 
Summary: MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING
GERARD AWANOU
Abstract. We consider the problem of partial hedging of an European deriva
tive under the assumption that the volatility is stochastic, driven by two diffusion
processes, one fast mean reverting and the other varying slowly. For options with
long maturities typically beyond 90 days, the singular perturbation analysis in [Par
tial Hedging in a Stochastic Volatility Environment, M. Jonsson and K.R. Sircar,
Mathematical Finance, 12, pp. 375409, 2002] ignores the slow factor. In this paper,
we investigate the full two factors model and show how an additional term can be
included in the approximate value functions and strategies.
1. Introduction
We consider the problem of shortfall risk minimization in stochastic volatility models
under the assumption that the volatility is driven by two diffusions, one slowly varying
and one fast mean reverting. Following the methodology of [3], the shortfall risk
minimization problem is transformed into a state dependent utility maximization
problem. The optimal strategies depend on the solution of a high dimensional HJB
equation satisfied by the value function of the utility problem. The PDE satisfied by
the Legendre transform of the value function is derived along with the one satisfied
by the smallest optimizer in the Legendre transform. That optimizer can be thought
as the inverse of marginal utility and is shown to be the price of a modified claim. We
