Perturbation Methods in Credit Derivatives. Colin Turfus

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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.