Bermudan-to-American Option Convergence and Time-Grid Error
Summary
The document asks how quickly the value of a Bermudan option approaches the value of an American option as exercise dates become more frequent. It describes the practical setting in which numerical methods use discrete exercise opportunities, and proposes comparing the two prices to establish an error bound. A suggested exercise policy rounds the American option’s optimal exercise time up to the next Bermudan date.
One answer gives an informal first-order illustration using a put well inside the early-exercise region, with zero carry costs. It compares immediate exercise with waiting one time step and argues that the value difference is proportional to the step size. This example is not a general analytical proof: it relies on a simplified setting and does not establish a global convergence rate near the exercise boundary or under other market assumptions. Another answer points to Richardson extrapolation as a possible numerical technique, without detailing its application.
Key ideas
- Discrete exercise dates make numerical American-option methods approximate Bermudan values.
- The proposed policy exercises at the first grid date after the American-optimal exercise time.
- A simplified deep-in-the-money put example suggests a price gap proportional to the time step.
- The example does not prove a general convergence rate across exercise boundaries and market conditions.
- Richardson extrapolation is mentioned as a numerical approach, but no procedure is given.
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# Convergence rate of Bermudan to American option
# 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
- Find the optimal exercise time of the American option, $\tau^*$, where $t_{h-1}<\tau^* < t_{h}$,
- 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?.
$$0\leq C^{A} - C^{B} \leq ... \approx O(\Delta t).$$
## Answer by Brian B (score 1)
https://quant.stackexchange.com/a/80269
You do not have to get very fancy to find a convergence rate $O(\Delta t)$ for Bermudan to American option value. Consider a put option that is well below the early exercise boundary. We can then ignore optionality, and view it as one where we can exercise either now or in time $\Delta t$. Let's also assume zero carry costs. Exercising now results in present value $ P_0=K-S $, exercising later results in (future expected) value $R_1=K-S e^{r \Delta t}$, which has present value $P_1 = K e^{-r \Delta t} - S$.
So, at any given moment, (especially the in-the-money region of) Bermudan option prices are lower than American option prices by an amount proportional to $\Delta t$.
At this point it is worth noting that almost all options actually traded OTC or on exchanges are Bermudan with daily exercise or sparser, and not American.
## Answer by dm63 (score 0)
https://quant.stackexchange.com/a/74359
Is this paper useful? Discussed usage of Richardson extrapolation for such purposes http://www.fin.ntu.edu.tw/~conference2002/proceding/5-4.pdfShown 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.