Lecture 25

Xiaoguang Wang

STAT 598W

April 24, 2014

(STAT 598W) Lecture 25 1 / 22

Outline

1 CRR Binomial Trees

2 Leisen-Reimer Binomial Tree

3 Flexible Binomial Tree

4 Trinomial Trees

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Outline

1 CRR Binomial Trees

2 Leisen-Reimer Binomial Tree

3 Flexible Binomial Tree

4 Trinomial Trees

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Exercise

Material: Rouah and Vainberg (2007)

Recall the Put-Call Parity (at time t = 0):

Put price = Call price + Ke−rT − S

Compute the price of an European Put Option.

Modify your code in order to compute the price of ATM AmericanCall option.

Compute the price of an ATM American Put Option. Can you use thePut-Call Parity here?

(STAT 598W) Lecture 25 4 / 22

CRR Binomial Tree

Recall:

In the CRR (Cox, Ross and Rubinstein 1979) Binomial tree theincrements are defined as:

u = eσ√

T/n

d =1

u.

There is a closed formula for the European Call option price:

Call Price = e−rTn∑

i=0

(n

i

)qiuq

n−id max (Suidn−i − K , 0)

(STAT 598W) Lecture 25 5 / 22

Exercise

Construct a VBA function EuroBinomial that computes the price of anEuropean Call or Put option with the following parameters:

Stock price: S

Strike price: K

Risk free rate: rf

Time period: T

Volatility: σ

Number of time steps: n

Whether a call or put is priced (String).

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Exercise

Construct a VBA function Binomial that computes the price of anEuropean or American Call or Put option with the previous parametersand:

String variable indicating whether an American or European option ispriced.

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Outline

1 CRR Binomial Trees

2 Leisen-Reimer Binomial Tree

3 Flexible Binomial Tree

4 Trinomial Trees

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Details

The CRR Binomial Tree requires many steps to converge to the BSOption formula.

Leisen and Reimer (1996) proposed an improvement using normalapproximations. (two different methods)

Only work for odd number of steps.

(STAT 598W) Lecture 25 9 / 22

Notation

Take the usual CRR parameters:

un = eσ√

T/n dn = 1/un

rn = erT/n pn =rn − dnun − dn

and define the new ones:

u = rnp′

p

d =rn − pu

1− p

(STAT 598W) Lecture 25 10 / 22

Notation

where:

p′ = h−1(d1)

p = h−1(d2)

h−1(z) =1

2+

Sign(d1)

2

√√√√√1− exp

−( z

n + 13 + 0.1

n+1

)2(n +

1

6

)d1 =

log(S/K ) + (r + σ2/2)T

σ√T

d1 =log(S/K ) + (r − σ2/2)T

σ√T

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Exercise

Implement the function LR with the same requirements as Binomial.

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Exercise

Compute the price of an European Put Option with parameters:

S = 30

Strike price: K = 30

Risk free rate: rf = 0.05

Time period: T = 0.4167

Volatility: σ = 0.3

Number of time steps: n = 10 to n = 200 in steps of size 10.

using the LR and CRR Binomial Tree. Analyze the convergence of bothmethods.

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Outline

1 CRR Binomial Trees

2 Leisen-Reimer Binomial Tree

3 Flexible Binomial Tree

4 Trinomial Trees

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Details

Tian (1999) proposed a model that allows CRR tree to be tiltedupward or downward. This means that the “center” of the treeincreases (λ > 0) or decreases (λ < 0) with the time.

The up and down moves are specified by:

u = eσ√dt+λσ2dtd = e−σ

√dt+λσ2dt

The tilt parameter can be chosen to ensure smooth convergence.This value is:

λ =log (K/S)− (2j0 − n)σ

√dt

nσ2dt

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Details

Where j0 is the integer part of:

log (K/S)− n log (d0)

log (u0/d0)

and u0 and d0 are the up and down moves when λ = 0 (CRR model).

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Exercises

Implement the function Flexible with the same requirements asBinomial and LR.

Compute the same European Put option as before. Analyze itsconvergence and compare it with the Binomial and LR methods.

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Extrapolation

This method can be improved by using this extrapolation formula:

Price.ext(n) = 2Price(n)− Price(n/2)

Exercise:Create the function FlexibleExt with the extrapolated version of theFlexible Tree. Note that the tree only admits an even number of steps.Analyze the convergence with our European put example.

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Outline

1 CRR Binomial Trees

2 Leisen-Reimer Binomial Tree

3 Flexible Binomial Tree

4 Trinomial Trees

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Details

Introduced by Boyle (1986). The asset price can go up, down or stayat the same price.

The structure is as follows:

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Details

The up movement is given by u = eσ√2dt and d = u−1.

Probabilities:

pu =

exp(12 rdt

)− exp

(−σ√

12dt

)exp

(σ√

12dt

)− exp

(−σ√

12dt

)

2

pd =

exp

(σ√

12dt

)− exp

(12 rdt

)exp

(σ√

12dt

)− exp

(−σ√

12dt

)

2

pm = 1− pu − pd .

(STAT 598W) Lecture 25 21 / 22

Exercises

Implement the function Trinomial with the same requirements asBinomial, LR, Flexible and FlexibleExt.

Compute the same European Put option as before. Analyze itsconvergence and do a comparison of all the Tree algorithms.

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