# Bermudan (Equity) Option (under lognormal model)

## Java Applet

Alternatively you may launch this applet using Java WebStart.
Minimum requirements: Java 1.6. Tested with Firefox and Safari. Source code for the underlying library is available at
finmath.net.

**IMPORTANT:** The applet is currently *self-signed*. Hence it will no longer run under the Java security level "high", which has become default. **In order to run the applet, you need to add the domain **`http://christian-fries.de` to the *Exception Site List*. To edit the *Exception Site List* open the Java control panel, select the `Security` tab, then press `Edit Site List`.

## Description

This tools illustrated Bermudan option pricing using Monte Carlo simulation.

It serves as a companion to Chapter 15 of Mathematical Finance.

Upon each exercise date T_{1}, T_{2},...,T_{n} the holder of the Bermudan has the right to either exercise and receive N_{i} * (S(T_{i}) - K_{i}) or receive a shorter Bermudan option on the remaining exercises T_{j }> T_{i}. The last option pays either N_{n} * (S(T_{n}) - K_{n}) or nothing. Setting notional negative will give you a put instead of a call.

### Evaluation Methods

Various methods to estimate the exercise boundary are implemented:

- Regression
- The method also known as Longstaff-Schwartz where the exercise boundary is estimated by a linear regression on some basis functions.
- Binning
- The definition space of the predictor variable (here the exercise value) is split into intervals (bins) on which a conditional expectation is calculated.
- Threshold optimization
- A root finder optimizes a scalar threshold used in a specific exercise criteria (e.g. a linear function of the intrinsic value).
- Perfect foresight
- The single value given by the future evolution of the path is used as "estimate" for the expectation. This gives a foresight bias an thus a wrong price (the strategy is super optimal).
- Analytic
- For an Bermudan with only two exercise date and a simple Black-Scholes type model the optimal exercise can be calculated analytically (using Black-Scholes Formula upon exercise).

### Exercise Boundary

Figure 1: Example of the estimation of the exercise boundary.

The exercise boundary is visualized in a scatter plot, see Figure 1. The picture shows the situation at a specific exercise date. Each dot represents the value upon hold depending on the value upon exercise for a specific simulation path. The conditional expectation is shown as a function of the value upon exercise (blue line). The exercise region (green) is defined as the paths having exercise values (yellow line) above the expected value upon hold (blue line).

© Copyright 2007,2013 Christian P. Fries