Feeds:
Posts

## LARGE DEVIATIONS FOR PRACTICAL VOLATILITY SURFACE ASYMPTOTICS

The large deviation principle for a family of random variables $X_n$ is the following bound in essence $P(X_n > x) \le \exp(-n I(x))$.  Nice application of this in stochastic volatility has been made for small maturities.  The idea is to rescale the volatility diffusion

$dY_t = \kappa (\theta - Y_t) dt + \nu \sqrt{Y_t} dw^2_t$

and rescale time $t\rightarrow \epsilon t$ and then apply the large deviation principle to $S_{\epsilon,t} = x\exp(-\frac{\epsilon}{2}\int_0^t Y_{\epsilon,s} ds + \sqrt{\epsilon}\int_0^t \sqrt{Y_{\epsilon,s}} dw_s)$.  This is done in detail in Feng-Forde-Fouque-ShortMaturityAsymptoticsHeston-2010.

We want to consider the Heston model with $dw^2_t$ replaced by a fractional Brownian motion with Hurst exponent $H \in (0,1)$ and replicate the analysis.  The scaling law for the fBM is $B^H(at) = a^H B^H(t)$.  Therefore $\kappa \rightarrow \kappa/\epsilon$, $\nu \rightarrow \nu \epsilon^{-H}$ and the correlation $\rho \rightarrow \rho \epsilon^{H-1/2}$.

The transformation $t\rightarrow \epsilon t$ with a fractional Brownian motion $dw^2_t$ that is uncorrelated with $dw^1_t$ is not problematic

$dY_{\epsilon,t} = \frac{\kappa}{\epsilon}(\theta - Y_{\epsilon,t}) dt + \nu \epsilon^{-H} \sqrt{Y_{\epsilon,t}} dw^2_t$