On the Robustness of Option Pricing

In this work we consider the following problem: Let (S (n) t )0 t T , n 2 N , be a sequence of stochastic processes describing the price of a stock during the time period [0; T ]. We assume that the distribution Pn of (S (n) t ) converges weakly to the distribution P of a stochastic process (St)0 t T . Secondly, we consider a European style derivative paying F (S) at time T if the stock price at time T is S. For n 2 N let F (n) 0 be an arbitrage free value of that derivative at time 0, assuming the underlying stock is Pn-distributed. Under which conditions does a subsequence of F (n) 0 converge to an arbitrage free price F0 of that derivative of a P-distributed stock? We will rst construct examples that show, that even if F (n) 0 and F0 are uniquely determined (as in the log-binomial and log-normal model), subsequences of F (n) 0 do not necessarily converge to F0. On the other hand we will observe that these examples allow \asymptotically arbitrages", i.e. strategies which lead asymptotically to risk-free gains. This will be the key observation toward formulating general conditions under which option prices are continuous under convergence in distribution.