Expected Delta hedging frequency as function of implied (and realized) volatility

I'm looking for a proxy (or some rule of thumb) that can create a link between the implied volatility, the realized volatility and the frequency of Delta hedging required to keep the Delta as close as possible to zero.

For example, let me short a straddle: it's likely I will have to Delta hedge it more frequently than a short strangle with very wide legs. What's the (unconditional) probability of having to Delta hedge it?

Can the same be said for the implied volatility, that is, if the same short straddle is built in a low volatility environment then the expected frequency of Delta hedging is lower than the high volatility environment?

If we assume that the implied volatility is a good forecast for the realized one, my guess is that the expected frequency of Delta hedging is an increasing function of moneyness and volatility: to put it simply, if I short a straddle on stock $$X$$ with 80% volatility it's almost sure that I will have to Delta hedge it at least once; on the contrary, if I short a 90/110 strangle on $$X$$ with 5% volatility, it might happen that I won't need to Delta hedge it before expiration.

I'm not able to help myself with standard Black & Scholes theory because it assumes that one can Delta hedge frictionless, for infinitely small increments, and without transaction costs, while, in reality, things are very different.

• It doesn't help with frequency, but the amount of expected delta re-hedging (I.e. turnover) can be inferred from Leland's result. Every 1bp of transaction cost result in approx 13bps wider vol, if you multiply this by your option's vega you get an estimate of the total delta re-hedging through the life. I've found this result useful in my time. – James Spencer-Lavan May 23 at 9:18

1 Answer

No easy answers to your question, and as far as I know there are no straightforward rules of thumb.

Depending on whether you can trade other options to hedge your target option, how I would initially go about it is to to make your portfolio at least gamma neutral and vanna neutral. This should stabilize your net delta for 'reasonable' moves in spot and implied vol. But then you have your third order greeks to consider. In that case I would simply plot (or run some optimization) to find the best combination that also minimizes your most relevant third order greeks (i.e. change in gamma wrt spot, change in gamma wrt to vol, change in vanna wrt to spot and change in vanna wrt vol and so forth).

There are some papers on optimal delta hedging with transaction costs. Such as

Zakamouline, Optimal Hedging of Option Portfolios with Transaction Costs

• May I ask you to reformulate your advice as if you were long Gamma/Vega? I already know Zakamouline's approach, I've implemented and traded it in the past. However, it didn't outperform those much simpler criteria from seasoned traders. Here the problem is different: selecting underlying, strike prices and expiry date such that you will have the largest probability of scalping a bit of Gamma from it. – Lisa Ann Apr 24 at 16:53