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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,

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Presentation on theme: "Online and Offline Selling in Limit Order Markets Aaron Johnson Yale University Kevin Chang Yahoo! Inc. Workshop on Internet and Network Economics December,"— Presentation transcript:

1 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

2 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

3 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

4 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

5 Limit Order Markets Trading one commodity 5

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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

24 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

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

26 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.

27 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.

28 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

29 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

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

31 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

32 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

33 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

34 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

35 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

36 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

37 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

38 Selling in Limit Order Markets 38 Price Volume

39 Selling in Limit Order Markets Price Volume

40 Selling in Limit Order Markets Price Volume

41 Selling in Limit Order Markets Price Volume

42 Selling in Limit Order Markets Price Volume

43 Selling in Limit Order Markets Price Volume

44 Selling in Limit Order Markets Price Volume

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

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

47 Selling in Limit Order Markets SALE: $ Price Volume

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

49 Selling in Limit Order Markets Price Volume Price Volume

50 Selling in Limit Order Markets Price Volume Price Volume

51 Selling in Limit Order Markets Price Volume Price Volume

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

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

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

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

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

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

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

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

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

61 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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