Why ATM Call Options have larger negative theta compare to OTM and ITM?

Question

1. Why do ATM Call Options have larger negative theta?

2. For the same ATM Call Options, why smaller expiration, the more negative the theta?

Can someone provide intuitions or explanations?

Answer 1

Why do ATM Call Options have larger negative theta?

Theta reflects the loss of time value each day (time decay). All options have zero time value at expiration. Before expiration, ATM options have the greatest time value (more than OTM and ITM) because they have the most uncertainty about expiring OTM or ITM. Since they have more time value to lose, ATM options have larger negative theta.

For the same ATM Call Options, why smaller expiration, the more negative the theta?

Time decay is nonlinear. The loss of time value is slow far from expiration, and becomes faster as expiration approaches. Mathematically, an ATM option with time to expiration T - t has a value ~ sqrt(T - t), so by differentiation, theta ~ -1/sqrt(T - t), which becomes more and more negative as t approaches T.