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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

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  • $\begingroup$ Is it american exercise? And/or is the underlier VIX? $\endgroup$ – StackG Oct 13 '20 at 9:49
  • $\begingroup$ @StackG why would the underlying specificalyl being VIX cause a difference here? $\endgroup$ – will Oct 13 '20 at 20:41
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    $\begingroup$ 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. $\endgroup$ – StackG Oct 13 '20 at 23:16
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    $\begingroup$ @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. $\endgroup$ – Jan Stuller Oct 26 '20 at 13:01
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Theta on a European Put option on a non-dividend paying stock is:

$$\Theta=-\frac{S_t \sigma}{2\sqrt{\tau}}N'(d_1)+rKe^{-r\tau}N(-d_2) $$

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 $+rKe^{-r\tau}$.

If you are short the deep ITM Put option, you are short the positive Theta, which means your Theta can be negative.

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.

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