Theta over Time

Quantitative Finance Level 2

Assuming constant volatility, what does the ATM theta against time graph look like?

( k k is a constant)

θ = k t \theta = \frac{k}{t} θ = k t \theta = \frac{ k } { \sqrt{t}} θ = k t \theta = kt θ = k t 2 \theta = \frac{ k } { t^2 } θ = k t \theta = k \sqrt{t} θ = k t 2 \theta = k t^2

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Calvin Lin by Calvin Lin

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1 solution

Chew-Seong Cheong
Mar 23, 2015

For ATM options:

V C = V P ≈ S σ t 2 π V_C=V_P \approx S\sigma \sqrt{\frac{t}{2\pi}}

Differentiating with respect to t t :

θ A T M = ∂ V ∂ t ≈ k t \theta_{ATM} = \frac {\partial V}{\partial t} \approx \frac {k}{\sqrt{t}}

See Straddle Approximation Formula .

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