Perturbation Methods in Credit Derivatives. Colin Turfus

One of these is that the training process for a machine learning‐based algorithm which replicates pricing functionality is prohibitively high for the high‐dimensional problem which is constituted by revaluing a bank's portfolio under diverse future market scenarios. Another problem is that the state space over which learning must take place is unbounded with respect to many of the relevant parameters, especially market parameters. Furthermore, regulators are very concerned that it be demonstrated in the model validation process that the internal model remains robust under extreme scenarios, such as those which occurred during the credit crunch of 2007. Consequently, a lot of effort is likely to have to be expended in the learning phase of the machine‐learning process in relation to scenarios of marginal importance at the edge of the state space, which scenarios may contribute little to the overall risk numbers.

      Encouragement is taken from the recent work of Antonov et al. [2020] which addresses the problem of how to handle the outer limits of the phase space in a machine‐learned representation of a pricing algorithm, effectively by substituting in an asymptotic representation of the pricing algorithm, assuming such is available, for points outside a core region of the phase space which is sampled fairly exhaustively in the learning process.

      The authors show how such asymptotic representations of the pricing function can be used effectively as a control variate for the learning process: rather than trying to learn the pricing algorithm itself, the learning process targets the difference between the exact pricing function and its asymptotic representation.

      It is recognised by the team in the bank that, rather than seeing machine learning as an alternative to the use of analytic/asymptotic formulae to achieve speed‐up of computational times, it would be potentially fruitful to view them as complementary partners. In particular, since some of the most challenging problems of implementing the machine learning strategy for real portfolios arise in addressing hybrid multi‐asset derivatives whose value depends on market data including rates curves, possessed of many degrees of freedom, as well as just spot values, some of the formulae presented in the current book and the strategies used to obtain them could prove a useful adjoint to brute‐force application of machine learning techniques.

2.8 INCORPORATING INTEREST RATE SKEW AND SMILE

Having successfully integrated analytic pricing functionality for Black–Karasinski rates modelling into the pricing library, in particular analytic formulae for cap, floor and swaption pricing, the hybrid derivatives traders are interested in whether it is possible also to incorporate interest rate skew and smile into analytic pricing formulae for rates options. They find that there are, unfortunately, no formulae in the book which address this problem, so they contact the author to ask if this is something that could be done. He informs them that, while this is difficult for the Black–Karasinski model, he has plans to extend the Hull–White model in precisely this way and hopes to publish analytic option pricing formulae in due course.¹

NOTE

      1 1 But if they are interested in expediting the calculation, he could consider offering consultancy services to that end.