Efficient Market Making via Convex Optimization, and a Connection to Online Learning

16 Pages·2016·0.14 MB·English

Efficient Market Making via Convex Optimization, and a
Connection to Online Learning
by J. Abernethy, Y. Chen and J.W. Vaughan
PresentedbyJ.DurajandD.Rishi
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 1/16
Outline
1 Motivation
2 Reasonable Market Maker properties
3 Characterization of Market Makers and Convex Duality
4 Market Maker Loss and Loss of Information
5 Connections to Online Learning
6 Connections to Market Scoring Rules
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 2/16
Motivation
Motivation
From economic theory: Arrow-Debreu Securities, promise an amount of
money/bundle of goods in a particular time if some event is realized
(contingent securities).
A securities market is complete if it offers a linearly independent set of
securities over the set of each possible contingency (i.e. for each mutually
exclusive and exhaustive state of the world).
In reality completeness not given:
1. Humans bad at estimating small and near-1 probabilities.
2. It may be computationally intractable to manage large sets of
securities.
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 3/16
Motivation
Motivation
New approach needed: here axiomatic framework for design of
automated market makers with desirable properties even when
completeness not given.
Reduction to Convex Optimization which is abundantly studied.
Method for designing computationally efficient markets.
Connections to Online Learning and Market Scoring rules.
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 4/16
ReasonableMarketMakerproperties
Reasonable Market Maker properties
Informally...
1. Prices in a market should be always well-defined.
2. Market incorporates information from the traders.
3. No Arbitrage.
4. Market is expressive: any set of consistent beliefs can be achieved
by prices in the market.
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 5/16
ReasonableMarketMakerproperties
Reasonable Market Maker Properties
Formally...
Let O be an outcome space and ρ : O→RK an arbitrary but efficiently
+
computable payoff function for the securities (complex market).
Cost function cost(r|r ,...,r ) gives cost of buying r given sequence of
1 t−1
previous trades r ,...,r .
1 t−1
Condition 1: Path Independence. For any r,r(cid:48),r(cid:48)(cid:48) with r = r(cid:48)+r(cid:48)(cid:48) one has
cost(r|r ,...,r ) = cost(r(cid:48)|r ,...,r )+cost(r(cid:48)(cid:48)|r ,...,r ,r(cid:48))
1 t 1 t 1 t
Implies existence of C : RK→R with
cost(r |r ,...,r ) = C(r +···+r )−C(r +···+r ).
t 1 t−1 1 t 1 t−1
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 6/16
ReasonableMarketMakerproperties
Reasonable Market Maker Properties
Condition 2: Existence of Instantaneous Prices. C is continuous and
differentiable everywhere on RK. In particular,
p = ∇C(q)
are the instantaneous prices at vector q of outstanding securities in the
market.
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 7/16
ReasonableMarketMakerproperties
Reasonable Market Maker Properties
Condition 3: Information Incorporation. For any q,r ∈ RK
C(q+2r)−C(q+r) ≥ C(q+r)−C(q)
Implies C is convex.
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 8/16
ReasonableMarketMakerproperties
Reasonable Market Maker Properties
Condition 4: No Arbitrage. For all q,r ∈ RK there exists o ∈ O with
C(q+r)−C(q) ≥ r·ρ(o)
If violated traders would make profits out of the Market Maker.
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 9/16
ReasonableMarketMakerproperties
Reasonable Market Maker Properties
Condition 5: Expressiveness. For any p ∈ ∆ and any (cid:15) > 0 there exists
|O|
some q ∈ RK for which
||∇C(q)−E [ρ(o)]|| < (cid:15)
o∼p
Any trader can move prices approximately to his beliefs.
byJ.Abernethy,Y.ChenandJ.W.Vaughan(PresentedPbreyseJn.tDatuiroanjfaonrdCDS.2R3i6srhi) 10/16