Delta hedging when volatility is stochastic

Question

From my understanding in a BSM world you can make a bet on volatility using options and delta hedging with the underlying.

If you think realized volatility of the underlying will be higher than the volatility implied by the option price you can buy the option, delta hedge with the underlying continuously and your PnL at the end is determined by the difference between IV & RV.

There's a pretty big assumption made in most of the papers I've read though - that volatility is constant. It's pretty clear that in reality volatility isn't a constant and can fluctuate over the course of the option duration.

My question is: how does a stochastic volatility affect the PnL when delta hedging?

Wilmott says in chapter 12.7 of his book: 'The argument that the final profit is guaranteed is not affected by having implied volatility stochastic, except insofar as you may get the opportunity to close the position early if implied volatility reaches the level of actual.'.

Can someone explain this to me? Seemingly he's saying that it doesn't actually matter whether volatility is constant or stochastic in order to profit from a difference in RV vs IV, but intuitively this doesn't make sense to me - if volatility is high in the earlier period (with more time to expiry and hence a lower gamma) and then low later when the gamma is higher, it seems that the final PnL would be affected significantly.

Answer 1

How to Delta Hedge and the PnL is indeed affect if real volatility is not constant (either deterministic ou stochastic). The exactly difference will depends on the model choose.

I don't recall this specific chapter in Wilmott book (are you talking about the condensed or full version? I may give it a check.), but notice something in what you said:

"(...)is not affected by having implied volatility stochastic(..)."

It means the final PnL is not affected by the option price in the market, but the assumption of knowing the average volatility in the period still holds. You already have the option (or shorted it) and you will delta hedge with the "true" volatility. The only thing with stochastic implied is that the PnL of closing the position could be equal or bigger than if you keep delta hedging until maturity.