Implied Expected Stock Return from Options at Any Strike

Summary

The document derives a risk-neutral expected simple stock return from European call and put prices with a shared strike that need not equal the current stock price. The key step is to express the terminal stock price change as the difference between call and put payoffs at strike K, plus the deterministic difference between K and the current spot price. Taking the discounted-payoff expectations then yields an option-price term and a strike adjustment, both scaled by the current stock price.

This extends the at-the-money expression by showing how to account for a non-at-the-money strike. The derivation assumes European option payoff pricing under the risk-neutral measure, with the stated discounting convention and conditional expectations. It describes an implied expected return under that measure; it does not establish a forecast of the stock’s realized return under the real-world probability measure.

Key ideas

• A call-minus-put payoff at a common strike equals the terminal stock price minus that strike.
• For a strike away from spot, the expected return formula needs an adjustment equal to the strike-minus-spot difference divided by spot.
• The call and put prices contribute through their price difference, scaled by discounting and current spot.
• The resulting expectation is risk-neutral and should not be confused with a real-world return forecast.

Tags

# Implied Expected Stock Return from European Option Prices


# 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.

## Answer by Gordon (score 3, accepted)

https://quant.stackexchange.com/a/27669

Note that \begin{align*} \frac{S_T-S_t}{S_t} &= \frac{S_T-K +K-S_t}{S_t}\\ &=\frac{(S_T-K)^+-(K-S_T)^+ +K-S_t}{S_t}. \end{align*} Then, \begin{align*} E\left(\frac{S_T-S_t}{S_t} \mid \mathcal{F}_t \right) &= \frac{e^{rT}}{S_t}(C_t-P_t)+ \frac{K-S_t}{S_t}. \end{align*} where \begin{align*} C_t &= e^{-rT} E\left((S_T-K)^+ \mid \mathcal{F}_t \right),\\ P_t &= e^{-rT} E\left((K-S_T)^+ \mid \mathcal{F}_t \right). \end{align*}

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.