Generalized delayed Black and Scholes Formula

TL;DR

This paper derives explicit formulas for European option prices where the underlying stock price follows a delayed differential equation, extending classical models and analyzing market properties like arbitrage and completeness.

Contribution

It introduces a generalized delayed Black-Scholes formula using martingale methods, extending previous models to include delay effects in the stock dynamics.

Findings

Derived explicit pricing formulas for delayed stock models

Identified conditions for market no-arbitrage and completeness

Analyzed arbitrage possibilities in extended models

Abstract

The mean objective of this paper is to derive an explicit formula for a price of an European option associated to the underlying delayed stock price which follows a linear differential equation with a general delay in the drift term. We use an equivalent martingale measure method based on Girsanov's property. Two of our model maintains the no-arbitrage property and the completeness of the market and can be considered as an extension some previous model introduced by Arriojas et al. in \cite{Aal}. The last one has a possible arbitrage property such that we can not obtain an unique price of an European option associated.