Option pricing under stochastic volatility and stochastic interest rate in the Spanish case

Among the underlying assumptions of the Black-Scholes option pricing model, the largest empirical biases are caused by those with a fixed volatility of the underlying asset and a constant short-term riskless interest rate. Only recently has attention been paid to the simultaneous effects of the stochastic nature of both variables on the pricing of options. Using a discrete approach an attempt is made to estimate the effects of a stochastic volatility and a stochastic interest rate in the Spanish option market; symmetric and asymmetric GARCH models were tried. The presence of in-the-mean and seasonality effects was allowed. Also estimated were the stochastic processes for the MIBOR90, a Spanish short-term interest rate, from 19 March 1990 to 31 May 1994 and the stochastic processes for the volatility of the returns of the most important Spanish stock index (IBEX-35) from 1 October 1987 to 20 January 1994. These estimators were used on pricing call options on the stock index, from 30 November 1993 to 30 May 1994. Hull-White and Amin-Ng pricing formulas were used. These prices were compared with actual prices and with those derived from the Black-Scholes formula, trying to detect the biases reported previously in the literature. Whereas the conditional variance of the MIBOR90 interest rate seemed to be free of ARCH effects, an asymmetric GARCH with in-the-mean and seasonality effects and some evidence of persistence in variance, IEGARCH(1,2)-M-S, was found to be the model that best represent the behaviour of the stochastic volatility of the IBEX-35 stock returns. All the biases reported previously in the literature were found. All the formulas overpriced the options in near-the-money case and underpriced the options otherwise. Furthermore, in most option trading, Black-Scholes overpriced the options and, because of the time-to-maturity effect, implied volatility computed from the Black-Scholes formula underestimated the actual volatility.