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Time decay versus a call's positive drift

Question

A call option has a positive expected return, yet its theta (time decay) is negative. How can both be true?

Short answer

Theta holds the stock price fixed, but the stock moves. By Itô, E[dC]=(Θ+μSΔ+12σ2S2Γ) dt\mathbb{E}[dC] = (\Theta + \mu S\Delta + \tfrac12\sigma^2S^2\Gamma)\,dt. The Black-Scholes PDE gives Θ=rC−rSΔ−12σ2S2Γ\Theta = rC - rS\Delta - \tfrac12\sigma^2S^2\Gamma, so E[dC]=[rC+(μ−r)SΔ] dt>0\mathbb{E}[dC] = [rC + (\mu - r)S\Delta]\,dt > 0: the stock's drift and the convexity gain from Γ\Gamma more than offset the decay.

Adapted from Crack, Heard on the Street, Q2.60, p.32; question and answer rewritten by SelfMadeQuant.com.

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