Perturbation Methods in Credit Derivatives. Colin Turfus
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Library of Congress Cataloging‐in‐Publication Data is available
Names: Turfus, Colin, author.
Title: Perturbation methods in credit derivatives : strategies for efficient risk management / Colin Turfus.
Description: Chichester, West Sussex, United Kingdom : John Wiley & Sons, 2021. | Series: Wiley finance series | Includes bibliographical references and index.
Identifiers: LCCN 2020029878 (print) | LCCN 2020029879 (ebook) | ISBN 9781119609612 (hardback) | ISBN 9781119609629 (adobe pdf) | ISBN 9781119609599 (epub)
Subjects: LCSH: Credit derivatives. | Financial risk management.
Classification: LCC HG6024.A3 T87 2021 (print) | LCC HG6024.A3 (ebook) | DDC 332.64/57–dc23
LC record available at https://lccn.loc.gov/2020029878
LC ebook record available at https://lccn.loc.gov/2020029879
Cover Design: Wiley
Cover Image: © MR.Cole_Photographer/Getty Images
Preface
This is a book about how to derive exact or approximate analytic expressions for semi‐exotic credit and credit hybrid derivatives prices in a systematic way. It is aimed at readers who already have some familiarity with the concept of risk‐neutral pricing and the associated stochastic calculus used to define basic models for pricing derivatives which depend on underlyings such as interest and FX rates, equity prices and/or credit default intensities, such as is provided by Hull [2018]. We shall set out models in terms of the stochastic differential equations which govern the evolution of the risk factors or market variables on which derivatives prices depend. However, we shall in the main seek to re‐express the model as a pricing equation in the form of a linear partial differential equation (PDE), more specifically a second order diffusion equation, using the well known Feynman–Kac theorem, which we shall use without proof.
Our approach will be mathematical in terms of using mathematical arguments to derive solutions to pricing equations. However, we shall not be concerned here about the details of necessary and sufficient conditions for existence, uniqueness and smoothness of solutions. In the main we shall take advantage of the fact that the equations we are addressing are already known to have well‐behaved solutions under conditions which have been well‐documented. Our concern will be to use mathematical analysis to infer analytic representation, either exact or approximate, of solutions. We shall in some cases seek to offer more rigorous justification of the methods employed. But our general approach will be to demonstrate that the results are valid either in terms of satisfying the specified pricing equation (exactly or approximately), or else replicating satisfactorily prices derived by an established method such as Monte Carlo simulation.
Our method combines operator formalism with perturbation expansion techniques in a novel way. The focus is different from much of the work in the literature insofar as:
Rather than deriving particular solutions for individual products with a specific payoff, we obtain first general solutions for pricing equations; in other words, pricing kernels. We then use these to produce prices for particular products simply by taking a convolution of the payoff function(s) with the kernel.
Rather than focussing on products whose value is contingent on spot variables such as FX or inflation rates, or equity or commodity prices, and building expansions based on the assumption of low variability of local and/or stochastic volatility, we consider mainly rates‐credit hybrid derivatives, taking the short rate and the instantaneous credit default intensity to be stochastic and building expansions based on the assumption of low rates and/or intensities. This latter assumption is almost always valid allowing simple expressions which are only first order, or at most second order, to be used with very high accuracy. Implementation of the derived formulae typically involve nothing more complicated than quadrature in up to two dimensions and fixed point iterative solution of one‐dimensional non‐linear equations, so are well suited to scripting languages such as Python, which was indeed used for most of the calculations presented herein.
As a consequence, we are able to derive many new approximate but highly accurate expressions for hybrid derivative prices which have not been previously available in the literature. These approximations are furthermore uniformly valid in the sense that they remain valid over any trade time-scale unlike many other popular asymptotic methods such as the SABR approximation of Hagan et al. [2015], the accuracy of which depends on an assumption of short time‐to‐maturity (low term variance). We are also able to point the reader in the direction of how to derive further results for models and products other than those considered explicitly here.
The essence of our approach is that we focus on models where the stochastic factors approximate to a good degree to being normally distributed (or lognormally, which simply means that the logarithm of the variable in question is normally distributed) and where interest rates and credit default intensities are taken to be governed by short‐rate models.1 This means that the pricing kernel can to leading order be expressed as a multivariate gaussian distribution (multiplied by a discount factor). Corrections need to be applied to this base representation to obtain a sufficiently accurate result. We show how in many cases this can be done exactly. In other cases, in particular where rates or credit intensities are lognormal rather than normal, one or two correction