1.4.7 Charm

Charm is the first derivative of delta with respect to time, and is often referred to as “delta decay.” Options that are in the money have a delta that naturally drifts to 1 over time and options that are out of the money have a delta that naturally drifts to 0. For the delta hedged portfolio, this means constant buying of the underlying for positive charm and constant selling for negative charm.

Summary of Terms

$V$ = Contract Value

$\Delta$ = Contract Delta

$\xi$ = Contract Charm

$S$ = Spot Price

$K$ = Strike Price

$\sigma$ = Implied Volatility

$\tau$ = Years to Expiration

$r$ = Risk Free Rate

$q$ = dividend yield

Calculation

\xi = \frac{\partial^2 V}{\partial S \partial \tau} = \frac{\partial \Delta}{\partial \tau}

\xi_{calls} = {qe^{-q\tau} \over 2}\bigg[1 + erf\bigg({d_+ \over \sqrt2}\bigg)\bigg] - \bigg({ e^{-q\tau} \over \sqrt{2\pi}}\bigg){2(r-q)\tau - d_- \sigma \sqrt{\tau} \over 2\tau \sigma \sqrt{\tau}}e^{-{1 \over 2}d_+^2}

\xi_{puts} = -{qe^{-q\tau} \over 2}\bigg[1 + erf\bigg({-d_+ \over \sqrt2}\bigg)\bigg] - \bigg({ e^{-q\tau} \over \sqrt{2\pi}}\bigg){2(r-q)\tau - d_- \sigma \sqrt{\tau} \over 2\tau \sigma \sqrt{\tau}}e^{-{1 \over 2}d_+^2}

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}

Charm market impact

The figure below shows how charm varies for a $100 strike contract as a function of spot. See the delta section for more details on market maker hedging impact. As illustrated, the charm for a contract is positive when above spot and negative when below spot. This means that all OTM contracts decay to 0 and all ITM contracts decay to 1. Note that delta is negative for puts, so a negative charm means it is gaining delta over time.

charm