Understand Theta and time decay

WHAT is Theta?

WHY negative for buyers? When you buy an option, time is your enemy — you paid for optionality that erodes. When you sell (write) an option, time is your friend: you collect that decay, so a short position effectively has positive theta.

WHY does time decay happen? (First principles)

An option's premium splits into two parts:

$\text{Premium} = \underbrace{\text{Intrinsic Value}}_{\max(S-K,\,0)\text{ for a call}} + \underbrace{\text{Time Value}}_{\text{extra paid for uncertainty}}$

• Intrinsic value doesn't decay — it only depends on where the spot $S$ is relative to strike $K$ today.
• Time value exists because there is still time for the stock to move in your favour. More time = more possible good outcomes = more you'll pay. As expiry nears, fewer future paths remain, so time value shrinks toward zero.

HOW fast? Deriving the shape of decay

Let's derive the key qualitative result from the Black–Scholes formula for a European call:

$C = S\,N(d_1) - K e^{-r\tau} N(d_2),\qquad d_{1,2}=\frac{\ln(S/K)+(r\pm\tfrac12\sigma^2)\tau}{\sigma\sqrt{\tau}}$

Here $\tau = T - t$ is time remaining to expiry. Theta is $\Theta = \partial C/\partial t = -\,\partial C/\partial \tau$. Differentiating gives (for a non-dividend call):

$\boxed{\Theta = -\frac{S\,N'(d_1)\,\sigma}{2\sqrt{\tau}} - rKe^{-r\tau}N(d_2)}$

Reading the formula from scratch:

• Why the $\sqrt{\tau}$ in the denominator? — Because uncertainty (a diffusion) grows like $\sqrt{\tau}$: a stock's standard deviation of moves scales with $\sqrt{\text{time}}$. Time value $\propto \sqrt{\tau}$ near the money, so its derivative carries a $\frac{1}{\sqrt{\tau}}$. This is the crucial insight.

Key behaviours to know

Common mistakes (steel-manned)

Flashcards

Connections

• The Greeks — theta is one of the five core sensitivities
• Understand Delta — directional sensitivity (spot), contrast with time
• Understand Gamma and its impact — high gamma ↔ high theta trade-off
• Understand Vega and volatility — vega feeds the time value that theta erodes
• Black-Scholes Model — source formula for $\Theta$
• Option Selling Strategies — "theta-gang," collecting decay
• Intrinsic vs Time Value — only time value decays

Concept Map

Hinglish (regional understanding)