2014/04/17 by Conforti, Giovanni, De Marco, Stefano, Deuschel, Jean-Dominique
#60H30 #91G80 #FOS: Economics and business #FOS: Mathematics #Pricing of Securities (q-fin.PR) #Probability (math.PR)
paper · doi:10.48550/arxiv.1404.4464
The one-dimensional SDE with non Lipschitz diffusion coefficient dXt = b(Xt)dt + σXtγ dBt, X0=x, γ<1 is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and S.~Violante. Marginal density expansions for diffusions and stochastic volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with the rescaled variable Xε:=ε1/(1-γ) X: while allowing to turn a space asymptotic problem into a small-ε problem with fixed terminal point, the process Xε satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise ε dB). We prove a pathwise large deviation principle for the process Xε as ε → 0. As it will become clear, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell--Freidlin theory. As for applications, the ε-scaling allows to derive exact log-asymptotics for path functionals of the process: while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving such an SDE as a component.