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The 2nd edition of this successful book has several new features.

EconPapers: Damiano Brigo and Fabio Mercurio: Interest Rate Models – Theory and Practice

The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous fanio matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced.

The old sections devoted to the smile issue in the LIBOR market model have been enlarged into a new chapter. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. The fast-growing interest for hybrid products has led to a new chapter.

Professional Area of Damiano Brigo’s web site

A special focus here is devoted to the pricing of inflation-linked derivatives. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives — mostly Credit Default Swaps CDSCDS Options and Constant Maturity CDS – are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.

It perfectly combines mathematical depth, historical perspective and practical relevance. The fact that the authors combine a strong mathematical finance background with expert practice knowledge they both work in a bank contributes hugely to its format.

I also admire the style of writing: If you are looking for one reference on interest rate models then look no further as this text will provide you with excellent knowledge in theory and practice.

Overall, this is by far the best interest rate models book in the market. Its main goal is to construct some kind of bridge between theory and practice in this field. From one side, the authors would like to help quantitative analysts and advanced traders handle interest-rate derivatives with a sound theoretical apparatus.

Please note that the first edition is out of print and the second will be available in March ISBN Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? Examples of calibrations to real market data are now considered.

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A Modeps Course Springer Finance. Advances in Financial Machine Learning. Marcos Lopez de Prado. Review From the reviews: Springer; 2nd edition August 2, Language: Try the Kindle edition and experience these great reading features: Share your thoughts with other customers. Write a customer review. Showing of 12 reviews.

Top Reviews Most recent Top Reviews. There was a interrst filtering reviews right now. Please try again later. The modeling of interest rates is now a multi-million 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 mfrcurio 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 integest 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 kodels a monograph, there are no exercises, intereat 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 inteest, 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 fbio 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.

All changes in the value of the portfolio can be shown to be entirely mecrurio to capital gains, with none resulting from the withdrawal or infusion of cash. 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 mercurko or mocels depending on the history of the stock price. Inferest bearer will obtain a payment at expiry, the size of which depends on the prior price history.

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. 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 inteerest 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 Cox-Ingersoll-Ross 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.

Fabio Mercurio

Of particular importance in this discussion is the role of the Radon-Nikodym 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. 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. Positive interest short-rate 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. If this value drops below a certain level, the firm is taken to be insolvent. The authors give intetest brief overview of structural models, emphasizing their similarities to barrier-free 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. 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 faibo a damiaho 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 intreest 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 well-known, and the authors discuss one of these, the control variate technique.

Also discussed is a innterest model where both interest rates and stochastic intensities are involved, fabi the authors show how to calibrate survival probabilities and discount factors moddls when there is no correlation between the interest rates and intensities. The calibration must then be brkgo 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.