Generalized delayed Black and Scholes Formula
Hubert Le Bi Golé, Auguste Aman
TL;DR
The paper addresses European option pricing when the underlying follows a generalized delayed SDE with drift depending on past values and volatility driven by history through $g(S(t-b))$. By constructing an equivalent martingale measure via Girsanov, it establishes no-arbitrage and, in Market I, market completeness, yielding a delayed Black-Scholes-type price for $t\in[T-\ell,T]$ and a conditional-expectation representation for earlier times that incorporates the delayed history through $\Psi$ and $\Theta$. It also demonstrates that a delayed risk-free asset model (Market II) can violate no-arbitrage, highlighting the sensitivity of pricing to the delay structure. Overall, the work extends Arriojas et al. by integrating delayed dynamics and history-dependent volatility into explicit option pricing and hedging formulas within a forward stochastic Volterra framework.
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.