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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 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, and used to infer the prices of a number of semi‐exotic derivatives in each case. Some evidence is provided of the favourable performance of approximate results against calculation performed by numerical schemes capable of delivering arbitrarily high precision.

Chapter 11 expands the envelope one step further, looking at a three‐factor model incorporating an FX rate and two interest rates, deducing an exact pricing kernel and using this to infer option prices. It is noted that the model considered is of Jarrow–Yildirim type so is applicable also to the pricing of inflation derivatives. A further turn of the handle in Chapter 12 also brings credit risk into the mix, resulting in a four‐factor model. A pricing kernel expansion is deduced and used to price a number of semi‐exotic credit derivatives. Most notably we revisit quanto CDS pricing (covered in the first instance in Chapter 10), now allowing interest rates to be stochastic as well as credit and FX rates.

The next two chapters of the book take us off in slightly different directions. First we look forward to the new risk‐free LIBOR replacement rates which are set in arrears on the basis of compounding daily (or overnight) rates (Chapter 13). This approach is intended to supplant the currently used multi‐curve frameworks where LIBOR rates embed a tenor‐dependent stochastic spread, the modelling of which is the subject of Chapter 14. In each of these cases we consider in the first instance how the pricing kernel for the short‐rate model is affected then look at how the integration with a Black–Karasinski credit model impacts the resulting hybrid kernel and assess the consequent impact on credit derivatives formulae.

The remaining chapters are devoted to applications of the methods and results herein expounded in various areas of contemporary interest in a risk management context. Chapter 15 looks at scenario generation where interest rate and credit curves need to be evolved alongside spot processes to allow risk measures such as market risk, counterparty exposure and CVA, depending on a projected distribution of future prices, to be calculated. In Chapter 16 we look at model risk, noting that our methods have utility here too, both in providing useful, easily implemented benchmarks for model validation purposes and for making quantitative assessments of the influence of model parameters and modelling assumptions on portfolio evaluations. Finally the newly evolving application of machine learning to problems in quantitative finance and the question of how asymptotic methods could complement this approach in practice are addressed in Chapter 17.

C. Turfus

London, 2020

Note

1 1 We exclude for the former reason rates (interest or credit) which are governed by a model of the CIR type defined by Cox et al. [1991) (where the underlying stochastic factor follows a distribution), and for the latter reason rates which are governed by either a HJM model of the type defined by Heath et al. [1992) or a LIBOR market model. Most of the standard models for spot underlyings are encompassed within the framework, the main exceptions being Lévy models and rough volatility models.