Monte Carlo Approximations of American Options that Preserve Monotonicity and Convexity

Abstract

It can be shown that when the payoff function is convex and decreasing (respectively increasing) with respect to the underlying (multidimensional) assets, then the same is true for the value of the associated American option, provided some conditions are satisfied. In such a case, all Monte Carlo methods proposed so far in the literature do not preserve the convexity or monotonicity properties. In this paper, we propose a method of approximation for American options which can preserve both convexity and monotonicity. The resulting values can then be used to define exercise times and can also be used in combination with primal-dual methods to get sharper bounds. Other application of the algorithm include finding optimal hedging strategies. Recently, two-stage Monte Carlo methods were developed by Rogers (2002), Andersen and Broadie (2004) and Haugh and Kogan (2004). For an interesting review of Monte Carlo methods, see Fu et al. (2001).

Optimal hedging in discrete time

“…Linear interpolations are used for points outside the grid at each time step. Since the expectations are always with respect to the same probability measure, only one sequence of random numbers may be used, using the ideas in Del Moral et al (2006Moral et al ( , 2012.…”