Estimates of Dynamic VaR and Mean Loss Associated to Diffusion Processes

Abstract

Let X_t be a stochastic process driven by a differential equation of the form dX_t=σ(t,X_t)dW_t+b(t,X_t)dt, t>0, and let X^⋆_s,t=sup_s≤u≤tX_u, be the maximum of the diffusion. In this work we obtain bounds for the tail distribution of X^*_s,t, define several dynamic VaR type quantiles for this process and give upper and lower bounds for both, the VaR quantile and the conditioned mean loss associated to it. The results we obtain are based in the change of time property of the Brownian Motion, and can be applied to a a large class of examples used in Finance, in particular where σ(t,X_t)=σ_tX^γ_t , where 0≤γ<1. The estimates we obtain are sharp. We discuss carefully the Geometric Brownian Motion, the Cox-Ingersoll-Ross and the Vasicek type models, and give an application to Russian options.