# Decreasing value of the Put option with increasing Time to maturity [closed]

Can you think of a situation when increasing the time to maturity lowers the value of a put option? If yes, show the example pls.

This situation can arise with some non-vanilla options. For example, a digital put option, which pays $$1$$ if the underlying price $$S$$ is below a strike $$K$$ at expiry, can exhibit "negative theta".

Assuming zero interest rate and dividend yield to keep it simple, the price is $$\\P = N(-d_2), \quad d_2 = \frac{\log \frac{S}{K}}{\sigma \sqrt{T}} - \frac{\sigma}{2}\sqrt{T}$$

where $$N$$ is the standard normal CDF, $$\sigma$$ is volatility, and $$T$$ is time-to-expiration.

We then have

$$\Theta = \frac{\partial P}{\partial T} = \frac{e^{-d_2^2/2}}{2\sqrt{2 \pi}\sqrt{T}}\left(\frac{\sigma}{2} + \frac{\log \frac{S}{k}}{\sigma T} \right)$$

When the term in parentheses is positive we have the usual situation where $$\Theta > 0$$ and the option value decreases as the time-to-expiration decreases.

However, we have $$\Theta < 0$$ under the condition where the option is in-the-money ($$S < K$$) and

$$\log \frac{S}{K} < -\frac{1}{2}\sigma^2 T$$

Here the value of the option can increase as the time-to-expiration decreases. Interpret this as -- the likelihood of the option expiring out-of-the-money due to random fluctuations in the underlying is diminished as the remaining life decreases -- however, there is no further upside if the underlying price falls further.