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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.

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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.

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