Stein's Lemma

Suppose $X$ is normally distributed and $g$ is some reasonably smooth function of $X$ — a payoff, a portfolio weight rule, anything. You often want $\text{Cov}(X, g(X))$, and computing it directly means grinding through an integral against the normal density every time. Stein's Lemma skips that grind: if $X \sim N(\mu, \sigma^2)$, then

$$\text{Cov}(X, g(X)) = \sigma^2 \, E[g'(X)] .$$

In plain English: the covariance between a normal variable and any smooth function of it equals the variance of the variable times the average slope of the function. A steeper function on average produces a bigger covariance; a flat function produces none, no matter how wild $X$ is.

Analogy. Think of $X$ as a lever position and $g(X)$ as a dial reading tied to that lever. How much the dial reading moves together with the lever depends only on how steeply the dial is geared at each position, averaged over where the lever tends to sit — not on the dial's shape in any more complicated sense.

Worked example

Let $X$ be a stock's log-return, $X \sim N(0, \sigma^2)$ with $\sigma^2 = 0.04$, and let $g(X) = X^2$ (a proxy for realized variance exposure). Then $g'(X) = 2X$, so $E[g'(X)] = 2E[X] = 0$, giving $\text{Cov}(X, X^2) = 0$. That matches intuition: a symmetric normal variable's sign carries no information about the magnitude of its square. Swap in $g(X) = e^X$ (a simple payoff), and $g'(X) = e^X$, so $\text{Cov}(X, e^X) = \sigma^2 E[e^X] = \sigma^2 e^{\sigma^2/2}$ — a closed form with no direct integration needed.

Where it shows up

Stein's Lemma is the quiet reason CAPM's beta, $\beta = \text{Cov}(R_i, R_m)/\text{Var}(R_m)$, has a clean interpretation when returns are normal: it reduces beta to an average sensitivity of the asset's payoff to the market, rather than a raw covariance number. The same identity underlies several option-Greek relationships, since it turns covariances involving a normal underlying into expectations of derivatives.