## Empirical Research

There are a number of problems in carrying out empirical research to test the Black-Scholes and other option pricing models. The first problem is that any statistical hypothesis about how options are priced has to be a joint hypothesis to the effect that (1) the option pricing formula is correct; and (2) markets are efficient. If the hypothesis is rejected, it may be the case that (1) is untrue; (2) is untrue; or both (1) and (2) are untrue. A second problem is that the stock price volatility is an unobservable variable. One approach is to estimate the volatility from historical stock price data. Alternatively, implied volatilities can be used in some way. A third problem for the researcher is to make sure that data on the stock price and option price are synchronous. For example, if the option is thinly traded, it is not likely to be acceptable to compare closing option prices with closing stock prices. This is because the closing option price might correspond to a trade at 1:00 p.m., while the closing stock price corresponds to a trade at 4:00 p.m.

Black and Scholes and Galai have tested whether it is possible to make excess returns above the risk-free rate of interest by buying options that are undervalued by the market (relative to the theoretical price) and selling options that are overvalued by the market (relative to the theoretical price).16 A riskless delta-neutral portfolio is assumed to be maintained at all times by trading the underlying stocks on a regular basis as described in Section 13.5. Black and Scholes used data from the over-the-counter options market where options are dividend protected. Galai used data from the Chicago Board Options Exchange (CBOE) where options are not protected against the effects of cash dividends. Galai used Black's approximation as described in Section 10.14 to incorporate the effect of anticipated dividends into the option price. Both of the studies showed that, in the absence of transactions costs, significant excess returns over the risk-free rate could be obtained by buying undervalued options and selling overvalued options. It is possible that these excess returns were available only to market makers, and that when transactions costs are considered, they vanish.

A number of researchers have chosen to make no assumptions about the process followed by stock prices and have tested whether arbitrage strategies can be used to make a riskless profit in options markets. Garman provides a very

16 See F. Black and M. Scholes, "The Valuation of Option Contracts and a Test of Market Efficiency," Journal of Finance, 27 (May 1972), 399-418; D. Galai, "Tests of Market Efficiency and the Chicago Board Options Exchange," Journal of Business, 50 (April 1977), 167-97.

efficient computational procedure for finding any arbitrage possibilities that exist in a given situation.17 One study by Klemkosky and Resnick which is frequently cited tests whether the relationship in Equation (7.9) is ever violated.18 It concludes that some small arbitrage profits were possible from using the relationship. These were due mainly to the overpricing of American calls.

Chiras and Manaster have carried out a study using CBOE data which compares the weighted implied standard deviation from options on a stock at a point in time with the standard deviation calculated from historical data.19 They found that the former provides a much better forecast of the volatility of the stock price during the life of the option. The study has been repeated by other authors using other data and has always given similar results. We can conclude that option traders are using more than just historical data when determining future volatilities. Chiras and Manaster also tested to see whether it was possible to make above-average returns by buying options with low implied standard deviations and selling options with high implied standard deviations. This strategy showed a profit of 10 percent per month. The Chiras and Manaster study can be interpreted as providing good support for the Black-Scholes model while showing that the CBOE was inefficient in some respects.

MacBeth and Merville have tested the Black-Scholes model using a different approach.20 They looked at different call options on the same stock at the same time and compared the volatilities implied by the option prices. The stocks chosen were AT&T, Avon, Kodak, Exxon, IBM, and Xerox, and the time period considered was the year 1976. They found that implied volatilities tended to be relatively high for in-the-money options and relatively low for out-of-the-money options. A relatively high implied volatility is indicative of a relatively high option price and a relatively low implied volatility is indicative of a relatively low option price. Therefore, if it is assumed that Black-Scholes prices at-the-money options correctly, it can be concluded that out-of-the-money call options are overpriced by Black-Scholes and in-the-money call options are underpriced by Black-Scholes. These effects become more pronounced as the time to maturity increases and the degree to which the option is in or out of the money increases. MacBeth and Merville's results are consistent with the displaced diffusion model when a > 1, the compound option model, the absolute diffusion model, and the stochastic volatility model when the stock price and volatility are negatively correlated.

17Garman, M. B., "An Algebra for Evaluating Hedge Portfolios," Journal of Financial Economics, 3 (October 1976), 403-27.

18R. C. Klemkosky and B. G. Resnick, "Put-call Parity and Market Efficiency," Journal of Finance, 34 (December 1979), 1141-55.

19D. Chiras and S. Manaster, "The Information Content of Stock Prices and Test of Market Efficiency," Journal of Financial Economics, 6 (September 1978), 213-34.

20See J. D. MacBeth and L. J. Merville, "An Empirical Examination of the Black-Scholes Call Option Pricing Model," Journal of Finance, 34 (December 1979), 1173-86.

Rubinstein has carried out a study similar to the MacBeth and Merville study, but using a far larger data set and a different time period.21 He looked at all reported trades on the 30 most active Chicago Board Option Exchange options classes between August 23, 1976 and August 31, 1978. Special care was taken to incorporate the effects of dividends and early exercise. Rubinstein compared implied volatilities of matched pairs of call options which differed either only as far as exercise price was concerned or only as far as maturity was concerned. He found that his time period could be conveniently divided into two subperiods: August 23, 1976 to October 21, 1977 and October 22, 1977 to August 31, 1978. For the first period, his results were consistent with those of MacBeth and Merville. However, for the second period, the opposite result from MacBeth and Merville was obtained; that is, implied volatilities were relatively high for out-of-the-money options and relatively low for in-the-money options. Throughout the entire period Rubinstein found that for out-of-the-money options, short-maturity options had significantly higher implied volatilities than long-maturity options. The results for at-the-money and in-the-money options were less clear cut.

No single alternative to the Black-Scholes model seems superior for both of Rubinstein's time periods. Indeed, it is difficult to imagine a model that leads to the changes in the biases that were observed between the first time period and the second time period. Possibly macroeconomic variables affect option prices in a way that is as yet not fully understood. At present, there does not seem to be any really compelling arguments for using any of the models introduced earlier in this chapter in preference to Black-Scholes.

A number of authors have researched the pricing of options on assets other than stocks. For example, Shastri and Tandon, and Bodurtha and Courtadon have examined the market prices of currency options;22 Shastri and Tandon in another paper have examined the market prices of futures options;23 Chance has examined the market prices of index options.24 The authors find that the Black-Scholes model and its extensions misprice some options. There appears to be some evidence, for example, that currencies follow jump processes. However, the mispricing was not sufficient in most cases to present profitable opportunities to investors when

21 See M. Rubinstein, "Nonparametric Tests of Alternative Options Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Options Classes from August 23, 1976 through August 31, 1978," Journal of Finance, 40 (June 1985), 455-80.

22See K. Shastri and K. Tandon, "Valuation of Foreign Currency Options: Some Empirical Tests," Journal of Financial and Quantitative Analysis, 21, (June 1986), 145-60; J. N. Bodurtha and G. R. Courtadon, "Tests of an American Option Pricing Model on the Foreign Currency Options Market," Journal of Financial and Quantitative Analysis, 22 (June 1987), 153—68.

23 See K. Shastri and K. Tandon, "An Empirical Test of a Valuation Model for American Options on Futures Contracts," Journal of Financial and Quantitative Analysis, 21 (December 1986), 377-92.

24See D. M. Chance, "Empirical Tests of the Pricing of Index Call Options," Advances in Futures and Options Research, 1, pt. A (1986), 141-66.

transactions costs and bid-ask spreads were taken into account. In their two papers, Shastri and Tandon point out that even for a market maker, some time must elapse between a profitable opportunity being identified and action being taken. This delay, even if it is only to the next trade, can be sufficient to eliminate the profitable opportunity.

In an interesting study, Lauterbach and Schultz investigated the pricing of warrants.25 They conclude that the biases are consistent with Figure 17.1(b). The constant elasticity of variance model with a = 0.5 gave a better fit to the data than did the Black-Scholes model. From Table 17.2 we see that Lauterbach and Schultz's results are also consistent with the the compound option pricing model, the displaced diffusion model where a > 1, and the stochastic volatility model where the stock price and interest rate are negatively correlated. Their results were found to persist throughout a 10-year time period.

### 17.7 HOW THE MODELS ARE USED IN PRACTICE

Practitioners when using Black-Scholes recognize that it is a less than perfect model. A common approach is to construct a matrix of volatilities, one dimension of the matrix being strike price, the other being time to maturity. For actively traded options, volatilities are implied from market prices. This provides some of the points in the matrix. The rest of the matrix is then determined using an interpolation procedure.

When the matrix shows that biases corresponding to Figure 17.1(d) are observed, practitioners sometimes refer to the phenomenon as the volatility smile. The reason for this will be clear from Figure 17.3, which plots implied volatility against strike price.

How important is the pricing model being used if practitioners are prepared to use a different volatility for every deal? It can be argued that the model is simply a tool for understanding the volatility environment and for pricing illiquid securities consistently with the market prices of actively traded securities. If practitioners stopped using Black-Scholes and switched to the constant elasticity of variance model, the matrix of volatilities would change and the shape of the smile would change—but arguably the prices quoted in the market would not change appreciably.

### 17.8 SUMMARY

The Black-Scholes model and its extensions assume that the probability distribution of the stock price at any given future time is lognormal. If this assumption is

25See B. Lauterbach and P. Schultz, "Pricing Warrants: An Empirical Study of the Black-Scholes Model and its Alternatives," Journal of Finance, 4 no. 4 (Sepember 1990), 1181-1210.

Strike Price Figure 17.3 The Volatility Smile incorrect, there are liable to be biases in the prices produced by the model. If the right tail of the true distribution is fatter than the right tail of the lognormal distribution, there will be a tendency for the Black-Scholes model to underprice out-of-the-money calls and in-the-money puts. If the left tail of the true distribution is fatter than the left tail of the lognormal distribution, there will be a tendency for the Black-Scholes model to underprice out-of-the-money puts and in-the-money calls. When either tail is too thin relative to the lognormal distribution, the opposite biases are observed.

A number of alternatives to the Black-Scholes model have been suggested. These include models where the future volatility of a stock price is uncertain, models where the company's equity is assumed to be an option on its assets, and models where the stock price experiences occasional jumps rather than continuous changes. The models can be categorized according to the biases they give rise to. It is interesting to note that biases arising from jumps become less pronounced as an option's life increases, while biases arising in other ways become more pronounced as the option life increases.

Generally, the empirical research that has been done is supportive of the Black-Scholes model. It is a model that has stood the test of time. Differences between market prices and the Black-Scholes prices have been observed. However, these differences have usually been small when compared to transactions costs.

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