One of the assumptions of the Black-Scholes model is that the volatility is constant for all strikes and maturities. The reality is that the implied volatility surface is not flat:

This has a significant implication on Greeks. For example, the true theta of an OTM put option is lower than estimated by the Black Scholes model. The reason is that over time the implied volatility of the out of the money put option is going up - the smile is getting bigger.

For example the SPY 2012 Dec 60 Put has an IV of 33.7%, while the 2011 Dec 60 Put has an IV of 35.4%. So over 1 year the IV goes up by 1.7%. The Black-Scholes model says that the theta is 0.0038. If we adjust for the increase in IV over time the theta drops to 0.0028 - a 30% reduction in theta.

On the other hand the IV of an ATM option is going down over time (at least based on the current IV surface), so the theta is underestimated.

To make things even more complicated, the IV surface is not static, it is changing all the time.

## Friday, April 23, 2010

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