School of Education, Culture and Communication
Division of Applied Mathematics
MMA707 Analytical Finance
“Bermudan Options with the
Binomial Model”
Autumn 2011
Sheila Farrahi sfi09001@student.mdh.se
Amirhossein Heydarizadeh ahh09001@student.mdh.se
Oluwayinka Ogunniyi ooi08001@student.mdh.se
Instructor:
Jan Röman
Bermudan Options with the Binomial Model
Abstract
Bermudan option is a type of option which can only be exercised at specific dates
between the issue date and maturity. In this report we designed a GUI using MATLAB to
compute the value of Bermudan option with binomial model and compare the result
with European and American ones.
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Bermudan Options with the Binomial Model
Contents
Introduction ………………………………………………………………………………………………………………4
Bermudan Option
Binomial Model……………………………………………………………………………………….5
American & European Options………………………………………………………….….….5
Example …………………………………………………………………………………………………7
Solution …………………………………………………………………………………………………9
Conclusion…………………………………………………………………………………………………………….…10
References ………………………………………………………………………………………………………………11
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Bermudan Options with the Binomial Model
Introduction
Like Bermudian islands which are located between Europe and America, Bermudan
options are a combination of American and European options. Unlike the European
options which can be exercised only at maturity
and American option that can be
exercised at any time, Bermudan can be exercised at predetermined dates up to
maturity.
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Bermudan Options with the Binomial Model
Bermudan Option
A Bermudan Option is a type of option with early exercise restricted to certain dates
during the life of the option. Bermudan Options have an “early exercise” date and
expiration date. Bermudan options act like both European options and the American
ones. It behaves like European options since it cannot be exercised at any time and it
behaves like American options due to the fact that it can be exercised at some specific
times. The value of Bermudan Option is always equal or greater than European and
equal or less than American ones.
Binomial Model:
Binomial model is a very popular model for option pricing, Binomial tree shows
different ways that stock price can move during option’s life time based on certain
probability of moving up or either down. There are different formulas for the probability
of up and down but Cox-Ross-Rubenstein formula is the most common model for the
binomial tree so in our model we used Cox-Ross-Rubenstein formula:
{
𝑢 = 𝑒 𝜎√∆𝑡
(𝑢𝑝 𝑓𝑎𝑐𝑡𝑜𝑟)
𝑑 = 𝑒 −𝜎√∆𝑡 (𝑑𝑜𝑤𝑛 𝑓𝑎𝑐𝑡𝑜𝑟)
𝑒 𝜎∆𝑡 − 𝑑
p=
(risk neutral probability of moving up)
u−d
𝑞 = 1 − 𝑝(Probability of moving down)
American and European options:
Since Bermudan option is a combination of European and American option we describe
here shortly these two options:
To calculate European put or call option we should start from the end of tree at time T
(maturity) and calculate the max (𝑘 − 𝑠𝑇 , 0) for put and max (𝑠𝑇 − 𝑘, 0) for call at each
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Bermudan Options with the Binomial Model
end node. Then it should be discounted back to calculate the option value by backward
induction at each node. The formula for European put and call option in the other nodes
is:
Vn = e-rΔt(pVu+(1-p)Vd)
Calculate American put or call option at the final nodes of the tree is exactly like the
European ones. But for the other nodes the formula is different as follow:
American put: Vn = max(𝑘 − 𝑠𝑇 , e-rΔt(pVu+(1-p)Vd))
American Call: Vn = max(𝑠𝑇
− 𝑘,e-rΔt(pVu+(1-p)Vd))
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Bermudan Options with the Binomial Model
Example:
Consider the tree below which is a 6-step binomial tree with T=1.5 year and the
Bermudan option can only be exercised once a year. As you see in the picture below the
red nodes are the nodes that Bermudan option can be exercised at that time. For those
nodes option value should be calculated like European ones due to the property of
European option that it can be exercised only at that time, for the rest of the nodes, they
should be calculated like American options that can be exercised at any time.
t=3 months t=6 months t=9 months t=1 year t=15 months t=1.5 year
∆t=3 months
Calculate like American
options
Calculate like European
options
Calculate like American
options
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Bermudan Options with the Binomial Model
Solution
We designed a GUI using MATLAB to calculate the value of Bermudan call and put option
not using any MATLAB build in functions according to the data given by a user and
compare it with the European and American options.
This is how the GUI looks:
When a user is inserting the data, the data is being checked to be valid, for example
settlement date cannot be greater than maturity. If there is any invalid input the user
will face an error message.
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Bermudan Options with the Binomial Model
For example for the given data, you can see the value of Bermudan put option which is
exactly lies between European and American put option.
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Bermudan Options with the Binomial Model
Conclusion
Bermudan Option is one of the popular kinds of option in financial world and in this
seminar, we considered a Bermudan Option with non-paying dividend using Binomial
model.
in comparing the price of the Bermudan option with the European and the American
Option, it was observed that it option price(Bermudan Option) lies in between the
European and the American Option prices, which is true from the mathematical model
presented above.
Conclusively, A Matlab GUI was designed to compute and compare the prices of this
options and a payoff diagram was shown.
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Bermudan Options with the Binomial Model
References


Hull, John C. Options, futures, and other derivatives.6th ed. Upper Saddle River,
N.J. : Pearson Prentice Hall, cop. 2006
Wilmott, Paul, The Mathematics of Financial Derivatives, Cambridge :
Cambridge Univ. Press, 1995
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