When does convergence of asset price processes imply convergence of option prices?

We consider weak convergence of a sequence of asset price models (S-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-n). (In the case of nonuniqueness of the risk-neutral measures the question of the "good choice" of (Q(n)) also arises.) In particular we investigate under which conditions the weak convergence of (P-n) to P implies the weak convergence of (Q(n):) to Q and thus the convergence of prices of derivative securities. The main theorem of the present paper exhibits an intimate relation of this question with contiguity properties of the sequences of measures (P-n) with respect to (: Q(n)), which in turn is closely connected to asymptotic arbitrage properties of the sequence (S-n) of security price processes. We illustrate these results with general homogeneous binomial and some special trinomial models.