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Theta - the cost of Gamma

The positive convexity from the gamma must be paid for, and it is paid through the time decay or theta.

Theta can be thought of as the amortized (non-linear) cost of the option, spread over its lifetime.

To make money over a single delta-hedging step --> necessary to make more on the gamma than is lost on the theta. 



Once the delta is hedged, on option trader is left with three main risks: Gamma, theta and vega. 

Daily Delta-neutral P&L = Gamma P&L + Theta P&L + Vega P&L + Other

Other: P&L from financing reverse delta position, P&L due to changes in IR, dividend expectations, high-order sensibilities. 


P l


Assuming zero IR, constant IV and negligible high-order sensibilities: P l1 

In our zero IR world, theta can be re-expressed in Gamma: Thetagamma  --> Heart of BS analysis

Plugging this into the previous equation --> P l2

First term in bracket = one-day stock return --> Squared, it can be interpreted as realized one-day variance.

Second term in bracket = squared daily IV --> daily implied variance.

Factor in front of bracket --> Dollar Gamma. 


Daily P&L of a delta-hedged position is driven by difference between RV and IV, multiplied by the Dollar Gamma. 

If delta-hedging daily, the strategy will profit if daily RV is greater than the daily IV. 


It can easily be extended to give P&L from delta-hedging over the lifetime of the option:

P l over lifetime


The only reason the exposure to volatility is non-constant is the changing Gamma.  

This equation is close to the payoff of a variance swap. 

It is a weighted sum of squared realized returns minus a constant that has the same role as a strike.

But in a variance swap the weights are constant, while here the weights depend on the option gamma through time.

This explains an option trading phenomenon known as path-dependency


Vega and other sensitivities

In reality, IV of the option will change throughout the lifetime of the option. 

This will affect the option P&L both directly and indirectly: 

- directly: volatility goes up --> option becomes more valuable

- indirectly: volatility moves --> hedging quantities dictated by the delta will also change 


Obviously if holding an option to maturity and delta-hedging at original IV: 

- vega P&L over lifetime of the option will be zero. 

- total P&L will be the gamma P&L vs theta P&L as described above. 


In practice since the changes in IV alter the hedging amounts --> actual effects more difficult to quantify.


Note that option prices are also sensitive to changes in IR (rho) and in dividends (mu). 





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