How Stochastic Volatility Affects Delta-Hedged Option P&L
Summary
The document asks whether stochastic implied volatility changes the profit from buying an option and delta hedging when realized volatility is expected to exceed implied volatility. It highlights the role of gamma over the option’s life: volatility realized when gamma is higher may contribute differently to hedge P&L than volatility realized when gamma is lower. The answer distinguishes the market’s changing implied volatility from the volatility governing the underlying’s realized path. It says stochastic implied volatility does not itself change the final P&L argument, provided the hedge uses the true volatility and the assumed average volatility over the holding period is known.
The answer also cautions that nonconstant realized volatility does affect delta-hedging P&L, with the precise effect depending on the chosen model. A changing implied volatility can offer an opportunity to close the position early, potentially for a result as good as or better than continuing to hedge through expiry. The exchange gives no derivation or numerical example, and its claims rely on assumptions about knowing the relevant volatility and hedging behavior.
Key ideas
- Stochastic implied volatility and stochastic realized volatility are distinct issues for delta-hedged option P&L.
- Nonconstant realized volatility affects hedge P&L, and the effect depends on the model.
- The answer assumes the hedger knows the true volatility and average volatility over the holding period.
- A favorable implied-volatility move may create an opportunity to close before expiry.
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# Delta hedging when volatility is stochastic # Delta hedging when volatility is stochastic 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 by André Bittencourt (score 1) https://quant.stackexchange.com/a/71693 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.
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