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Online and Offline Selling in Limit Order Markets Aaron Johnson Yale University Kevin Chang Yahoo! Inc. Workshop on Internet and Network Economics December, 17 th

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Limit Order Markets Match buyers with sellers Electronic Communication Networks (ECNs) – NASDAQ – Instinet – NYSE-Euronext Prediction Markets – Intrade – Iowa Electronic Markets Market makers Market orders, fill or kill, cancellation 2

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Results Reservation price algorithm for online selling has competitive ratio e log(R), R = p max /p min. (improves O(logR logN) of [KKMO04]) Optimal selling offline is NP-Hard. PTAS for offline selling when number of prices is constant. Extend PTAS to offline buying 3

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Related Work [EKKM06] Even-Dar, Kakade, Kearns, and Mansour. (In)Stability properties of limit order dynamics. ACM EC [KKMO04] Kakade, Kearns, Mansour, and Ortiz. Competitive algorithms for VWAP and limit order trading. ACM EC [LPS07] Lorenz, Panagiotou, and Steger. Optimal algorithms for k-search with applications in option pricing. ESA

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Limit Order Markets Trading one commodity 5

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) BUY1$3 6

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book BUY5$1 BUY2$2 BUY1$3 SELL1$5 SELL3$7 SELL10$10 7

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL1$5 SELL3$7 SELL10$10 8

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL1$5 SELL3$7 SELL10$10 BUY2$6 9

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL1$5 SELL3$7 SELL10$10 BUY2$6 10

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL0$5 SELL3$7 SELL10$10 BUY1$6 1 $5 11

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL3$7 SELL10$10 BUY1$6 12

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL3$7 SELL10$10 BUY1$6 13

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL3$7 SELL10$10 BUY1$6 14

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Limit Order Problems Sequence of orders Volume to trade Insert orders to maximize value at given volume 15 General Problem Options Online / Offline / Probabilistic Buy / Sell / Both Exact volume / Volume constraint

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Limit Order Problems Sequence of orders Volume to trade Insert orders to maximize value at given volume 16 General Problem Options Online / Offline / Probabilistic Buy / Sell / Both Exact volume / Volume constraint

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Limit Order Problems Sequence of orders: (σ 1,…, σ n ) : σ i = Volume to sell : N Insert sell orders to maximize revenue. – Output (σ 1,τ 1,σ 2,τ 2,…, σ n,τ n ), τ i =. – Σ i v τ iN – Maximize revenue earned from τ i sales. 17 Offline Selling

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Limit Order Problems Results 1.Problem is NP-Hard, even when there are only three prices in sequence. 2.Problem with two prices is linear-time solvable. 3.Exists a Polynomial-Time Approximation Scheme when number of prices is constant. 18 Offline Selling

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Limit Order Problems Results 1.Problem is NP-Hard, even when there are only three prices in sequence. 2.Problem with two prices is linear-time solvable. 3.Exists a Polynomial-Time Approximation Scheme when number of prices is constant. 19 Offline Selling

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1.Give a canonical form for optimal solutions to case when input sequence has only three prices. 2.Form leads to algorithm for two-price case. 3.Reduce K NAPSACK to three-price instance. 1.Easy to see that solutions to K NAPSACK instance give solutions to three-price instance. 2.Canonical form guarantees that a solution to three-price selling gives a solution to K NAPSACK. 20 Proving Hardness

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Optimal Offline Selling Lemma 1: We can assume that all sales at the highest price i) are made by the algorithm and ii) have sell orders that are placed at the beginning. Lemma 2: We can assume that all sell orders at the lowest price that are inserted by the algorithm are placed immediately after the last sale made by the algorithm at a higher price. 21

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Two-Price Offline Algorithm With only two prices for orders (high and low), use this algorithm: At the beginning, place a sell order for volume N at the high price. If volume sold is N, return this. Else, After each high-price sale, calculate value of inserting sell order for remaining volume at low price. Return the maximum sequence. 22

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Three-Price Offline Selling Three prices for orders (high: p h, medium: p m, and low: p l ). Lemma 3: We can assume that the algorithm inserts any medium-price orders i) immediately after high-price sales and ii) such that they are tight, i.e., increasing the volume would reduce the volume of high-price sales. Theorem 1: Three-price offline selling is NP- Hard. 23

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Reducing K NAPSACK to 3-Price Selling K NAPSACK – n items (w i, v i ) – Capacity C – Value V – Find subset S [n] such that i S w i C and i S v i V 24

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Reducing K NAPSACK to 3-Price Selling Let σ i be the sequence Let α = (σ 1, σ 2, …, σ n ).

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Reducing K NAPSACK to 3-Price Selling Let σ i be the sequence Let α = (σ 1, σ 2, …, σ n ). Step High price Med. price Canonical Optimum Alg. Order Vol. Alg. Sale Vol.

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Reducing K NAPSACK to 3-Price Selling Let σ i be the sequence Let α = (σ 1, σ 2, …, σ n ). Step Alg. Order Vol. High price Med. price At start place high sell. Canonical Optimum Alg. Sale Vol.

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Reducing K NAPSACK to 3-Price Selling Let σ i be the sequence Let α = (σ 1, σ 2, …, σ n ). Step Alg. Sale Vol. Alg. Order Vol. High price Med. price At start place high sell. After high sales, medium sell volumes 0 and a i +w i are tight. More is not optimal. Canonical Optimum

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Reducing K NAPSACK to 3-Price Selling Let ω be the sequence With initial high sale, books at start of ω just have low buys. This is maintained. Canonical Optimum Let σ = (α, ω). Let i (l) be revenue after σ i with l fewer initial low buys. 29 n (l)= p m 2 ( i w i )+p l (C-l) : lC p m 2 ( i w i )-p m (l-C) : lC n lC

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Reducing K NAPSACK to 3-Price Selling 30 Step Alg. Sale Vol. Alg. Order Vol. High price Med. price i C σiσi

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Reducing K NAPSACK to 3-Price Selling 31 Step Alg. Sale Vol. Alg. Order Vol. High price Med. price i-1 C Inserting a medium sell decreases later low buys by w i and increases revenue by (k) v i. i S if medium after σ i. i wiwi (k)vi(k)vi σiσi

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Reducing K NAPSACK to 3-Price Selling At beginning of σ 0, l=0. Can set p m, p l to ensure that should not shift by more than C. Can set a i, b i to ensure that medium insertion of a i +w i provides (k) v i revenue but more is not profitable. K NAPSACK solution leads to stated 3-price solution. Canonical form guarantees optimal solution in form that can be converted to a K NAPSACK solution. 32

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Conclusions Prove optimal competitive ratio for reservation price algorithm for online selling of e log(R), R = p max /p min. Optimal selling offline is NP-Hard. PTAS for offline selling when number of prices is constant. Limit order markets are a basic market mechanism with many open problems. 33 Online / Offline / Probabilistic Buy / Sell / Both Exact volume / Volume constraint

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL3$7 SELL10$10 BUY1$6 SELL1$7 34

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL3$7 SELL10$10 BUY1$6 SELL1$7 35

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Limit Order Markets Trading one commodity Order 1.BUY/SELL 2.Volume 3.Price o Lowest (SELL) o Highest (BUY) Buy book / Sell book Matching algorithm 1. Match new order with existing orders. 2.Remaining volume goes on a book. BUY5$1 BUY2$2 BUY1$3 SELL1$7 SELL10$10 BUY1$6 SELL3$7 36

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Offline Selling 1.Inserting sell orders affects the possible revenue gained later in the sequence. In fact, it can only lower it. 2.Inserting a sell order of volume V can cause at most volume V change in the books later in the sequence. Thus, the sales change by at most volume V. 37 Main Observations

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Selling in Limit Order Markets 38 Price Volume

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Selling in Limit Order Markets Price Volume

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Selling in Limit Order Markets Price Volume

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Selling in Limit Order Markets Price Volume

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Selling in Limit Order Markets Price Volume

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Selling in Limit Order Markets Price Volume

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Selling in Limit Order Markets Price Volume

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Selling in Limit Order Markets SALE: $3 Price Volume

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Selling in Limit Order Markets SALE: $3 4. Price Volume

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Selling in Limit Order Markets SALE: $ Price Volume

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Selling in Limit Order Markets 48 Price Volume SALE: $ SALE: $5 1

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Selling in Limit Order Markets Price Volume Price Volume

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Selling in Limit Order Markets Price Volume Price Volume

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Selling in Limit Order Markets Price Volume Price Volume

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Selling in Limit Order Markets Price Volume SALE: $ Price Volume

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Selling in Limit Order Markets Price Volume SALE: $ Price Volume

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Selling in Limit Order Markets Price Volume SALE: $ Price Volume

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Selling in Limit Order Markets Price Volume SALE: $ Price Volume

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Selling in Limit Order Markets SALE: $ Price Volume SALE: $ SALE: $ Price Volume

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Selling in Limit Order Markets SALE: $ Price Volume SALE: $ SALE: $ Price Volume

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Selling in Limit Order Markets SALE: $ Price Volume SALE: $ SALE: $ Price Volume

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Selling in Limit Order Markets SALE: $ Price Volume SALE: $ SALE: $ Price Volume

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Selling in Limit Order Markets SALE: $ SALE: $5 Price Volume SALE: $ SALE: $ SALE $3 Price Volume

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Selling in Limit Order Markets Lemma 0 ([KKMO04]): Inserting a unit-volume sell order results in at most one less sale from the original sell orders. 61

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