P&L of Delta Hedged Option Position

QGenius · August 30th, 2005, 5:06 pm

The following theoretical approximation to the daily P&L of a delta-hedged-option-position is well known. Could any of the experienced traders around here confirm whether it is really accurate in practice as well:P&L for [t,t+1] = 1/2 * Gamma * S(t)^2 * { [(S(t+1)-S(t))/S(t)]^2 - [ImpliedVol(t)^2]/225 }Can anyone provide some information on how well this approximation works in the real world ? Thx.

Blazes · August 31st, 2005, 11:15 am

Should be pretty easy to test I would think. Just look at historic prices for single stock or index options and the underlying and you should have all the information you need (easily available on Bloomberg). One problem I suppose is that trader may rebalance more than once a day. When I traded options I re-balanced more or less frequently depending on the product and a range of other things that were not scientific!

MForde · August 31st, 2005, 12:18 pm

If u compute the effective local vol surface in the Dupire 96 sense initially, + then dynamically delta-hedge the call to maturity under the associated local vol modelthe P&L (if S is a process for which we can apply Ito's lemma) isP&L = int 1/2 S_t^2 Gamma_t (sigma_t^2 - E(sigma_t^2 | S_t) dt, t=0..Twhere Gamma_t is the gamma under the local vol modelThe expectation of the integrand at any point t, given the filtration at time zero, is zero even when we condition on S_t, which could be kryptonite to the argument that delta-hedging under local vol models is bad

QGenius · September 1st, 2005, 4:08 pm

Thanks for the replies guys. Actually I was able to find an answer to what was bothering me in a paper of Carr and Madan.

wstguru · September 2nd, 2005, 12:41 pm

QuoteOriginally posted by: QGeniusThanks for the replies guys. Actually I was able to find an answer to what was bothering me in a paper of Carr and Madan.what was it then? am interested

DogonMatrix · September 2nd, 2005, 5:53 pm

QuoteOriginally posted by: QGeniusThe following theoretical approximation to the daily P&L of a delta-hedged-option-position is well known. Could any of the experienced traders around here confirm whether it is really accurate in practice as well:P&L for [t,t+1] = 1/2 * Gamma * S(t)^2 * { [(S(t+1)-S(t))/S(t)]^2 - [ImpliedVol(t)^2]/225 }Can anyone provide some information on how well this approximation works in the real world ? Thx.I thinl actually that the formual is wrong if the assumption of stochastic volatility is true (for which there is also a closed form formula for the dHedge PL). In other words if you believe that Black Scholes describes fairly well the trading world where you are 'leaving' in then it is right, other wise it is grossly wrong. But It's tautology since that's probably true of every thing that has to do with options: the discrepancy between the real world and the formula will have to do with how bad the assumptions you made are.

QGenius · September 2nd, 2005, 7:24 pm

I did make one error : instead of ImpliedVol(t) I should have put HedgeVol.Dogon Matrix : It turns out that this is in fact a precise formula , even in the case of stochastic volatility! The KEY is exactly the previous correction I mentioned.

zhongx · September 2nd, 2005, 8:43 pm

sorry, what is the definition of P&L? (the value chage of the delta-hedged position?)where I can find the reference of the theoretical approximation you gave in the first place?

spacemonkey · September 2nd, 2005, 8:55 pm

This is not a precise formula, not for the Black-Scholes case nor for the stochastic volatility case. It is the first order term in an infinite expansion of the true error. In the SV case, there should be an extra term corresponding to the discrete time hedging error from the options, or whatever you are using to hedge the PL from the volatility. If you aren't hedging the volatility component then the error from that will dominate the expansion, and then the formula is useless. If you hold the option and a perfect delta-hedge then there is no PL. The formula is an approximation to the PL that you will get from hedging in discrete time instead of continuous time, ignoring transaction costs. Which paper from Carr and Madan?

DogonMatrix · September 3rd, 2005, 1:36 am

That's right. I responded too fast.

QGenius · September 3rd, 2005, 8:05 am

The paper I mentioned was Carr,Madan(2002) Towards a Theory of Volatility Trading.For those still interested , let me clear any confusion and start from a CLEAN slate :1. Assume a stochastic volatility process : any non-ridiculous dynamics2. Say, at time t=0 you sell a European Option for a certain implied vol hereafter called hedge vol.3. Suppose that irrespective of what is happening to the volatility in the market you delta hedge your position according to Black and Scholes Model throughout the life of the option and always based on the original hedge volThe result at time t=T will be a P&L or Hedge Error given by : Integral from t=0 to t=T of 1/2 * Dollar Gamma * (hedge vol ^ 2 - instantaneous vol ^ 2) * dtNote : You can find a proof in the paper mentioned.This is what I was trying to say but I was a bit sloppy in the beginning...

Paul · September 3rd, 2005, 11:02 am

Some further thoughts on hedging with different vols attached.The theoretical profit turns out to be less in practice because of transaction costs. But most importantly, hedging error is usually so large (because you have to hedge discretely) that the profit is often swamped.P

zhongx · September 4th, 2005, 3:30 am

Isn't the PnL is always negative and equal to the option price C(0) assuming a constant vol which is also used for delta, according to Hull's book?

QGenius · September 4th, 2005, 6:40 am

zhongx: check out Paul's lecture above. It has many examples on this issue.

DogonMatrix · September 6th, 2005, 12:25 pm

QuoteOriginally posted by: DogonMatrixThat's right. I responded too fast.Actually let me correct myself. QuantGenius formula is not correct for stochastic vol. One clue: where is the vol convexity term ? None of the parameter of the stochastic vol appear here ( drift of vol or vol of vol).