Hello, I am looking for some article or info regarding the impact in the hedging P&L of a vanilla option depending on the path the underlying took during the life of the option. I mean, if you buy a vanilla call at 20% vol and after that the stock moves with 30% vol during the life of the option, you will get a different P&L if the stock moves close to the strike than if it had moved very far form the strike (you could even lose money in this case).Can anybody help?Thanks

Hi Mok,I think "Dynamic Hedging" by Taleb treats this matter briefly. And Steve Allen's book, as far as I remember, discuss it more in detail.Hope it helps to some extend.

I carried out a few experiments on exactly this question. I've attached a few jpg's showing how the P&L is affected by misspecified hedge parameters. In all cases I used a call option with spot and strike = $100, risk-free-rate of 5%, a hedging vol of 20%, and time to maturity of 250 days with rebalancing every 5 days. The suffix on the figure indicates the realized vol, e.g. pl25 means a realized vol of 25%. Notice that the P&L becomes significantly skewed when the realized vol and hedging vol mismatch - but the skewness switches sign (as you may expect) as the underlier becomes more volatile relative to the hedging vol. One thing I find curious is even when the realized and hedging vol are identical the skewness is not identically zero - it's small but consistently negative (of course it does go to zero as the hedging frequency increases). Anyone have an explanation for this? - mayaro

Hm.. seems that my upload didn't work. Let's try this again...-mayaro

the bible for teaching vol trading to non-quants is "Buying and Selling Volatility" by Kevin B. Connolly. It discusses these issues from a practical standpoint. It's a really quick and easy read. Highly recomended for the begining trader or a quant who doesn't have experience tradng. Very basic but it answers your question really well. I've read some more techinical papers with lots of backtesting and stuff, one from lehman on a related ussue of how frequently one should hedge optimally, but in general I have heard that "delta hedging is more an art than a science". If you knew before hand where exactly a stock was going to go there are easier ways to make money than delta hedging options!

In practice, the first question is whether you're dynamically hedging a long or a short option position. They are very dissimilar in nature. Short vol trader has to over-hedge in the direction of his risk or be forced to trade with stops. Long vol traders can under-hedge in trends and use limits during mean-reverting situations. There is no "theoretically optimal" way to delta hedge. It is all about risk management and market dynamics. Good vol traders have developed an intuitive feel for when to capture gamma and when to let it run. Hope this helps.

QuoteOriginally posted by: MokHello, I am looking for some article or info regarding the impact in the hedging P&L of a vanilla option depending on the path the underlying took during the life of the option. I mean, if you buy a vanilla call at 20% vol and after that the stock moves with 30% vol during the life of the option, you will get a different P&L if the stock moves close to the strike than if it had moved very far form the strike (you could even lose money in this case).Can anybody help?ThanksHye Mok,If you hedge a short call at times t_i, when S=S_i, thenP&L = Sum_i ( 1/2*Gamma(S_i,t_i)*S_i^2*(vol_r(t_i)^2-vol_implied^2)) (I assume r=q=0)This means that the cost of your hedge will increase if the realized vol is higher than implied vol, but the impact will be weighted by Gamma*S^2. This means you could lose money even if average realized vol is lower to implied vol: this could happen if the stock is very volatile when you're very gamma short (close to the strike) and then goes to a zone which is more gamma flat and is not very volatile.If you ever come accross a good article, tell us.