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 :
- What similarity form? And into which equation? (Not clear at all to me…)
- 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!