FROM DISCRETE- TO CONTINUOUS-TIME FINANCE: WEAK CONVERGENCE OF THE FINANCIAL GAIN PROCESS'

Conditions suitable for applications in finance are given for the weak convergence (or convergence in probability) of stochastic integrals. For example, consider a sequence Sn of security price pro- cesses converging in distribution to S and a sequence 8" of trading strategies converging in distri- bution to 0. We survey conditions under which the financial gain process J 0" dSn converges in distribution to J 0 dS. Examples include convergence from discrete- to continuous-time settings and, in particular, generalizations of the convergence of binomial option replication models to the Black- Scholes model. Counterexamples are also provided. Although a large part of financial economic theory is based on models with continuous- time security trading, it is widely felt that these models are relevant insofar as they char- acterize the behavior of models in which trades occur discretely in time. It seems natural to check that the limit of discrete-time security market models, as the lengths of periods between trades shrink to zero, produces the effect of continuous-time trading. That is one of the principal aims of this paper. If {(P, 6.)) is a sequence of security price processes and trading strategies converging in distribution to some such pair (S, 6), we are con- cerned with additional conditions under which the sequence {! 6: dS;} of stochastic in- tegrals defining the gains from trade converges in distribution to the stochastic integral J Bt dS,. Conditions recently developed by Jakubowski, MCmin, and Pagks (1989) and Kurtz and Protter (1991a, b) are restated here in a manner suitable for easy applications in finance, and several such examples are worked out in this paper. The paper gives parallel conditions for convergence of gains in probability. In short, this paper is more of a "user's guide" than a set of new convergence results. A good motivating example is Cox, Ross, and Rubinstein's (1979) proof that the Black- Scholes (1973) option pricing formula is the limit of a discrete-time binomial option pricing formula (due to William Sharpe) as the number of time periods per unit of real time goes to infinity. Aside from providing a simple interpretation of the Black-Scholes formula, this connection between discrete- and continuous-time financial models led to a standard technique for estimating continuous-time derivative asset prices by using nu- merical methods based on discrete-time reasoning. One of the examples of this paper is