A U T U M N 2 0 0 9
INTRODUCTION TO BINOMIAL MODELS
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Numerical Methods in Finance (Implementing Market Models)
©Finbarr Murphy 2007
Agenda Page
Introduction to Binomial Model
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Lecture Objectives Binomial Models Understand simple 1-step binomial models Price an option using a 1-step binomial model
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Understand multiplicative binomial process
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Binomial Models American options on a non-dividend asset are never
exercised early, so they can be valued using the Black-Scholes Formula But, it can be optional to early exercise American
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calls and puts where the underlying asset pays a dividend There are no closed-form solutions to these options,
we must use numerical techniques E.g. Binomial Trees
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Binomial Models Binomial Models assume that the underlying asset
follows a binomial process At any time, the asset price can change to one of two
possible values The asset price follows a binomial distribution
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E.g. Binomial Trees
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Binomial Models
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The asset price starts on the left and takes one of two steps as time moves forward
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Binomial Models
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Turn on the side and notice the binomial distribution
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Binomial Models
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Compare with a Gaussian Distribution
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Binomial Models Consider an asset with a current price S which follows a
binomial process During a time period Δt, the asset price can go up to uS
or down to dS
uS
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S
dS
Δt
This is known as a Multiplicative Binomial Process 9
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Binomial Models Now, consider a call option on this asset at the same
nodes Recall again that the value of a European call option at
its expiration is given by
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cT = max( ST − K ,0 ) uS Cu = max(uS-K,0)
S C
dS Cd = max(dS-K,0)
Δt t=0
t=T
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Binomial Models We construct a portfolio at t=0, such that at maturity
(t=T), the value of the portfolio will be the same whether the asset price goes to uS or dS Let the portfolio consist of a short position in the call
option and a long position in Δ units of the asset
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Portfolio = ΔS – C at t=0 At maturity (t=T) the portfolio is worth ΔuS – Cu or ΔdS – Cd Rearranging, we have
Cu − C d ∆= (u − d ) S
Eq 2.1.1 11
©Finbarr Murphy 2007
Binomial Models As the portfolio is riskless, it must grow at the risk free
rate of interest, therefore
e r∆t ( ∆S − C ) = ∆uS − Cu = ∆dS − Cd
Eq 2.1.2
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Where r is the risk free rate of interest
Now, combining Eq 2.1.1 and Eq 2.1.2 we have
C = e − r∆t [ pCu + (1 − p ) Cd ] where
e r∆t − d p= u−d
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Binomial Models
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Jump to 2-step and then n-step
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Binomial Models Now, we need to work with Binomial Trees using
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mathematical software. It helps to visualise how we can fit the tree into a matrix
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Binomial Models We have transformed the tree into a ½ matrix
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Continuing this process, add nodes
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Binomial Models MS Excel isn’t powerful enough but we can use it for
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testing purposes. Examine the example below;
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Binomial Models In the last slide, we saw how we can use a generic Excel
reference array to calculate any value of the tree. Excel isn’t scalable. We might need to include thousands
of nodes and these will get more complicated as we shall see.
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We use a programming language such as MatLab
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Recommended Texts Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing derivative
models, 1st ed., John Wiley and Sons Ltd. — Chapter 2
Additional/Useful Hull, J. (2009) Options, futures and other derivatives, 7th ed.,
Prentice Hall
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— Chapters 11
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