A Comparative Study of GARCH (1,1) and Black-Scholes Option Prices

This paper examines the behaviour of European option price (Duan (1995)) and the Black-Scholes model bias when stock returns follow a GARCH (1,1) process. The GARCH option price is not preferenceneutral and depends on the unit risk premium (λ) as well as the two GARCH (1,1) process parameters (α1 , β1). In general, the GARCH option price does not seem overly sensitive to these parameters. Deep-out-ofthe-money and short maturity options are an exception. The variance persistence parameter, γ = α1 + β1, has a material bearing on the magnitude of the Black-Scholes model bias. The risk preference parameter, l, on the other hand, determines the so called “leverage effect” and can be important in determining the direction of the Black-Scholes model bias. Consequently, a time varying risk premium (l) may help explain a general underpricing or overpricing of traded options (Black (1975)). Consistent with "volatility smile" and similar to the bias noted by Merton (1976), deep-out-of-themoney and deep-in-the-money (at-the-money) options with a very short time to expiration are found to be underpriced (overpriced) by the Black-Scholes model. The direction of striking price bias for longer maturities is mostly influenced by GARCH option valuation parameters, a result that could be useful in resolving conflicting striking price biases observed empirically. This paper makes a novel attempt to decompose the Black-Scholes model bias into components related to three important features of GARCH option valuation: level of the unconditional variance of the locally risk-neutral return process, relative level of the initial conditional variance, and path dependence of the terminal stock price distribution. An analysis of their behaviour sheds light on the making of the overall systematic biases mentioned above as well as the time to maturity bias reversal phenomenon (Rubinstein (1985) and Sheikh (1991)). One modification to the Black-Scholes model that corrects only the unconditional variance bias does not improve accuracy enough to justify the additional input requirement. Another modification corrects the unconditional variance bias and the conditional variance bias, but not the path dependence bias. This latter modification, which we call the Pseudo-GARCH or PGARCH formula, performs rather well when the impact of a given period’s variance innovation is low (small α1) but nearly permanent (γ close to 1.0). In these empirically relevant situations, the Black-Scholes type PGARCH formula offers practical approximations to the theoretically correct simulated GARCH prices.