The Galai Studies

As in the Black-Scholes study, Galai created hedged portfolios of options and stock and used these portfolios to study the correspondence between the Black-Scholes model price and actual market prices for options.15 In contrast to the Black-Scholes study, Galai used listed options data from the Chicago Board Options Exchange. With options trading on an exchange, Galai had access to daily price quotations. Therefore, he was able to compute the rate of return on the hedged option-stock portfolio for each option for each day. He also adjusted the hedge ratio each day to maintain the neutral hedge—neutral in the sense that a change in the stock price would not change the overall value of the combined optionstock position. Comparing market prices to Black-Scholes model prices, Galai assumed that he sold overpriced options and bought underpriced options each day.

Galai's results showed that this strategy could earn excess returns. In other words, his initial results seemed to be inconsistent with an efficient market. However, this apparent result disappeared when Galai considered transaction costs. If transaction costs were only 1 percent, the apparent excess returns disappeared. Most traders outside the market face transaction costs of 1 percent or higher. However, market makers can transact for less than 1 percent transaction costs. This suggests that market makers could have followed Galai's strategy to earn excess returns. Yet even market makers face some additional transaction costs implied by their career choice. For instance, the market maker must buy or lease a seat on the exchange and the market maker must forego alternative employment. When Galai brought these additional implicit transaction costs into the analysis, the market maker's apparent excess returns diminished or disappeared. At any rate, Galai's results showed that Black-Scholes model prices closely match actual market prices for options.

The Bhattacharya Study

Mihir Bhattacharya used an approach like the Black-Scholes and Galai studies to analyze the correspondence between actual market prices and theoretical prices.16 Bhattacharya discussed the adherence of market prices to theoretical boundaries implied by no-arbitrage conditions. We focus on one of his three boundaries. As we discussed in Chapter 3, a call option should be worth more than its exercise value if time remains until expiration. Bhattacharya compiled a sample of 86,000 transactions and examined them to determine if immediate exercise was profitable. He found 1100 such exercise opportunities, meaning that the stock price exceeded the exercise price plus the call price. As we argued in Chapter 3, such a price relationship should not exist. However, these exercise opportunities assumed that the exercise could be conducted without transaction costs. When Bhattacharya considered transaction costs, these apparently profitable exercise opportunities disappeared. The apparent violation of the boundary condition was observed only because trans action costs were not considered. This means that traders could not exploit the deviation from the boundary condition to make a profit.

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