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.