# Convergence rate of Bermudan to American option

When trying to value an American option we often use grid-based methods (e.g. Monte Carlo in combination with Longstaff Schwartz; or Finite Difference Methods). As such, we are in fact estimating the value of a Bermudan option with discrete time points where we can exercise the option, i.e. $$0=t_0 < t_1 < ... < t_n = T$$.

However, as the time grid gets finer the value of the Bermudan option converges to the American option. I have heard that the rate of convergence is of order $$O(\Delta t)$$. How can this be shown analytically?

I am especially considering the following exercise strategy

1. Find the optimal exercise time of the American option, $$\tau^*$$, where $$t_{h-1}<\tau^* < t_{h}$$,
2. Exercise the Bermudan option at the first time point after $$\tau^*$$, i.e. at time $$t_{h}$$.

My starting point would be to have a look at the price difference between the two options and continue from there to find an upper bound of order $$O(\Delta t)$$... but how?.

$$0\leq C^{A} - C^{B} \leq ... \approx O(\Delta t).$$

## 1 Answer

Is this paper useful? Discussed usage of Richardson extrapolation for such purposes http://www.fin.ntu.edu.tw/~conference2002/proceding/5-4.pdf

• It seems to be within the same topic from what I can tell about Schmidt's Inequality (if you are looking at that section?). However, I am not entirely sure if that result can be applied to obtain the order of convergence? Jan 14, 2023 at 13:19
• Because you put this as an answer, it would be much better if you can explain in sufficient detail how the question is answered in the referenced paper in case the link is removed or changed. Jan 14, 2023 at 23:14