BFS 2002

Poster Presentation

Black-Scholes formula for security markets with delayed response

Yuriy Kazmerchuk, Anatoly Swishchuk, Jianhong Wu

The subject of this work is the following Stochastic Delay Differential Equation (SDDE) \begin{equation} \frac{dS(t)}{S(t)}=rdt+\sigma(t,S_t)dW(t), \end{equation} where $S_t=\{S(t+\theta),\theta\in [-\tau,0]\}$, $\tau>0$, $\sigma(\cdot,\cdot)$ is a continuous function of time $t$ and segment of stock price path $S(t)$ on the interval $[t-\tau,t]$ to reflect the reality that responses are usually delayed, which is normally ignored in the literature.
We show that a continuous time equivalent of GARCH(1,1) model gives rise to a stochastic volatility model with delayed dependence on stock value. Then we derive an analogue of Ito's lemma for this type of SDDE and we obtain an integral-differential equation for functions of option price with boundary conditions specified according to the type of option to be priced. In the case of vanilla call option, we obtain a closed-form solution and the results are directly transferable to European puts through the use of put-call parity. We observe for a sample set of parameters, that the original Black-Sholes price overvalues in-the-money call options due to ignorance of the delay.