# Negative theta for a short put

I am getting a negative theta for a short put deal Is it possible and if yes then under what conditions. Kindly explain

I am just learning these concepts so my question may sound vague to some of you but please help

• Is it american exercise? And/or is the underlier VIX? Commented Oct 13, 2020 at 9:49
• @StackG why would the underlying specificalyl being VIX cause a difference here?
– will
Commented Oct 13, 2020 at 20:41
• Theta is partial derivative wrt. $t$ (ie. The rate of change if nothing else changes). Usually we think of this as measuring time decay of the option. With VIX options though, which are options on the vol, if nothing else is changing with time, then the volatility is actually decreasing, which can cause incteases in put prices. It's an interesting effect - I can expand on it in a separate answer if you're interested. Commented Oct 13, 2020 at 23:16
• @StackG: I would be interested! Put options on VIX in my opinion do deserve a stand-alone answer, pls feel free to post one below. I am sure many others would take interest also. Commented Oct 26, 2020 at 13:01

$$\Theta=-\frac{S_t \sigma}{2\sqrt{\tau}}N'(d_1)+rKe^{-r\tau}N(-d_2)$$
For deep in-the-money Puts, $$d_1$$ and $$d_2$$ go to negative infinity: consequently, the term $$N'(d_1)$$ goes to zero, whilst the term $$N(-d_2)$$ goes to 1. Therefore, deep ITM puts can have a positive Theta, with a limit equal to $$+rKe^{-r\tau}$$.
For completeness: $$\tau$$ is time to maturity, $$K$$ is strike, $$\sigma$$ is vol, $$S_t$$ is the value of the underlying at the point in time when Theta is computed, $$r$$ is the risk-free rate. $$N'(d_1)$$ is the Standard Normal PDF with $$d_1$$ being the domain, whilst $$N(-d_2)$$ is the Standard Normal CDF with $$-d_2$$ being the domain.