Why Deep In-the-Money European Puts Can Have Positive Theta

Summary

The note explains why a long European put can have positive theta under Black–Scholes, despite the usual intuition that long options lose value as expiry approaches. A deep in-the-money put can trade below its intrinsic value when interest rates are positive, giving it negative time value. As expiry nears, its price must converge to intrinsic value, so this negative time value diminishes and the option can gain value over time.

The explanation identifies the corresponding case for calls: deep in-the-money European calls with positive dividend yield can also be below intrinsic value and exhibit positive theta. These conditions are also associated with circumstances in which early exercise may be sensible for American options. The note offers intuition rather than a full derivation, and the behavior depends on option type, moneyness, rates, dividends, and exercise style; it does not imply that long options generally have positive theta.

Key ideas

• A deep in-the-money European put can have negative time value when interest rates are positive.
• As expiry approaches, the put converges to intrinsic value, allowing its value to rise over time.
• Deep in-the-money European calls with positive dividend yield can show the analogous behavior.
• Positive theta in these cases does not change the usual tendency of long options to lose time value.

Tags

# Positive theta on a long put?


# Positive theta on a long put?












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

Given the following parameters:





- Risk-free rate: $0.03$



- 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 by Ezy (score 10, accepted)

https://quant.stackexchange.com/a/42613

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

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

- 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)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.