professor.
Daniel C
4.0 out of 5 stars
it is nevertheless useful for someone who has a need for the underlying ...
3 April 2016 
Published on Amazon.comVerified Purchase
A solid, widely accepted reference on fixed income modeling. Written more from an academic's than practitioner's perspective, it is nevertheless useful for someone who has a need for the underlying theory.
Richard Ren
5.0 out of 5 stars
Applicationbased but it still contains useful proof of formulas
29 August 2016 
Published on Amazon.comVerified Purchase
Very clearlywritten book. Applicationbased but it still contains useful proof of formulas.
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