# Strange use of dynamical programming principe

I am in a finance seminar and yesterday evening we had a lecture from a quant in a big bank about shortcomings of Heston model.

He was deriving the Heston PDE. (I know how to derive the Heston PDE when you set up a portfolio with cash and two options etc, but I have a problem with this new method.) He took a portfolio (self-financed for sure but he did not mention it) where we sold an option and we delta-hedge it with a "functional" quantity $$\Delta$$.

Noting $$S$$ the underlying and $$V$$ the variance he noted $$U(t,S_t,V_t)$$ the P&L of that portfolio. (He did not mention why that P&L is necessarily a function of $$S$$ and $$V$$, I guess it has to do with the fact that the Heston model is a markovian model, but I don't succeed it proving that the P&L must have this form. Same question for $$\Delta$$ I guess.)

After that he defined $$m(t,S_t,V_t) = \mathbf{E}\left[ \left. U(t,S_t,V_t) \right| \mathscr{F}_t \right]$$ and $$W(t,S_t,V_t) = \mathbf{V}ar\left[ \left. U(t,S_t,V_t) \right| \mathscr{F}_t \right]$$ and he said that he was going to derive PDE's satisfied by $$m$$ and $$W$$ and that from these PDE's he will then derive a PDE satisfied from the option's price.

To derive the PDE for $$m$$ he wrote that we start to write the dynamical programming principle as follows : $$m(t,S_t,V_t) = \mathbf{E}\left[ \left. \left( U(t+dt,S_t + dS, V_t + dV) + \Delta \left( dS_t - rS_t dt \right) \right) e^{-rT} \right| \mathscr{F}_t \right].$$

I don't understand at all what he meant by that.

• It is possible to formulate the Dynamic Hedging problem as a Control Theory problem (that is dynamic optimization over time). The goal is to choose $\Delta$ to maximize the discounted P&L of the hedged position. As mentioned by Bob Jansen the equation above can be derived from Bellman's Principle. Dec 14, 2022 at 6:52