On the Uniqueness of Martingales with Certain Prescribed Marginals

Abstract

This note contains two main results. (i) (Discrete time) Suppose that S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Cox-Ross-Rubinstein binomial tree model.) Then S is a geometric simple random walk. (ii) (Continuous time) Suppose that S = S 0 e σ X−σ 2 X /2 is a continuous martingale whose marginal laws agree with a geometric Brownian motion. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Black-Scholes model with volatility σ > 0.) Then there exists a Brownian motion W such that X t = W t + o(t 1/4+ε ) as t ↑ ∞ for any ε > 0.