Preprint A411/2005
On the Asymptotics of Fast Mean-Reversion Stochastic Volatility Models.

Jorge Zubelli | Souza, Max

**Keywords: **
stochastic volatility | quantitative finances | mathematical methods in finances

We consider the asymptotic behavior of options under stochastic
volatility models for which the volatility process fluctuates on a much faster
time scale than that defined by the risk-less interest rate. We identify
the distinguished asymptotic limits and, in contrast with previous studies,
we deal with small volatility-variance (vol-vol) regimes. We derive the
corresponding asymptotic formulae for option prices, and find that the
first order correction displays a dependence on the hidden state and
a non-diffusive terminal layer. Furthermore, this correction cannot
be obtained as the small variance limit of the previous calculations.
Our analysis also includes
the behavior of the asymptotic expansion, when
the hidden state is far from the mean. In this case, under suitable
hypothesis, we show that the solution behaves as a constant volatility
Black-Scholes model to all orders. In addition, we derive an
asymptotic expansion for the implied volatility that is uniform in
time. It turns out that the fast scale plays an important role in such
uniformity. The theory thus obtained yields a more complete picture
of the different asymptotics involved under stochastic
volatility. It also clarifies the remarkable independence on the state of the
volatility in the correction term obtained by previous authors.