= Option theta
{title2=$\Theta=\partial_tV$}
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> gives
$$
\Theta=\rho(V-xV_x)-\frac12\sigma^2x^2V_{xx}.
$$
The 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 $\Theta\leq0$ when $\rho\geq0$; a <concave> payoff with a long bond position has $\Theta\geq0$. Negative <interest rates> need not obey these signs: the affine payoff $x-K$ has $\Theta=-\rho Ke^{-\rho(T-t)}$. For a fixed payoff function, remaining-maturity sensitivity is $-\Theta$.
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