Why a Forward Price Is Not the Stock’s Expected Future Value

Summary

The document presents an attempted derivation of a stock forward price as an expectation under the maturity forward measure. The derivation changes measure to the risk-neutral measure, substitutes expressions involving a zero-coupon bond and the money market account, then assumes a constant interest rate to simplify the expectation. It ends with a value that includes a volatility adjustment and asks where the reasoning fails.

The question highlights the need to handle the Radon–Nikodym density and discounting consistently when moving between measures. Under deterministic rates, the forward price for a non-dividend-paying stock is tied to spot and financing, while a risk-neutral expectation of the future stock price is a different quantity. No response or correction is included, and the setup does not discuss dividends or other carry terms, so the derivation is an unresolved prompt rather than a complete pricing explanation.

Key ideas

• A forward price is expressed as an expectation under the maturity forward measure.
• Changing to the risk-neutral measure requires the correct measure density.
• A risk-neutral expected future stock price should not be confused with a forward price.
• The attempted derivation is unresolved and does not address dividends or other carry terms.

Tags

# What is incorrect with the following derivations of forward price?


# What is incorrect with the following derivations of forward price?












We understand that the forward price of a stock S is $K=E^{Q_T}[S_T]$ where $E^{Q_T}$ denotes expectation under the T-forward measure. I have the following derivation that produces incorrect results, but couldn't figure out what is wrong with it. Can anyone help me take a look?

$$\begin{equation} \begin{split} E^{Q_T}[S_T] &= E^{Q}[S_T \frac{P(T,T,T)/P(0,0,T)}{B(T)/B(0)}] \\ &=E^Q[S_T \frac{1}{B(T)P(0,0,T)}] \\ &=E^Q[S_T] \text{ (assuming constant interest rate)} \\ &=S_0e^{rT + 0.5\sigma^2T} \\ \end{split} \end{equation}$$ $$ \text{B(t) denotes money market account and P(t,S,T) denotes a zero coupon bond. } B(t)=e^{\int_0^t r(u)du} $$

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