When Does Convergence of Asset Price Processes Imply Convergence of Option Prices?

We consider weak convergence of a sequence of asset price models "(S-super-n)" to a limiting asset price model "S". A typical case for this situation is the convergence of a sequence of binomial models to the Black-Scholes model, as studied by Cox, Ross, and Rubinstein. We put emphasis on two different aspects of this convergence: first we consider convergence with respect to the given "physical" probability measures "(P^n)" and second with respect to the "risk‐neutral" measures "(Q^n)" for the asset price processes "(S-super-n)". (In the case of nonuniqueness of the risk-neutral measures the question of the "good choice" of "(Q-super-n)" also arises.) In particular we investigate under which conditions the weak convergence of "(P-super-n)" to "P" implies the weak convergence of "(Q-super-n)" to "Q" and thus the convergence of prices of derivative securities. Copyright Blackwell Publishers Inc 1998.