BFS 2002 

Poster Presentation 
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
integraldifferential 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 closedform solution
and the results are directly transferable to European puts through the use of putcall parity.
We observe for a sample set of parameters, that the original BlackSholes price overvalues inthemoney
call options due to ignorance of the delay.
http://www.math.yorku.ca/Who/Grads/yorik/poster.pdf