The exact Taylor formula of the implied volatility

In a model driven by a multi-dimensional local diffusion, we study the behavior of implied volatility {\sigma} and its derivatives with respect to log-strike k and maturity T near expiry and at the money. We recover explicit limits of these derivatives for (T,k) approaching the origin within the parabolic region |x-k|^2 < {\lambda} T, with x denoting the spot log-price of the underlying asset and where {\lambda} is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for implied volatility within the parabola |x-k|^2 < {\lambda} T. In order to include important models of interest in mathematical finance, e.g. Heston, CEV, SABR, the analysis is carried out under the assumption that the infinitesimal generator of the diffusion is only locally elliptic

Analytical approximation of the transition density in a local volatility model

We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.option pricing, analytical approximation, local volatility

Analytical expansions for parabolic equations

We consider the Cauchy problem associated with a general parabolic partial differential equation in $d$ dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provide convergence results for arbitrarily large time intervals.Comment: 23 page

Pricing approximations and error estimates for local L\'evy-type models with default

We find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar L\'evy-type stochastic processes. We derive rigorous error bounds for the approximate solutions. We also provide numerical examples illustrating the usefulness and versatility of our methods in a variety of financial settings.Comment: 36 pages, 4 figures, 1 table

Intrinsic Taylor formula for Kolmogorov-type homogeneous groups

We consider a class of ultra-parabolic Kolmogorov-type operators satisfying the Hormander's condition. We prove an intrinsic Taylor formula with global and local bounds for the remainder given in terms of the norm in the homogeneous Lie group naturally associated to the differential operator

Asymptotics for dd-dimensional L\'evy-type processes

We consider a general d-dimensional Levy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Levy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (2010), Pagliarani and Pascucci (2013) and Lorig et al. (2013a) for Markov diffusions to Markov processes with jumps.Comment: 20 Pages, 3 figures, 3 table

Black-Scholes formulae for Asian options in local volatility models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.Asian Options, Degenerate Diffusion Processes, Transition Density Functions, Analytic Approximations, Option Pricing