121 Arbitrage Equilibrium

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The one-period model

State prices and arbitrage

Arbitrage Pricing with Multiple StatesPricing derivatives

Henrik Olejasz Larsen

University of Copenhagen, 2012

Henrik Olejasz Larsen Arbitrage Pricing
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Outline

1 The one-period model

States of the world

SecuritiesPortfolios

Arbitrage

2 State prices and arbitrage

The fundamental theorem of arbitrage pricingAn example

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States of the world

Securities

Portfolios

Arbitrage

Outline


States of the world


Arbitrage



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States of the world

Securities

Portfolios

Arbitrage

A two period economy, t= 0, 1

Att= 1the true state of the economy is revealed as one ofa finite set ofSpossible states ={1, . . . , S}

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States of the world

Securities

Portfolios

Arbitrage

Outline


States of the world


Arbitrage




S f h ld
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States of the world

Securities

Portfolios

Arbitrage

Nsecurities traded at t= 0

Securities characterized by their pay off or dividend at t= 1dependent on the revealed state of the economy

D=

d11 . . . d1S...

. . . ...

dN1 . . . dNS

The securities can be traded att= 0at pricesq RN


St t f th ld
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States of the world

Securities

Portfolios

Arbitrage

Outline


States of the world


Arbitrage




States of the world
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States of the world

Securities

Portfolios

Arbitrage

"There is no ignorance, there is knowledge"

A portfolio of the Nsecurities is represented by RN

A positive element in is called alongposition in the

relevant security, a negative is called ashortposition

The portfoliohas the price qatt= 0

and the dividend at t= 1ofD RS

Investors may disagree on the probabilities of the states in

, but we assume that all on agree on Dand that everystate is possible


States of the world
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States of the world

Securities

Portfolios

Arbitrage

Outline


States of the world


Arbitrage




States of the world
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States of the world

Securities

Portfolios

Arbitrage

Definition (Arbitrage opportunity)

An arbitrage opportunity (or just an arbitrage) is a portfolio

RN such that

q

0andD

> 0orq < 0andD 0

Remark

The definition is equivalent to defining an arbitrage

opportunity as a portfolio RN with paymentsm= (q,D) R RS that hasm> 0



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The fundamental theorem of arbitrage pricing

An example

Outline


States of the world


Arbitrage



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An example

Definition (State prices)

A state price vector is a strictly positive vector RS such thatq

=D

Theorem (Fundamental theorem of arbitrage pricing)

The pair(q,D)is arbitrage free if and only if there is a stateprice vector

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An example

Lemma (Separating Hyperplane Theorem)

Suppose two convex setsA,B RN are disjoint. Then there is alinear functionalFonRN and a valuec R such thatF(x)c

for allx AandF(x)cfor allx B

Proof.

See ?, Theorem M.G.2.

Alinear functionalis just a linear map on a vector space to thescalars (here a dot product with a constant vector)

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An example

For general illustration - this is not our K or M!

K

M

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An example

Proof (if)

If we have strictly positive state prices then non arbitrage

follows directly


Th i d d l Th f d l h f bi i i
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An example

Proof (only if)

LetL, K,Mbe as in the remark above.

By the lemma since K is convex there is a hyperplane U in

L such thatMUandU K= 0

The hyperplane separates such that there is a linearfunctionalF: L R with kernelN(F) = Usuch thatF(x)> 0x K\ {0}.

Write an element(v, c) Lwithv R andc RS asv(1, 0) +Sj=1cj(0, ej)where(1, 0), (0, e1), . . . , (0, eS)is thecanonical basis ofL. Let=F(1, 0)and=

F(0, e1), . . . , F(0, eS)

. Since the basis vectors are all

inK, they are strictly positive.

NowF(v, c) =vF(1, 0) +cF(0, e) =v +c


Th i d d l Th f d t l th f bit i i
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An example

SinceMis in the kernel of F v +c= 0for(v, c)M.Thusq= D. Simplify by dividing by and getq=D, where =/.

To interpret the, note that by taking a portfolioi = (0, . . . , 1, . . . , 0) with just one security iwe getqi =

Ss diss

Thus are state prices as in the theorem.


The one period model The fundamental theorem of arbitrage pricing


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An example

With no bid-ask spread the situation looks like

K

M

(1,)


The one period model The fundamental theorem of arbitrage pricing
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An example

With a bid-ask spread the state prices are not uniquely given

M

(1,)


The one-period model The fundamental theorem of arbitrage pricing
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An example

Outline


States of the world

Securities

Portfolios

Arbitrage




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The one period model



An example

Back to the Binomial

LetS= 2and N= 3

Let securities be such that such thatq= (1, S0,f0) and

D=

er

Su fuer Sd fd

With arbitrage free prices there exists state prices(1, 2)such that

q= D

12


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The one period model



An example

Back to the Binomial 2

Suppose thatf0 is unknown

IfSu=Sdthe matrix

D12=

er

Suer Sd

is invertible

We find

f0=

fufd

D1

12

1

S0


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p


g p g

An example

Outline


States of the world

Securities

Portfolios

Arbitrage




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p


g p g

An example

Definition (Redundant securities)

A security that has payoffs that can be replicated by a portfolio

of other securities (called a replicating portfolio) are said to be

redundant.

Remark (Pricing of redundant securities)

In an arbitrage equilibrium a redundant security must have the

same price as the replicating portfolio




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State prices and arbitrage An example

Definition (Complete markets)

If payoff that can be achieved by portfolios

x RS| RN

has the full dimension Sthe security markets are complete




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Arrow-securities

Definition (Arrow-securities)

A security that has a pay-off of 1in a state jis called thej-Arrow security

Remark (State prices as prices of Arrow securities)

Suppose we have an arbitrage equilibrium where are stateprices. Then a price of aj-Arrow security ofj will be arbitrage

free (i.e. if we add this security to the set of securities and giveit this price we will still have an arbitrage equilibrium)



S i d bi


A l
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Note that arbitrage free prices are necessary for a solution

to the investors portfolio optimization problem, thus for ageneral equilibrium


Appendix

R f
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References

Bibliography

Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green.

Microeconomic Theory. Oxford University Press, 1995.

http://goforward/http://find/http://goback/

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121 Arbitrage Equilibrium

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