Interest Rate Models — Theory and Practice. With Smile From Short Rate Models to HJM. Front Matter. Pages PDF · One-factor short-rate models. The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for. Germany, the. ) Brigo D. and Mercurio F., Interest Rate Models: Theory and Practice (, 2nd ed. ) Buff R., Uncertain Volatility Models-Theory and Application.
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“assets” of interest rate theory, and it is their behaviour we are trying to model. . In practice, the payout of an interest rate derivative is specified in terms of one. The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably. Interest Rate Models - Theory and Practice. With Smile, Inflation and Credit. Bearbeitet von. Damiano Brigo, Fabio Mercurio. Neuausgabe Buch. LVI,
We then present a method of ours for extending pre-existing timehomogeneous models to models that perfectly calibrate the initial yield curve while keeping free parameters for calibrating volatility structures. Our method preserves the possible analytical tractability of the basic model. The reader, however, will have to adapt the model from intensity to interest rates on her own. We then show how to extend the Dothan and EV models, as possible alternatives to the use of the popular BK model.
We start by explaining the importance of the multi-factor setting as far as more realistic correlation and volatility structures in the evolution of the interest-rate curve are concerned. First,we apply our above deterministic-shift method for extending preexisting time-homogeneous models to the two-factor additive Gaussian case G2.
As usual, our method preserves the analytical tractability of the basic model. We introduce the general framework and point out how it can be considered the right theoretical framework for developing interest-rate theory and especially no-arbitrage.
We report conditions on volatilities leading to a Markovian process for the short rate.
This is important for implementation of lattices, since one then obtains linearly-growing recombining trees, instead of exponentiallygrowing ones.
We then introduce the Ritchken and Sankarasubramanian framework, which allows for Markovianity of an enlarged process, of which the short rate is a component.
The related tree Li, Ritchken and Sankarasubramanian is presented. This chapter presents one of the most popular families of interest-rate models: the market models. Now, the cap and swaption markets are the two main markets in the interest-ratederivatives world, so compatibility with the related market formulas is a very desirable property.
However, even with rigorous separate compatibility with the caps and swaptions classic formulas, the LFM and LSM are not compatible with each other. Still, the separate compatibility above is so important that these models, and especially the LFM, are nowadays seen as the most promising area in interest-rate modeling. We start the chapter with a guided tour presenting intuitively the main issues concerning the LFM and the LSM, and giving motivation for the developments to come.
We introduce several new parametric forms for instantaneous correlations, and we deal both with full rank and reduced rank matrices.
We consider their impact on swaptions prices, and how, in general, Monte Carlo simulation should be used to price swaptions with the LFM instead of the LSM. We point out that terminal correlation depends on the particular measure chosen for the joint dynamics in the LFM. These formulas clarify the relationship between instantaneous correlations and volatilities on one side and terminal correlations on the other side.
We develop a formula for transforming volatility data of semi-annual or quarterly forward rates in volatility data of annual forward rates, and test it against Monte Carlo simulation of the true quantities. This is useful for joint calibration to caps and swaptions, allowing one to consider only annual data.
In this chapter, we start from a set of market data including zero-coupon curve, XXIV Preface caps volatilities and swaptions volatilities, and calibrate the LFM by resorting to several parameterizations of instantaneous volatilities and by several constraints on instantaneous correlations. Swaptions are evaluated through the analytical approximations derived in the previous chapter. We examine the evolution of the term structure of volatilities and the ten-year terminal correlation coming out from each calibration session, in order to assess advantages and drawbacks of every parameterization.
No optimization is necessary in general and the calibration is instantaneous. However, if the initial swaptions data are misaligned because of illiquidity or other reasons, the calibration can lead to negative or imaginary volatilities. We show that smoothing the initial data leads again to positive real volatilities. The interpolation of missing quotes in the original input swaption matrix seems to heavily affect the subsequent calibration of the LIBOR model.
We test this new method and see that practically all anomalies present in earlier cascade calibration experiments are surpassed. Partial tests had already been performed at the end of Chapter 7. We conclude that the above formulas are accurate in non-pathological situations. We also plot the real swap-rate distribution obtained by simulation against the lognormal distribution with variance obtained by the analytical approximation.
The two distributions are close in most cases, showing that the previously remarked theoretical incompatibility between LFM and Preface XXV LSM where swap rates are lognormal does not transfer to practice in most cases.
Interest Rate Models Theory and Practice
We also test our approximated formulas for terminal correlations, and see that these too are accurate in non-pathological situations. Chapter Local-Volatility Models. Local-volatility models are based on asset dynamics whose absolute volatility is a deterministic transformation of time and the asset itself.
Their main advantages are tractability and ease of implementation. We then illustrate the class of density-mixture models proposed by Brigo and Mercurio and Brigo, Mercurio and Sartorelli, providing also an example of calibration to real market data. A seemingly paradoxical result on the correlation between the underlying and the volatility, also in relation with later uncertain parameter models, is pointed out. The local-volatility models in this chapter are meant to be calibrated to the caps market, and to be only used for the pricing of LIBOR dependent derivatives.
Chapter Stochastic-Volatility Models. When the instantaneous correlation between a forward rate and its volatility is zero, the existence itself of a stochastic volatility leads to smile-shaped implied volatility curves. Explicit formulas for both caplets and swaptions are usually derived by calculating the characteristic function of the underlying rate under its canonical measure.
Chapter Uncertain Parameters Models. Uncertain-volatility models are an easy-to-implement alternative to stochastic-volatility models. The volatility, therefore, is not constant and one assumes several possible scenarios for its value, which is to be drawn immediately after time zero. To account for skews in implied volatilities, uncertain-volatility models are usually extended by introducing uncertain shift parameters.
Besides their intuitive meaning, uncertain-parameters models have a number of advantages that strongly support their use in practice. As a drawback, future implied volatilities lose the initial smile shape almost immediately. This can further support their use in the pricing and hedging of interest rate derivatives.
We will derive caps and approximated swaptions prices in closed form. We will then consider examples of calibration to caps and swaptions data.
Maybe, for once we could try to be kind. Most of these are products that are found in the market and for which no standard pricing technique is available.
The differences are based on realistic behaviour, ease of implementation, analytical tractability and so on. For each product we present at least one model based on a compromise between the above features, and in some cases we present more models and compare their strong and weak points.
We add numerical examples for Bermudan swaptions. Further, in this new edition we consider target redemption notes and CMS spread options. Again, most of these are products that are found in the market and for which no standard pricing technique is available. A market quanto adjustment and market formulas for basic quanto derivatives are also introduced. To this end, we will review and use i the Jarrow and Yildirim model, where both nominal and real rates are assumed to evolve as in a onefactor Gaussian HJM model, ii the Mercurio application of the LFM, and iii the market model of Kazziha, also independently developed by Belgrade, Benhamou and Koehler and by Mercurio.
Examples of calibration to market data will also be presented. Chapter Introducing Stochastic Volatility. In this chapter we add stochastic volatility to the market model introduced in Chapters 16 and Chapter Introduction and Pricing under Counterparty Risk.
We present a guided tour to give some orientation and general feeling for this credit part of the book. The need for dynamical models of dependence is pointed out.
This is the only part of the book where we mention multi-name credit derivatives. The book focuses mostly on single name credit derivatives. We also introduce constant maturity CDS, a product that has grown in popularity in recent times. This product presents analogies with constant maturity swaps in the default free market.
Finally, we close the chapter with counterparty risk pricing in interest rate derivatives. The counterparty risk pricing formula of Brigo and Masetti for non-standard swaps and swaps under netting agreements is only hinted at. Chapter Intensity Models.
In this new chapter we focus completely on intensity models, exploring in detail also the issues we have anticipated in the earlier chapter in order to be able to deal with CDS and notions of implied hazard rates and functions. Default is not induced by basic market observables but has an exogenous component that is independent of all the default free market information. Monitoring the default free market interest rates, exchange rates, etc does not give complete information on the default process, and there is no economic rationale behind default.
This family of models is particularly suited to model credit spreads and in its basic formulation is easy to calibrate to Credit Default Swap CDS or corporate bond data. The basic facts from probability are essentially the theory of Poisson and Cox processes. We then move to time-inhomogeneous Poisson processes, that allow to model credit spreads without volatility.
This approach allows us to take into account credit spread volatility. In all three cases of constant, deterministictime-varying and stochastic intensity we point out how the Poisson process structure allows to view survival probabilities as discount factors, the intensity as credit spread, and how this helps us in recycling the interest-rate technology for default modeling. We then analyze in detail the CDS calibration with deterministic intensity models, illustrating the notion of implied hazard function with a case study based on Parmalat CDS data.
We explain the fundamental idea of conditioning only to the partial information of the default free market when pricing credit derivatives.
We also explain how to simulate the default time, illustrating the notion of standard error and presenting suggestions on how to keep the number of paths under control. We show how to calibrate the SSRD model to CDS quotes Preface XXXI and interest rate data in a separable way, and argue that the instantaneous correlation has a negligible impact on the CDS price, allowing us to maintain the separability of the calibration in practice even when correlation is not zero.
We present some original numerical schemes due to Alfonsi and Brigo for the simulation of the SSRD model that preserve positivity of the discretized process and analyze the convergence of such schemes. We also introduce the Brigo-Alfonsi Gaussian mapping technique that maps the model into a two factor Gaussian model, where calculations in presence of correlation are much easier. We analyze the mapping procedure and its accuracy by means of Monte Carlo tests.
As an exercise we price a cancellable structure with the stochastic intensity model. Numerical examples of implied volatilities from CDS option quotes are given, and are found to be rather high, in agreement with previous studies dealing with historical CDS rate volatilities Hull and White.
Interest Rate Models - Theory and Practice (eBook)
We give some hints on modeling of the volatility smile for CDS options, based on the general framework introduced earlier. This formula is based on a sort of convexity adjustment and bears resemblance to the formula for valuing constant maturity swaps with the LIBOR model, seen earlier in the book.
The adjustment is illustrated with several numerical examples. We present a few interestrate models that are particular in their assumptions or in the quantities they XXXII Preface model, and that have not been treated elsewhere in the book.
We do not give a detailed presentation of these models but point out their particular features, compared to the models examined earlier in the book. The appendix treats equity-derivatives valuation under stochastic interest rates, presenting us with the challenging task of modeling stock prices and interest rates at the same time.
Precisely, we consider a continuoustime economy where asset prices evolve according to a geometric Brownian motion and interest rates are either normally or lognormally distributed.
Explicit formulas for European options on a given asset are provided when the instantaneous spot rate follows the Hull-White one-factor process. It is also shown how to build approximating trees for the pricing of more complex derivatives, under a more general short-rate process. There is, of course, a dearth of good mathematics teachers [ We quickly introduce the related Ito and Stratonovich integrals, and introduce the fundamental Ito formula.
These schemes are essential when in need of Monte Carlo simulating the trajectories of an Ito process whose transition density is not explicitly known. We include two important theorems: the Feynman-Kac theorem and the Girsanov theorem. The Girsanov theorem in particular is used in the book to derive the change of numeraire toolkit. Given its importance in default modeling, we also introduce the Poisson process, to some extent the purely jump analogous of Brownian motion.
Appendix D: a Useful Calculation. This appendix reports the calculation of a particular integral against a standard normal density, which is useful when dealing with Gaussian models. Appendix E: a Second Useful Calculation.
This appendix shows how to calculate analytically the price of an option on the spread between two assets, under the assumption that both assets evolve as possibly correlated geometric Brownian motions.
In this appendix we answer a number of frequently asked questions concerning the book trivia and curiosities. It is a light appendix, meant as a relaxing moment in a book that at times can be rather tough. Appendix H: Talking to the Traders. This is the ideal conclusion of the book, consisting of an interview with a quantitative trader. Several issues are discussed, also to put the book in a larger perspective.
A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced. 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. Examples of calibrations to real market data are now considered.
A special focus here is devoted to the pricing of inflation-linked derivatives. The three final new chapters of this second edition are devoted to credit. 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 CDS , CDS 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. Skip to main content Skip to table of contents. Advertisement Hide. Definitions and Notation.
Pages No-Arbitrage Pricing and Numeraire Change. Front Matter Pages One-factor short-rate models. Two-Factor Short-Rate Models. Including the Smile in the LFM. Local-Volatility Models. Stochastic-Volatility Models.Intensity Models. The appendix treats equity-derivatives valuation under stochastic interest rates, presenting us with the challenging task of modeling stock prices and interest rates at the same time. The 2nd edition of this successful book has been extensively updated and expanded.
We also plot the real swap-rate distribution obtained by simulation against the lognormal distribution with variance obtained by the analytical approximation. The 2nd edition of this successful book has several new features. Given its importance in default modeling, we also introduce the Poisson process, to some extent the purely jump analogous of Brownian motion.
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