How Hedge Volatility Choices Affect Delta-Hedged Option P&L

# Delta Hedging with fixed Implied Volatility to get rid of vega?


# Delta Hedging with fixed Implied Volatility to get rid of vega?












I'm wondering if i should use a floating IV or a fixed IV to delta hedge my options every day.

I've read this post but would like different information : Delta Hedging with fixed Implied Volatility or floating Implied Volatility?

I have a view on future realized volatility for an option $\sigma_{e}$ (e for expected). When i look at an option, if its implied volatility $\sigma_{i0}$ is lower than $\sigma_{e}$ then i want to buy the option and delta hedge at a certain frequency (daily for example).

I've always delta hedged using a floating IV which is changing daily but i realized this may not best thing to do.

If we decompose the variation of a PNL between 2 hedging periods we have :

$$ dPNL = \vartheta * d\sigma + \theta*dt + 0.5 * dS^2*\Gamma $$

I've considered as well known that the daily PNL of a delta hedged between option is given by : $$ dPNL = 0.5 *(\sigma_{i}^2-RV^2)\Gamma * S * dt $$ with $RV$ = realized volatility = $ ds/S $

My concern with this is that i have the feeling that we are only looking at $\Gamma$ and $\theta$ influences in the case we are delta hedging using floating IV every day.

Looking at this formula, PNL at maturity should not be IV path dependant, however running simulations i get to very different PNL at maturity using a floating IV. So is this formula only true when using a fixed IV to hedge daily ?

My intuition is the following : If we want to get rid of the vega effect then we need to replicate the same option with a constant IV so that the vega does not have any effect on PNL.

But does this mean that we can use any IV to hedge the option ? This makes no sense to me ? I would like to have a mathematical proof of this coming from the Greek PNL decomposition above if possible ? What IV should i use in my situation ?

## Answer by julien (score 5)

https://quant.stackexchange.com/a/34620

You should have a look at the following paper:

Ahmad, Riaz and Paul Wilmott (2005) "Which free lunch would you like today, Sir? Delta hedging, volatility arbitrage and optimal portfolios," Wilmott Magazine, Nov. 2005, pp. 64—79

which tackles this exact issue.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/34619

My intuition is that your expected p/l from delta hedging is the same, regardless of what vol you use at each step (since this just changes your series of spot transactions, each of which has zero expected value). However if you hedge at a vol much lower than the IV, or a vol which is much higher than the IV, your eventual p/l will be more risky than just using the IV at each step. Meaning, it will have a wider distribution. For example, using a zero vol to hedge would result in no delta hedging p/l on any day except the days you cross the strike, which is highly path dependent. I'm not sure how to prove that explicitly.