# Implied Expected Stock Return from European Option Prices

We can calculate the expected stock return (under the measure $Q$) from at-the-money ($K=S_t$) option prices as:

$$E\left(\frac{S_T-S_t}{S_t}\right)=\frac{e^{rT}}{S_t}(C_t-P_t)$$

The result is mainly based on the fact that $$(S_T-S_t)^+-(S_t-S_T)^+=S_T-S_t$$ and $C_t=e^{-rT}E((S_T-K)^+)$.

I am looking for an expression of stock return using options with $K\neq S_t$.

My first approach was to equate the two sides and determine the difference:

$$(S_T-K)^+-(K-S_T)^+\stackrel{!}{=} S_T-S_t+\left[(S_T-S_t)^+-(S_t-S_T)^+-((S_T-K)^+-(K-S_T)^+)\right]$$

Maybe it would be possible to rearrange this term to get a sum of option payoffs plus a deterministic part (i.e. a bond).

Please let me know if you find a solution.

• Thank you, can you explain how you come from the first to second step, i.e. why exactly the term $+K-S_t$? – emcor Jun 17 '16 at 14:47
• @emcor: Do you mean this step: $S_T-S_t =S_T-K+K-S_t$? – Gordon Jun 17 '16 at 15:00
• +1 for the answer. @emcor, this is merely an application of call-put parity. Since $E_t((S_T-S_t)/S_t) = (F(t,T) - S_t)/S_t$ and by call-put parity the fair forward price is $F(t,T) = K + (C(t,T)-P(t,T))e^{rT}, \forall K$. Just plug it in and you're done. – Quantuple Jun 17 '16 at 19:24