Risk-Neutral Dynamics of Forward Prices and Discounting
Summary
The document asks whether a forward price under the risk-neutral measure follows the same drift as a spot price, and examines pricing a payoff tied to a forward maturing after the payoff date. It sets out a geometric Brownian motion model for spot and applies risk-neutral valuation to a spot-based payoff, then attempts the same calculation for a forward.
Its main lesson is that the calculation depends on the chosen numeraire and measure: a forward price for a fixed delivery date is generally a martingale under the relevant forward measure, while under the money-market risk-neutral measure its drift depends on rates and carry. The document raises the distinction but does not resolve it; in particular, it does not provide assumptions about interest rates, dividends, or storage costs needed to specify the forward dynamics. Treat its proposed drift and pricing expression as an open question, not a demonstrated result.
Key ideas
- A spot price under the risk-neutral measure has drift determined by the risk-free rate and carrying costs.
- A forward price’s drift depends on the pricing measure and the numeraire.
- A forward price for a fixed delivery date is a martingale under its associated forward measure.
- Pricing a payoff on a forward requires specifying interest rates and other carry assumptions.
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Full text
# What is the dynamic of the forward price process under $\mathbf{Q}$?
# What is the dynamic of the forward price process under $\mathbf{Q}$?
Let me define the Spot price process of an underlying as follows: $$dS_{t}=\mu_{S}S_{t}dt+\sigma_{S}S_{t}dW_{t},$$ where $\left(W_{t}\right)_{t\geq0}$ is an appropriate Wiener-process, so $\left(S_{t}\right)_{t\geq0}$ is a simple geometric Brownian motion.
Let me define the following payoff: $$X=S_{T}-K$$ In this case the price of $X$ is $$\begin{align*} Price\left(X\right) & =\mathbb{E}^{\mathbf{Q}}\left[e^{-rT}\left(S_{T}-K\right)\right]=e^{-rT}\mathbb{E}^{\mathbf{Q}}\left[S_{0}e^{\left(r-\frac{1}{2}\sigma_{S}^{2}\right)T+\sigma_{S}W_{T}^{\mathbf{Q}}}-K\right]=\\ & =\mathbb{E}^{\mathbf{Q}}\left[S_{0}e^{-\frac{1}{2}\sigma_{S}^{2}T+\sigma_{S}W_{T}^{\mathbf{Q}}}\right]-e^{-rT}K=S_{0}-e^{-rT}K, \end{align*}$$ where $\left(W_{t}^{\mathbf{Q}}\right)_{t\geq0}$ is a Wiener process under $\mathbf{Q}$, and I used the fact that $e^{-\frac{1}{2}\sigma_{S}^{2}T+\sigma_{S}W_{T}^{\mathbf{Q}}}$ is a martingale under $\mathbf{Q}$.
My question is why we can't use the same procedure for forward prices: $$dF\left(t,\tilde{T}\right)=\mu_{F}F\left(t,\tilde{T}\right)dt+\sigma_{F}F\left(t,\tilde{T}\right)dW'_{t},$$ where $\left(W'_{t}\right)_{t\geq0}$ is an appropriate Wiener-process, so $F\left(t,\tilde{T}\right)$ follows a simple geometric Brownian motion again.
Let me define the following payoff (for $T<\tilde{T}$): $$Y=F\left(T,\tilde{T}\right)-K$$ In this case I would say, the price of $Y$ is $$\begin{align*} Price\left(Y\right) & =\mathbb{E}^{\mathbf{Q}}\left[e^{-rT}\left(F\left(T,\tilde{T}\right)-K\right)\right]=e^{-rT}\mathbb{E}^{\mathbf{Q}}\left[F_{0}e^{\left(r-\frac{1}{2}\sigma_{F}^{2}\right)T+\sigma_{F}W_{T}^{\mathbf{Q}}}-K\right]=\\ & =\mathbb{E}^{\mathbf{Q}}\left[F_{0}e^{-\frac{1}{2}\sigma_{F}^{2}T+\sigma_{F}W_{T}^{\mathbf{Q}}}\right]-e^{-rT}K=F_{0}-e^{-rT}K, \end{align*}$$ but I'm not quite sure this is true. (I'm not sure this calculation is correct, since I'm not sure about I can simply change $\mu_{F}$ to $r$.)
So in short my question: is the forward price dynamic under $\mathbf{Q}$ looks like $$dF\left(t,\tilde{T}\right)=rF\left(t,\tilde{T}\right)dt+\sigma_{F}F\left(t,\tilde{T}\right)dW_{t}^{\mathbf{Q}}$$ just like in the case for Spot prices? If not, then why do they differ when basically everything is the same?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.