the limits of arbitrage

The Limits of Arbitrage

ANDREI SHLEIFER and ROBERT W.VISHNY商学院周美 & 杜慧卿

ABSTRACT

In reality ， almost all arbitrage requires capital, and is typically risky.

Professional arbitrage is conducted by a relatively small number of highly speciali-zed investors using other’s capital.

Arbitrage becomes ineffective in extreme circumstances.

Anomalies in financial markets.

Fundamental concepts

“The simultaneous purchase and sale of the same, or essentially similar, security in two different markets for advantage-ously different prices.”

No risk and need no capital.

“The simultaneous purchase and sale of the same, or essentially similar, security in two different markets for advantage-ously different prices.”

No risk and need no capital.

Realistic arbitrage are more complex

Require capital good faith moneyIn short run, one may lose moneyDifferent trading hours, settlement dates,

and delivery terms.If prices are moving rapidly, the value

may differ additional risks

Require capital good faith moneyIn short run, one may lose moneyDifferent trading hours, settlement dates,

and delivery terms.If prices are moving rapidly, the value

may differ additional risks

PBA( performance-based arbitrage)

More commonly, conducted by relatively few professional, highly specialized investors;

Outside resourcesBrains and resources are separatedAllocate funds based on past returns

I. An agency model of limited arbitrage

Assume ：① Three types: noise traders, arbitrageurs,

and investors in arbitrage funds

② Fundamental value is V;

③ Three time periods

tp

Assume ：① Three types: noise traders, arbitrageurs,

and investors in arbitrage funds

② Fundamental value is V;

③ Three time periods

④ , ,

⑤

tptS

tt pSVt ][)(QN tF


Arbitrageurs‘ demand 22)2( pFQA Arbitrageurs‘ demand Aggregate demand equals the unit

supply:

22)2( pFQA

222 FSVp

Arbitrageurs‘ demand Aggregate demand equals the unit

supply:Not fully invest,

22)2( pFQA

222 FSVp

1D


supply:Not fully invest, ,

22)2( pFQA

222 FSVp

1D

11)1( pDQA 111 DSVp 11)1( pDQA 111 DSVp


supply:Not fully invest, ,Market segment, T investors with $1, so

22)2( pFQA

222 FSVp

1D

11)1( pDQA 111 DSVp

TF 2


supply:Not fully invest, ,Market segment, T investors with $1, so

22)2( pFQA

222 FSVp

1D

11)1( pDQA 111 DSVp

TF 2


Compete in the price the charge;Assume marginal cost constant, so

competition drives price to marginal costBayesians, allocate funds according to

past performance( PBA );An increasing function:

})()(*){(* 111121112 FDFppFDGFF

Compete in the price the charge;Assume marginal cost constant, so

competition drives price to marginal costBayesians, allocate funds according to

past performance( PBA );An increasing function:

})()(*){(* 111121112 FDFppFDGFF


Benchmark: zero returnA linear function: aaxxG 1)(

Benchmark: zero returnA linear function: ,with aaxxG 1)( 1a

Benchmark: zero returnA linear function: ,withThen

aaxxG 1)( 1a

)1()1()}()(*{ 12111111212 ppaDFFaDFppDaF

Benchmark: zero returnA linear function: ,withThenIf , gain funds; or ,lose funds The higher is , the more sensitive to

past performance

aaxxG 1)( 1a

)1()1()}()(*{ 12111111212 ppaDFFaDFppDaF

21 pp

a


An arbitrageur’s optimization problem:

q


q ;

1-q12 SSS


q ;

1-q ,12 SSS

02 S Vp 202 S Vp 2


q ;

1-q , .

Maximize:

12 SSS

02 S Vp 2

})1()*

{(*)(

})1()*

(){1(

1111

21

2

1111

1

FaDFp

pD

p

Vq

FaDFp

VDaqEW


q ;

1-q , .

Maximize:

12 SSS

02 S Vp 2

})1()*

{(*)(

})1()*

(){1(

1111

21

2

1111

1

FaDFp

pD

p

Vq

FaDFp

VDaqEW

II. Performance-based Arbitrage and Market Efficiency

The case of 1aThe case of

first order condition:

1a

0)1()1(-121

2

1

p

V

p

pq

p

Vq）（

The case of


Inequality ：

1a

0)1()1(-121

2

1

p

V

p

pq

p

Vq）（

11 FD

The case of


Inequality ：；Equality ：

1a

0)1()1(-121

2

1

p

V

p

pq

p

Vq）（

11 FD

11 FD

The case of


Inequality ：；Equality ： .

The initial displacement is large and will recover with a high probability; if they fall, it can’t be large.

1a

0)1()1(-121

2

1

p

V

p

pq

p

Vq）（

11 FD

11 FD

The case of


Inequality ：；Equality ： .

The initial displacement is large and will recover with a high probability; if they fall, it can’t be large. fully invested at 1

1a

0)1()1(-121

2

1

p

V

p

pq

p

Vq）（

11 FD

11 FD


Proposition 1:For a given , , , ; and , there is a such that, for ,

, and for , .

Proposition 2: At the corner solution( ),

, , and . At theinterior solution, , , and .

It shows that arbitrageurs ability to bear mispricing is limited, larger shocks, less efficient.

V 1S S 1F

a *q *qq

11 FD *qq 11 FD

11 FD

011 dSdp 02 dSdp 01 dSdp

011 dSdp 02 dSdp 01 dSdp


Uncertainty of the effect: a higher a could make market less

efficient, by withdrawing funds; A higher a will make prices adjust quickly

by giving more funds after a partial reversal of the noise shock.


Consider about the extreme circumstances:

two ways: 21 FD

Consider about the extreme circumstances:

two ways: ？？Proposition 3 ： If arbitrageurs are fully inv

ested at , and noise trader mispe-rceptions deepen at , then, for ,

, and .

Fully invested arbitrageurs may face equity withdrawals and liquidate.

21 FD 2211 pFpD

1t

2211 pFpD

1t

2t

2211 pFpD

1t

2t 1a

2211 pFpD

1t

2t 1a21 FD

2211 pFpD


Proposition 4: At the fully invested equili-brium, and .12 dSdp 02

2 dadSpd12 dSdp 022 dadSpd

Proposition 4: At the fully invested equili-brium, and .

This shows: prices fall more than one for one with the noise trader shock at time 2,when fully invested at time 1.

A market driven by PBA can be quite in-effective in extreme circumstances.

12 dSdp 022 dadSpd

III. Discussion of Performance-based Arbitrage

We are uncertain about the significance of PBA

Funds decline with a lag Contractual restrictions expose more risk Agency problem inside Arbitrageurs are risk-averse

So the efficiency of arbitrage will be limited.

III. Discussion of Performance-based Arbitrage

PBA supposes that all arbitrageurs have the same sensitivity, and will invest all funds when asset mispriced.

In reality, they differ. resources indepen-dent and invest more when price diverge further; not need to liquidate.

Big shock need more to eliminate, if not, mispricing gets deeper.

Little fresh capital available to stabilize.So PBA is likely to be important .

IV. Empirical Implications

A. Which markets attract arbitrage reso-urces?

large funds concentrated in a few markets, bond markets & foreign exch-ange market;

the ability to ascertain value;

specialized arbitrageurs avoid volatile;

short horizons may be more relevant


B. Anomalies• higher historical returns.• EMH: compensation for higher risk impl-

ausible: large number of diversified arbitrageurs.

• few specialized arbitrageurs care about total risk fundamental or idiosyncratic.

• failing to recognize price-revisal .


• In extreme circumstances, lose enough money and liquidate;

• Investors become knowledgeable about the strategies, diminish withdrawals; but it will be slowly for investors take action.

V. Conclusion

PBA may not be fully effective in bringing prices to fundamental values;

Specialized professional arbitrageurs may avoid extremely volatility;

The avoidance suggests a different approach to understanding persistent excess returns.

Thank you !

the limits of arbitrage

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