A pathwise dynamic programming approach to nonlinear option pricing
Abstract
In this talk, we present a pathwise method to construct confidence
intervals on the value of some discrete time stochastic dynamic
programming equations, which arise, e.g., in nonlinear option pricing
problems such as credit value adjustment and pricing under model
uncertainty. Our method generalizes the primal-dual approach, which is
popular and well-studied for Bermudan option pricing problems. In a
nutshell, the idea is to derive a maximization problem and a
minimization problem such that the value processes of both problems
coincide with the solution of the dynamic program and such that
optimizers can be represented in terms of the solution of the dynamic
program. Applying an approximate solution to the dynamic program, which
can be precomputed by any algorithm, then leads to `close-to-optimal'
controls for these optimization problems and to `tight' lower and upper
bounds for the value of the dynamic program, provided that the algorithm
for constructing the approximate solution was `successful'. We
illustrate the method numerically in the context of credit value
adjustment and pricing under uncertain volatility.
The talk is based on joint work with C. Gärtner, N. Schweizer, and J.
Zhuo.