Adjoint Expansions in Local Lévy Models

Abstract

We propose a novel method for the analytical approximation in local volatility models with Lévy jumps. The main result is an expansion of the characteristic function in a local Lévy model, which is worked out in the Fourier space by considering the adjoint formulation of the pricing problem. Combined with standard Fourier methods, our result provides efficient and accurate pricing formulae. In the case of Gaussian jumps, we also derive an explicit approximation of the transition density of the underlying process by a heat kernel expansion: the approximation is obtained in two ways, using PIDE techniques and working in the Fourier space. Numerical tests confirm the effectiveness of the method.

Asymptotic expansions for degenerate parabolic equations

“…Thus, if we set C 1 = max Σ(T −εr 2 ,T ,0,r)|u − u| then by(15) and the maximum principle, we have:|u − u| ≤ C 1 v, in H T − εr 2 , T , 0, r and also |u −ū N | ≤ |u − u| + | u −ū N | ≤ C 1 v + | u −ū N |, in H T − εr 2 , T , 0, δr .Therefore the thesis follows from(16) and the asymptotic estimate of Theorem 2.1 applied to ∂ t + A. Finally, if u admits the Feynman-Kac representation (3), then u L ∞ ([0,T ]×D) ≤ e T a L ∞ ([0,T ]×D) ϕ L ∞ (D) , and a similar estimate holds for u.…”

“…Since the pioneering papers [9] and [10], several different approaches have been proposed in the literature (see, for instance, [8,11,1,15,16]). Only recently however, rigorous error bounds have been proved under the restrictive assumption of non-degeneracy of the generator of the underlying stochastic process.…”

Two-Dimensional Fourier Cosine Series Expansion Method for Pricing Financial Options

“…Local volatility models, for example, typically do not yield analytic functions ϕ, but recent research in [25] proposes a second-order approximation formula, so that an approximate characteristic function may be derived.…”