EXPECTED RETURN IN OPTION PRICING
One of the most mystifying features of modern option pricing is the irrelevance of the expected return of the underlying stock.
One of the most mystifying features of modern option pricing is the irrelevance of the expected return of the underlying stock. This fact, that an option's price does not depend on the stock's expected return, baffles students and seasoned practitioners alike. If calls are bullish instruments and puts are bearish, why do their prices not depend on the expectation of which way stocks are moving? This article presents an intuitive explanation. The explanation uses little math and requires only familiarity with option payoff diagrams.
Consider a portfolio that combines a call option, a short position in one share, and a risk-free bill whose face value equals the call's strike price. We can depict the performance of such a portfolio by drawing its payoff diagram. To do so, we simply draw the payoff diagrams for each of the portfolio's components, and then add them up. This is shown in figure 1. The shape of the portfolio's payoff diagram should be familiar to anyone who has seen payoff diagrams. It is the same as the payoff diagram of a put. What we have here is the fastest and simplest proof of put-call parity. A portfolio consisting of a call, a short stock, and a risk-free bill has the same payoff as a put. Consequently, that portfolio replicates the put, and must be valued the same as the put.
Now consider the portfolio that combines a call with a put. Let both options be at-the-money--a straddle. The straddle's payoff diagram is presented in figure 2.
Suppose a straddle was traded as a single instrument. Without relying on mathematics or on formal option pricing formulas, consider what variables should be relevant in pricing it. Clearly, this instrument becomes more valuable the farther from today's price the stock price might bounce. Thus, volatility is certainly a relevant factor. The amount of time left to expiry is relevant, since more time implies more opportunity for the stock to move into the profitable region. Today's stock price and the strike price must be known so that we can determine how much in-the-money the straddle already is. Lastly, the interest rate is relevant, since it represents the opportunity cost of funds tied up in the straddle.
What about the expected return or expected direction of the stock price? If you already knew the five variables listed above, would you pay money to learn more about the expect return? The answer is no. For the straddle to be profitable, it does not matter whether the stock moves up or down, just so long as it moves far. Consequently, we see that expected return is not relevant in the pricing of a straddle.
The irrelevance of the expected return for pricing the straddle carries over to the pricing of calls and puts. Since a put can be replicated by combining a call, a short stock, and a risk-free bill, it follows that a straddle can be replicated by two calls, a short stock, and a risk-free bill. The five variables needed to price the straddle are: volatility, time to expiry, the risk-free rate, the stock price, and the strike price. If we know these five variables, we not only know the price of the straddle, but we know the price of the stock and risk-free bills. We can subtract the latter two prices from the former to ferret out the price of the call. By analogy, suppose we know that the cake consists of flour, sugar, milk, and eggs. If we know the price of the cake, flour, sugar, and milk, we can figure out the price of the eggs. Since we did not need to know the expected return of the straddle, we similarly do not need to know expected return to price a call option.
Perhaps now the irrelevance of expected return in option pricing will not seem so counter-intuitive. Since puts can be converted into calls, calls into puts, and each can be converted into a straddle, it becomes clear that the value of neither puts nor calls can depend on a forecast of which way the market is moving. Volatility, not expected return, is all we need to know about the stock's price dynamics.
This week's Learning Curve was written bySteven Feinstein, assistant professor of finance,Babson College, Babson Park, MA