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

Efficient Market Making via Convex Optimization, and a Connection to Online Learning by J. Abernethy, Y. Chen and J.W. Vaughan Presented by J. Duraj and D. Rishi 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 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. 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. 4 / 16

Reasonable Market Maker properties 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. 5 / 16

Reasonable Market Maker properties Reasonable Market Maker Properties Formally... Let O be an outcome space and ρ : O R K + an arbitrary but efficiently computable payoff function for the securities (complex market). Cost function cost(r r 1,..., r t 1 ) gives cost of buying r given sequence of previous trades r 1,..., r t 1. Condition 1: Path Independence. For any r, r, r with r = r + r one has cost(r r 1,..., r t ) = cost(r r 1,..., r t ) + cost(r r 1,..., r t, r ) Implies existence of C : R K R with cost(r t r 1,..., r t 1 ) = C(r 1 + + r t ) C(r 1 + + r t 1 ). 6 / 16

Reasonable Market Maker properties Reasonable Market Maker Properties Condition 2: Existence of Instantaneous Prices. C is continuous and differentiable everywhere on R K. In particular, p = C(q) are the instantaneous prices at vector q of outstanding securities in the market. 7 / 16

Reasonable Market Maker properties Reasonable Market Maker Properties Condition 3: Information Incorporation. For any q, r R K Implies C is convex. C(q + 2r) C(q + r) C(q + r) C(q) 8 / 16

Reasonable Market Maker properties Reasonable Market Maker Properties Condition 4: No Arbitrage. For all q, r R K there exists o O with C(q + r) C(q) r ρ(o) If violated traders would make profits out of the Market Maker. 9 / 16

Reasonable Market Maker properties Reasonable Market Maker Properties Condition 5: Expressiveness. For any p O and any ɛ > 0 there exists some q R K for which C(q) E o p [ρ(o)] < ɛ Any trader can move prices approximately to his beliefs. 10 / 16

Characterization of Market Makers and Convex Duality Characterization of Market Makers and Convex Duality Theorem (Characterization of Complex Markets) Under Conditions 2-5, C must be convex with (up to technicalities) { C(q) : q R K } = convhull(ρ(o)). (1) Moreover, any convex differentiable function C : R K R respecting (1) must also satisfy conditions 2-5. 11 / 16

Characterization of Market Makers and Convex Duality Characterization of Market Makers and Convex Duality Theorem is important not only for characterization but also design of automated markets. Theorem (Duality Characterization) Assume image of ρ is bounded in R K. Then for any cost function C : R K R satisfying Conditions 2-5, up to technicalities, there exists a strictly convex function R : R K R {, + } with C(q) = sup x q R(x). x convhull(ρ(o)) Furthermore, for any convex R on convhull(ρ(o)), if R is strictly convex on its domain then the cost function defined by the conjugate C = R satisfies Conditions 2-5. Finally, the price vector is given by C(q) = arg max x q R(x). x convhull(ρ(o)) 12 / 16

Market Maker Loss and Loss of Information Market Maker Loss and Loss of Information Bid-ask spread of security bundle r, given some q : BA(q, r) = (C(q + r) C(q)) (C(q) C(q r)). Information Loss in markets without a continuum of quantities: bid-ask spread for the smallest trading unit. Define market depth parameter at q as β = 1 λ C (q). inverse of largest eigenvalue of 2 C(q). Worst case market depth β = inf q R K β(q). One has the bound BA(q, r) r 2 β. 13 / 16

Market Maker Loss and Loss of Information Market Maker Loss and Loss of Information Market Maker s worst-case Monetary Loss: ( wmml = sup q R K sup(ρ(o) q) C(q) + C(0) o O ). Result: If C is so that its R is bounded, then so is Market Maker s Loss. If C is twice-differentiable. wmml const β. There is a tension between lower bound of worst-case loss and upper bound of bid-ask spreads. 14 / 16

Connections to Online Learning Connections to Online Learning Recall the FTRL algorithm: w t = arg min{l t 1 w + 1 w K µ R(w)}. with K convex, compact decision set, R strictly convex differentiable function, µ > 0. Syntactically similar to cost function markets: 1. K O, w x, R R, L t = L t+1 + l t q t = q t 1 + r t. 2. Moreover, updating rules are similar as are period costs to learner/market maker. 15 / 16

Connections to Market Scoring Rules Connections to Market Scoring Rules Given a strictly convex, continuous conjugate R and a proper regular scoring rule s with R(x) = n x i s i (x), i=1 s i (x) = R(x) R(x) x + R(x) x i. one has the following relations: 1. Trade in cost function market q q, x(q) x(q ) gives same profit as in MSR market moving the probabilities x(q) x(q ). 2. Given any probability vector x for which s i (x) is finite for all i in the MSR market there always exists q with C(q) = x in the cost function market. 3. If initial probability x 0 in MSR market is equal to initial price vector C(0) in the cost function market, then both markets have same worst-case loss. 16 / 16