Forward price vs. futures price - Wilmott

Question

I am reading Paul Wilmott's book PWOQF2, and there is something I don't get in his derivation of the convexity adjustment between forward and futures prices (chap. 30).

He models $S$ and $r$ following SDEs $$dS_t = \mu S_t dt +\sigma S_t dX_1$$

$$dr_t = u(r,t)dt + w(r,t) dX_2$$

$$d\langle X_1, X_2 \rangle_t = \rho dt$$

under the physical measure, the risk-neutral measure dynamics being the same up to the market price of risk term $\lambda$.

He shows the well-known result for forward price, i.e.

$$\text{Forward price} = \frac{S}{Z}$$ where $Z$ is the relevant zero coupon bond price. Until then everything's fine.

He then writes the futures price as $F(S, r, t) = \frac{S}{p(r,t)}$, where $p$ is some kind of discount factor. Following his usual routine, we get the pricing PDE for a derivative depending on $S$ and $r$: $$\frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 F}{\partial S^2}+\rho\sigma Sw\frac{\partial^2 F}{\partial S \partial r} + \frac{1}{2}w^2\frac{\partial^2 F}{\partial r^2} + rS\frac{\partial F}{\partial S} + \left( u - \lambda w \right)\frac{\partial F}{\partial r} = 0$$

And then he derives a PDE for $p$: $$\frac{\partial p}{\partial t} + \frac{1}{2}w^2\frac{\partial^2 p}{\partial r^2} + \left( u - \lambda w \right)\frac{\partial p}{\partial r} - rp \underline{-w^2\frac{\left(\frac{\partial p}{\partial r}\right)^2}{q} + \rho\sigma\beta\frac{\partial p}{\partial r}} = 0$$ commenting "Just plug the similarity form into the equation to see this".

My questions are :

1. What similarity form? And into which equation? (Not clear at all to me…)
2. Do you guys have any ideas where the $q$ and $\beta$ in the underlined terms (the famous convexity adjustment) come from? They never appear in the equations given at the beginning of the section (yes, the long summary was about that)

Thanks a lot for your help!

Answer 1

Modulo the two $\beta$ and $q$ errors, the proof is not that complicated actually. The similarity solution is simply $F(S, r, t) = \frac{S}{p(r, t)}$ and it has to be input into the previous pricing PDE. We thus replace : $$\frac{\partial F}{\partial t} = -\frac{S}{p^2}\frac{\partial p}{\partial t}$$ $$\frac{\partial F}{\partial S} = \frac{1}{p}$$ $$\frac{\partial^2 F}{\partial S^2} = 0$$ $$\frac{\partial F}{\partial r} = -\frac{S}{p^2}\frac{\partial p}{\partial r}$$ $$\frac{\partial^2 F}{\partial r^2} = -\frac{S}{p^4}\left[p^2 \frac{\partial^2 p}{\partial r^2}-2p\left( \frac{\partial p}{\partial r} \right)^2 \right] \equiv -\frac{S}{p^3}\left[p \frac{\partial^2 p}{\partial r^2}-2\left( \frac{\partial p}{\partial r} \right)^2 \right]$$ $$\frac{\partial^2 F}{\partial S \partial r} = -\frac{1}{p^2}\frac{\partial p}{\partial r}$$ $$\Rightarrow -\frac{S}{p^2}\frac{\partial p}{\partial t} - \frac{\rho\sigma Sw}{p^2}\frac{\partial p}{\partial r} - \frac{1}{2} w^2 \frac{S}{p^3}\left[p \frac{\partial^2 F}{\partial r^2}-2\left( \frac{\partial p}{\partial r} \right)^2 \right] + r\frac{S}{p} - (u - \lambda w)\frac{S}{p^2}\frac{\partial p}{\partial r} = 0$$ You multiply by $-\frac{p^2}{S}$ and we're done. $$\frac{\partial p}{\partial t} + \frac{1}{2} w^2 \frac{\partial^2 p}{\partial r^2} + (u - \lambda w) \frac{\partial p}{\partial r} - rp \underline{- \frac{w^2}{\color{red}{p}} \left( \frac{\partial p}{\partial r} \right)^2 + \rho\sigma \color{red}{w} \frac{\partial p}{\partial r}} = 0$$