Essay: Using Black and Scholes Model for Pricing Derivatives

Sample Essay

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Since the revolutionary paper of Black and Scholes written in early 70s, there has been a huge amount of debate and controversy following it. The formula and the method introduced by the two renowned authors are very complex in nature but still have not been able to predict and explain the market prices completely. James D. Macbeth and Larry J. Merville in their pioneering study ‘An Empirical Examination of the Black-Scholes Call Option Pricing Model’ published in 1979 have examined more than twelve thousand option prices and were surprised at discovering remarkable congruence in the results for options on six separate stocks. The two authors claimed their study to be ‘one of the most extensive empirical examinations of option prices’ to be made public to that date. While unearthing this kind of discoveries does deserve such an accolade, a few shortcomings in the study should also be noted. A major one was that the researchers analyzed only six underlying securities’ daily prices during a one year period. Hence, the research’s conclusions can’t be considered very well established. (Macbeth and Merville 1979)

If we assume that the Black and Scholes model based at the money options’ prices are accurate and these options have at least ninety days to expiry, then the Macbeth and Merville study finds (a) For in the money options, the Black and Scholes model based prices are on average less (greater) than market prices for in the money (out of the money) options, (b) The degree to which the option is in the money (out of the money) positively correlates with the degree to which the Black and Scholes model underprices (overprices) the option. As the time to expiration reduces, the Black and Scholes model’s positive correlation turns into negative (However, an out of the money option with less than ninety days to expiry is an exception in this case) and (c) On an average basis, the prices calculated by Black and Scholes model for an out of the money option with not more than or equal to ninety days to expiry are above the market price levels. A stable and reliable relationship can, however, be not established between the degree to which it does so and the degree of option’s moneyness and time to expiry. “We emphasize that our results are exactly opposite to those reported by Black, wherein he states that deep in the money (out of the money) options generally results also conflict with Merton’s statement that practitioners observe Black and Scholes model prices to be less than market prices for deep in the money as well as deep out of the money options. We propose that these conflicting empirical observations may, at least in part, be the result of a non-stationary variance rate in the stochastic process generating stock prices”. (Macbeth and Merville 1979)

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