A Continuous-time Garch Model for Stochastic Volatility with Delay

We consider a (B, S)-security market with standard riskless asset B(t) = B0ert and risky asset S(t) with stochastic volatility depending on time t and the history of stock price over the interval [t − τ, t]. The stock price process S(t) satisfies a stochastic delay differential equation (SDDE) with past-dependent diffusion coefficient. We state some results on option pricing in such a market and its completeness. We derive a continuous-time analogue of GARCH(1,1) model for our past-dependent volatility. We then show that the equation for the expected squared volatility under the risk-neutral measure is a deterministic delay differential equation, and we construct the solutions for such an equation. We also construct numerical solutions and develop estimation procedures to the option pricing problem, and show the comparison of numerical results.