Tail asymptotics for diffusion processes, with applications to local volatility and CEV-Heston models

We simplify the Norris-Stroock[NS91] result for the tail asymptotics of the fundamental solution to the Cauchy problem for a parabolic equation with uniformly elliptic coefficients, when the operator is time-independent. Specifically, we show that we can replace their general Energy functional with 1 2 d(x, y), where d(., .) is the distance function on R associated with the Riemmanian metric a. We compare these two quantities qualitatively and numerically, by solving the Euler-Lagrange equation. We show how the Norris-Stroock result can also be used to characterize the large-strike smile asymptotics for a Dupire-type[Dup94] local volatility model, using the right-wing-tail formula of Benaim&Friz[BF06 I]. We also derive a similar result for the CEV process (Theorem 1.6), and a CEV process evaluated at independent stochastic time τ (ω, t)(Theorem 2.1). This result is applicable to the CEV-Heston model introduced by Atlan&Leblanc[AL05]. Finally, we show that if we wish to use an extended version of the Carr-Lee[CL04] methodology to infer the characteristic function of τ (t) from an observed single-maturity smile under the time-changed CEV model, then the tails of the distribution function of τ (t) must have sub-exponential behaviour. Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom. Email: [email protected] . The author would like to thank Prof E.B.Davies, Prof. Peter Laurence, Kostas Manolarakis, Prof. James Norris, Dr Vladimir Piterbarg, Prof D.W.Stroock, and especially Dr Alan Lewis for guidance and related discussions.