When Long Put Options Can Have Positive Theta

Summary

The document explains why a long put can sometimes have positive theta, meaning its value can rise as time passes. The sign depends on the relationship among the option price, intrinsic value, interest rates, and remaining maturity; it is not determined solely by whether the position is long or the put is out of the money. In a deep in-the-money case, discounting the strike-related payoff can make the put's value increase as expiration approaches, particularly under suitable rate conditions.

The answers give illustrative Black–Scholes examples, including a scenario with negative rates and a long-dated put under high positive rates. These cases show that positive theta is possible, but they do not imply it is typical for an out-of-the-money put. The original question mentions both Black–Scholes and Crank–Nicolson calculations, yet the replies focus on option economics rather than diagnosing implementation details. Results depend on the contract inputs and rate assumptions.

Key ideas

• A long put can have positive theta under some combinations of rates, strike, spot, volatility, and maturity.
• Discounting can make a put's value rise as expiration approaches when the payoff is likely to be dominated by the strike.
• The examples demonstrate possibility rather than a general rule for out-of-the-money puts.
• A surprising computed theta should be interpreted alongside the pricing assumptions and option value relative to intrinsic value.

Tags

# Negative theta for long OTM put?


# Negative theta for long OTM put?












after a few years following the forum, I have a question to ask.

After running the model we use for getting the greeks of options, I got a very odd result for otm long put.

i got a positive theta..

has anyone any idea on why? this is tested and implement with BS and Crank Nicolson

## Answer by Kevin (score 4)

https://quant.stackexchange.com/a/49735

That is quite possible. You have negative time value and a positive theta if the option price is below the intrinsic value.

Look at deep ITM put options, the stock price is basically so low, the chance of it rising is negligible and the option price is the discounted payoff. This has a positive theta since the longer the time of maturity, the lower the option price in this case.

Set $S=1$, $K=100$, $r=1%$, $q=0$ and $\sigma=0.25$. The put price is 98.00 and the theta is $-0.99$. If you instead set $r=-0.01$, then you get a positive theta and a higher put price.

## Answer by Chris Taylor (score 4)

https://quant.stackexchange.com/a/49736

This is possible if the option is long-dated and interest rates are high enough.

For example, a five-year put struck at \$90 where the spot is \$100 (so it is in the money with respect to the spot price) with implied volatility 20% and interest rates 10% has a theta of \$0.19.

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