Option Pricing: The Multi Period Binomial Model

Gurzuf, Crimea, June 2001 1

Option Pricing:The Multi Period Binomial Model

Henrik Jönsson

Mälardalen University

Sweden


Contents

• European Call Option

• Geometric Brownian Motion

• Black-Scholes Formula

• Multi period Binomial Model

• GBM as a limit

• Black-Scholes Formula as a limit


European Call Option

• C - Option Price• K - Strike price• T - Expiration day• Exercise only at T• Payoff function, e.g.

400 420 440 460 480 500 520 540 560 580 6000

10

20

30

40

50

60

70

80

90

100

s

g(s)

K=

KsKssg ,0max][)(


Geometric Brownian Motion

S(y), 0y<t, follows a geometric Brownian motion if

• independent of all prices up to time y

•

)(

)(

yS

ytS

ttNyS

ytS 2,~)(

)(ln


Black-Scholes Formula

The price at time zero of a European call

option (non-dividend-paying stock):

where

)()()0( tKeSC rt

t

SKtrt

)0(ln22


The Multi Period Binomial Model

i

i

i

i

i p

pprob

dS

uSS

11

1

0S

0dS

0uS

0udS

0

2Su

0

2Sd

0

2dSu

0

3Su

0

2Sud

0

3Sd

i

S

i=1,2,…

Note:

• u and d the same for all moments i

• d < 1+r < u, where r is the risk-free interest rate



• Let

• Let (X1, X2,…, Xn) be the vector describing the outcome after n steps.

• Find the set of probabilities P{X1=x1, X2 =x2,…, Xn =xn}, xi=0,1, i=1,…,n, such that there is no arbitrage opportunity.

1

1

0

1

ii

ii

dSSif

uSSifi

X i=1,2,…



• Choose an arbitrary vector (1, 2, …, n-1) • If A={X1= 1, X2= 2, …, Xn-1= n-1} is true

buy one unit of stock and sell it back at moment n

• Probability that the stock is purchased qn-1=P{X1= 1, X2= 2, …, Xn-1= n-1}

• Probability that the stock goes up pn= P{Xn=1| X1= 1, …, Xn-1= n-1}



i

i

i

i

i p

pprob

dS

uSS

11

1

}1,1,01{

}1,1,0{

)1,1,0(

32144

3213

XXXXPp

XXXPq

Example: 0S

0uS

0dS

0

2dSu

0

3Su

0

2Su

0

2Sud

0udS

0

3Sd

0

2Sd

i

S

1 2 3 n=4



• Expected gain =

• No arbitrage opportunity implies

qn-1[pn(1+r)-1uSn-1+(1- pn) (1+r)-1dSn-1-Sn-1]

du

drpn

1

r = risk-free interest rate



• (1, 2, …, n-1) arbitrary vector

• No arbitrage opportunity

X1,…, Xn independent with

P{Xi=1}=p, i=1,…,n

du

drp

1 Risk-free interest rate r the

same for all moments i



Limitations:• Two outcomes only • The same increase &

decrease for all time periods

• The same probabilities

Qualities:• Simple mathematics• Arbitrage pricing• Easy to implement


Geometric Brownian Motion as a Limit

The Binomial process:

rateinterest freerisk period one n

rt1, and

du

dp

ed

eu nrt

n

t

n

t

n 2,..., 1,j ,1

))1(()(

p

pprob

d

u

n

tjS

n

tjS


0S

0dS

0uS

0udS

0

2Su

0

2Sd

0

2dSu

0

3Su

0

2Sud

0

3Sd

S

in

tn

t2 tnt3

The Binomial Process


GBM as a limit

Let

and , Y ~ Bin(n,p)

n

jjXY

1

))1(()(0

))1(()(1

n

tjdS

n

tjSif

n

tjuS

n

tjSif

jX


GBM as a Limit

The stock price after n periods

where

W

ntY

n

t

n

Y

YnY

eS

eeS

dd

uS

duStS

)0(

)0(

)0(

)0()(

2

ntYn

tW 2


GBM as a Limit

Taylor expansion

gives

n

t

n

ted

n

t

n

teu

n

t

n

t

21

21

2

2

422

11 nt

nt

nrt r

du

dp


GBM as a limit

Expected value of W Variance of W

tr

rnt

pnt

ntEYn

tEW

nt

nt

)2

(

)42

(2

)2

1(2

2

2

tpnpn

t

VarYn

tVarW

22

2

)1(4

2

EY = np

VarY = np(1-p)


GBM as a limit

By Central Limit Theorem

nasttrNS

tSW 2

2

,)2

(~)0(

)(ln

ntYn

tweStS W 2,)0()(


GBM as a limit

The multi period Binomial model becomes geometric Brownian motion when n → ∞, since

• are independent

•

,,...,1,)1(

nj

n

tjS

n

tjS

ttrN

S

tS 22

,)2

(~)0(

)(ln


B-S Formula as a limit

• Let , Y ~ Bin(n,p)

• The value of the option after n periods = where S(t)= uY dn-Y S(0)

n

iiXY

1

K]-E[S(t))(1 C n-

n

rt

max[S(t)-K,0] = [S(t)-K]+

• No arbitrage


B-S formula as a limit

The unique non-arbitrage option price

As n → ∞

ntYwKeSEn

rt

Kdd

uSE

n

rtC

ntW

n

n

Yn

2)0(1

)0(1

,)0(e)

2(rt-

2

KeSEC

Xttr

X~N(0,1)



where X~N(0,1) and

AX

xttr

Xttr

dxxfKeS

KeSEC

)()0(e

)0(e

)2

(rt-

)2

(rt-

2

2

)0(

ln2

1:

2

AS

Krtt

txx



21

rt-

X

)2

(rt-

)2

(rt-

)0(

)(e-(x)dxf)0(e

)()0(e

2

2

KIeIS

dxxfKeS

dxxfKeS

rt

A AX

xttr

AX

xttr



)ln( where)(

)}ln(:{

and where

)}ln(:{ where

)0(21

)0(21

221

)0(2122

21

1

2

2

2

2

22

SK

t

SK

t

B

y

SK

tA

xxtt

trt

trttyyB

txydxe

rttxxAdxeI

(·) is the N(0,1) distribution function



)ln( where)(

)}ln(:{ where

)0(21

)0(212

21

2

2

2

2

SK

t

SK

tA

x

trtt

rttxxAdxeI



nastKeSC rt )()()0(

t

SKtrt

)0(ln22

where

Option Pricing: The Multi Period Binomial Model

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