Quantitative Finance Asked by user40929 on October 27, 2021
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
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.
Answered by Arshdeep on October 27, 2021
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.
Answered by dm63 on October 27, 2021
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