Introduction || Monte Carlo Simulation || Binomial Method || Perpetual Bermudans || Known Names & Variants || References || Advanced Readings || Pricing Models

Bermudan options are similar in style to American style options in that there is a possibility of early exercise, but instead of a single exercise date there are predetermined discrete exercise dates. They are commonly used in interest rate and FX markets but we generalise them in this case for any type of options.

The main difficulty in determining suitable valuation for Bermudan options comes in the form of the boundary problem. Because of multiple exercise dates, determining the boundary condition in order to solve the pricing problem can be difficult. By determining the optimal exercise strategy and the respective boundary condition, one can generally use simulation methods to determine suitable prices for Bermudans.

In addition to having predetermined exercise dates, Bermudans also have a 'early exercise date' characteristic which we will say is where t = E. This date lies before the predetermined exercise dates, but after t = 0. Between 0 and E, the Bermudan behaves like a European contract and no exercise is possible; and between t = E and t = T, the Bermudan has American style traits. The price of a Bermudan is typically between that of an American option and a European option.

Because a this option has both European and American style properties (but with more American style exercise properties) it was named a Bermudan - in the sense that Bermuda lies between the two regions but closer to the USA.

Lattice Methods

Since the introduction of the binomial lattice in 1979 by Cox, Ross & Rubinstein for use in option pricing, almost all types of path dependent options can be effectively priced without much computation time.

In the case of Bermudan options, a modified version of the CRR tree can be implemented.

Because of the 'early exercise date' characteristic which sets a boundary between European style and American style characteristics, we can simply build a binomial tree which considers a European style exercise up until t = E at which point, the option can be compared to American style options - albeit often with discrete exercise dates:

By considering the above illustration, one can picture the use of a standard binomial tree to value Bermudans.

With the same concept in mind, you may also considering using a modified trinomial tree ala Boyle (1986) by separating the exercise regions to account for both European and American traits.

Monte Carlo Simulation

By far the most popular method for pricing Bermudans is the use of Monte Carlo simulation.

Perpetual Bermudans

In the unique case of perpetual Bermudan options, exact analytical solutions can actually be found.

See for example Boyarchenko & Levendorskii (2002).

Other Known Names / Variants

Bermudan Swaptions
Quasi-American Options
Rainbow Bermudan

Additional/Useful List of resources

Papers:

Andersen, L., “A Simple Approach to Pricing of Bermudan Swaptions in the Multi-Factor Libor Market Model,” Working Paper, General Re Financial Products (1998)
Berridge, S., Schumacher, H., "An Irregular Grid Method for Solving High-Dimensional Problems In Finance", International Conference on Computational Science, 2, (2002)
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Boyarchenko, S.I., & Levendorskii, S.Z., "Pricing of Perpetual Bermudan Options", Journal of Quantitative Finance, 2, 432-442 (Dec '02)
Boyle, P., "Option Valuation Using a Three-Jump Process", International Options Journal 3, pages 7-12 (1986)
Carr, P., & Yang, G., "Simulating Bermudan Interest Rate Derivatives", Working Paper, 1997
Cox, J., Ross, S. & Rubinstein M. “Option Pricing: A Simplified Approach." Journal of Financial Economics, 7. (Sep '79).
Douady, R., "Bermudan Option Pricing With Monte-Carlo Methods", Quantitative Analysis in Financial Markets, 2002
Hull, J., "Options, Futures & Other Derivatives", 5th Edition 2002
Joshi, M., & Theis, J., "Bounding Bermudan Swaptions In A Swap-Rate Market Model" Working Paper, 2002
Longstaff, F., & Schwartz, E., "Valuing American Options by Simulation: A Simple Least-Squares Approach", Review of Financial Studies, 14, 1, 113-147 (2001)
Meyer, M., "Monte Carlo Pricing of Bermudan Options"
Pedersen, M., “Bermudan Swaptions in the Libor Market Model,” Working Paper, SimCorp A/S (1999)
Schweizer, M., "On Bermudan Options", Advances in Finance & Stochastics, pp. 257-269 (2002)

Advanced Readings

Herzberg, F.S., "A Brief Note on the Soundness of Bermudan Option Pricing via Cubature", University of Oxford Working Paper (2006)
Jackel, P., "Monte Carlo in the BGM/J Framework: Using a Non-Recombining Tree to Design a New Pricing Method for Bermudan Swaptions", RBS Quantitative Research Centre Working Paper, 2000