Consider a delta hedged option postion. Futhermore assume that I can perfectly forecast realized volatility over the life of the option.
Vol I buy the option at = Implied Vol (IV)
Realized volatility over the life of the option = Realized Vol (RV) Futhermore, suppose RV > IV
Now, there are 2 ways in which I can monetize RV being greater than IV.
Method 1-> I remark the vol of the option to RV (realize the pnl as vega PnL today).
Then, given that I am heding the option using the correct realized volatility, my cumulative delta hedging pnl at expiry will be known, and should perfectly offset my theta.
In this case, PnL realized = Vega x (RV-IV)
This pnl will be a linear function of RV.
Method 2-> I do not remark my vol, and delta hedge the option using the IV as the marked vol.
In this case, of course, my pnl will be path dependent, but the expected pnl would be =
0.5 x $Gamma x (RV-IV)
The gamma PnL, of course, is a quadratic function of RV.
My questions are ->
a. Is the pnl in method 1 = expected PnL in method 2 ?
b. If yes, how is the PnL in method 1 a linear function of RV, while the PnL in method 2 a quadratic function of RV.
Elaborating on question b->
A common heuristic seems to be.
I pay 100cents for a swaption with IV=2bp/day
If realized vol = 2.1bp/day, total PnL = 10c
If realized vol = 2.2bp/day, total PnL = 30c
So it's not a linear function of realized vol. But if I remark to RV and realized the PnL as a vega pnl, the pnl will be a linear function of RV (since an atm straddle is a linear function of volatility).