Characterization of Option Prices as Gamma Gains

The question then is, how does a trader "characterize" an option using these hedging gains? First of all, in liquid option markets the order flow determines the price and the trader does not have to go through a pricing exercise. But still, can we use these trading gains to represent the frame of mind of an options trader?

The discussion in the previous section provides a hint about this issue. The trader buys or sells an option with strike price K. The cash needed for this transaction is either borrowed or lent. Then the trader delta hedges the option. Finally, this hedge is adjusted as the underlying price fluctuates around the initial St0 .

According to this, the trader could add the (discounted) future gains (payouts) from these hedge adjustments and this would be the true time-value of the option, besides interest or other expenses. The critical point is that these future gains need to be calculated at the initial gamma, evaluated at the initial St0, and adjusted for passing time.

We can explain this statement. First, for simplicity assume interest rates are equal to zero. We then let the price of the vanilla call be denoted by C (St,t). Then by definition we have

This will be the future value of the option if the underlying ended up at the St0 at time T. Now, this value is equal to the initial price plus how much the time value has changed between t0 and T,

J to

Now, we know from the Black-Scholes partial differential equation that

Substituting and reorganizing equation (42) above becomes

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