# 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

• This is weird. Did you also get negative option value?
– Vim
Commented Nov 15, 2019 at 17:05
• It becomes much clearer if you break up theta into partial derivatives. You have components from rates, fwd curves, volatility (ie change in implied vol as you decrease time to maturity), etc etc etc, and then theta due to loss of optionality. It is this last term that people traditionally think about theta as being, but the others all contribute as well. You can think of the last term as how much money you expect to make from locking in profits by repeatedly delta hedging over one day.
– will
Commented Nov 16, 2019 at 9:04

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.

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.