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Hull-White on Derivatives

  • Published: August 1999
  • Region: World
  • 356 Pages
  • Incisive Media

A classic collection of the writing of John Hull and Alan White

-Hull and White's classic analysis of the impact of stochastic volatility on the pricing and hedging of options
-Examines the valuation of interest-rate options and the problem of how to build a no-arbitrage model of the term structure of interest rates

Preface

Stochastic Volatility
Introduction
The Pricing of Options on Assets with Stochastic Volatitlities
An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility
Hedging the Risks from Writing Foreign Currency Options

Numerical Procedures
Introduction
Valuing Derivative Securities Using the Explicit Finite Difference Method
The Use of the Control Variate Technique in Option Pricing
Efficient Procedures for Valuing European and American Path-dependent Options

Credit Risk
Introduction
Assessing Credit Risk in a Financial Institution’s Off-balance
Sheet Commitments
The Impact of Default Risk on the Valuation of Options and Other Derivative Securities

Term Structure Models: Theory
Introduction
Pricing Interest Rate Derivative Securities
Bond Option Pricing Based on a Model for the Evolution of Bond Prices
The Pricing of Options on Interest Rate Caps and Floors Using the Hull-White Model

Term Structure Models: Implementation
Introduction
Single-factor Interest Rate Models and the Valuation of Interest Rate Derivative Securities
Numerical procedures for Implementing Term Structure Models
Single-Factor Models
Numerical Procedures for Implementing Term Structure Models
Two-Factor Models
Using Hull-White Interest Rate Trees

Index

Reviewed by Bruno Dupire - Nikko Europe

John Hull's book, Options, Futures and Other Derivative Securities, is one of the rare hits of financial literature. Hull's more advanced work, always written with Alan White, has appeared as academic papers in various publications. Hull-White on Derivatives organises these papers into five main themes, each with an added introduction that justifies the purchase of the book even for those who already have the 15 papers in it.

1. Stochastic volatility. Hull & White made clear that when the dynamics of volatility do not depend on the spot, the option price is an average of Black-Scholes prices with different volatilities, which generates a mild symmetric smile. To get market skews, one needs to introduce correlation between changes in spot and changes in volatility. Analytical expansions (for square root volatility processes) and hedging (no major changes in delta) are considered.

2. Numerical procedures. Links between trinomial trees and explicit discretisations of partial differential equations are shown. Tricks such as changing variables to normalise the volatility and changing the branching scheme to follow the drift are detailed. The idea of control variate techniques is presented: if you have a good approximation, try to estimate the error (through Monte Carlo, for instance) and add it to the approximation. The last paper explains the use of two-dimensional trees to compute the price of an option whose path-dependency is captured by a single variable.

3. Credit risk. The loss incurred in case of default of a counterparty is interpreted as an option payout and is valued following option pricing principles. The discount that should be demanded of a risky counterparty which sells an option is calculated through implied default probabilities read from risky bond prices.

4. Term-structure models: theory. Hull & White present a class of yield curve models, the most popular of which is the extended Vasicek model (a Markov Gaussian model on the short-term rate, which is fitted to the initial yield curve). Analytical formulas are given for yield curve reconstruction and option prices.

5. Term-structure models: implementation. This makes a nice case for the use of trinomial trees. Their application to the two-dimensional case is described as well as procedures to calibrate them to market prices.

Hull & White have profoundly influenced the derivatives market, not least because they have a flair for addressing the right questions at the right time (any researcher knows that it is often harder to identify the problem than to solve it once it is well posed), a straightforward approach and a clear pedagogical talent.

Hull & White are not the most theoretical people but their familiarity with practitioners makes their work quite usable. Some of their contributions carve in stone what some banks have been working on for years (trinomial trees, Gaussian yield curve models, numerical schemes for path dependent options, etc).

For instance, it does not deal with powerful tools such as changes of numeraire and measures, calibration to volatility smiles and low discrepancy sequences. It must be added, however, that this is not the aim of this book, which is a remarkable piece of work all the same.

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