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 influenced by the availability of analytic solutions, even of analytic approximations such as SABR [Hagan et al., 2002]. For example Brigo and Mercurio [2006] observed of the short‐rate model of Black and Karasinski [1991] that
the rather good fitting quality of the model to market data, and especially to the swaption volatility surface, has made the model quite popular among practitioners and financial engineers. However,…the Black–Karasinski (1991) model is not analytically tractable. This renders the model calibration to market data more burdensome than in the Hull and White (1990) Gaussian model, since no analytic formulae for bonds are available.
It is undoubtedly true that the relative tractability of the Hull–White model has been an important factor resulting in its much wider adoption as an industry standard.
No single reason can be cited to account for the relatively limited use to which analytic approximations are put. Practitioners' views vary greatly depending on the types of models they are looking at and what they are using them for. A number of factors can be pointed to, as we shall elaborate in the following section. For the moment we make the following observations, specifically comparing analytic pricing with a Monte Carlo approach.
There is a general distrust by financial engineers of methods involving any kind of approximation. The fact that, if results involve power series‐like constructions, it may not be possible to guarantee arbitrage‐free prices in 100% of cases is often cited as a reason to avoid use of such approximations in pricing models intended for production purposes. Furthermore, it can be more work to assess the error implicit in a given approximation than it is to compute prices in the first place.
While analytic methods are computationally more efficient, they appear
