Suitability of Black & Scholes Model for Pricing Derivatives and The Usage of Greeks

Suitability of Black & Scholes Model for Pricing Derivatives and The Usage of Greeks
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December 06, 2010
Using Black and Scholes Model for Pricing Derivatives: A Discussion
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)
Geske’s compound option model is another call option pricing model which overtly takes account of a non-stationary variance rate. If the variance rate for day t, ot2 is incorporated in the Black and Scholes model, then the market prices and the prices calculated by Black and Scholes model will equate, assuming that the market prices are given by Geke’s model. “Thus, the compound option model provides theoretical justification for our methodology of computing implied variance rates on a daily basis, but it still does not explain the systematic differences we observe in implied variance rates.” (Macbeth and Merville 1979)
There has been a lot of conflicting literature since the introduction of the Black and Scholes model. Such conflicts could be lessened if we accept, on a hypothetical basis, that market prices should correspond to the compound option pricing model. Chiras and Manaster have presented a to-be-tested profitable options trading strategy in which they would sell the option having a market price above the Black and Scholes model price and buy the option which has a market price lesser than that of Black Scholes model. The difference in terms of weight of the implied variance values between the studies of Chiras and Manaster and that of Macbeth and Merville results in different outcomes of the Black and Scholes models of the two research groups. Though the differences are not significant, the price calculated by the model of Chiras and Manster comes out to be above than that of Macbeth and Merville. Apparently, the reason appears to be the deep selling of in the money option and deep out of money buying by Chiras and Manaster. Though the price is greater than that of Macbeth and Merville, it does not necessarily mean that the return would be greater as well when this strategy is employed. (Macbeth and Merville 1979)
Now that we have seen how the Black and Scholes model can give different result when elaborate studies use different assumptions and variables, let us go deeper into the pricing of options from the original perspective of Black and Scholes with their paper named ‘The Pricing of Options and Corporate Liabilities’. “We have done empirical tests of the valuation formula on a large body of call-option data. These tests indicate that the actual prices at which options are bought and sold deviate in certain systematic ways from the values predicted by the formula. Option buyers pay prices that are consistently higher than those predicted by the formula. Option writers, however, receive prices that are at about the level predicted by the formula. There are large transaction costs in the option market, all of which are effectively paid by option buyers.” (Black F. and Scholes M. 1973)
The concerns of the academia about data choice, model specification, and potential cross-equation correlations are accounted for in a study by Rahman A., Kryzanowski L. and Sim A named ‘Simultaneous Estimation of the Parameters of the Black-Scholes Option Pricing Models’. A set of nonlinear and unrelated regression equations are set as the model thus allowing the authors to take account of cross-equation correlations. “Using the Quadratic Hill Climbing algorithm, the incorporation of cross-correlations into the estimation procedure of the Black and Scholes model yields a better fit (lower MSPE), better out-of-sample forecasts (lower MSFE), and more efficient parameter estimates for options of shorter duration (< 3 months). Thus, it would appear that the Borch conjecture is true for short options. Since the parameter estimates are quite unrealistic for longer options, the empirical findings appear to provide little support for either the implications of the Borch thesis or for the Black and Scholes model itself. A possible explanation is that infrequent trading of longer options leads to a potential lack of simultaneous trades across options on the same security so that the sample cross-correlation is obtained with error.” (Rahman A., Kryzanowski L. and Sim A. 1987)
So far, we have only seen the work of the researchers who tend to agree with Black and Scholes model by and large. Now the picture will get interesting as we come to the other side of the table. We’ll present the researchers who heavily criticize the Black and Scholes model and sometimes, totally reject it. One of such works is that of Moore L. and Juh S. They have conducted a very interesting study of looking into the past in the terms of how the financial world survived before the Black and Scholes model was introduced. Was everyone making wrong decisions? Did no logic prevail in those times? And the authors have not just gone a few decades back, rather they have gone sixty years before the Black and Scholes model was introduced thus naming their work ‘Derivative Pricing 60 Years before Black-Scholes: Evidence from the Johannesburg Stock Exchange’. Daily data for warrants traded on the Johannesburg Stock Exchange was collected by the authors. The data pertained to the period from 1909 and 1922, and for a broker's call option quotes on stocks from 1908 to 1911. This data was used to see how different the prices would have been had the Black and Scholes model was present at that time. The striking conclusion comes out to be that even sixty year before the introduction of this revolutionary Black and Scholes model, investors did have that intuitive grasp required for derivatives pricing. “In this paper we present a new data set of 15 warrants traded in the early 20th century on the JSE and of a broker's quotes of call options written on 112 stocks. Using modern derivative pricing theory, we assess how well investors priced these securities. We find that, in general, warrant prices were surprisingly accurate in the pre- Black-Scholes era, with an average absolute percentage mispricing of 23.7% when we use a perfect foresight measure of volatility and 27.4% when we use a historical measure of volatility. Comparing these warrants to derivatives trading between 2001 and 2003 on the same exchange and using the same methods to compute volatility, we find that early twentieth-century investors mispriced in a comparable way using a historical measure of volatility and outperformed modern JSE investors using a perfect foresight measure of volatility.” (Moore L. and Juh S. 2006)
Thus, in a mind-blowing conclusion, the authors have utterly rejected the idea that the development of modern theory has done any good to the performance of South African investors, at least. While the study is limited to South Africa, it is a very serious observation and has fueled the controversy over the Black and Scholes model even more. The authors further go ahead to claim...