Читать книгу Perturbation Methods in Credit Derivatives - Colin Turfus - Страница 8

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 to be intrinsically less scalable than Monte Carlo methods from a development and implementation standpoint. Whereas the Monte Carlo implementation of a model mainly involves the simulation of the underlying variables, with different products merely requiring different payoffs to be applied, each product variant tends to have a different analytic formula with limited scope for reuse with reference to other products. Also, if an additional stochastic factor is included in a Monte Carlo method, this can often be handled as an incremental change, while in the case of analytic methods, they will often break down completely when an additional risk factor is added.

 Another argument that is not infrequently heard against the introduction of new analytic results is that it is just too much trouble to integrate them into pricing libraries which are already quite mature. An accompanying argument may be that, since the libraries of financial institutions are already written in highly optimised C++ code, any gains that might be made are only likely to be marginal.

 There is also a suspicion concerning the utility of perturbation methods insofar as, while the most interesting and challenging problems in derivatives pricing occur where stochastic effects have a significant impact on the pricing, most perturbation approaches have some kind of reliance on the smallness of a volatility parameter, usually a term variance.¹ But, for this parameter to have a significant impact on pricing it cannot be too “small”, so we are led to the expectation that we will need a large number of terms in any approximating series to secure adequate convergence in many cases of importance.

 A more recent argument which the author has encountered in a number of conversations with fellow researchers is that, insofar as more efficient ways are sought to carry our repetitive execution of pricing algorithms, the strategy adopted in the future will increasingly be to replace the time-consuming solution of SDEs and PDEs not with analytic formulae but with machine-learned algorithms which can execute orders of magnitude faster (see for example Horvath et al. [2019]). The cost of adopting this approach is a large amount of up-front computational effort in the training phase where the full numerical algorithm is run multiple times over many market data configurations and product specifications to allow the machine-learning algorithm to learn what the “right answer” looks like so that it might replicate it. There will also be a concomitant loss of accuracy. But if, as is often the case, the requirement is to calculate prices for a given portfolio or the CVA associated with a given “netting set” of trades with a given counterparty over multiple scenarios for risk management or other regulatory purposes, the upfront cost can be amortised against a huge amount of subsequent usage of the machine-learned algorithm. Since machine-learning approaches are a fairly blunt instrument, there is not the need to customise the approach to the particular problem addressed, as would be necessary if perturbation methods were used instead as a speed-up strategy wherein some accuracy is traded for speed.

 Finally, there is not uncommonly a perception that, unlike with earlier analytic options pricing formulae which were deduced using suitable application of the Girsanov theorem, with which financial engineers tend to be familiar, perturbation‐based methods are by comparison something of a dark art. Many of the results are derived using Malliavin calculus or Lie theory, with which relatively few financial engineers are familiar, and often presented in published research papers in notation which is relatively opaque and quite closely tied in to the method of derivation. Other derivations are performed using methodologies and notations borrowed from quantum mechanics or other areas of theoretical physics, areas with which a contemporary financial engineer is unlikely to be familiar. There is, furthermore, not a clearly defined body of theory which the practitioners of perturbation analysis seek to rely on; books which offer a unified approach to perturbation methods applicable to a range of problems in derivatives pricing such as Fouque et al. [2000], Fouque et al. [2011] and Antonov et al. [2019] are few and far between.