Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory

Abstract

We study the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated Black-Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.Mathematics Subject Classification (1991): 65M12, 35K55, 49L25

Modelling and Numerical Valuation of Power Derivatives in Energy Markets

“…For modelling spikes Merton suggested the Poisson process which is presumed to be independent of the Brownian part. According to [3] the modified one-factor SDE becomes…”

Numerical valuation of options with jumps in the underlying

“…Note that this assertion is related to the iterative method and not to the stability of the solution. Note also that conditions arising from explicit methods are in general worse, since they demand k = O(h 2 ); see [6].…”

“…More general models based on Lévy processes are also solved numerically in [1] by the ADI finite difference method combined with the fast Fourier transform and in [25] by a finite element method that gives a compressed sparse matrix in a convenient wavelet basis. An explicit method was used in [6] to solve Merton's model and a convergence theory for explicit schemes and CFL conditions were given for a general family of integro-differential Cauchy problems. Recently, we came across [12], where the value of American options using Merton's model is found implicitly by the penalty method.…”

A Semi-Lagrangian Approach for American Asian Options under Jump Diffusion

“…In addition, sufficient conditions to ensure convergence of a discrete numerical scheme to the viscosity solution, is given in [5]. Finally, an application of the results in [5] to the case of European options with jump diffusion is given in [12,13]. Typically when pricing continuously observed arithmetic average Asian option, a two dimensional problem must be solved (2.4).…”

“…In the fully implicit case, it is straightforward to prove l ∞ stability and monotonicity, which are important properties of discrete schemes for option pricing [5,34,13,12].…”