Call Auctions with Contingent Orders

1. Call Auctions with Contingent Orders Yuzhao Wu E-mail:hbxfwtt@163.com Tel:86-10-13810359471

2. Index • Abstract • Introduction • Call Auctions • CACO • CACOP algorithm • Properties • Market Clearing • Non-Discriminatory Pricing • Discriminatory Pricing • Future Works • Call Auctions & Market Clearing? • References

3. Abstract • Call auctions with contingent orders(CACO) is a mechanism which can be widely used in stock exchanges. In this talk, we will show some of its properties and try to make a combine with market clearing. Market clearing is a problem which has been proved to be NP-complete in discriminatory situation.

4. Introduction • Traditional stock exchanges mechanism: call auctions at the opening and closing of trading sessions • Problem: opening volatility stock price changes tend to be higher at the opening interval than other trading time during a day • Mechanism with contingent orders investor who holds stock A and would like to buy stock B only if she sells stock A [Isa E. Hafalir and SerkanImisiker, 2011]

5. Call Auctions • Current call auction mechanism investors’ orders: individual stock, price and quantity • Call auction(CA): values and quantities pair (v,q), the price of stock p buyers: would like to buy the stock for q units only if v≤p sellers: would like to sell the stock for q units only if v≥p call auction price(CAP): one of the prices that maximize the total sales- the minimum of “total quantities demanded by buyers with values v≤p” and “total quantities supplied by sellers with values v≥p"

6. Call Auctions with Contingent Orders • Call Auctions with Contingent Orders(CACO) • buy orders • sell orders • contingent orders investor who holds stock A and would like to buy stock B (at most price x) only if she sells stock A (q units for at price at least y) • Call Auctions with Contingent Orders Price (CACOP)

7. CACOP algorithm • Stage 1: Compute all the buy orders and sell orders to get price vector p1with CA mechanism. If any of the contingent orders is executed with p1, put the buy orders corresponding to it into the auction book and moves to Stage 2. Otherwise ends with the resulting CACOP p1. • Stage k: Compute all the buy orders and sell orders to get price vector pk(in the updated auction) with CA mechanism . If any of the contingent orders is executed with pk, put the buy orders corresponding to it into the auction book and moves to Stage k+1. Otherwise ends with the resulting CACOP pk.

8. Example for CACOP algorithm(1) Stage 1:By CA mechanism: Stock A: Price-.99 Volume-19,000 Stock B: Price-.50 Volume-12,000

9. Example for CACOP algorithm(2) 1.01 9000*0.51/1.01 =4544 .52 12000*0.99/0.52= 22846 Stage 2:By CA mechanism: Stock A: Price-.99 Volume-19,000 Stock B: Price-.52 Volume-22,846

10. Example for CACOP algorithm(3) Stage 3:By CA mechanism: Stock A: Price-1.00 Volume-19,544 Stock B: Price-.52 Volume-22,846

11. Incentives Properties of CA(1) • Proposition 1 There are no remaining matching buy and sell orders at the CACO. Moreover, the trading volume of CACO is greater than or equal to the volume with CA. • Proof Just follows the definition. • Remark 1 Since there are no remaining matching buy and sell orders at the CACO, ceteris paribus, the volatility after opening CACO will be lower than that of opening CA.

12. Incentives Properties of CA(2) • Proposition 2 In a CA(CACO), given any type announcement of other investors, an investor is never better off (compared to truthful announcement) by announcing another type, unless with that announcement she becomes a price setter (that is, CAP is equal to her announced value). • Examples All prices in [0.9, 0.6] maximize sales (3000 quantities) and 0.8 will be chosen as CAP. However, if buyer 2 and both sellers announce their types truthfully, buyer 1 has a strict incentive to announce her value as 0.7, as with that deviation, the price would be 0.7 and she would be strictly better off.

13. Questions Remaining • Truthful? • Complexity? n is the number of orders • For each stage, we run O(n2) time.(In fact, O(nlogn)) • For each stage, we should at least execute one contingent order, otherwise the algorithm will end. Therefore, we run at most O(n) stages. • Thus we get the algorithm run in O(n2logn). • Right? Optimal?

14. Market Clearing • In classic economic theory of supply and demand carves(called partial equilibrium theory [Mas-Colell, Whinston, & Green 1995]): the market is cleared at some per-unit price for which supply equals demand • Auctioneer: achieve profit from the same supply/demand curves by reducingthe number of units traded, and charging one per-unit price to the buyers while paying a lower per-unit price to the sellers. [TuomasSandholm & SubhashSuri, 2002]

15. The Market Model • n sellers and m buyers • seller: supply curve s(p) for price p. s and p are non-negative. • buyer: submits a demand curved(p). d and p are non-negative. • We assume that supply/demand curves are piecewise linear.

16. Non-Discriminatory Pricing • Non-discriminatory market: two clearing prices, one for all the sellers p*ask, and one for all the buyers p*bid. • For , maximize the auctioneer’s profit: • Given n piecewise linear curves each of which has at most k pieces, market clearing can be solved in time O(nklog(nk))

17. Non-Discriminatory Pricing(2)

18. Non-Discriminatory Pricing(3) • Since all buyers are cleared at the same price p*bid, we can infer quantities sold to each individual buyer by evaluating their curves at p*bid together. The same holds for the sellers. • q* is either in q*i or breakpoints for each piece. • We can scan them in O(nklog(nk)) for doing a sorting in O(nklog(nk)) for initial and then check all the O(nk) points each for a O(1) time.

19. Non-Discriminatory Pricing(4)

20. Discriminatory Pricing • Discriminatory market: the market can clear each seller and each buyer at a distinct unit price. • For , maximize the auctioneer’s profit: • Discriminatory market clearing is NP-complete.

21. Discriminatory Pricing(2) • Reduce from the knapsack problem [Garey & Johnson 1979], and applies to the restricted case of one seller and multiple buyers. • Knapsack problem {(s1, v1), (s2, v2),…, (sn,vn), Z} Z: the knapsack capacity siand vi: the size and value of item I Goal: maximum value with total size at most Z.

22. Discriminatory Pricing(3) • Seller: Z units of the goods with a step function bid (0,Z) (the seller can sell at most Z goods at any price but no more than Z at any price) • Buyer i: with step function bid (vi/si, si) • Goal: choose a subset of buyer bids maximizing the total revenue which is not more than Z, which is equivalent to a solution of the knapsack.

23. Discriminatory Pricing(4) • With linear supply/demand curves: discriminatory markets can be cleared in polynomial time. • Theorem In a multi-buyer, multi-seller discriminatory price exchange with linear supply/demand curves, a profit maximizing clearing can be determined in O(NlogN)time, where N is the number of participants.

24. Future Works • Call Auctions & Market Clearing? • Non-Discriminatory / Discriminatory • Contingent Orders • Auctioneer • Complexity? • Rightness? • Mixed Mechanism?

25. References • [1]Isa E. Hafalir and SerkanImsiker(2011), “Call Auctions with Contingent Orders”, in SSRN Electronic Journal [1556-5068] • [2]TuomasSandholm, and SubhashSuri(2002), “Optimal Clearing of Supply/Demand Curves”. in Proceedings of the 13th Annual International Symposium on Algorithms and Computation (ISAAC), Vancouver, Canada. • [3]TuomasSandholm, and SubhashSuri(2001), ”Market Clearability”, in Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI). • [4]jayant R, Andrew J and Ho S.Lee(2000), “Computational aspects of clearing continuous call double auctions with assignment constraints and indivisible demand”, in IBM Research Report RC21660 • [5]Evan Gatev, William N, and K. Geert(2006), “Pairs Trading: Performance • of a Relative-Value Arbitrage Rule”, in The Review of Financial Studies 1v19n3