Theta

The value theta of a portfolio [also called the time decay of the portfolio] is the sensitivity of the portfolio with respect to the time to maturity. Formally,

$$\Theta _{\Pi }=\frac{\partial \Pi }{\partial T}$$.

Thetas are usually not important on their own, since the time to maturity is considered to be deterministic, and therefore it does not represent any kind of uncertainty. Nevertheless, for a european call option, $\Theta$ is always negative [the time value always decreases] and is given by^8.1

$$\Theta _{C}=-\frac{S\mathcal{N}^{\prime }\left( d_{1}\right) \sigma }{2\sqrt{%% T}}-rKe^{-rT}\mathcal{N}\left( d_{2}\right)$$

The pattern of theta across moneyness levels and time to maturity is given in figure 8.3

Figure 8.3: Behavior of a Call option Theta. Part (a) gives the behavior of the theta on a three options with specifications $(K,\tau,r,\sigma)$ equal to $(1,0.2,0,0.5)$ for the solid line, $(1,0.2,0,1)$ for the thin line, and $(1,0.5,0,0.5)$ for the dashed line. Part (b) gives the behavior of the theta for a contract which is in-the-money [solid line], at-the-money [thin line], and out-of-the-money [dashed line].