A CLOSED-FORM OPTION VALUATION FORMULA IN MARKOV JUMP DIFFUSION MODELS

To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address the leptokurtic feature of the asset return distribution, volatility smile and the effects of volatility clustering phenomenon. However, analytical tractability remains a problem for most of the alternative models. In this paper, we propose a Markov jump diffusion model, that can not only incorporate both the leptokurtic feature and volatility smile, but also present the economic features of volatility clustering and long memory. To evaluate derivatives prices, we apply Lucas’s general equilibrium framework to provide closed form formulas for option and futures prices. When the jump size follows a specific distribution, for instance a lognormal distribution and a default probability, we write explicit analytic formulas for the equilibrium prices. Through these formulas, we illustrate the effect of jumps, via stochastic intensity, on implied volatility and volatility surface as well as sensitivity analysis in stock option prices