Risk-Neutral Pricing

I have previously introduced the concept of the "risk premium" in investing, which holds that "risky" investments should have a higher expected rate of return than "non-risky" investments. In that earlier entry, I noted that Utility Theory provides a framework for explaining why the risk premium should exist. In this entry, I will try to explain a useful way of quantifying the disposition of risk premium: "risk-neutral pricing."

Typically, the risk premium will be spoken of in terms of "expected rate of return," as in, "the expected rate of return on common stocks is around 6%, while the expected rate of return on cash is around 1%." The higher expected rate of return stated here for stocks as opposed to cash indicates a higher risk premium for stocks.

However, when option securities are considered, this way of thinking runs aground. We might consider a call option, for example, that pays in dollars the amount (if any) by which a particular stock index exceeds a given value at close of trading on a particular day, or nothing if the stock index is less than or equal to the value. But now how do we come up with an expected rate of return for this option?

If we look at the Put-Call Parity equation,

     C - P = S - X * exp(-rT),

we see that on the right hand side of the equation are terms, S and X * exp(-rT), that should have the expected returns of stocks and of cash, respectively (cash being a risk-free asset, subject to the risk-free interest rate of r).

So what if we treat the call as the difference between the index value -- discounted at the expected returns on stocks -- and a riskless payment of the strike price -- discounted at the risk-free interest rate -- on the expiration date?

This method falls apart pretty quickly. Suppose we look at just a piece of the call using this perspective: the portion from where the index equals the strike price to where it's the strike price plus one. Let's suppose also that the strike price is 1000 and the expiration date is one year in the future. So the value of the stock piece, using the expected rates of return above, will be between 1000 and 1001 discounted by 6%, or between about 943.40 and 944.34. But the value of the risk-free piece will be 1000 discounted by 1%, or about 990.10. This suggests that the value of this portion of the call option will be between -46.70 and -45.76: negative values for something that can only ever result in positive cash flow! No rational person will give you money to accept the possibility that he might give you more money later on.

The way around this is to discount everything at the risk-free interest rate: both the stock portion and the strike price portion. This gives the value of the portion of the call option we're looking at as being between 0 and 0.99, which is very consistent with the value at the end of the year being between 0 and 1.

But where does the risk premium appear?

The risk premium appears in adjustments to the probabilities of different outcomes. In particular, prices will be determined using probabilities that are different from actual expectations.

In looking at the portion of the call option where the final index value is between the strike price and the strike price plus one, I never considered the probability of this outcome: I essentially took each potential outcome -- from the index being equal to the strike price up to being that value plus one -- as given, and looked at what the value of the option would be under that outcome. To get an actual value of the call option, I'd have to assign a probability to each of the outcomes, and generate a weighted average for the option's value under all possible outcomes. In essence, the risk premium on stocks is created because the pricing probability assumes that high returns on stocks are less likely than actual expectations.

So, why use the risk-less interest rate rather than some other rate?

First of all, it would probably be more accurate to use actual expectations, and then have a different expected rate of return apply to each possible outcome. But this would create a mess: how do you figure out what expected rate of return to apply to a given outcome, and from where (without a LOT of work) do you get actual expectations?

Risk-neutral pricing takes advantage of the fact that prices based on uncertain events are based on the product of the probability of given outcomes and the expected rate of return that applies to that outcome. So rather than (more accurately) using actual expectations for the probabilities and some yet-to-be-determined expected rate of return for each outcome, risk-neutral pricing uses a single expected rate of return (the risk-free interest rate) and adjusted probabilities to come up with the same price that would result from the theoretically more accurate method of using actual expectations and variable expected rates of return.

So, again, why use the risk-less interest rate rather than some other rate?

The class of securities that by definition earn the risk-free interest rate -- riskless securities -- place some constraints on the adjusted pricing probabilities: the price indicated, under whatever pricing algorithm we use, for a fixed amount paid under all possible outcomes must be consistent with the risk-free interest rate. If we try to do this using some other interest rate, we will have to have our adjusted probabilities sum to less than 1 if we use an interest rate lower than the risk-free interest rate, and we will have to have our adjusted probabilities sum to more than 1 if we use an interest rate higher than the risk-free rate. The adjusted probabilities would not be proper probability distribution functions, because a proper probability distribution function will sum to exactly 1. And, if a function that did not sum to 1 was used, it would create immediate arbitrage potential in either selling a payoff under all outcomes if the function sums to less than 1, or buying a payoff under all outcomes if the function sums to more than 1

So, it is ultimately simplicity that drives us to use a single interest rate and adjusted probabilities rather than variable expected rates of returns and actual expectations. Then, having limited ourselves in this way out of concern for simplicity, we are limited to using the risk-free interest rate as our single expected rate of return for the reason given above.

Finally, the reason this method is called "risk-neutral pricing" is that, using the adjusted pricing probabilities, all securities are assumed to have an equal expected rate of return, which is consistent with a risk-neutral market.