Huberman and Stanzl

8/14/2019 Huberman and Stanzl

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Price Manipulation

and Quasi ArbitrageBy Gur Huberman And Werner Stanzl

Summer 2013Radhika Nangia


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Introduction

"Assuming no arbitrage is compelling because the presence of

arbitrage is inconsistent with equilibrium when preferences

increase with quantity. More fundamentally, the presence of

arbitrage is inconsistent with the existence of an optimal

portfolio strategy for anycompetitive agent who prefers more toless, because there is no limit to the scale at which an individual

would want to hold the arbitrage position. Therefore, in

principle, absence of arbitrage follows from individual rationality

of a single agent. One appeal of results based on the absence of

arbitrage is the intuition that few rational agents are needed tobid away arbitrage opportunities, even in the presence of a sea

of agents driven by `animal spirits'." (P.H. Dybvig and S.A.Ross,

1987)


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Model

Single asset traded over periods over (0,1]. The asset can be bought or sold at times (equally

spaced) , 1 ,where 1/.

is exogenously fixed. In each period , the initial price is , . The trader trades , on his own. Price Impact

, (Immediate effect):

, , ,(,). Price Update ,(Permanent effect):

+, , , , .


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Model

Competitive Liquidity Providers

Given: A set of price impact and price updatefunctions as a consequence of asymmetric

information , inventory costs , etc. Not dependent on the equilibrium of market

Price change due to public news , is revealedthe beginning of period

Trades from the trading crowd : , , and , are i.i.d ,independent and have 0

expected values.


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Model

Price dynamics

, , , , , , ., , ,(, ,). Fixed transaction costs for trades is 0 Conditions for pure arbitrage :

Traders know the prices they can trade at any time

Investments can be scaled without affectingprices.

Both these assumptions are relaxed!


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Model

Round trip is a sequence of trades , =

if

, 0= .Hence, actual no. of trades are (,) . Profit of round trip :

, ,,=

((,)).

Risk Neutral Price Manipulation : Round trip trade ,withexpected value , 0.

An unbounded (risk neutral) Price manipulation: A sequenceof round trip trades {,}= with lim ,

Quasi Arbitrage : An unbounded price manipulation {,}= that satisfies lim

, /Std[ , ] .


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Model

Market Viability: Depends on whether optimaldemand exists!

Traders optimal demand :

argmax, ((,

)) .

A market is weakly viableif the optimal demand ofthe trader exists .

A market is strongly viable if the optimal demandexists uniquely and is zero !

Limited price manipulation possible in a weaklyviable market

Weak viability No quasi arbitrage !


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Model

Expected Price-Impact function:,() [,( ,)] Expected Price-Impact function:

,() [,( ,)]

, 0 , 0 0 , 0 , 0 0 Market Classification :

: Prices have first moments and is fixed i.e. PriceManipulation

:Prices have first moments and ,( ) ,()fornonnegative i.e. Unbounded price manipulation

:Prices have both first and second moments and variances donot grow faster than linearly as i.e. Quasi arbitrage


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Absence of Price Manipulation

CLAIMS

The Expected price update function must be symmetric i.e.

, ,().This yields , 0forsufficiently smaller

.

is continuous everywhere except possibly at the origin i.e.lim , 0when lim . is linear since 1 or 1 induces

price manipulation!

U is linear. Using . =0.


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Proposition 1

Suppose that any trade size is allowed for the trader and

that either :

i. , 0 1orii. The crowds trades are normally distributed.

Then , the absence of price manipulation in requires Uto be linear with nonnegative slope L(R)-a.e. The linearity

of U with nonnegative slope is also implied by each, theabsence of unbounded price manipulation in and theabsence of quasiarbitrage in .


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Theorem 1

Defn : A function: is quasilinear if it has therepresentation on , 0 , ( )-a.e.,where the - Borel Measurable function : satisfies

, , , 0

Theorem

The absence of price manipulation in requires U to be quasilinear. The quasi-linearity of U is also a consequence of each ,

the absence of unbounded price manipulation in , and theabsence of quasiarbitrage in .


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Proposition 2

If either no price manipulation or no quasi arbitrage holds and if

fixed costs c(k) are proportional to , 1then the followingtwo conditions hold

i.

for

>

< 0 ,

ii. 0and 0.


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Sufficient condition for NoPM-NoQA

Proposition 3:

Let Ube linear with nonnegative slope,L(R)a.e. and assume

the crowds trades have a normal distribution. If

fornonnegative

, then (NoPM)-

(NoQA) are all satisfied.

Proposition 4:

Suppose (i)

satisfies the condition given in Proposition 3 and

(ii)the crowds trades are normally distributed .Then , linearity of

U ( with nonnegative slope L(R)a.e.) is equivalent to the

absence of price manipulation in .It is also equivalent toNoUM and NoQA.


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Corollary

Corollary

Assume that the conditions (i) and (ii) in proposition 4 are met.

Then, the absence of quasi arbitrage in market ischaracterized by the strong viability of

.If the trader is risk

neutral , the strong viability of market is equivalent to theabsence of unbounded price manipulation in .


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Time dependent price impact

Liquidity ( first derivative of price impact and price update

function) is varying across time .

, , , , , , .

, , ,(, ,).

For sequences {,}= and {,}= , where , , 0 The price update slope can have any sign but the price impact

slope should only be positive!

Unbounded price manipulation can be implemented in finitely

many trades.


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Theorem 2

Fixing an arbitrary . For all sufficiently large prices ,, noprice manipulation in () is characterized by the positivesemidefiniteness of the matrix

2,,,

,23,3,

,3,24,

,3,4,

, 3, 4, 2,

For any given price update sequence there exists a price-

impact sequence that preserves the absence of price

manipulation.

Non linear ,time dependent price update functions may lead

to a chaotic shapes but not necessarily price manipulation.


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Kyle Model

Black(1995) conjectured that the Kyle Model allowed for

arbitrage opportunities for uninformed agents if they pretend

to be informed.

Kyle showed that for the case of normally and i.i.d trading

volume of the crowd, this game has a unique linearequilibrium where price evolves according to

, , , , , , is endogenously determined Theorem 2 is applicable . For any , Kyles slopes are almost constant and hence price

manipulation is infeasible. Hence , Kyles equilibrium is

strongly viable.


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Proposition 5

Suppose the price update functions {,}= are symmetric ,i.e.,, , 0, 0, and monotonein thesense that , , 0.Then , given the pricedynamics , , , , , ,, normallydistributed ,s and the condition that lim(,), 1 exists ,the absence of unbounded price manipulation in implies thatthe expected price update functions converge pointwise to a linear

function on any interval , 1 , 0


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Some other key points

For a multiple asset case : we can incorporate cross priceimpact functions

All results in theorem 1, and propositions 1 and 4 regardingthe absence of price manipulation in

are literally true for

the multi asset case. Only we impose a restriction of notemporary price impacts.

Gain-Loss Ratio : [+]/[], where is thepayoff of a zero cost portfolio!


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Thank You

Huberman and Stanzl

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