An approximation scheme for Black-Scholes equations with delays

Abstract

This paper addresses a finite difference approximation for an infinite dimensional BlackScholes equation obtained by Chang and Youree (2007). The equation arises from a consideration of an European option pricing problem in a market in which stock prices and the riskless asset prices have hereditary structures. Under a general condition on the payoff function of the option, it is shown that the pricing function is the unique viscosity solution of the infinite dimensional Black-Scholes equation. In addition, a finite difference approximation of the viscosity solution is provided and the convergence results are proved.

Joint distributions for stochastic functional differential equations

“…The process X = X (t) ; −r ≤ t ≤ T is called the delayed Black-Scholes model in mathematical finance (cf. [1,4,5,6,13]).…”

“…In this section, we shall apply our studies to the option pricing of the asset price dynamics model with delayed effects. See [1,4,5,6,13] on details. Let A i ∈ C 1 0+,b (R ; R) (i = 0, 1) with the uniformly elliptic condition of the form: there exists a positive constant C 12 with…”

“…It seems us to be very natural to study the models described by (1) stated above, because the models with their past histories often appear in finance, physics, biology and industry, etc. One of the typical examples in mathematical finance is the delayed Black-Scholes model studied in [1,4,5,6,13], which will be mentioned in Section 5. On the other hand, the Malliavin calculus is applicable to the study on the densities for the solution to (1).…”