Facoltà di scienze economiche

An option pricing formula for the GARCH diffusion model

Ravanelli, Claudia ; Barone-Adesi, Giovanni (Dir.) ; Chesney, Marc (Codir.) ; Vanini, Paolo (Codir.)

Thèse de doctorat : Università della Svizzera italiana, 2003 ; 2003ECO001.

In this thesis, we derive an analytical closed-form approximation for European option prices under the GARCH diffusion model, where the price is driven by a geometric process and the variance by an uncorrelated mean reverting geometric process. This result has several important implications. First and foremost, these conditional moments allow us to obtain an analytical closed-form approximation... Plus

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    Summary
    In this thesis, we derive an analytical closed-form approximation for European option prices under the GARCH diffusion model, where the price is driven by a geometric process and the variance by an uncorrelated mean reverting geometric process. This result has several important implications. First and foremost, these conditional moments allow us to obtain an analytical closed-form approximation for European option prices under the GARCH diffusion model. This approximation can be easily implemented in any standard software package. As we will show using Monte Carlo simulations, this approximation is very accurate across different strikes and maturities for a large set of reasonable parameters. Secondly, our analytical approximation allows to easily study volatility surfaces induced by GARCH diffusion models. Thirdly, the conditional moments of the integrated variance implied by the GARCH diffusion process generalize the conditional moments derived by Hull and White (1987) for log-normal variance processes. Finally, the conditional moments of the integrated variance can be used to estimate the continuous time parameters of the GARCH diffusion model using high frequency data. The thesis is organized as follows. Chapter 1 introduces stochastic volatility option pricing models and discusses in details the GARCH diffusion model and its properties. Chapter 2 presents the analytical approximation formula to price European options under the GARCH diffusion model. Using Monte Carlo simulations, we verify the accuracy of the approximation across different strike prices and times to maturity for different parameter choices. We investigate differences between option prices under the GARCH diffusion and the Black and Scholes model. Then, we qualitatively study implied volatility surfaces induced by the GARCH diffusion. Chapter 3 studies the accuracy of the inference results on the GARCH diffusion model based on the Nelson's theory. Using such a procedure, we fit the GARCH diffusion model to daily log-returns of Deutsche Mark versus US dollar exchange rates. Chapter 4 gives some concluding remarks.