An option pricing formula for the GARCH diffusion model

We derive analytically the first four conditional moments of the integrated variance implied by the GARCH diffusion process. From these moments we obtain an analytical closed-form approximation formula to price European options under the GARCH diffusion model. Using Monte Carlo simulations, we show that this approximation formula is accurate for a large set of reasonable parameters. Finally, we use the closed-form option pricing solution to shed light on the qualitative properties of implied volatility surfaces induced by GARCH diffusion models. aInstitute of Finance, University of Southern Switzerland, Lugano, CH. bMathematical Institute, University of Oxford, Oxford, UK. ∗Corresponding author: Claudia Ravanelli, University of Southern Switzerland, Institute of Finance, Via Buffi 13, CH-6900, Lugano, tel: +41 (0)91 912 47 86. E-mail: [email protected]. Giovanni Barone-Adesi and Claudia Ravanelli gratefully acknowledge the financial support of the Swiss National Science Foundation and NCCR FINRISK. The authors thank Jerome Detemple and Loriano Mancini for their valuable suggestions.