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2001 | Buch

Definitions and Notation Kapitel lesen Erstes Kapitel lesen

Interest Rate Models Theory and Practice

verfasst von: Damiano Brigo, Fabio Mercurio

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Finance

Enthalten in: Professional Book Archive

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Über dieses Buch

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

The fast-growing interest for hybrid products has led to new chapters. 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.

Inhaltsverzeichnis

Frontmatter

Models: Theory and Implementation

Frontmatter
1. Definitions and Notation
Abstract
In this first chapter we present the main definitions that will be used throughout the book. We will introduce the basic concepts in a rigorous way while providing at the same time intuition and motivation for their introduction. However, before starting with the definitions, a remark is in order.
Damiano Brigo, Fabio Mercurio
2. No-Arbitrage Pricing and Numeraire Change
Abstract
The fundamental economic assumption in the seminal paper by Black and Scholes (1973) is the absence of arbitrage opportunities in the considered financial market. Roughly speaking, absence of arbitrage is equivalent to the impossibility to invest zero today and receive tomorrow a nonnegative amount that is positive with positive probability. In other words, two portfolios having the same payoff at a given future date must have the same price today. By constructing a suitable portfolio having the same instantaneous return as that of a riskless investment, Black and Scholes could then conclude that the portfolio instantaneous return was indeed equal to the instantaneous risk-free rate, which immediately led to their celebrated partial differential equation and, through its solution, to their option-pricing formula.
Damiano Brigo, Fabio Mercurio
3. One-factor short-rate models
Abstract
The theory of interest-rate modeling was originally based on the assumption of specific one-dimensional dynamics for the instantaneous spot rate process r. Modeling directly such dynamics is very convenient since all fundamental quantities (rates and bonds) are readily defined, by no-arbitrage arguments, as the expectation of a functional of the process r.
Damiano Brigo, Fabio Mercurio
4. Two-Factor Short-Rate Models
Abstract
However, motivation never harmed anyone (least of all, pure mathematicians), [ L.C.G. Rogers and D. Williams, in Chapter III.4 of “Diffusions, Markov Processes and Martingales”, Vol. 1, 1994, Wiley and Sons
In the present chapter we introduce two major two-factor short-rate models. Before starting with the actual models, we would like to motivate two-factor models by pointing out the weaknesses of the one-factor models of the previous chapter. This is the purpose of this introductory section.
Damiano Brigo, Fabio Mercurio
5. The Heath-Jarrow-Morton (HJM) Framework
Abstract
Modeling the interest-rate evolution through the instantaneous short rate has some advantages, mostly the large liberty one has in choosing the related dynamics. For example, for one-factor short-rate models one is free to choose the drift and instantaneous volatility coefficient in the related diffusion dynamics as one deems fit, with no general restrictions. We have seen several examples of possible choices in Chapter 3. However, short-rate models have also some clear drawbacks. For example, an exact calibration to the initial curve of discount factors and a clear understanding of the covariance structure of forward rates are both difficult to achieve, especially for models that are not analytically tractable.
Damiano Brigo, Fabio Mercurio
6. The LIBOR and Swap Market Models (LFM and LSM)
Abstract
In this chapter we consider one of the most popular and promising families of interest-rate models: The market models.
Damiano Brigo, Fabio Mercurio
7. Cases of Calibration of the LIBOR Market Model
Abstract
In this chapter we present some numerical examples concerning the goodness of fit of the LFM to both the caps and swaptions markets, based on market data. We study several cases based on different instantaneous-volatility parameterizations. We will also point out a particular parameterization allowing for a closed-form-formulas calibration to swaption volatilities and establishing a one to one correspondence between swaption volatilities and LFM covariance parameters.
Damiano Brigo, Fabio Mercurio
8. Monte Carlo Tests for LFM Analytical Approximations
Abstract
In this chapter we test the analytical approximations leading to closed-form formulas for both swaption volatilities and terminal correlations under the LFM, by resorting to Monte Carlo simulation of the LFM dynamics. We first explain what kind of rates we are dealing with, and then move to the volatility part. Section 8.2 gives a plan of the tests on the swaption-volatility approximations and the subsequent section presents results in detail. In particular, we plot, in several cases, the real swap-rate probability density as implied by the LFM dynamics versus a lognormal density characterized by our analytically approximated volatility. We thus measure indirectly the discrepancy between the LFM swap-rate distribution and the lognormal-distribution assumption for the swap rate, as implied instead by the swap market model LSM.
Damiano Brigo, Fabio Mercurio
9. Other Interest-Rate Models
Abstract
In this chapter we introduce brief sketches of some of the models that are known in the literature and that have not been included in the previous chapters. All models are arbitrage free, and we will not discuss no-arbitrage implications further. Instead, we synthetically explain in what these models differ from the previous models and what are their original features. We also give references for the readers who might wish to deepen their knowledge of a specific approach. Clearly, presenting all the models that have been proposed in the literature is a huge task. We only present a few, without any claim to completeness of the treatment. Indeed, there are certainly several other relevant and worthy models that have not been included here, and we make the excuse that a choice is necessary since it is impossible to do justice to all the models appeared over the years. The reader interested in models that have not appeared in this book can also check other books on interest rate models such as for example James and Webber (2000).
Damiano Brigo, Fabio Mercurio

Pricing Derivatives in Practice

Frontmatter
10. Pricing Derivatives on a Single Interest-Rate Curve
Abstract
In this chapter, we present a sample of financial products we believe to be representative of a large portion of the interest-rate market. We will use different models (mostly the LFM and the G2++ model) for different problems, and try to clarify the advantages of each model. All the discounted payoffs will be calculated at time t = 0.
Damiano Brigo, Fabio Mercurio
11. Pricing Derivatives on Two Interest-Rate Curves
Abstract
In this chapter, we explain how one can model both a first (domestic) and a second (foreign) interest-rate curve, each by a two-factor additive Gaussian short-rate model, in order to Monte Carlo price a quanto constant-maturity swap and similar contracts, which we will present in the following sections.
Damiano Brigo, Fabio Mercurio
12. Pricing Equity Derivatives under Stochastic Rates
Abstract
The well consolidated theory for pricing equity derivatives under the Black and Scholes (1973) model is based on the assumption of deterministic interest rates. Such an assumption is harmless in most situations since the interestrates variability is usually negligible if compared to the variability observed in equity markets. When pricing a long-maturity option, however, the stochastic feature of interest rates has a stronger impact on the option price. In such cases it is therefore advisable to relax the assumption of deterministic rates.
Damiano Brigo, Fabio Mercurio

Appendices

Frontmatter
A. A Crash Introduction to Stochastic Differential Equations
Abstract
This book uses continuous time stochastic calculus as a mathematical tool for financial modeling. In this appendix we plan to give a quick (informal) introduction to stochastic differential equations (SDEs) for the reader who is not familiar with this field. These notes are far from being complete or fully rigorous, in that we privilege the intuitive aspect, but we give references for the reader who is willing to deepen her knowledge on such matters.
Damiano Brigo, Fabio Mercurio
B. A Useful Calculation
Abstract
Let M, V and K be real numbers with V and K positive.
Damiano Brigo, Fabio Mercurio
C. Approximating Diffusions with Trees
Abstract
In this appendix, we show how to approximate a diffusion process with a tree. The general procedure we outline is used throughout the book in the tree construction for both one-factor and two-factor short-rate models. In the one-factor case, the tree is constructed by imposing that the conditional local mean and variances at each node are equal to those of the basic continuous-time process. The geometry of the tree is then designed so as to ensure the positivity of all branching probabilities. In the two-factor case, instead, we first construct the trees for the two factors along the procedure that applies to one-factor diffusions. We then construct a two-dimensional tree by imposing that the tree marginal distributions match those of the two factors’ trees and by imposing the correct local correlation structure so as to preserve the positivity of all branching probabilities as well.
Damiano Brigo, Fabio Mercurio
D. Talking to the Traders
Abstract
In this appendix, we would like to reproduce an hypothetical conversation between a trader in interest rate derivatives and a quantitative analyst eager to get acquainted with some specific market practice. This virtual interview reflects our personal experience of interaction with traders, considering some of the traders’ opinions we have collected over the years.
Damiano Brigo, Fabio Mercurio
Backmatter
Metadaten
Titel
Interest Rate Models Theory and Practice
verfasst von
Damiano Brigo
Fabio Mercurio
Copyright-Jahr
2001
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-04553-4
Print ISBN
978-3-662-04555-8
DOI
https://doi.org/10.1007/978-3-662-04553-4