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In Black Scholes framework, assuming zero interest rates and realized volatility to be same as implied volatility, gamma pnl is exactly same and opposite of theta pnl. So if I buy an option and delta hedge then I make money on gamma but lose on theta and these two offset each other.

Then how do I recover option price from delta hedging i.e. shouldn't my pnl be equal to the option price paid?

Note: I realize if you hedge discretely rather than continuously there will be a hedging error, but please ignore this error for the purpose of this question.

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  • $\begingroup$ If you were to delta hedge continuously and on a costless basis, then your payoff at expiry would match that of a vanilla option. That is not the same as the pnl equalling the price paid, instead the expected pnl of the strategy would be the same as the option value. $\endgroup$
    – will
    May 15, 2019 at 21:30

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The theta PnL here is the option price paid (for the time-value of the option); it is just a greek word for it with an extra feature showing how the option premium continously declines with the passage of time. Say that you buy an out of the money option and then the market just dies. You then get noting but theta losses. They will add up to the premium you paid and lost.

On the other hand, the gamma PnL is paid to you on the side, not on the option premium, but from the trading activities in the underlying you carry out your hedging account.

With your assumptions above:

Lost premium from time decay = theta costs = gamma gains

so what you lose on premium payment you gain on your gamma trading account and you break even as you expect!

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An important assumption in BS is that you have to do instantaneous hedging, i.e, an infinitesimal move. In reality, you can't. Therefore you won't recover option price -- instead your price pnl minus the option price will be +- around zero.

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  • $\begingroup$ Thanks for helping, but does that mean theta pnl only partially offsets Gamma pnl and not fully even if implied vol = realized vol? Because assuming interest rates are zero, there is no other source of making money. $\endgroup$
    – InnocentR
    May 16, 2019 at 19:40
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If you perfectly hedge (infinitesimal moves), theta will offset gamma but if you do periodic hedges for finite moves, you would have gamma slippage and then you end up in a distribution of Pnl around zero.

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  • $\begingroup$ Assuming continuous hedging, if theta offsets gamma (implied = realized vol case) then how do we make money on the hedge? As it is the pnl of the hedge that offsets the option premium. Please ignore differences due to periodic vs continuous for this question. $\endgroup$
    – InnocentR
    May 16, 2019 at 19:54
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Isn't that exactly how you recover the option price? You paid the option price $C$, conduct delta-hedging continuously with initial capital $-C$ (zero cash to start with). At the end, the option expires and has payoff $X$, your hedging portfolio should then end up with $-X$. The pnl from you hedging portfolio is $-X-(-C)=C-X$ (in cash). You get the payoff $X$ from the option, so your final cash position will be $C-X+X=C$, and you have recovered the option price (certainly the net pnl is zero).

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