Forward Price Dynamics Under Physical and Risk-Neutral Measures

Summary

The document derives the dynamics of a stock forward when the underlying stock pays a constant dividend yield. Starting from a geometric Brownian motion for the stock, it states the forward price as the spot price adjusted by the financing and dividend yields over the contract's remaining life, then gives the resulting drift and volatility under the physical measure.

The answer confirms this expression when interest rates are deterministic and clarifies the measure change: forwards are martingales under the matching forward measure, while futures are martingales under the risk-neutral measure. With deterministic rates, forwards and futures coincide, so the forward is also driftless under the risk-neutral measure. The exchange is brief and does not show the Ito derivation or address stochastic rates, dividend uncertainty, or contract details beyond the stated constant-yield setup.

Key ideas

• A stock forward with constant dividend yield is spot multiplied by the financing and dividend adjustment through maturity.
• Under the physical measure, the forward's drift reflects the stock's expected return and the carry adjustment.
• For deterministic interest rates, forward and futures prices coincide.
• A forward price is a martingale under its corresponding forward measure, while a futures price is a martingale under the risk-neutral measure.

Tags

# Ito's lemma for a Forward


# Ito's lemma for a Forward












I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$ $dS_{t} = S_{t}(\mu dt + \sigma dW_{t})$ $\\$ and $r$ is interest rate.

Can someone verify that the Ito's process of $F$ is the following:$\\$

$F_{t} = S_{t}\exp{(r-y)(T-t)}\\$

$dF_{t} = F_{t}((y-r+\mu)dt + \sigma dW_{t})$

If the above is correct then how would this change under a risk neutral measure?

## Answer by river_rat (score 1)

https://quant.stackexchange.com/a/54116

Your equation looks ok. If interest rates are deterministic then forwards (being the same as futures) are driftless under the risk neutral measure. Otherwise, Forwards are driftless (i.e. martingales) under the corresponding forward measure while futures are martingales under the risk neutral measure.

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