Why Expected Hedging P&L Is Independent of the Hedge Volatility
Summary
The discussion examines a result from an option-hedging paper: although realized-volatility and implied-volatility hedges can produce different pathwise profit-and-loss expressions, the expected total P&L is argued not to depend on the hedge choice. One answer gives an intuitive explanation: rebalancing trades occur at market prices, so those transactions do not change expected P&L at the time they are made.
A second answer sketches a risk-neutral derivation. It identifies a gamma-related term in the P&L and says the remaining drift and Brownian-motion terms combine into a martingale under a measure change, giving them zero expectation; the resulting expected contribution matches the expression in the alternative hedge case. This is a compact forum explanation, not a full proof. It relies on the paper’s setup and risk-neutral assumptions, and does not spell out the conditions needed for the measure change, integrability, or equality of present-value expressions in general.
Key ideas
- The choice of hedge volatility can change realized P&L paths without changing expected P&L under the stated setup.
- The discussion interprets rebalancing trades at market prices as having no expected value at execution.
- A risk-neutral measure change is used to treat drift and Brownian terms together as a zero-mean contribution.
- The argument leaves technical assumptions and a full derivation unstated.
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# Hedging with different volatility (Ahmad and Wilmott paper) # 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 ## Answer by dm63 (score 5) https://quant.stackexchange.com/a/45680 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. ## Answer by Arshdeep (score 2) https://quant.stackexchange.com/a/55790 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.
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