Theta, Volatility, and the Rent on Gamma

More volatility makes your option decay faster — the daily rent you pay for holding gamma.

The intuition (right where it counts)

The argument in one breath:

more vol ⇒ the option is worth more ⇒ more time value to lose ⇒ theta (the decay) grows.

With $r=q=0$ an at-the-money option is pure time value, so raising the vol input inflates exactly the thing that melts away as the clock ticks. More to lose per day means a bigger decay. Over every volatility you will ever trade, this is correct: theta grows with vol.

Why it is exactly true: theta is gamma, priced up

Theta is not a separate risk. The Black–Scholes equation with $r=q=0$ says

$$\frac{\partial V}{\partial t} + \tfrac12\,\sigma^{2}S^{2}\,\frac{\partial^{2}V}{\partial S^{2}} = 0 \qquad\Longrightarrow\qquad \boxed{\;\Theta \;=\; -\,\tfrac12\,\sigma^{2}S^{2}\,\Gamma\;}$$

Theta and gamma are the same quantity seen from two sides: the daily decay is your gamma multiplied by $\tfrac12\sigma^{2}S^{2}$. Turn up $\sigma$ and, for the same gamma, the decay scales with $\sigma^{2}$. Explicitly, for an at-the-money option,

$$\Theta_{\text{ATM}} \;=\; -\,\frac{S\,\sigma\,\varphi(d_1)}{2\sqrt{\tau}},\qquad \varphi(x)=\tfrac{1}{\sqrt{2\pi}}e^{-x^2/2}.$$

(Sign convention: theta is negative for a long option — you lose value as time passes. "Theta grows" means it becomes more negative: faster decay.)

The numbers

Daily time decay of an at-the-money call ($S=100$, one year, $r=q=0$; annual theta $/365$):

Through the whole realistic range, decay climbs with vol — the intuition, confirmed.

The twist worth knowing

It does not climb forever. $|\Theta|$ peaks around $\sigma\sqrt{\tau}\approx 2$ (about 200% for a one-year option) and then falls back toward 0 at extreme vol. The reason is the boxed identity: $\Theta = -\tfrac12\sigma^{2}S^{2}\Gamma$, and gamma collapses (the flat-delta death from the gamma note) faster than $\sigma^{2}$ can inflate it. So at infinite vol, gamma and theta both vanish — the option is so wide there is no sharp time value left to bleed. You will never trade there; the reversal is a limit curiosity, not a working regime. Everywhere real, more vol means more decay.

The payoff: theta is the rent you pay for gamma

The identity is the whole economics of a long option in one line. Holding gamma earns you money on moves — the gamma P&L over a move is $\tfrac12\Gamma(\Delta S)^2$. Holding it costs you the theta, $\tfrac12\sigma^{2}S^{2}\Gamma$ per unit time, and those are the same $\Gamma$. You break even when what the market realises matches what you paid in implied vol: long gamma wins if realised volatility beats implied, and bleeds if the market stays quiet. Theta is simply the daily invoice for that bet.

Takeaway. More vol inflates the time value your option loses each day, so theta grows — exactly as expected, across every real volatility. And it is not a coincidence: $\Theta=-\tfrac12\sigma^{2}S^{2}\Gamma$, so theta is just gamma priced up by $\tfrac12\sigma^{2}S^{2}$ — the daily rent on your gamma.

Assumes $r=q=0$ and a flat volatility surface (the exam convention); $S=100$ in the table, and theta shown per calendar day (annual $/365$). Non-zero carry adds terms but not the mechanism.