Option Pricing - Theta

The theta measures the change in the option price with respect to a change in time.

               change in the option price

Theta =    ----------------------------

               change in time

The theta value is a measure of the effect time has on the option price. The value of the option premium should decrease with lime when all other variables are held constant, so the theta is always negative. However, this change is not linear but becomes much greater as the option nears expiration. The theta is greatest near expiration and becomes less pronounced the further the option is to expiration. For example, if the future and volatility do not change, the time decay in one day will be much greater for an option with one day versus one year to expiration. The change in price of the option with time is also partly a function of whether the option is at the money or away from it.

The theta is related to the gamma because positions with a positive gamma will have a negative theta. This simply means the price of the option will erode due to time decay. The position will incur a loss if there is no significant movement in time. For example, a trader holding a positive gamma position such as a straddle will lose money with time if the market does not move. The premium of the options will decay with time. However, the trader with a negative gamma such as short the straddle will make money if the market does not move.

A positive gamma is always offset by negative theta. A negative gamma is helped by time decay. The option trader must always realize there is a constant trade-off between being long options and watching them decay, or being short options and concerned about a significant price move. It is always necessary to understand the relationship of gamma and theta.