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  1. Online and Offline Selling in Limit Order Markets Aaron JohnsonYale University Kevin ChangYahoo! Inc. Workshop on Internet and Network Economics December, 17th 2008

  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

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

  4. Related Work • [EKKM06] Even-Dar, Kakade, Kearns, and Mansour. (In)Stability properties of limit order dynamics. ACM EC 2006. • [KKMO04] Kakade, Kearns, Mansour, and Ortiz. Competitive algorithms for VWAP and limit order trading. ACM EC 2004. • [LPS07] Lorenz, Panagiotou, and Steger. Optimal algorithms for k-search with applications in option pricing. ESA 2007.

  5. Limit Order Markets • Trading one commodity

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

  7. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book SELL 10 $10 SELL 3 $7 SELL 1 $5 BUY 1 $3 BUY 2 $2 BUY 5 $1

  8. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 SELL 1 $5 BUY 1 $3 BUY 2 $2 BUY 5 $1

  9. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 SELL 1 $5 BUY 2 $6 BUY 1 $3 BUY 2 $2 BUY 5 $1

  10. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 BUY 2 $6 SELL 1 $5 BUY 1 $3 BUY 2 $2 BUY 5 $1

  11. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 BUY 1 $6 SELL 0 $5 1 $5 BUY 1 $3 BUY 2 $2 BUY 5 $1

  12. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 BUY 1 $6 BUY 1 $3 BUY 2 $2 BUY 5 $1

  13. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 BUY 1 $6 BUY 1 $3 BUY 2 $2 BUY 5 $1

  14. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 BUY 1 $6 BUY 1 $3 BUY 2 $2 BUY 5 $1

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

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

  17. Limit Order Problems Offline Selling • Sequence of orders: (σ1,…, σn) : σi = <B/S,vi,pi> • Volume to sell : N • Insert sell orders to maximize revenue. • Output (σ1,τ1,σ2,τ2,…, σn,τn), τi = <S,vτi,p τi>. • Σivτi≤N • Maximize revenue earned from τi sales.

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

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

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

  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.

  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.

  23. Three-Price Offline Selling • Three prices for orders (high: ph, medium: pm, and low: pl). 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.

  24. Reducing Knapsack to 3-Price Selling Knapsack • n items (wi, vi) • Capacity C • Value V • Find subset S [n] such that iSwi ≤ C and iS vi ≥ V

  25. Reducing Knapsack to 3-Price Selling • <B, pm, ai+wi> • <S, pl, wi> • <B, pl, wi> • <S, pm, ai> • <B, ph, ai+bi>. Let σibe the sequence Let α = (σ1,σ2,…,σn).

  26. Reducing Knapsack to 3-Price Selling Canonical Optimum • <B, pm, ai+wi> • <S, pl, wi> • <B, pl, wi> • <S, pm, ai> • <B, ph, ai+bi>. Let σibe the sequence Let α = (σ1,σ2,…,σn). High price Alg. OrderVol. Med. price Step Alg. Sale Vol.

  27. Reducing Knapsack to 3-Price Selling Canonical Optimum • <B, pm, ai+wi> • <S, pl, wi> • <B, pl, wi> • <S, pm, ai> • <B, ph, ai+bi>. Let σibe the sequence • At start place high sell. Let α = (σ1,σ2,…,σn). High price Alg. OrderVol. Med. price Step Alg. Sale Vol.

  28. Reducing Knapsack to 3-Price Selling Canonical Optimum • <B, pm, ai+wi> • <S, pl, wi> • <B, pl, wi> • <S, pm, ai> • <B, ph, ai+bi>. Let σibe the sequence • At start place high sell. • After high sales, medium sell volumes 0 and ai+wi are tight. More is not optimal. Let α = (σ1,σ2,…,σn). High price Alg. OrderVol. Med. price Step Alg. Sale Vol.

  29. Reducing Knapsack to 3-Price Selling Let ω be the sequence Canonical Optimum • <S, pl, iwi-C> • <B,pm, pm(iwi)>. • With initial high sale, books at start of ωjust have low buys. This is maintained. Let σ = (α, ω). Let i(l) be revenue after σiwith l fewer initial low buys. n n(l)= pm2(iwi)+pl(C-l) : l≤C pm2(iwi)-pm(l-C) : l≥C C l

  30. Reducing Knapsack to 3-Price Selling High price Alg. OrderVol. Med. price σi Step Alg. Sale Vol. i C

  31. Reducing Knapsack to 3-Price Selling High price Alg. OrderVol. Med. price σi Step Alg. Sale Vol. i-1 • Inserting a medium sell decreases later low buys by wiand increases revenue by (k) vi. • iS if medium after σi . i (k)vi wi C

  32. Reducing Knapsack to 3-Price Selling • At beginning of σ0,l=0. • Can set pm, pl to ensure that  should not shift by more than C. • Can set ai, bi to ensure that medium insertion of ai+wi provides (k) vi revenue but more is not profitable. • Knapsack solution leads to stated 3-price solution. • Canonical form guarantees optimal solution in form that can be converted to a Knapsack solution.

  33. Conclusions • Prove optimal competitive ratio for reservation price algorithm for online selling of e log(R), R = pmax/pmin. • 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. Online / Offline / Probabilistic Buy / Sell/ Both Exact volume / Volume constraint

  34. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 SELL 1 $7 BUY 1 $6 BUY 1 $3 BUY 2 $2 BUY 5 $1

  35. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 3 $7 SELL 1 $7 BUY 1 $6 BUY 1 $3 BUY 2 $2 BUY 5 $1

  36. Limit Order Markets • Trading one commodity • Order • BUY/SELL • Volume • Price • Lowest (SELL) • Highest (BUY) • Buy book / Sell book • Matching algorithm • Match new orderwith existing orders. • Remaining volumegoes on a book. SELL 10 $10 SELL 1 $7 SELL 3 $7 BUY 1 $6 BUY 1 $3 BUY 2 $2 BUY 5 $1

  37. Offline Selling Main Observations • Inserting sell orders affects the possible revenue gained later in the sequence. In fact, it can only lower it. • 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.

  38. Selling in Limit Order Markets Volume 1 -1 1 2 3 4 5 Price

  39. Selling in Limit Order Markets <B, 2, $1> Volume 1 -1 1 2 3 4 5 Price

  40. Selling in Limit Order Markets <B, 2, $1> Volume 1 -1 1 2 3 4 5 Price

  41. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> Volume 1 -1 1 2 3 4 5 Price

  42. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> Volume 1 -1 1 2 3 4 5 Price

  43. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> <B, 1, $4> Volume 1 -1 1 2 3 4 5 Price

  44. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> <B, 1, $4> Volume 1 -1 1 2 3 4 5 Price

  45. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> <B, 1, $4> SALE: $3 Volume 1 -1 1 2 3 4 5 Price

  46. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> <B, 1, $4> SALE: $3 <S, 2, $5> Volume 1 -1 1 2 3 4 5 Price

  47. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> <B, 1, $4> SALE: $3 <S, 2, $5> <B, 1, $5> Volume 1 -1 1 2 3 4 5 Price

  48. Selling in Limit Order Markets <B, 2, $1> <S, 1, $3> <B, 1, $4> SALE: $3 <S, 2, $5> <B, 1, $5> SALE: $5 Volume 1 -1 1 2 3 4 5 Price

  49. Selling in Limit Order Markets Volume <B, 2, $1> <S, 1, $3> <B, 1, $4> <S, 2, $5> <B, 1, $5> 1 -1 1 2 3 4 5 Price <S, 2, $1> <B, 2, $1> <S, 1, $3> <S, 1, $3> <B, 1, $4> <S, 1, $5> <S, 2, $5> <B, 1, $5> Volume 1 -1 1 2 3 4 5 Price

  50. Selling in Limit Order Markets Volume <B, 2, $1> <S, 1, $3> <B, 1, $4> <S, 2, $5> <B, 1, $5> 1 -1 1 2 3 4 5 Price <S, 2, $1> <B, 2, $1> <S, 1, $3> <S, 1, $3> <B, 1, $4> <S, 1, $5> <S, 2, $5> <B, 1, $5> Volume 1 -1 1 2 3 4 5 Price