Gurzuf, Crimea, June 2001 1

The Arbitrage Theorem

Henrik Jönsson

Mälardalen University

Sweden

Gurzuf, Crimea, June 2001 2

Contents

• Necessary conditions

• European Call Option

• Arbitrage

• Arbitrage Pricing

• Risk-neutral valuation

• The Arbitrage Theorem

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Necessary conditions

• No transaction costs

• Same risk-free interest rate r for borrowing & lending

• Short positions possible in all instruments

• Same taxes

• Momentary transactions between different assets possible

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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][)(

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Arbitrage

The Law of One Price:

In a competitive market, if two assets are equivalent, they will tend to have the same market price.

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Arbitrage

Definition:• A trading strategy that takes advantage of two or

more securities being mispriced relative to each other.

• The purchase and immediate sale of equivalent assets in order to earn a sure profit from a difference in their prices.

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Arbitrage

• Two portfolios A & B have the same value at t=T

• No risk-less arbitrage opportunity They have the same value at any time tT

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Arbitrage Pricing

The Binomial price model

0S

0dS

0uS

prob.

q

1-q

0du 0q1

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Arbitrage Pricing

1+r < d: 1+r > u:

Action at time 0

t=0 t=T

Borrow S -(1+r)S

Buy stock -S dS

Return 0 >0

Action at time 0

t=0 t=T

Lend -S (1+r)S

Sell stock S -uS

Return 0 >0

r = risk-free interest rate

d < (1+r) < u

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Arbitrage Pricing

Equivalence portfolioCall option

BS 0

BrdS )1(0

BruS )1(0

r = risk-free interest rate

C

],0max[ 0 KuSCu

],0max[ 0 KdSCd

(t=T)(t=0) (t=T)(t=0)

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Arbitrage Pricing

Choose and B such that

)1)((

)()1(

)1(0

0

0

rdu

dCuCB

Sdu

CC

CBrdS

CBruS

ud

du

d

u

No Arbitrage Opportunity BSC 0

Gurzuf, Crimea, June 2001 12

Arbitrage Pricing

10,)(

)1(where

)1()1(

)(

)1(

)(

)1()1(

)1)(()(

1

1

0

0

pdu

drp

CppCr

Cdu

ruC

du

drr

rdu

dCuCS

Sdu

CCC

du

du

uddu

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Risk-neutral valuation

p = risk-neutral probability

pdudrq )()1(

0S

0dS

0uS

prob.

q

1-q

000 )1()1( SrdSqquS

( p = equivalent martingale probability )

Expected rate of return = (1+r)

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Risk-neutral valuation

Expected present value of the return = 0

C = (1+r)-1[pCu + (1-p)Cd]

( p = equivalent martingale probability )

Price of option today = Expected present value of option at time T

Risk-neutral probability p

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The Arbitrage Theorem

• Let X{1,2,…,m} be the outcome of an experiment

• Let p = (p1,…,pm), pj = P{X=j}, for all j=1,…,m

• Let there be n different investment opportunities

• Let = (1,…, n) be an investment strategy (i pos., neg. or zero for all i)

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The Arbitrage Theorem

• Let ri(j) be the return function for a unit investment on investment opportunity i

1

1r1(1)

1r1(2)

1r1(m)

prob.

p1

p2

pm

Example: i=1

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The Arbitrage Theorem

• If the outcome X=j then

n

iii jr

1)( investment ofReturn

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The Arbitrage Theorem

Exactly one of the following is true: Either

a) there exists a probability vector p = (p1,…,pm) for which

or

b) there is an investment strategy = (1,…, m) for which

n 1,...,i allfor 0)()(1

m

jijip jrpXrE

m 1,...,j allfor 0)(1

n

iii jr

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The Arbitrage Theorem

Proof: Use the Duality Theorem of Linear Programming

Primal problem Dual problem

n

ijiji

n

iii

mjbxats

xc

1,

1

,...,1,..

max

mjy

nicyats

yb

j

m

jijji

m

jjj

,...,10

,...,1,..

min

1,

1

If x* primal feasible & y* dual feasible then

• cTx* =bTy*

• x* primal optimum & y* dual optimum

If either problem is infeasible, then the other does not have an optimal solution.

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The Arbitrage Theorem

Proof (cont.): Primal problem Dual problem

1 and

n, 1,...,i ),( where

,...,1,0..

max

,...,1,)(..

max

,1

,

1

1,

1

11

1

jn

iji

n

iiji

n

n

inii

n

a

jra

mjats

mjjrts

m 1,...,j ,0

1

n 1,...,i,0)(..

0min

m 1,...,j ,0

1

n1,...,i,0..

0min

1

1

1,1

1,

j

m

jj

n

iji

j

m

jjjn

m

jjji

y

y

yjrts

y

ya

yats

Gurzuf, Crimea, June 2001 21

The Arbitrage Theorem

• Dual feasible iff y probability vector under which all investments have the expected return 0

• Primal feasible when i = 0, i=1,…, n,

cT* = bTy* = 0 Optimum! No sure win is possible!

Proof (cont.):

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The Arbitrage Theorem

Example:

• Stock (S0) with two outcomes

• Two investment opportunities:• i=1: Buy or sell the stock

• i=2: Buy or sell a call option (C)

)2(

)1(

1prob

0

0

1

j

j

p

p

dS

uSS

Gurzuf, Crimea, June 2001 23

The Arbitrage Theorem

Return functions:• i=1:

• i=2:

2)1(

1)1()(

0

1

0

0

1

0

1 jifSrdS

jifSruSjr

2)1(][

1)1(][)(

1

0

1

0

2 jifCrKdS

jifCrKuSjr

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The Arbitrage Theorem

Expected return• i=1:

• i=2:

(1) )1()1()( 000

1

1 SdSppuSrjrEp

(2) ])[1(][)1()( 00

1

2 CKdSpKuSprjrEp

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The Arbitrage Theorem

• (1) and the Arbitrage theorem gives:

• (2), (3) & the Arbitrage theorem gives the non-arbitrage option price:

1

du

drp

])[1(][)1( 00

1 KdSpKuSprC

(3)