Fast and reliable pricing of American options with local volatility

Since Black and Scholes published their seminal paper [2] in 1973, the pricing of options by means of deterministic partial differential equations or inequalities has become standard practise in computational finance. An option gives the right (but not the obligation) to buy (call option) or sell (put option) a share for a certain value (the exercise price K) at a certain time T (exercise date). On the exercise day T , the value of an option is given by its pay–off function φ(S) = max(K − S, 0) =: (K − S)+ for put options and φ(S) = (S −K)+ for call options. In contrast to European options which can only be exercised at the expiration date T , American options can be exercised at any time until expiration. Due to this early exercise possibility the evaluation of American options is formulated as an optimal stopping problem: The holder of the American option has to decide, whether his gain by immediately exercising the option exceeds the current value of the option. In the original paper of Black and Scholes it is assumed, that the risk–less interest rate and the volatility are constant. Meanwhile, financial practise has led to a number of local volatility models, where the volatility is a given deterministic function of time and space [4]. While existence, uniqueness and discretization is well understood (cf., e.g., [1, Chapter 6]), the efficient and reliable solution of