Dynamic Programming and the Markov View of Hedged Option P&L
Summary
The question concerns a proposed derivation of the Heston partial differential equation using the P&L of a short option position hedged with a quantity of the underlying. It asks why that P&L and hedge quantity can be represented as functions of time, spot price, and variance, and how conditional expected P&L and its variance could satisfy differential equations. The suggested framework treats the Heston state variables as a Markov process and invokes a dynamic programming principle to relate current value to its evolution over a short time step.
The only answer tentatively identifies the idea as Bellman's principle of optimality. It does not derive the equations, explain the hedge's dependence on the state, or verify the displayed discounting and time-step expression. As a result, this exchange offers a pointer to a general concept rather than a worked method for deriving a pricing PDE; the proposed interpretation remains uncertain.
Key ideas
- The question frames option hedging P&L as a function of time, spot price, and variance under a Heston model.
- A Markov state representation may motivate expressing conditional quantities in terms of those variables.
- Dynamic programming relates current values to expected future values over a short interval.
- The answer only tentatively links the proposed approach to Bellman's principle and supplies no derivation.
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# Strange use of dynamical programming principe
# 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.
## Answer by Bob Jansen (score 1)
https://quant.stackexchange.com/a/74069
I think it's hard for anyone that wasn't there to answer but if I had to guess I think he referred to Bellman's principle of optimality.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.