# 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?

• The forward price process is a martingale (no drift term). Because it is the ratio of two traded assets (stock divided by bank account).
– dm63
Jan 19 at 2:24
• The forward price is a martingale or the discounted forward price? Under which measure? Jan 19 at 22:24
• The forward price is a martingale under the measure associated with the zero coupon bond maturing on the delivery date of the forward.
– dm63
Jan 20 at 5:15
• I thought discretized price processes are martingales under $\mathbf{Q}$. But it turns out it is only true for spot processes? Not for forward? So spot processes are martingales under $\mathbf{Q}$, but the forward prices not necessary martingales? (I know there can be an other $\mathbf{Q}'$ where the forward price is martingale as well... Jan 22 at 13:05