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Expected Return, Fair Value, Beta of Put Option & Return on Put Option

You are concerned that the stock market will decline over the next year, so you are considering buying a one-year put option on the S&P 500 Index. Assume that the Index has a value today of 1,400. One year from now, it will be either 1,800 or 1,000. The put option has an exercise price of 1,200. The risk-free rate is 5%. (Note: Treat the S&P Index simply as a stock with a price that equals its level, in dollars. So, for example, an index level of 1,000 can be considered to represent a stock worth $1,000.)

a) If the true probability that the market goes up next year is 0.65, and the probability that it goes down is 0.35, what is the expected return on the market? (Ignore dividends.)
b) What is the fair value for the put option?
c) What is the beta of the put option (assuming that the S&P Index represents "the market" in the CAPM)?
d) What is the expected return for the put option? Does the CAPM "work" for the put option? In other words, does its expected return fit what the CAPM predicts? Justify your answer.

Solution Preview

a) If the true probability that the market goes up next year is 0.65, and the probability that it goes down is 0.35, what is the expected return on the market? (Ignore dividends.)

When the market goes up, the return to the Index is Rh = 1800 / 1400 - 1 = 29%
When the market goes down, the return to the Index is Rl = 1000 / 1400 - 1 = -29%
Then the the expected return on the market = 0.65 * Rh + 0.35 * Rl = 8.57%

b) ...

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