Pricing the Passport option

Question

Suppose underlying asset $S$ $$dS = \mu Sdt + \sigma Sd W$$ our portfolio $\pi$ consist with $q(t)$ stock $S$ and cash $\pi - qS$ at time $t,$ so we have $$d\pi = r(\pi− qS) dt + q dS.$$ with $|q|\leq 1.$

And assume the final payment of our passport option is $$V(\pi,s, T) = \max\{\pi,0\}.$$ Use the hedge portfolio, we can obtain the PDE $$V_t + \dfrac{1}{2}\sigma^2s^2 V_{ss} + q\sigma^2s^2V_{s\pi} + \dfrac{1}{2}q^2\sigma^2s^2V_{\pi\pi} + rsV_s + r\pi V_{\pi} -rV = 0.$$

We want to choose $q(t)$ to maximize $V.$

One thing I confuse here, why does author only easily maximize the terms containing $q$ in the PDE? i.e $$\max\limits_{|q|\leq 1}\ \left(q\sigma^2s^2V_{s\pi} + \dfrac{1}{2}q^2\sigma^2s^2V_{\pi\pi}\right) $$

That is the book Paul Wilmott on Quantitative Finance page 455

Answer 1

You maximize the terms in $q$ in the PDE because this is a consequence of the Bellman principle of optimality in dynamic programming. The intuition is that the global optimal strategy $\{q_t\}_{0 \leq t \leq T}$ is locally optimal such that (under the risk neutral measure because the option is dynamically hedged) $$V_t = \max_{|q_t|\leq 1}e^{-r dt}E_t\left[V_{t+dt} \right]$$ that is $$ V(\pi, S, t) = \max_{|q|\leq 1}e^{-r dt} E_t\left[V(\pi+ d\pi, S+dS, t+dt) \right] $$ along with the stochastic dynamics (under the risk neutral measure) $$ dS = rS dt + \sigma S dW $$ $$ d\pi=r(\pi - qS) dt + qdS $$ and the terminal condition $V(\pi, S, T) =\max(\pi, 0)$. In words, you start from the end and go back in time, finding the optimal strategy at each time step conditional on the state you're in.

You then apply Ito's lemma to $V(\pi+ d\pi, S+dS, t+dt)$ to obtain the author's result.