Expected value of a call one day from now

Question

A call option is trading at price $c$ today. What do you expect its price to be tomorrow?

Short answer

More than $c$ . Under the real-world measure an option earns a positive expected return, higher than the stock's because it is levered: $\mu_c = r + \frac{S\Delta}{c}(\mu - r)$ , so tomorrow's expected price is about $c(1 + \mu_c/365)$ . Under the risk-neutral measure it would grow at $r$ . Theta being negative does not contradict this.

Adapted from Crack, Heard on the Street, Q2.59, p.32; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Market Model and Risk-Neutral Valuation
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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