This post concludes my day-to-day review of the workshop on Computational Methods that took place at Fields last week.
As could be inferred from his informal presentation on GPUs the previous day, Mike Giles does heavy duty numerical computations with a vengeance. He described a Monte Carlo method implemented on different levels of resolutions in order to achieve a prescribed accuracy and showed how to apply it to challenging exotic option pricing and sensitivity calculations, as well as a stochatic PDE example.
Peter Forsyth started by explaining what GMWB (guaranteed minimum withdraw benefits) are and introduced a penalty method to price them numerically by solving the associated HJB equations. Interestingly, his results were not sensitive to the size of the penalty term, which silences a common criticism to such methods. Even more interestingly, we learned at the end of the talk that policyholders in Hamilton tend to exercise their guarantees optimally (well, at least in one historical example), thereby wreaking havoc for insurance companies, which routinely underprice such policies betting on consumer's sub-optimality.
Nizar Touzi described his probabilistic approach for solving fully nonlinear PDEs. As it is well-known, probabilistic methods can be used to solve second order linear parabolic PDEs via the Feynman-Kac formula, which can then be approximated by Monte Carlo. For quasi-linear PDEs, a similar approach leads to a numerical scheme for solving BSDEs. What Nizar showed is how the scheme can be generalized to the fully nonlinear case, leading to a scheme that involves both Monte Carlo and finite-differences, but without relying directly on BSDEs.
Phillip Protter brought the workshop to conclusion with a talk on absolutely continuous compensators. Now, these are objects that show up naturally in the study of totally inaccessible stoping times (such as those used in the majority of reduced-form credit risk models), so the motivation for studying them is perfectly understandable. Not the same for the machinery necessary to appreciate the results... As another participant observed, it is likely that about 2 people in the audience knew what the Levy system of a Hunt process is (don't count me as one of them). In the end, the message I got from the talk is that any good old totally inaccessible stopping time that has any chance of making into a financial model will likely have an absolutely continuous compensator - but it is nice to know that there are people proving such results. Before I finish, I used to say that Phillip is the Woody Allen of mathematics, but the analogy is no longer valid: Woody is not that funny anymore, whereas Phillip is in great form.