Journal of Applied Probability

On the uniqueness of martingales with certain prescribed marginals

Michael R. Tehranchi

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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=S0eσ X2X〉/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.

Article information

Source
J. Appl. Probab., Volume 50, Number 2 (2013), 557-575.

Dates
First available in Project Euclid: 19 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1371648961

Digital Object Identifier
doi:10.1239/jap/1371648961

Mathematical Reviews number (MathSciNet)
MR3102500

Zentralblatt MATH identifier
1291.60084

Subjects
Primary: 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter 91B25: Asset pricing models

Keywords
Fake Brownian motion binomial tree model geometric Brownian motion weak convergence to Brownian motion

Citation

Tehranchi, Michael R. On the uniqueness of martingales with certain prescribed marginals. J. Appl. Probab. 50 (2013), no. 2, 557--575. doi:10.1239/jap/1371648961. https://projecteuclid.org/euclid.jap/1371648961


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