A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion

David Baker · Marc Yor

25 citations0/0 files storedElectronic Journal of Probability

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Stochastic processes and financial applications Probability and Statistical Research Probability and Risk Models

Abstract

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion.This provides a short proof of the recent result due to P. Carr et al that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion is replaced by a stable subordinator.

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1996G.Samorodnitsky,M.S.Taqqu:Stable non-Gaussian Random Processes--Stochastic Models with Infinite Variance 1995Arbitrage possibilities in Bessel processes and their relations to local martingales 2002Making Markov martingales meet marginals: with explicit constructions 1996STABLE NON-GAUSSIAN RANDOM PROCESSES: STOCHASTIC MODELS WITH INFINITE VARIANCE 2008On the qualitative effect of volatility and duration on prices of Asian options

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