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.

  • $\begingroup$ Is $T$ the maturity of the option? $\endgroup$ – Daneel Olivaw Jun 26 '18 at 14:11
  • $\begingroup$ @DaneelOlivaw I believe so $\endgroup$ – Wolfy Jun 26 '18 at 14:32

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.

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