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Why a Bermudan Option Is Worth More Than Its Best European Counterpart

Article Quant Q&A · Author: Artem Kazantsev
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Summary

A Bermudan option can be exercised on several specified dates. Its value is not generally the highest value among European options with matching individual exercise dates. Taking that maximum amounts to choosing a single exercise date at the outset and committing to it, before the future market state is known. A Bermudan holder can instead wait, observe how conditions develop, and exercise at a later allowed date if that is more valuable.

The replies explain this added flexibility with an analogy in which a payoff can be accepted after each of several sequential chances, or deferred to the next chance. That ability to make the decision later gives the Bermudan option additional value relative to a fixed-date European choice. The discussion is intuitive rather than a formal pricing derivation; it provides no model assumptions, numerical comparison, or valuation method. The exact amount of the value advantage depends on the option and pricing assumptions.

Key ideas

  • A Bermudan option permits exercise at several specified dates.
  • The maximum European value represents a fixed exercise-date choice made at inception.
  • A Bermudan holder can condition the exercise decision on later information.
  • This timing flexibility makes the Bermudan option more valuable than the maximum individual European value.
  • The discussion gives intuition but no quantitative pricing procedure.

Tags

OptionsDerivatives pricingVolatility
Full text
# Valuation of Bermudan option as maximum of relevant European options


# Valuation of Bermudan option as maximum of relevant European options












Assume I need to price a Bermudan option which can be exercised at following dates: $t_1$, $t_2$, ..., $t_n$. I think that the price of such an option will be maximum of the prices of European options with maturities $t_1$, $t_2$, ..., $t_n$. Am I right or wrong?

## Answer by Antoine Conze (score 4)

https://quant.stackexchange.com/a/37446

You are wrong. Using the maximum of the prices of the European options is equivalent to choosing (and making that choice final) on $t=0$ the date $t_i$ on which you will exercise. As such a choice would be sub-optimal, you would be giving up value. Therefore the Bermuda option is worth more than the maximum of the prices of the European options.

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/54969

Without the math, look at it this way: I give you a die to toss. You can toss it thrice and take the payoff as the number on the die. On each turn you can either accept the payoff or move on. At the last throw, you must accept the payoff.

Your logic would state that this product has value equivalent to tossing a die once and only once (here, all 'Europeans' are equally valuable, so your 'bermudan (3 tosses)' should equal the most expensive european (any one of the 3 tosses). You can see that this is wrong. To convince yourself more, think what would happen if I was allowed to toss the coin an infinite number of times.

Shown 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.