1.4.4 Theta

Theta is the derivative of the option price with respect to time. It is generally quoted in how much the price will drop over one day, and provides a sense of the rate of decay of an options value as it approaches expiration.

Summary of Terms

$V$ = Contract Value

$\Theta$ = Contract Theta

$S$ = Spot Price

$K$ = Strike Price

$\sigma$ = Implied Volatility

$\tau$ = Years to Expiration

$r$ = Risk Free Rate

$q$ = dividend yield

Calculation

\Theta = \frac{\partial V}{\partial \tau}

\Theta_{calls} = -{S\sigma e^{-q\tau} \over \sqrt{8\pi\tau}}e^{-{1 \over 2}d_+^2} - rKe^{-r\tau}{1 \over 2}\bigg[1 + erf\bigg({d_- \over \sqrt2}\bigg)\bigg] + qSe^{-q\tau}{1 \over 2}\bigg[1 + erf\bigg({d_+ \over \sqrt2}\bigg)\bigg]

\Theta_{puts} = -{S\sigma e^{-q\tau} \over \sqrt{8\pi\tau}}e^{-{1 \over 2}d_+^2} + rKe^{-r\tau}{1 \over 2}\bigg[1 + erf\bigg({d_- \over \sqrt2}\bigg)\bigg] - qSe^{-q\tau}{1 \over 2}\bigg[1 + erf\bigg({d_+ \over \sqrt2}\bigg)\bigg]

d_+ = {1 \over \sigma \sqrt{\tau}}\bigg[\ln\bigg({S \over K}\bigg) + \bigg(r + {\sigma^2 \over 2}\bigg)\tau\bigg]

d_- = d_+ - \sigma\sqrt{\tau}

Interactive Chart

[coming soon…]