An extended Stein's lemma for asset pricing

Abstract

Stein's lemma is extended to the case where asset returns have skewed and leptokurtic distributions. The risk premium is still the negative of the covariance of the excess return with the log stochastic discount factor. The risk-neutral distribution has a simple form but is a nontrivial transformation of the physical distribution.

Notes

¹Stein's lemma says that if x and y have a bivariate normal distribution and h(y) is a differentiable function such that E[|h′ (y)|] < ∞, then Cov[x, h(y)] = Cov(x, y)E[h′(y)]. Here we have x = Z, y = ln M and h(y) = exp(y) so h′(y)=M. Therefore, Cov(Z_i , M) = Cov(Z_i ,lnM)E(M). See, e.g. Cochrane (2001).

²More generally, for an asset with random payoff X, the current price is P = E(X) E(M) + Cov(X, ln M)E(M).