Early Exercise of American Digital Puts at Nonnegative Rates

Summary

The document explains why an American digital put can be valued through the probability that the underlying asset reaches a specified minimum by expiry. The key reasoning is about exercise timing: when interest rates are nonnegative, the response says it is optimal to exercise as soon as the option is in the money, since waiting does not increase the payoff.

Under that condition, the dynamic stopping problem reduces to assessing whether the underlying has crossed the relevant threshold at any point before expiry, then discounting the resulting payoff. The questioner had considered Monte Carlo simulation, but the accepted answer gives an exercise argument rather than a numerical procedure or worked calculation. The conclusion depends on nonnegative rates and the particular digital payoff structure; the document does not address negative rates, transaction costs, or other contract variations.

Key ideas

• For nonnegative interest rates, the response argues that an in-the-money American digital option should be exercised immediately.
• The option value can be expressed using the probability that the underlying minimum reaches the strike threshold.
• The exercise-timing argument reduces the dynamic stopping problem to a threshold-crossing event.
• The document does not cover negative rates or alternative digital option specifications.

Tags

# Develop a pricing formula for an American digital put option


# Develop a pricing formula for an American digital put option












This problem comes from concepts and practice of mathematical finance by Joshi Chapter 8 problem 9.

> Develop a pricing formula for an American digital put option

Joshi's solution - He states that we simply need to compute $$e^{-rT}\mathbb{P}\left(m_{T}^{s} \geq k \right)$$

where $m_{T}^{s}$ is denoted as the minimum up to time $T$. I really do not understand where we arrives at this conclusion at all or how to solve the problem in any other way than just using Monte Carlo simulation since we are dealing with a dynamic stopping problem.

## Answer by Antoine Conze (score 0, accepted)

https://quant.stackexchange.com/a/40498

Obviously when interest rates are non negative it is optimal to exercise an American digital option as soon as it is in the money (you will not get more by waiting). The conclusion follows.

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.