A Delayed Black and Scholes Formula II

TL;DR

This paper extends the Black and Scholes option pricing formula to models with stochastic functional differential equations featuring variable delays, providing explicit pricing formulas and ensuring market completeness.

Contribution

It introduces explicit option pricing formulas for stock models with variable delays in stochastic functional differential equations, expanding previous fixed-delay models.

Findings

Explicit formulas for options with variable delays

Market completeness maintained under new models

Framework for future empirical testing

Abstract

This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic delay equation with fixed delays in the drift and diffusion terms. In this article, we look at models of the stock price described by stochastic functional differential equations with variable delays. We present a class of examples of stock dynamics with variable delays that permit an explicit form for the option pricing formula. As in [A.H.M.P], the market is complete with no arbitrage. This is achieved through the existence of an equivalent martingale measure. In subsequent work, the authors intend to test the models in [A.H.M.P] and the present article against real market data.