A Delayed Black and Scholes Formula

Abstract

In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic functional differential equation. We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. Furthermore, the model maintains the completeness of the market. The derivation of the option-pricing formula is based on an equivalent martingale measure.

Delay geometric Brownian motion in financial option valuation

“…As a result, the discounted asset price process e −rt S(t) is a martingale and the model is arbitrage-free. The uniqueness of the equivalent martingale measure guarantees that the market is complete and therefore appropriate hedging strategies can also be obtain, for more details see Arriojas et al [3]. Hobson & Rogers [13] also observed that their model produces the right smiles and skews in the resulting implied volatility plots as in Figure 1.…”

“…It is here where one can observe a link between Arriojas et al [3] and Hobson & Rogers [13] since both approaches require the same structure for the unique (in each case) equivalent martingale measure dP dP := exp…”

“…This idea was developed in [20] and used recently by [3]. However, we will need the local Lipschitz condition in the next sections when we study the sensitivity of the time lag.…”

“…In [3], a closed form expression is produced for pricing European call options. One could in principle prove Theorem 3.4 by a more direct argument using the aforementioned formula.…”

“…However, given that the SDDE (1.1) could provide a good approximation to the SFDE (1.2) while the former is also much simpler and closer to practitioners' views than the latter, we concentrate our study on the SDDE (1.1). A recent paper by Arriojas et al [3], where a delay Black-Scholes formula is established through the existence of a unique equivalent martingale measure, also uses an SDDE to model asset price processes. One further observes that there exists a link between the way the risk-neutral measure is obtained in both papers, [3] & [13], that results in a complete and arbitrage market with the correct smiles and skews.…”