Introduction To Binomial Models: Autumn 2 0 0 9

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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