When a Digital Put Can Lose Value as Expiry Gets Longer

Summary

The answer gives an example of an option whose value can fall as its time to expiry increases: a cash-or-nothing digital put, which pays a fixed amount if the underlying finishes below its strike. Under the simplifying assumptions of zero interest rates and zero dividend yield, it expresses the option price through the normal cumulative distribution function and derives the sign of sensitivity to maturity.

For an in-the-money digital put, the maturity sensitivity is negative when the underlying is sufficiently far below the strike relative to volatility and time. In that region, extending the option’s life adds a chance that price fluctuations will carry the underlying back above the strike, eliminating the payout; further declines do not increase the fixed payoff. The example shows that maturity effects depend on payoff shape. It concerns a digital option and the stated assumptions, so it should not be generalized directly to standard vanilla puts or markets with rates and dividends.

Key ideas

• A cash-or-nothing digital put pays a fixed amount when the underlying finishes below the strike.
• With zero rates and dividends, its maturity sensitivity can be negative for some in-the-money states.
• The negative sensitivity occurs when the underlying is sufficiently far below the strike relative to volatility and time.
• A fixed payout creates different maturity behavior from a vanilla put, whose payoff grows as the underlying falls.

Tags

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


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












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.

## Answer by RRL (score 3)

https://quant.stackexchange.com/a/42447

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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