Appl Math Optim 41:377–385 (2000)
2000 Springer-Verlag New York Inc.
Arbitrage Opportunities for a Class of Gladyshev Processes
A. Dasgupta and G. Kallianpur
Department of Statistics, University of North Carolina,
Chapel Hill, NC 27599-3260, USA
Abstract. Geometric versions of a class of Gaussian processes are investigated as
possible models for stock prices. Arbitrage opportunities are constructed for these
processes showing that option pricing is not possible with these models.
Key Words. Geometric fractional Brownian motion, Gaussian processes, Arbi-
AMS Classiﬁcation. 60G15, 60H05, 90A09.
Geometric Brownian motion is one of the most used models in stochastic ﬁnance and a
complete theory of option pricing can be built for it. However, practitioners have ques-
tioned the validity of using geometric Brownian motion to model stock price processes.
Among the alternative models proposed, geometric fractional Brownian motion is an
important one. A basic assumption in stochastic ﬁnance is that the price process should
not allow arbitrage or riskless gain, because then option pricing becomes meaningless.
The main object of this paper is to show that if geometric versions of a class of Gaussian
processes, of which fractional Brownian motion is a particular one, are used to model
stock prices, then the market contains arbitrage opportunities.
An example of an arbitrage opportunity for geometric fractional Brownian motion
has been constructed by Rogers . In this example using information from the entire
past of fractional Brownian motion one chooses stopping times to invest judiciously in
such a way that starting from unit wealth one can make a positive gain with probability
one. Rogers also indicates howsimilar results can be proved for a largerclass of processes
This research was supported by AFOSR Contract No. F49620-95-1-0138.