Rohini Kumar

Large deviations for multi-scale jump-diffusion processes

We obtain large deviation results for a two time-scale model of jump-diffusion processes. The processes on the two time scales are fully inter-dependent, the slow process has small perturbative noise and the fast process is ergodic. Our results extend previous large deviation results for diffusions. We provide concrete examples in their applications to finance and biology, with an explicit calculation of the large deviation rate function.

Large deviations for the boundary local time of doubly reflected Brownian motion

We compute a closed-form expression for the moment generating function fˆ(x;λ,α)=1λEx(eαLτ), where Lt is the local time at zero for standard Brownian motion with reflecting barriers at 0 and b, and τ∼Exp(λ) is independent of W. By analyzing how and where fˆ(x;⋅,α) blows up in λ, a large-time large deviation principle (LDP) for Lt/t is established using a Tauberian result and the Gartner–Ellis Theorem.

Effect of Volatility Clustering on Indifference Pricing of Options by Convex Risk Measures

Abstract In this article, we look at the effect of volatility clustering on the risk indifference price of options described by Sircar and Sturm in their paper (Sircar, R., & Sturm, S. (2012). From smile asymptotics to market risk measures. Mathematical Finance. Advance online publication. doi:10.1111/mafi.12015). The indifference price in their article is obtained by using dynamic convex risk measures given by backward stochastic differential equations. Volatility clustering is modelled by a fast mean-reverting volatility in a stochastic volatility model for stock price. Asymptotics of the indifference price of options and their corresponding implied volatility are obtained in this article, as the mean-reversion time approaches zero. Correction terms to the asymptotic option price and implied volatility are also obtained.

Large-time option pricing using the Donsker-Varadhan LDP - correlated stochastic volatility with stochastic interest rates and jumps

We establish a large-time large deviation principle (LDP) for a general mean-reverting stochastic volatility model with non-zero correlation and sublinear growth for the volatility coefficient, using the Donsker-Varadhan[DV83] LDP for the occupation measure of a Feller process under mild ergodicity conditions. We verify that these conditions are satisfied when the process driving the volatility is an Ornstein-Uhlenbeck(OU) process with a perturbed (sublinear) drift. We then translate these results into large-time asymptotics for call options and implied volatility and we verify our results numerically using Monte Carlo simulation. Finally we extend our analysis to include a CIR short rate process and an independent driving Lévy process. ‡