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}

Theta market impact

The figure below shows the value of theta for an options contract with a $100 strike. Note that contracts that are far in the money or far out of the money have very small amounts of theta. Most of the nominal value decline over time occurs on contracts where the strikes are nearest spot. However, for the trader who is primarily concerned with percent returns, should be evaluating the theta decay relative to the options contract price.

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The figure below shows the relative percent decline in value over time for a $100 strike call contract as a function of underlying spot price, which is just theta divided by the contract value. The relative value decay of the call thus occurs most strongly for far out of the money contracts, and reduces to zero for deep in the money contracts. Thus, in the money contracts are relatively insulated from theta decay, while out of the money contracts are especially sensitive to it.

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