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On the Uniqueness of Martingales with Certain Prescribed Marginals

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  30 January 2018

Michael R. Tehranchi
Michael R. Tehranchi*
Affiliation:
University of Cambridge
*
∗ Postal address: Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: [email protected]
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Abstract

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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=S0eσ X-σ2〈 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 Xt=Wt+o(t1/4+ ε) as t↑∞ for any ε> 0.

Keywords

Fake Brownian motionbinomial tree modelgeometric Brownian motionweak convergence to Brownian motion

MSC classification

Primary: 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter 91B25: Asset pricing models
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 557 - 575
Copyright
© Applied Probability Trust 

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