# Delta Hedging with fixed Implied Volatility or floating Implied Volatility?

When delta hedging an option until expiry at implied volatility, is it better to rehedge using the fixed implied volatility given by the option price upon its purchase (or sale), or to rehedge using the changing implied volatility given by the market price of the option?

What are the consequences of each option?

For example: On day 1 you buy an option at 20% implied volatility, thinking realised volatility will be higher. You use implied vol of 20% to calculate the delta of the option to hedge its sensitivity to movement in the underlying.

On day 2, the underlying has moved and the market now prices the option you bought with a 25% implied volatility. You look to rebalance your hedge. Do you use 20% or 25% implied volatility to calculate your delta? And Why?

Thank you!

Generally speaking, in the real world, you'd always want to use the correct implied vol. But you should think of your question in terms of:

(1) Vega mark-to-market (m2m) PnL vs. theta/gamma profile
(2) Change in risk and PnL due to higher order risks (vanna, volga)

Vega mark-to-market PnL vs. theta/gamma profile

In a simple, pure Black Scholes world implied volatility is of course constant. Your PnL, should you delta hedge an option to expiry in isolation, is dependent upon both the level and path of realised volatility. That's because the gamma of the option, i.e. the frequency/magnitude with which you need to delta hedge, is highest near the strike of the option, closer to expiry, and with a lower implied volatility. Hull has nice plots of these relationships.

Ask yourself: if you buy an option, delta hedge it to expiry, can you lose more money than your premium? (See bottom) This is a simple, intuitive question that people often get wrong because complex mathematics have supplanted their ability to use common sense and reasoning.

In your example, if you've bought your option at 20% and the market has repriced to 25%, you can either:

• raise the implied vol you're using for the option price/risk which will mean instant positive vega PnL, a change in delta on top of your gamma (dDelta/dVol, or vanna), but also therefore a higher theta bill and lower gamma going forward. you can lock in the vega PnL by selling the option back or a similar option.
• ignore the change in implied volatility if you intend to hold the option to maturity, which means the vega m2m PnL is somewhat irrelevant.

If it's your intention to hold and hedge the option to maturity, you can calculate the delta with any implied volatility you like. Broadly speaking, a higher vol (positive vega PnL upfront, higher theta bill, less gamma) would benefit you if realised changes in the underlying ended up being evenly distributed in time and magnitude, and a lower vol (negative vega PnL upfront, lower theta bill, more gamma) would ultimately benefit you if the realised changes were large and centered around the strike near expiry. You can convince yourself of this by running some simulations with delta hedging done using different implied vol levels.

Change in risk and PnL due to higher order risks (vanna, volga)

If you're using a more realistic stochastic vol model for a market making book or portfolio, your main concern is that (a) the model is arbitrage free and (b) your greeks for a larger group of trades is consistent.

If you are hedging one option within a portfolio and do not account for the vanna/volga contributions to your change in delta, that is you do not remark the implied vol of that one option, then your portfolio delta is inaccurate, i.e. you've effectively chosen not to mark your portfolio to market and have delayed taking higher order PnL and hedging the associated change in risk.

On market making and particularly exotic trading books, these higher order contributions to your delta drive a significant percentage of overall PnL, so using the correct implied volatility for your delta is as important as using the correct spot price for the underlying.

If you buy an option, delta hedge it to expiry, can you lose more money than your premium?

Yes! A simple example: What is your PnL if you buy a delta-hedged 3 month 37.00 (5% delta) call option on INTC for 0.05 (spot is 35). The stock trades sideways and then rallies instantly to 36.99 into expiry and then the call expires worthless. You've lost your premium, the option has expired worthless, and you've lost money on your delta hedge.

• Very interesting post. When you say "you can calculate the delta with any implied volatility you like", it seems that you are saying that in the BS model, if you hold the opton until maturity, the expected P&L at maturity is independent of the vol used to delta hedge, is that right?
– AFK
Oct 26, 2015 at 20:06
• Almost, in the model the expected P&L at maturity is very, very close with different vols used. The max/min, and standard deviation of profits will vary, though (higher vol used, higher max profit). Key practical differences are the P&L profiles of mark-to-market vs. mark-to-model until expiry, and of course the non-continuity of hedging and returns in the real world. A very nice, succinct paper is "Which Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios" by Ahmad and Wilmott. It has great detail on returns with different hedging vols. Oct 27, 2015 at 14:48