Option theta ($\Theta ={\partial }_{t}V$)

The derivative of an option's value with respect to calendar time at fixed current underlying price and other parameters. In the dividend-free Black-Scholes model, the Black-Scholes equation givesThe first term is the interest earned on the bond value of the replicating portfolio, while the second uses option gamma. A convex payoff with a short bond position has $\Theta \le 0$ when $\rho \ge 0$; a concave payoff with a long bond position has $\Theta \ge 0$. Negative interest rates need not obey these signs: the affine payoff $x-K$ has $\Theta =-\rho K{e}^{-\rho (T-t)}$. For a fixed payoff function, remaining-maturity sensitivity is $-\Theta$.