# 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 ?

• The paper referenced by @julien in one of the answers will indeed be useful. Note that if you choose to hedge with a fixed IV your prices won't be marked to the market anymore, which may lead to some problems with your risk department. – Quantuple Jun 9 '17 at 8:57
• I second the recommendation by @julien. See also Question IX in Carr's "FAQs in Option Pricing Theory" as well as this related question: quant.stackexchange.com/questions/33371. – LocalVolatility Jun 9 '17 at 12:10
• There is no such thing as "getting rid of Vega" if you (dynamically) trade only the underlying AFAIK, you would need some other vol sensitive asset... – noob2 Jun 9 '17 at 15:59
• I don't mean vega hedge my portfolio, my formula is inexact. I mean having a PNL at maturity that is not IV path dependent. If i delta hedge with floating IV, PNL at maturity is IV path depedent, while it's not if i use a constant IV over the life of the option. – Clement Jun 9 '17 at 18:06

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.

• Thx for the paper, i read it but there are a few things I don't understand: 1/ Page 67, beginning of the page, the author is detailing the decomposition of the pnl through sensitivities : Basically he writes : $$dV^i = \Theta^i dt + \Delta^i dS + 0.5 \sigma^2 S^2 \Gamma^i dt$$ Actually i would decompose $dV^i$ as follows : $$dV^i = \Theta^i dt + \Delta^i dS + 0.5 \sigma^2 S^2 \Gamma^i dt + vega * d \sigma^i$$ What is wrong with my reasonning ? Even if we are using a constant volatility, price of market is moving and implied volatility changes should impact prices variations. – Clement Jun 13 '17 at 22:48
• 2/ Page 67, case 2 : I feel like the author is going from $dV^i = 0.5 (\sigma^2-\sigma*^2) S^2 \Gamma^i dt$ to $0.5 (\sigma^2-\sigma*^2) \int_{t0}^{t} e^{-r(t-t0)}S^2 \Gamma^i dt$ just with actualization and integration but implied volatility $\sigma*$ is changing after every rebalancing, he cannot integrate it like a constant ? Maybe the point i'm missing here is that the author is supposing implied volatility constant but that's a big assumption that is a key in my understanding of this problem. My question is more about floating IV vs fixed IV than which fixed IV to use. – Clement Jun 13 '17 at 22:59
• Haven't read the paper in a while but iirc, volatility is assumed constant. Which means dsigma = 0. The price is moving due to passage of time, change of underlying but that can still happen with constant implied volatility. I think it also covers your second comment: Implied vol is considered constant. This paper basically tries to measure the impact of using one fixed IV (the one you bought the option at) or another (your expectation of future realized). – julien Jun 20 '17 at 0:04

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.

• Expected (under the risk neutral measure) PnL is always the same whether you hedge or not: this is because the (discounted) hedging portfolio is a martingale. As you pointed what will depend on the hedging strategy is the PnL distribution. – Antoine Conze Jun 9 '17 at 7:49