I have a question about the following derivation in this pdf (sample chapter from Bergomi - Stochastic Volatility Modeling). He derives the PnL for a delta hedged position as

$$PnL = -[P(t+\delta,S+\delta S)-P(t,S)] + rP(t,S)\delta t + \Delta(\delta S - rS \delta t + q S\delta t)$$ where $r$ is the interest rate, $q$ the repo rate including dividend yield and $P(t,S)$ denotes the price of the option with underlying $S$ at time $t$. He chooses $\Delta = \frac{\partial P}{\partial S}$. Then he wants to expand the PnL in powers of $\delta S$ and $\delta t$ just looking at $\delta t, \delta S$ and $\delta S \delta S$ terms. His result is

$$ PnL = -(\frac{dP}{dt}- rP +(r-q)S \frac{dP}{dS})\delta t - \frac{1}{2}S^2\frac{d^2P}{dS^2}(\frac{\delta S}{S})^2$$

I'm not sure how he gets that one. If I expand $P$ in the terms mentioned above I find

$$P(t+\delta,S+\delta S) = P(t,S) + \frac{dP}{dt}\delta t + \frac{dP}{dS}\delta S + \frac{1}{2}*\frac{d^2P}{dS^2}(\delta S)^2$$

Doing this for the three terms of $P$ above I dont get the correct result. So how do you get his formula

  • $\begingroup$ Well there you go for the $P(t,S) - P(t+\delta,S+\delta S)$ par of the $PnL$ but now you have to add the remaining part (see your first equation), with $\Delta = \partial P/\partial S$. It's as easy as that $\endgroup$ – Quantuple Nov 18 '16 at 15:36
  • $\begingroup$ I'm not even sure this question should remain open actually. $\endgroup$ – Quantuple Nov 18 '16 at 15:53

$\require{cancel}$ $$\text{PnL} = -[P(t+\delta t,S+\delta S)-P(t,S)] + rP(t,S)\delta t + \Delta(\delta S - rS \delta t + q S\delta t)$$ Assuming a pure diffusion, at the order 1 as $\delta t \to 0$ $$P(t+\delta,S+\delta S) = P(t,S) + \frac{\partial P}{\partial t}\delta t + \frac{\partial P}{\partial S}\delta S + \frac{1}{2}\frac{\partial^2P}{\partial S^2}(\delta S)^2$$ Plugging that back in the first equation and using the identity $\Delta = \frac{\partial P}{\partial S}$ gives: \begin{align} \text{PnL} &= -\frac{\partial P}{\partial t}\delta t \cancel{- \frac{\partial P}{\partial S}\delta S} - \frac{1}{2} S^2 \frac{\partial^2P}{\partial S^2}\left(\frac{\delta S}{S}\right)^2 + rP\delta t + \cancel{\Delta\delta S} - (r - q) S \Delta \delta t \\ &= -\left( \frac{\partial P}{\partial t} - rP + (r - q) S \frac{\partial P}{\partial S} \right) \delta t + \frac{1}{2} S^2 \frac{\partial^2P}{\partial S^2}\left(\frac{\delta S}{S}\right)^2 \end{align}


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