Appl Math Optim 43:47–62 (2001)
2001 Springer-Verlag New York Inc.
Asset Pricing with Stochastic Volatility
and J. Xiong
Center for Stochastic Processes,
Department of Statistics, University of North Carolina,
Chapel Hill, NC 27599-3260, USA
Department of Mathematics, University of Tennessee,
Knoxville, TN 37996-1300, USA
Abstract. In this paper we study the asset pricing problem when the volatility
is random. First, we derive a PDE for the risk-minimizing price of any contingent
claim. Secondly, we assume that the volatility process σ
is observed through an
observation process Y
subject to random error. A price formula and a PDE are then
derived regarding the stock price S
and the observation process Y
Finally, we assume that S
is observed. In this case we have a complete market
and any contingent claim is then priced by an arbitrage argument instead of by
Key Words. Asset pricing, Stochastic volatility, Nonlinear ﬁltering.
AMS Classiﬁcation. Primary 90A12, Secondary 60G44, 60H05, 60H10, 60H30.
We consider a market consists of a stock S
and a bond B
governed by the following
= a(t, S
dt + σ
The research of the ﬁrst author was supported by Army Research Ofﬁce Grant No. DAAL 0392G0008.
This research was carried out at the Center for Stochastic Processes in the Department of Statistics, University
of North Carolina.