An Option Pricing Model with Memory

TL;DR

This paper develops an option pricing model where stock dynamics depend on the entire past trajectory, using stochastic functional differential equations, and derives fair prices under no-arbitrage conditions.

Contribution

It introduces a novel memory-dependent stock model with a gap approximation and provides explicit option pricing formulas ensuring market completeness.

Findings

Derived explicit option pricing formulas with memory effects.

Established market completeness and no-arbitrage conditions.

Closed the memory gap to obtain strong solutions.

Abstract

We obtain option pricing formulas for stock price models in which the drift and volatility terms are functionals of a continuous history of the stock prices. That is, the stock dynamics follows a nonlinear stochastic functional differential equation. A model with full memory is obtained via approximation through a stock price model in which the continuous path dependence does not go up to the present: there is a memory gap. A strong solution is obtained by closing the gap. Fair option prices are obtained through an equivalent (local) martingale measure via Girsanov's Theorem and therefore are given in terms of a conditional expectation. The models maintain the completeness of the market and have no arbitrage opportunities.