black scholes - Positive theta on a long put?

I am trying to hand-price options under the Black-Scholes model.

Given the following parameters:

• Stock price: $12.53$


• Strike price: $14.00$


• Risk-free rate: $0.03$


• Annualized Volatility: $0.10$


• Time until expiry in years = $.238095$

The put will have a positive theta of $0.354295$. It has a very high probability of ending up ITM (using delta as an approximation, $\Delta = -0.982251$).

What is the intuition behind this behavior? I thought for long options theta is always negative as a long option loses it's extrinsic value over time. I could see a short option having a positive theta, but a long option? This behavior seems unintuitive.

Answer

If a european option value becomes lower than intrinsic value it gets negative time value.

In this circumstance the theta becomes positive because as time approaches to expiry the option value has to converge to intrinsic value.

For european options there are 2 circumstances that can lead to the option value being lower than intrinsic value

1. deep ITM puts in presence of positive interest rates $r>0$


2. deep ITM calls in presence of positive dividend yield $q>0$

Note that those are the 2 circumstances under which it makes sense for an american option to be exercised early.

For more details you can check the actual formula for theta on the wikipedia page dedicated to greeks

Greeks formulas (wp)