Expected value of a function of a random variable

Typically, a random variable is just a risk factor that will affect some managerially more relevant outcome linked to cost or profit. This link may be represented by a function; hence, we are interested in functions of random variables. Given a random variable X and a function, like g(x) = x², or g(x) = max{x, 0}, we define a new random variable g(x). Not surprisingly, the expected value of a function of a random variable is

It is fundamental to really understand the expression in (6.5). We should consider each value x_i in the support, compute the corresponding value g(x_i) of the function, multiply the result by probability p_i, and add everything up. This is the expected value of the function, which is not the function of the expected value. We should not calculate E[X] and then evaluate function g(x) for x = E[X] since, in general

In other words, we cannot commute the two operators, i.e., the expect; E[·] and the function g(·), as the following counterexample shows.

Example 6.6 Let X be a discrete random variable with support {−1, +1}. Both values have probability 0.5, so

Now consider function g(x) = x². The expected value of g(X) is

This is definitely not the same as the function of the expected value:

There is a case in which we may commute expectation and function, as suggested by Property 6.6: If we have linear affine function h(x) = αx + β, then it is true that

but this is a very peculiar situation. The following example illustrates the point again in a more practically relevant setting.

Example 6.7 As we have seen in Section 3.1, a European-style call option is a financial contract giving you the right, but not the obligation, to purchase a given asset (e.g., a stock share) at a fixed price (called the strike price), at a given date (the maturity of the option). Note that the investor holding the option is free to choose if she wants to exercise the option at maturity or not.⁷ The other party of the contract, the option writer, is forced to sell the underlying asset if the option holder exercises the option. In contrast, the forward contracts we dealt with in Section 1.3.1 are more symmetric, since both sides of the contract are forced to carry out their obligations, i.e., to respectively buy and sell the underlying asset at the agreed forward price.

Say that we hold an option with strike price €40, written on a stock share whose price now is S₀ = €35, maturing in 5 months. The stock price in five months, S₅, is a random variable. If it turns out that S₅ > €40, then the holder can exercise the option and buy at 40€ the asset, which can then be sold for S₅, with a payoff S₅ − €40. More generally, if S_T is the underlying asset price at maturity and K is the strike price, the payoff for the option holder is

Note that the option payoff cannot be negative, since the holder will not exercise the option if the asset price is below the strike price.⁸

Real-life probability distributions of stock share prices are rather complicated, but let us assume that the distribution of is fairly well approximated by a set of eight equally likely scenarios:

What is the expected value of the option payoff at maturity? Since each value in this discrete support has probability , symmetry suggests that E[S₅] = 37.5. We see that g(37.5) = max{37.5 − 40,0} = 0, but this is not what we want. We should first calculate the option payoff in each scenario:

Then, the expected value of the option payoff is

For an arbitrary function g(·), we cannot say anything about the relationship between E[g(X)] and g(E[X]). A notable exception is a convex function.⁹

THEOREM 6.8 (Jensen’s inequality) If g(x) is a convex function, then

It is quite instructive to see a proof for the simple case of a support consisting of two values, x₁ and x₂, with probabilities p and 1 − p, respectively, since this sheds some light on the connection between discrete expectations and convex combinations. The expected value of X is

which is a linear combination of values x₁ and x₂, with nonnegative weights that add up to 1. Hence, we are just taking a convex combination, and this also applies to a support consisting of more than two points. Using convexity of the function, we see that