Comparison of Black Scholes and Heston Models for Pricing Index Options

This paper studies the performance of Heston Model and Black-Scholes Model in pricing index options. I have compared the two models based on 1074 call option prices of S&P 500 on 1st November, 2016. I have calibrated the parameters of the Heston Model by non-linear least square optimization using call option prices from a period of 20 days (3rd October, 2016 to 31st October, 2016). The in-sample data had a total of 25,392 call options and thus 20 strike prices for each time-to-maturity. We observe that both Heston Model and Black Scholes Model under-price in-the-money options and over-price out-of-the money options, but the degree of error is different. Black Scholes Model slightly outperforms Heston Model for short term ITM, DITM and ATM options where Heston Model is unable to capture the high implied volatility. But Heston Model starts to give better estimates for ITM, DITM and ATM options as the time-to-maturity increases. For OTM and DOTM options, Heston Model significantly outperforms Black Scholes Model. In most of the cases, the implied volatility calculated from Heston model prices is found to be less than that calculated from market prices for different combinations of moneyness and time-to-maturity