Why a Short Deep In-the-Money Put Can Have Negative Theta

Summary

The document explains how a short put can have negative theta under the Black-Scholes setting described: a European put on a non-dividend-paying stock. Theta for the long put combines a time-value decay component with a positive interest-rate component. For a deep in-the-money put, the normal density term approaches zero while the relevant cumulative probability approaches one. The interest-rate term therefore dominates, and the long put’s theta can become positive, approaching the discounted rate contribution described in the answer.

A trader short that put has the opposite theta exposure, so the short position can have negative theta. This result depends on the stated option type, underlying dividend assumption, and model inputs; it is not a claim that every short put has negative theta. The document defines the variables and normal-distribution terms but gives no numerical example or treatment of other pricing models.

Key ideas

• A European put on a non-dividend-paying stock can have positive theta when it is deeply in the money.
• In that limit, the put’s volatility-related theta term becomes small.
• The interest-rate contribution can dominate the long put’s theta.
• A short position reverses the option’s theta exposure and can therefore have negative theta.
• The result depends on the assumptions and inputs used in the pricing model.

Tags

# Negative theta for a short put


# 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

## Answer by Jan Stuller (score 8, accepted)

https://quant.stackexchange.com/a/58644

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