# 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 Nov 15 '19 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 Nov 16 '19 at 9:04

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.

• OP asks about long OTM puts though. – Vim Nov 15 '19 at 17:04
• I've edited to make it clear that this can happen for OTM puts when interest rates are high and the option is long-dated. – Chris Taylor Nov 15 '19 at 17:55
• I finally got the point. When ttm is long and rate is high the effect of deferral of cash receipt of the long put party is so large, so that as time goes by and the effect vanishes the time value actually goes up first (but eventually go down to zero when close to maturity). For short-dated or low rates though this won't happen and time value will always decline. – Vim Nov 16 '19 at 7:50

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.