# Pricing of Limit Orders in the Electronic Security Trading System Xetra

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1 Pricing of Limit Orders in the Electronic Security Trading System Xetra Li Xihao Bielefeld Graduate School of Economics and Management Bielefeld University, P.O. Box D Bielefeld, Germany April 2004 Abstract In this paper we propose a theoretical framework for the pricing in auctions of limit orders in the Xetra System, the Electronic Security Trading System operated by German Stock Exchange(Gruppe Deutsche Börse). The price determination process is described and the volume maximizing price is formulated with an existence proof for such a price in our theoretical framework. The fundamental trading principles in Xetra are investigated and the price in Xetra, called Xetra Price, is derived from the volume maximizing price. The author wants to thank Dr. Jan Wenzelburger for his patience and precious advice.

4 Pricing of Limit Orders in the Electronic Security Trading System Xetra 3 where A D i = [0, a i ] and d i > 0 for every i = 1,..., I. 1 A D i (p) is the characteristic function such that: { 1 when p A D i, 1 A D i (p) := 0 when p R + \ A D i. d i is the amount that the agent wants to buy for this security, and a i is the highest price that this agent wants to pay for buying per unit of the security. Any price no higher than a i will trigger this bid to be executed. The aggregate demand function is defined as the sum of the individual demand functions: I Φ D : R + R +, p L D i (p). (2) Without loss of generality, let a I >... > a 2 > a 1 > 0, then we have the following lemma: Lemma 1. For a I >... > a 2 > a 1 > 0, the aggregate demand function Φ D (p) is a non-increasing function. More specifically, Φ D (p) = i=1 I α i 1 Ai (p) (3) i=0 where α i = I k=i+1 d k, for i = 0, 1,..., I 1, α I = 0; and A 0 = [0, a 1 ], A i = (a i, a i+1 ], for i = 1,..., I 1, A I = (a I, + ). Proof : Since a I >... > a 2 > a 1 > 0, noticing that I i=0 A i = R + and A i A j = i, j {0,..., I} with i j, it is obvious that {A 0,..., A I } is the partition of R +. Considering the value of Φ D (p) on its domain R + is equivalent to considering the value of Φ D (p) on the partition {A 0,..., A I }. p A 0 implies that all bids are executable since A 0 = A D 1 A D 2... A D I. The corresponding aggregate bids volume is α 0 = I k=1 d k. p A i with i = 1,..., I 1 implies that bids from i + 1 to I are executable since A i A D k = for k = 1,..., i and A i A D i+1... A D I. The corresponding aggregate bids volume is α i = I k=i+1 d k with i = 1,..., I 1. p A I implies that no bids can be executable since A i A D k = for k = 1,..., I. The corresponding aggregate bids volume is α I = 0. Therefore, we obtain the specific form of the aggregate demand function. Observing d k > 0 for k = 1,..., I, since α i = I k=i+1 d k for i = 0, 1,..., I 1 and α I = 0, we have: α 0 > α 1 >... > α I = 0. (4) This implies that the aggregate demand function is a non-increasing function. This proves this lemma.

5 Pricing of Limit Orders in the Electronic Security Trading System Xetra Supply-to-Sell (Asks) Schedule Each ask j can be stated as a price-quantity pair (b j, s j ) and treated as an individual supply function. We define the individual supply function that represents an ask as a step function which is analogous to the individual demand function: L S j (p) : R + R +, p s j 1 B S j (p) (5) where Bj S = [b j, ) and s j > 0 for every j = 1,..., J. 1 B S j (p) is the characteristic function such that: { 1 when p Bj S, 1 B S j (p) := 0 when p R + \ Bj S. s j is the amount that the agent wants to sell for this security, and b j is the lowest price that this agent wants to sell for each unit of the security. Any price that is no lower than b j will trigger this ask to be executed. The aggregate supply function is defined as: Φ S (p) : R + R +, p J L S j (p). (6) j=1 Without loss of generality, let b J following lemma: >... > b 2 > b 1 > 0, then we have the Lemma 2. For b J >... > b 2 > b 1 > 0, the aggregate supply function Φ S (p) is a non-decreasing function. More specifically, Φ S (p) = J β j 1 Bj (p) (7) j=0 where β 0 = 0, β j = j k=1 s k, for j = 1,..., J; and B 0 = [0, b 1 ), B j = [b j, b j+1 ), for j = 1,..., J 1, B J = [b J, + ). Proof : Since b J >... > b 2 > b 1 > 0, noticing that J j=0 B j = R + and B j B k = j, k {0,..., J} and k j, it is obvious that {B 0,..., B J } is the partition of R +. Considering the value of Φ S (p) on its domain R + is equivalent to considering the value of Φ S (p) on the partition {B 0,..., B J }. p B 0 implies that no asks can be executable since B 0 Bk S = for k = 1,..., J. The corresponding aggregate asks volume is β 0 = 0. p B j with j = 1,..., J 1 implies that asks from 1 to j are executable since B j Bk S = for k = j + 1,..., J, B j B1 S, and B j Bj S... B1 S with j > 1. The corresponding aggregate asks volume is β j = j k=1 s k.

6 Pricing of Limit Orders in the Electronic Security Trading System Xetra 5 p B J implies that all asks are executable. The corresponding aggregate asks volume is β J = J k=1 s k. Therefore, we obtain the specific form of the aggregate supply function. Observing s k > 0 for k = 1,..., J, since β 0 = 0 and β j = j k=1 s k for j = 1,..., J, we have: β J >... > β 1 > β 0 = 0 (8) This implies that the aggregate supply function is a non-decreasing function. Therefore, we have proved this lemma Executable Order Volume and Surplus Let p R +, when the aggregate demand Φ D (p) is greater than the aggregate supply Φ S (p), only part of Φ D (p) can be satisfied while all Φ S (p) could be executed and vice versa. This implies that only the minimum of Φ D (p) and Φ S (p) is executable. Therefore, the executable order volume is defined as the trading volume function: Φ V : R + R +, p min {Φ D (p), Φ S (p)}. (9) The highest executable order volume V max trading volume function defined by: is the maximum value of the V max := max {Φ V (p) p R + }. Notice that the existence of V max is trivial: the image of the trading volume function Φ V is a finite set because the images of Φ D and Φ S have finitely many values. Therefore, V max = max {Φ V (p) p R + } exists. For convenience of notation, we also define the set of volume maximizing prices as Ω := {p R + Φ V (p) = V max }. The excess demand for each p R + is as usual defined by the excess demand function: Φ Z (p) : R + R, p Φ D (p) Φ S (p). (10) The surplus in Xetra is defined as the positive difference between the aggregate demand Φ D (p) and the aggregate supply Φ S (p). The absolute value of the excess demand Φ Z (p) thus is the surplus. 2.2 Price Determination Phase Traders submit their bids and asks during the call phase. When an order enters into the order book, it is labelled with a time tag. Orders are executed by price/time priority in Xetra, thus the time tag attached in each order determines

7 Pricing of Limit Orders in the Electronic Security Trading System Xetra 6 the ranking of execution in the order book. Each order is corresponding to a position in the sequence of execution in the order book. After a minimum time period the call phase stops with a random end. The call phase is followed by the price determination phase. During the price determination phase, the order book is closed. New orders are not allowed to enter. Therefore, the status of the order book is given and fixed in this phase. In other words, all the bids (a i, d i ) i=1,...,i and asks (b j, s j ) j=1,...,j as well as their execution sequence in the order book are given. Given all the bids (a i, d i ) i=1,...,i and asks (b j, s j ) j=1,...,j, the Xetra Price P Xetra derived from the highest executable order volume prices is determined by the pricing mechanism using some specific matching rules. These matching rules are indicated in page 32 of German Stock Exchange (Gruppe Deutsche Börse 2003), which are the following: Rule 1: The auction price is the price with the highest executable order volume and the lowest surplus in the order book. Rule 2: If more than one candidate price satisfies Rule 1, there are two cases: Rule 2.1: If the surplus for the price satisfied with Rule 1 is on the demand side, then the auction price is stipulated according to the highest candidate price. Rule 2.2: If the surplus for the price satisfied with Rule 1 is on the supply side, then the auction price is stipulated according to the lowest candidate price. Rule 3: If more than one candidate price satisfies Rule 1 and Rule 2, a reference price P ref designated by Xetra is included as an additional criterion. 4 There are several cases with the reference price included. Rule 3.1: The auction price is the highest candidate price if the reference price is higher than the highest candidate price. Rule 3.2: The auction price is the lowest candidate price if the reference price is lower than the lowest candidate price. Rule 3.3: The auction price is equal to the reference price if the reference price lies between the highest candidate price and the lowest candidate price. Rule 4: If Rule 1 to Rule 3 fail, there exists no auction price. Notice that Rule 4 implies that there could be no executable order volume in Xetra, in this case V max = 0 and there exists no Xetra Price. First we provide a description of the set of volume maximizing prices in the following proposition: 4 This could be the case of the existence of a surplus of supply for one set of candidate prices and a surplus of demand for another set of candidate prices or the case of no surplus.

8 Pricing of Limit Orders in the Electronic Security Trading System Xetra 7 Proposition 1. If V max > 0, then Ω = [p, p], with p : = max {p R + Φ D (p) V max } (11) p : = min {p R + Φ S (p) V max } (12) Notice that p is the best bid price and p is the best ask price. When the best bid price is greater than the best ask price, that is p p, the order book is crossed and there exists some executable transaction. Analogously, p < p implies an uncrossed order book and no transaction takes place. Proof: In order to establish Proposition 1 the following lemma is needed. Lemma 3. If V max > 0, then p and p are well defined with p p. The proof of Lemma 3 is provided in the appendix. This lemma shows that there exists some strictly positive highest executable order volume only if the order book is crossed. We will show that Ω = [p, p] in two steps: STEP 1: To prove [p, p] Ω Since Φ S (p) is a non-decreasing function and Φ D (p) a non-increasing function, we have: Φ S (p) Φ S (p) V max for p p, Φ D (p) Φ D (p) V max for p p. By definition of Φ V (p), we thus have: Φ V (p) V max for p [p, p]. Notice that V max = max {Φ V (p) p R + }, thus we have: Φ V (p) = V max when p [p, p]. This shows [p, p] Ω. STEP 2: To prove Ω [p, p] Let p Ω be arbitrary. By the definition of Ω and Φ V (p), we have: Φ V (p) = V max Φ S (p) V max and Φ D (p) V max p p and p p p [p, p] Thus, we have proved that Ω [p, p]. Therefore, from Step 1 and Step 2 we conclude that: Ω = [p, p].

9 Pricing of Limit Orders in the Electronic Security Trading System Xetra 8 Notice that if p = p, Ω is reduced into one point, there exists a unique volume maximizing price. In this case the Xetra Price P Xetra is exactly: P Xetra = p = p. As we can see from Proposition 1, there could be more than one volume maximizing price for V max > 0. Therefore, some specific matching rules have to be applied for stipulating a unique auction price as the Xetra Price from the set of volume maximizing prices [p, p]. Deriving from the matching rules in Xetra, we have the following theorem to describe the price determination process in Xetra under the situation of a strictly positive highest executable order volume: Theorem 1. If V max > 0, then p when Φ Z [p,p] > 0, P Xetra = p when Φ Z [p,p] < 0, max{p, min{p ref, p}} otherwise. (13) Proof: In the case of V max > 0, only Rule 1 to Rule 3 are considered. By Proposition 1, Rule 1 states that Xetra Price must lie in Ω = [p, p], the set of volume maximizing prices. Rule 2.1 implies the Xetra Price is the highest price in Ω when the surplus for any price in Ω is on the demand side. Using excess demand function, Rule 2.1 can be restated as P Xetra = p when Φ Z [p,p] > 0. Analogous to Rule 2.1, Rule 2.2 implies P Xetra = p when Φ Z [p,p] < 0. When the surplus for any price in Ω is not satisfied with Rule 2, reference price P ref is introduced and Rule 3 is considered. Rule 3.1 states that P Xetra = p when P ref p. Rule 3.2 states that P Xetra = p when p P ref. Rule 3.3 states that P Xetra = P ref when p P ref p. Combining Rule 3.1 to Rule 3.3, we have P Xetra = max{p, min{p ref, p}} when the surplus for any price in Ω is not satisfied with Rule 2. Therefore, Rule 1 to Rule 3 indicates P Xetra is described in equation (13). Theorem 1 formalizes the pricing mechanism in Xetra. Given all the bids (a i, d i ) i I and asks (b j, s j ) j J, a unique price P Xetra is determined by this theorem. References Gruppe Deutsche Börse (2003): The Market Model Stock Trading for Xetra, Frankfurt a. M.

10 Pricing of Limit Orders in the Electronic Security Trading System Xetra 9 Krybus, S. (2001): Zum Handelssystem Xetra: Preisbildung, Mengenrationierung und Portfolioentscheidung im Xetra-Auktionsmodell, Master s thesis, Fakultäten für Mathematik und Wirtschaftswissenschaften, Bielefeld University. Sharpe, W., G. Alexander & J. Bailey (1999): Investments. Prentice Hall, 6 edn. A Appendix A.1 Proof of Lemma 3 We are going to prove Lemma 3 in this appendix. Proof: V max must be equal to some α i0 or some β j0. Therefore, we have two cases: CASE I: Let V max = α i0 > 0: Claim 1: There exists some j {1,..., J} such that β j 1 < α i0 β j. Claim 1 holds with the following argument: Since V max = α i0, there exists some p, such that Φ D ( p) Φ S ( p) and Φ V ( p) = min {Φ D ( p), Φ S ( p)} = Φ D ( p) = α i0 > 0. Let β j = Φ S( p). Notice that j > 0 since β j = Φ S( p) Φ D ( p) = α i0 > 0 = β 0. Let j = min{ j {1,..., J} β j α i0 }. We have β j 1 < α i0 β j since β j 1 < β j. Therefore, by assumption we have proved that claim 1 holds. By the definition of Φ D (p) and Φ S (p), there exists A i0 = (a i0, a i0 +1] corresponding to α i0 and B j = [b j, b j +1) corresponding to β j. In this case, V max = α i0. Noticing that Φ D (p) is a non-increasing function, equation (11) can be rewritten as: p = max {p Φ D (p) α i0 } p = max{ p p A i0 } p = a i0 +1,

11 Pricing of Limit Orders in the Electronic Security Trading System Xetra 10 and noticing Φ S (p) is a non-decreasing function and α i0 β j from claim 1, equation (12) can be restated as: p = min {p Φ S (p) α i0 } p = min{ p Φ S (p) = β j } p = min{ p p B j } p = b j. Now we need to prove p p, that is to prove a i0 +1 b j. By contradiction, if p < p, that is a i0 +1 < b j. We have A i0 = (a i0, a i0 +1] [0, b j ) since 0 < a i0 < a i0 +1 < b j. Thus, α i0 Φ D (p) p [0,bj ). Also Φ S (p) p [0,bj ) = {β 0, β 1,..., β j 1} since [0, b j ) = B 0 B 1... B j 1. Since β j 1 < α i0 β j and β j 1 >... > β 1 > β 0 = 0, we have α i0 > β j 1 >... > β 0. Thus, by definition of Φ V (p), we have Φ V (p) p [0,bj ) < α i0. Hence, Φ V (p) p Ai0 < α i0 since A i0 [0, b j ). This contradicts V max = α i0 since β j 1 < α i0. CASE II: Let V max = β j0 > 0: Claim 2: There exists some i {0, 1,..., I 1} such that α i +1 < β j0 α i. Claim 2 holds with the following argument: Since V max = β j0, there exists some p, such that Φ S ( p) Φ D ( p) and Φ V ( p) = min {Φ D ( p), Φ S ( p)} = Φ S ( p) = β j0 > 0. Let αĩ = Φ D ( p). Notice that ĩ < I since αĩ = Φ D ( p) Φ S ( p) = β j0 > 0 = α I. Let i = max{ĩ {0, 1,..., I 1} αĩ β j0 }. We have α i +1 < β j0 α i since α i < α i +1. Therefore, by assumption we have proved that claim 2 holds. By the definition of Φ D (p) and Φ S (p), there exists B j0 = (b j0, b j0 +1] corresponding to β j0 and A i = [a i, a i +1) corresponding to α i. In this case, V max = β j0. Noticing that Φ D (p) is a non-increasing function and α i +1 < β j0 α i from claim 2, equation (11) can be rewritten as: p = max {p Φ D (p) β j0 } p = max{ p Φ D (p) = α i } p = max{ p p A i } p = a i +1,

12 Pricing of Limit Orders in the Electronic Security Trading System Xetra 11 and noticing that Φ S (p) is a non-decreasing function, equation (12) can restated as: p = min {p Φ S (p) β j0 } p = min{ p p B j0 } p = b j0. Now we need to prove p p, that is to prove a i +1 b j0. By contradiction, if p < p, that is a i +1 < b j0, we have B j0 = (b j0, b j0 +1] (a i +1, + ) since a i +1 < b j0 < b j0 +1 < +. Thus, β j0 Φ S (p) p (ai +1,+ ). Also Φ D (p) p (ai +1,+ ) {α i +1,..., α I } since (a i +1, + ) = A i +1 A I. Since α i +1 < β j0 α i and α i +1 >... > α I = 0, we have β j0 > α i +1 >... > α I. Thus, by definition of Φ V (p), we have Φ V (p) p (ai +1,+ ) < β j0. Hence, Φ V (p) p Bj0 < β j0 since B j0 (a i +1, + ). This contradicts V max = β j0 since α i +1 < β j0. From CASE I and CASE II, we have proved Lemma 3.

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