A Stochastic Approximation for Fully Nonlinear Free Boundary Parabolic Problems

When the option pricing problem is of several dimensions, for example, basket options, deterministic methods such as finite difference are almost intractable; because the complexity increases exponentially with the dimension and one almost inevitably needs to use Monte Carlo simulations. Moreover, many problems in finance, for example, pricing in incomplete markets and portfolio optimization, lead to fully nonlinear PDEs. Only very recently there has been some significant development in numerically solving these nonlinear PDEs using Monte Carlo methods, see, for examplpe, [1–6]. When the control problem also contains a stopper, for example, in determining the super hedging price of an American option, see [7], or solving controller-and-stopper games, see [8], the nonlinear PDEs have free boundaries. For solving linear PDEs with free boundaries, that is, in the problem of American options, Longstaff–Shwartz [9], introduced a stochastic method in which American options are approximated by Bermudan options and least squares approximation is used for doing the backward induction. The major feature in [9] is the tractability of the implementation for the scheme proposed in terms of the CPU time in high dimensional problems. The most important feature of this model that facilitates the speed is that the number of paths simulated is fixed. Simulating