Perturbation Methods in Credit Derivatives. Colin Turfus

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:

      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 terms need to be added to a leading order pricing kernel formula. The prices of derivatives are then obtained by taking a convolution of the pricing kernel with the associated payoff functions, which task is typically a standard one.

      We start off in Chapter 1 by discussing why perturbation methods are not currently seen as “mainstream” quantitative finance, concluding that some of the reasons are seen on closer inspection to be invalid, while others, despite having some validity, do not apply to the methods set out in this book, which seeks to pioneer a new approach with wider applicability. We seek to justify this claim in the remainder of the book, starting with Chapter 2, which is dedicated to case studies illustrating how the approach we propose allows flexible response to evolving needs in a risk management context. In Chapter 3, we set out the mathematical approach and core tools which we will make use of throughout. We apply these in Chapters 4 and 5 to the construction of pricing kernels for the popular Hull–White and Black–Karasinski short‐rate models, respectively, using these kernels to derive important derivative pricing formulae; as exact expressions in the former case and as perturbation expansions in the latter.

We then turn our attention to hybrid and multi‐factor models, devoting Chapter 6 to setting out a generic framework for handling models with multiple factors following the Ornstein–Uhlenbeck processes, the detailed calculation associated with which method turns out to depend only on the (stochastic) discounting model employed. We set out the details for both Hull–White and Black–Karasinski discounting models. The next four chapters deal with two‐factor hybrid models: rates‐equity; rates‐credit; credit‐equity; and credit‐FX. Kernels are deduced, either exact or as perturbation expansions,

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