Perturbation Methods in Credit Derivatives. Colin Turfus
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.
Acknowledgments
The author is grateful to co‐researcher Alexander Shubert for his important contribution in implementing in Python the asymptotic formulae presented in Chapter 15 and in preparing the associated graphs.
CHAPTER 1 Why Perturbation Methods?
1.1 ANALYTIC PRICING OF DERIVATIVES
How important are analytic formulae in the pricing of financial derivatives? The way you feel about this matter will probably determine to a large degree whether this book will be of interest to you. Current opinion is undoubtedly divided and perhaps for good reasons. On the one hand, presented with the challenge of some new financial calculation, financial engineers these days are likely to spend considerably less time looking for analytic solutions or approximations than, say, twenty years ago, citing the ever‐increasing power and speed of computational resources at their disposal. On the other hand, where known analytic solutions exist, those same financial engineers are unlikely to eschew them and to persist doggedly in replicating the known solution using a Monte Carlo engine or a finite difference method.
So, it might be suggested, the resistance to analytic solutions that we observe is not to their use as such when they are already available, but to making the effort to find (and implement) them. One of the reasons for this is a perception that, given the huge amount of research effort that has been invested into finding solutions over the past few decades, most of the interesting and useful solutions have been found and published. It is the experience of the author that the reaction to the announcement of discovery of a new and interesting analytic solution tends to be indifference or scepticism rather than interest. At the same time, it is often assumed (correctly?) that such effort as is being invested into finding analytic solutions is these days directed mainly towards approximate solutions, most particularly using perturbation methods, which area continues to be a reasonably fertile ground for research effort, at least in academic institutions. We shall look more closely at the areas which are attracting attention below.
It is of interest to ask then why, despite the continuing effort being invested on the theoretical side into the development of analytic approximations, the take‐up in practice appears to be relatively limited, certainly compared to the heyday of options pricing theory when the choice of models made by practitioners was significantly