Sublinear and Locally Sublinear Prices - Slides

8/14/2019 Sublinear and Locally Sublinear Prices - Slides

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Sublinear and Locally Sublinear Prices

Alessandro Plasmati

Universit Commerciale L. BocconiMaster of Science in Finance

Relatore: Prof. Erio Castagnoli

July, 2008

Alessandro Plasmati Sublinear and Locally Sublinear Prices 1/16


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Table of Contents.1 Setting.2 Linear Prices - No Arbitrage.3 From Linear Prices to Sublinear Prices.4 Sublinear Prices

No Arbitrage and Super-replications

Frictions on the Riskless Asset

.5 Locally Sublinear PricesSublinear Price Increments

Super-replications

Restricting L to L+

The Pricing Functional with Multiple Price Changes

Deriving and Interpreting c

.6 Conclusions



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SettingOne Period Discrete Time Model

= {1, 2, . . . , i, . . . , m} is the set of states of the world

The nfinancial assets y1,y2, . . . ,yj, . . . ,yn are traded on themarket

yij represents the payoff of asset jif state ioccurs

Y=yij

is the m nmatrix collecting all the yijs

A vector a Rn will represent the number of units bought orsold for each of the nassets

is the row vector containing the prices of the traded securities,

i.e. = [(y1), (y2), . . . , (yj), . . . , (yn)]



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Linear Prices - No Arbitrage

Assumption: Linear functionals map the payoffs of a given randomvariable y in the vector space M Rm to prices (y) R.

Any linear functional can represented by a vector :

F(y) =

mi=1

yiF(ei) = y

Properties of a Linear Functional

.1 Additivity: F(x + y) = F(x) + F(y) = x + y = (x + y)

.2 Homogeneity: F(x) = F(x) = ( x)

Theorem (Fundamental Theorem of Finance)Prices = 0 do not allow for arbitrages of the first and of the second kindif and only if there exists (at least) one (row) vector Rm++ that solvesthe linear system Y= .



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From Linear Prices to Sublinear PricesMarket Incompleteness

Whenever the market is incomplete (i.e. Yis singular), there exists

z / M which can not be replicated by any portfolio a Rn

: thesystem Y a = z does not admit solution.

As a result, any non-replicable contingent claim has not a linearprice. The system

Y=

is solved by infinitely manyvectors , collected in the set L.

Solution: determine the minimum and maximum price consistentwith no arbitrage, if any, as

.1 Fa(z) = supL++ z = minyM{(y) : y z}

.2 Fb(z) = F(z) = infL++ z = minyM{(y) : y z}By the Hahn-Banach Theorem, the functional F(z) = sup

L z isthe unique sublinear extension on the whole Rm of the family oflinear functionals generated by the infinitely many vectors .



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Sublinear Pricesbid-ask spreads possibly for all the traded assets on the market

A functional Fis sublinear if, for any x, y M and , 0,

F(x+ y) F(x) + F(y)

The properties of a sublinear functional:

Subadditivity: F(x + y) F(x) + F(y)

Positive Homogeneity: F(x) = F(x)

Different prices to buy and sell a given security yj are allowed: askprices are collected in the price vector a, bid prices in b. We solvethe system of linear inequalities

b Y a

The set L collects all the solutions such that

L = { : b Y a}



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Sublinear PricesNo Arbitrage and Convenient Super-Replications

The distinction between bid and ask prices creates the possibility that

a random variable z be conveniently super-replicated, so that thereexists a Rn such that

Y a z and F(Y a) < F(z)

The properties of the pricing functional are used to assess the internal

coherence of the market:positivity y 0 F(y) 0 ask nai

monotonicity y 0 F(y) 0 bid nsri

The interval [Fb(y),Fa(y)] generates three possible cases:

[Fb(y),Fa(y)] R+: no convenient super-replications; [Fb(y),Fa(y)] overlaps with R+ but it is not included in it: only

no-arbitrage holds;

[Fb(y),Fa(y)] R+: there are arbitrages and super-replications.



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Sublinear PricesFrictions on the Riskless Asset

The sublinear pricing functional Fcan be rewritten as:

F(y) = maxpP+

Bp y = maxpP+ Ep

[By]

Bdepends on p, unless the riskless asset is replicable, i.e. 1 M: inthis case,

F(y) = BmaxpP+

Ep[y]

In a sublinear setting, prices are maximal discounted expectations offuture payoffs.

Problem: When the riskless asset is affected by frictions, are buyingprices discounted with the buying price of the discount factor Ba?The answer is that it does not always happen.

Interpretation: the market acts as if a separation between securitiesbought with own capitaland securities bought with speculative capitalexists. Therefore, its either

F(y) = BaEp[y] or F(y) = BbEp[y]



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Locally Sublinear PricesBeyond Sublinear Prices: Removal of Positive Homogeneity

Simple Observation: imposing different prices for the purchase andfor the sale of a security is equivalent to setting a portfolio K= 0

whose components signal the point where the price system changesfrom b to a.

Idea: set arbitrary values for the portfolio where prices change, sothat for a given K= [k1 k2] the price system changes from

to ,with for K 0.

Prices increasing with trade size are not positive homogeneous.

Example: assume n= m= 2 and K= [10 20]. Ksplits the a1a2 plane (of

quantities) and the a1a2 plane (ofquantity increments) in four regions

10

20 K

A1

A3

A2

A4

a1

a2

0

translate K

A1

A3 A4

A2

a1

a2

0



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Locally Sublinear PricesSublinear Price Increments

After the translation ofKto the origin, the price system changes in 0,exactly as in the sublinear case.

Intuition: since we are in the plane of quantity increments withrespect to K, price increments should be sublinear. This result canbe proven.

For each of the four regions Ai of the plane a (possibly) different pricevector i applies. As in the standard linear case, we can find

i Y= i for i= 1, . . . , 4

The Fundamental Theorem of Finance can be applied to check for thepresence of arbitrages and super-replications.

If we define y = Y K, and L is the convex hull generated by all thei, the pricing functional is given by

(y) = (y) + maxL+

(y y)

sublinear increments



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Locally Sublinear PricesSuper-Replications

Example: Suppose that in A3 there exists 31

< 0 and that the random

variable x A3. For any y x, it must be (y) < (x) because 31

< 0. We

can repeat the argument until we reach another area where prices arecoherent.

x

A1

A3

A2

A4

y1

400

50

100

400

3

y(140, 180)

y2

A1

A3

A2

A4

y1

400

50

100

400

3

y2

Therefore, we conclude that, when convenient super-replications are possible

in a given area, any r.v. y in that area is conveniently super-replicated by the

random variables with a greater payoff in the negative state, up to the edge of

an adjacent area in which super-replications are not possible. Alessandro Plasmati Sublinear and Locally Sublinear Prices 11/16


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Locally Sublinear PricesRestricting L to L+

The minimum cost super-replicating portfolio(s) delivers the price of

the r.v. when super-replications are not possible. Analogously, we canprice y with

(y) = (y) + maxL+

(y y)

The convex hull L contains some negative state-prices: in order to

eliminate them, we need to derive L+ = L Rm

+ by considering theintersections of the segments connecting the vectors with the axes.

L1

2

2(1.3,0.4)

1(0.1, 0.2)

3(0.05,0.65)

4(1.15, 0.05) L+

1

2

2(1.3,0.4)

1(0.1, 0.2)

3

(0, 0.625)

3

(0, 0.5)

2

(0.5,0)

2

(1.16667, 0)

3(0.05, 0.65)

4(1.15, 0.05)



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Locally Sublinear PricesThe Pricing Functional with Multiple Price Changes

In a market with multiple price changes, let us fix vertex yr of generic

Region r. The pricing functional in the 2m areas of which yr is avertex is

(y) = (yr) + max (y yr)

where the maximum refers to all the areas of which yr is a vertex.

Moreover, if we consider also all the functionals relative to the otherys, their prices are all smaller than or equal to . In fact, those pricesare derived from price vectors surrounding ys which are smaller withrespect to the positive components ofy ys and greater with respectto the negative components ofy ys.

Therefore, we can conclude that

(y) = maxr

(yr) + max

Lr (y yr)



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Locally Sublinear PricesDeriving and Interpreting c

We can manipulate the functional in the following way:

(y) = maxr

(yr) + maxLr

(y yr)

= maxr

maxLr

(yr) + (y yr)

= maxr

maxLr

(yr) + y yr

= maxr

maxLr

y + c

= maxL

y + c

where L = Lr and c = (yr) yr< 0.The pricing functional is convex because it is the maximum of a familyofaffine functionals. The constant c is the value of the functional inthe origin: c depends on the whole layout of the different gridpoints, but also on the prices of the securities.



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Locally Sublinear PricesDeriving and Interpreting c

Interpretation:.1 Of all the possible prices supplied by the intermediaries, the

market sets the most expensive price as the the market price.

.2 The way in which the different intermediaries set their prices isinfluenced by the incremental quantities they are willing tosupply: if they supply securities when the buyer has reachedtheir level of incremental supply, the price per unit is higher.

.3 The higher expense per unit is partially subsidized by theintermediary by discounting a fixed amount c from the total

price..4 Since market prices are set in a conservative way, a large c willbe applied only in a situation where quantities are also large.


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Conclusions

Locally sublinear models are versatile and, at a higher level ofcomplexity, they can accommodate a representation of a market in

which assets are not infinitely liquid, thereby removing positivehomogeneity from prices.

More qualitative insights can be gained by considering:

.1 Transparency: a transparent price system would be structured with asufficiently large number of grid points such that

cs variationsincrease proportionally with size increments.

.2 Liquidity: the liquidity of a given asset is an indication of the ease withwhich it can be bought or sold. In this respect, it is one of thecomponents of bid-ask spreads, and in our setting a lack of liquiditywould be signaled by prices increasing more than linearly with respect

to quantities.

.3 Market Depth: the constants c indicate if and how variations in pricesmake large trades difficult: for instance, we could compare two marketsbased on values of the different c, provided that they have the same K.


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Sublinear and Locally Sublinear Prices - Slides