# Hedging with different volatility (Ahmad and Wilmott paper)

In their paper they show that: - if you hedge with the realised volatility, the present value of the total p&l is the difference between the option value based on the realised volatility and the option value based on the implied volatility (this makes total sense) - if you hedge with the implied vol, the present value of the total p&l is equal to 1/2 times the integral of the gamma cash times the difference between the square of each volatility (this also makes sense)

They also comment that the expectation of the total p&l does not depend on the volatility. How do you prove that? does it mean that the 2 present value calculated in the 2 cases are equal? can we prove that mathematically? to compute the expectation in the second case, they derive a pde but dont give a closed form solution. should not it be equal to the difference of the black-scholes prices calculated with the 2 volatilities? There is something i dont understand. thank you for your help.

Chris

The choice of hedging strategy cannot affect the expected p/l, because hedging just consists of doing at-market purchases or sales of the underlying, each of which have zero expected value at the time of transacting.

• thank you for your answer but i am not sure to understand. can you elaborate more? May 18 '19 at 15:40
• I'm not sure how to simplify further. Perhaps I could point out that the purpose of hedging is to reduce the variance (unpredictability) of the overall p/l. This happens when the model being assumed gets as close to market behavior as possible. Thus if you hedge an option by inputting a volatility not representative of the marketplace, you will have greater variance of p/l. None of these choices affect the average p/l. As an extreme example, if you leave an option completely unhedged, your expected p/l is the initial premium, but it won't be very predictable.
– dm63
May 19 '19 at 12:17
• thank you. this is clear. however i still have a question regarding the formula for the expectation of rhe pnl. Let us asssume that the realised vol is different and greater than the implied vol. if we hedge with the realised vol the expected pnl is equal to the difference between the black-scholes price using the realised vol and the BS price using the implied vol. if we hedge with the implied vol, the expected val is the integral of the difference of the square of rhe 2 vol times S^2 times Gamma time some disconted factor. How do we prove that the expected pnl are the same in the 2 cases? May 19 '19 at 13:47

As the other answer says, expected PnL does not depend on hedging portfolio, so you can hedge with whatever vol, expected PnL is the same.

In this particular case, you can simply observe that in the paper, the PnL for the case of hedging with actual vol has the gamma term, and the other two terms combine to form a brownian motion under the risk neutral measure (Girsanov's theorem), so the expectation of the 2nd and 3rd term are 0 in the risk neutral measure.

$$d(PnL)= Gamma term + (X(u-r)/s)dt+ XdW$$, for some $$X$$. $$u$$ is the drift, $$r$$ is the risk free rate, $$s$$ is the vol. $$W$$ is brownian motion in the real world. By girsanov, expectation of 2nd and 3rd term combined is 0 under the risk neutral measure.

You are again left with the gamma term, same as the second case.