Dark Pool Trading Strategies, Market Quality and Welfare Online ppendix Sabrina Buti Barbara Rindi y Ingrid M. Werner z December, 015 Uniersity of Toronto, Rotman School of Management, sabrina.buti@rotman.utoronto.ca y Bocconi Uniersity and IGIER, barbara.rindi@unibocconi.it z Fisher College of Business, Ohio State Uniersity, werner.47@osu.edu
ppendix B B.1. Extension 1: and small traders Under this framework we allow for two di erent types of traders: at each trading round t, a trader may arrie either with a two-share (large) order with probability, or with a one-share (small) order with probability 1. s arriing with a large order can trade up to two shares, j = f0; 1; g, and we call them large traders (LT), whereas traders arriing with a small order can only trade one share and we call them small traders (ST). B.1.1. The benchmark framework ( B) The strategies aailable to traders in this framework are presented in Table B1. s in the original model, both large and small traders can submit market orders to the rst two leels of the price grid, ' M (j; p z i ); they can post limit orders to the rst leel, ' L(j; p z 1 ), and they can choose not to trade, '(0). In addition, a large trader can split his order and combine the two di erent strategies, ' ML (1; p z i ; 1; pz 1 ), taking adantage of the better execution probability of a one-unit limit order. 1 In this case, only part of the large trader s pro ts depends on the probability of the order being executed in the following trading rounds: e t [' ML (1; p z i ; 1; p z 1)] = t [' M (1; p z i )] + e t [' L (1; p z 1)] : (44) Moreoer, the large trader can submit marketable orders that are large market orders walking up or down the book in search of execution, ' M (; p z ): For example, pro ts from a marketable sell order for the large trader are gien by: t [' M (; p B )] = (p B 1 + p B ) t : (45) s in the preious framework, at each trading round t the arriing risk-neutral trader selects the optimal order submission strategy which maximizes his expected pro ts, conditional on the state of the LOB, b t, on his type represented by his personal aluation of the asset, t, and on whether he is a small or a large trader. The large trader chooses: max ' and the small trader still soles problem (7). e t [' M (j; p z i ); ' M (; p z ); ' ML (1; p z i ; 1; p z 1); ' L (j; p z 1); '(0) j t ; b t ] ; (46) [Insert Table B1 here] B.1.. Limit order book and continuous dark pool ( L&C) In this framework only large traders hae access to the CDP and their action space includes, in addition to the dark pool orders presented for the original model, the possibility to split the order between the LOB and the CDP, by combining a one-unit dark pool order with either a market order, ' MD (1; p z i ; 1; ep Mid;t), or a limit order, ' LD (1; p z 1 ; 1; ep Mid;t): ' MD (1; p z i ; 1; ep Mid;t ) = t [' M (1; p z i )] + e t [' D (1; ep Mid;t )]; (47) ' LD (1; p z 1; 1; ep Mid;t ) = e t [' L (1; p z 1)] + e t [' D (1; ep Mid;t )] : (48) 1 Because small traders trade only one unit, the execution probability of the rst unit of a limit order is higher. We omit the subscript i for the leel of the book since the order will be executed at di erent prices. 1
t each trading round the fully rational risk-neutral large trader chooses the optimal order submission strategy which maximizes his expected pro ts, conditional on his aluation of the asset, t, and his information set, t, respectiely: max ' e t ['M (j; p z i ); 'M (; pz ); 'ML(1; p z i ; 1; p z 1); 'MD(1; p z i ; 1; ep Mid;t ); (49) 'D(j; p Mid;t ; p z i ); 'D(j; ep Mid;t ); 'LD(1; p z 1; 1; ep Mid;t ); 'L(j; p z 1); '(0) j t ; t ] : s before, small traders sole problem (7), and shape their strategies depending on the expected state of the CDP. B.1.3. Results In commenting the results from the model with both large and small traders and a CDP, we focus only on those ndings which di er from our initial framework with only one-share trades. Oerall the results from the original model concerning order ows, market quality and welfare continue to hold. Note that by comparing books with one and two shares on the best ask price, we can now di erentiate between the e ects of spread and depth. Moreoer, we de ne a new order ow indicator, olume creation (V C), as the aerage LOB plus dark pool olume, V L&C t V B t : V C = 1 T, in excess of the total LOB olume in the B framework, P (Vt L&C t V B t ) ; (50) where we measure olume in each period t by weighing the ll rate, F R t ; by the order size, q t : Z V L&C;B P t = Pr(a)E t q t ' n a f ( t ) d t : (51) a=w ;N Notice that with a unitary trade size, T C and V C coincide. Order ows. When a CDP is introduced along side a LOB, we obsere OM and T C (Fig. B1). Moreoer, since orders may di er in size, we can also di erentiate T C from olume creation (V C) (Fig. B). The e ects of the introduction of a dark pool on order ows are all weaker when traders can choose between large and small size than when all orders were one share. The reason is that starting from both an empty book and an empty dark pool, it takes longer to obsere a book with two rather than one share at the best ask or bid price. For this reason traders hae less incentie to opt for dark trading. 3 Consider the new framework ealuated at t with three periods remaining, and compare books with di erent amounts of liquidity, b t = [00], [10], and [0]. Fig. B1 shows that OM at t is strongly increasing in the depth of the book. Dark orders become more attractie not only when depth increases, but also when spread becomes smaller. ll else equal, a smaller spread increases competition for the proision of liquidity and makes the dark pool option more attractie for limit orders. smaller spread also decreases the market order price improement in the dark pool which reduces the incenties to send market orders to the dark, but this e ect is compensated by the increased execution probability generated in the dark pool by the migration of the limit orders. [Insert Fig: B1 and Fig: B here] 3 Comparing Tables and B it is eident that starting from b t1 = [00], the probability to obsere b t = [10] in the framework with small (N and W ) traders is equal to a' L (1; p 1 )+(1 a)' L (1; p 1 ) = 1 0:4066+ 1 0:4066 = 0:4066, whereas the probability to obsere b t = [0] in the framework with large and small traders is equal to a' L (; p 1 ) = 1 0:1791 = 0:0895. 0
While OM increases as liquidity builds, T C and V C decrease when either spread narrows or depth builds up in the book with one or two shares on the best ask side. s we measure olume by weighing the ll rates by the size of the orders executed, V C has a pattern similar to T C, howeer, because the aerage size of the orders that migrate from the LOB to the dark pool is larger than the aerage size of the orders executed on the LOB, all the e ects generated by the migration of orders to the dark pool are magni ed when measured by olume rather than by ll rate, and so is the change in V C compared to the reduction in T C. Market quality. s in the initial framework, our results show that the introduction of a dark pool has a negatie e ect on the liquidity of an empty LOB as both the inside spread and the depth at the best bid and o er worsen (Fig. B3, Panel ). The e ects at work are ery much similar to those discussed for the original model. t the beginning of the trading game when both the book and the dark pool open empty, the introduction of a dark pool makes traders switch from limit to market orders anticipating a future reduction in the execution probability of limit orders (Table B). 4 s before, the resulting increased liquidity demand and reduced liquidity supply widen spread and reduce depth at the inside quotes. Howeer, we note that the magnitude of the e ect is smaller with large and small traders compared to the original model where all traders were small. Consider again the new framework ealuated at t with three periods remaining, and compare books with di erent amounts of liquidity, b t = [00], [10] and [0]. Fig. B3 shows that while the negatie e ect on spread decreases with the liquidity of the book, the e ect on depth rst improes and then, as traders more heaily moe to the dark pool, further deteriorates. [Insert Fig: B3 and Table B here] When liquidity builds up in the LOB and the book opens with one or two shares at the best ask price, two e ects occur. On the one hand, when the book becomes deeper traders generally tend to use more market than limit orders and so fewer traders switch from limit to market orders when a dark pool is introduced. This rst e ect attenuates the negatie impact on liquidity of the switch from limit to market orders. On the other hand, howeer, as liquidity builds up and the queue at the top of the LOB becomes longer, more limit orders are attracted to the dark pool and this e ect reduces liquidity of the LOB, especially depth. When the book moes from being empty to haing one share at the best ask price, the rst e ect dominates and both spread and depth improe (the impact of the dark pool introduction is less negatie). When instead depth builds up substantially and the book opens with two shares at the best ask price, the second e ect dominates and while spread improes, depth drops due to the migration of limit orders. The reason for why spread improes but depth deteriorates when the book opens with two shares at the best ask price is that a minimum of two shares at the best ask is required for the introduction of the dark pool to generate a migration of limit orders -not only a switch from market to limit orders. So the intense migration of the two-unit limit orders from the ask side does not impact spread heaily as the book already has two shares at p 1, whereas it does impact depth substantially. Welfare. In this new framework small traders play the role of N traders of the initial model, and large traders play the role of W traders with the di erence that they may submit orders of two shares. The conclusions are therefore ery similar to the ones of the preious model and are illustrated in Fig. B3, Panel B. s before, the e ect of the introduction of a dark pool on the welfare of small traders is mainly drien by the ariation in the spread. traders also do not bene t from the introduction of a dark pool when both the book and the dark pool open empty because both spread and depth deteriorate. 4 More precisely, traders switch from two-unit limit orders to two-unit market orders, and from two-unit limit orders to one-unit limit and one-unit market orders. 3
Finally, consider the new framework ealuated at t with three periods remaining, and compare books with di erent amounts of liquidity, b t = [00], [10] and [0]. Fig. B3, Panel B, shows that the negatie e ect on the welfare of small traders declines as book liquidity at t increases. The reason is that spread deteriorates and then partially recoers due to the introduction of the dark pool. Howeer, as the book becomes deeper and at the same time some liquidity builds up in the dark pool, large traders start using the dark pool intensiely and they now bene t from the access to dark pools so that een aggregate welfare increases. Yet note that in this new model large traders gains from trade outweigh small traders losses only when liquidity builds up substantially and the book opens with two shares on the ask side. B.. Extension : LOB and periodic dark pool ( L&P ) We extend the framework with large and small traders by assuming that the dark enue has periodic execution. P DP is organized like an opaque crossing network where time priority is enforced. In this trading enue, orders are crossed at the end of the trading game only if a matching number of orders on the opposite side has been submitted to the dark pool prior to the cross. The execution price is the spread mid-quote preailing on the LOB at the end of period which we indicate with ep Mid. Similarly to the L&C framework, the large traders action space includes, in addition to the orders presented for the benchmark model, the possibility to submit orders to buy or to sell the asset on the P DP, as shown in Table B1. In particular, large traders can split their orders between the LOB and the P DP, or they can submit their entire order to the P DP. Their optimal trading strategy is determined by soling at each round the following optimization problem: max ' e t ['M (j; p z i ); 'M (; pz ); 'ML(1; p z i ; 1; p z 1); 'MD(1; p z i ; 1; ep Mid ); (5) 'D(j; ep Mid ); 'LD(1; p z 1; 1; ep Mid ); 'L(j; p z 1); '(0) j t ; t ] : traders still sole problem (7), howeer they now condition their strategies not only on their own t and on the state of the LOB but also on the inferred state of the P DP. The game is soled as before by backward induction starting from. B.3 Proof of Proposition 4 We rst proide an example of how the extended ersion of the model with both small and large traders is soled, and then we discuss the results obtained. B.3.1. B framework n example of the extensie form of the game is presented in Fig. B4. Because the model is similar to the one with unitary trade size, for breity we proide an example of how the model is soled only for the last two periods of the trading game. From now onwards, we assume that for large traders the optimal order size is j = max [' j b t ], since @ e t (')=@j 0 due to agents risk neutrality. j Following Fig. B4, at we present as an example b t4 = [0], and focus on large trader s pro ts: t4 [' M (; p B )] = (p B t4 ) = (1 3 t4 ) (53) t4 [' M (; p 1 ) ] = ( t4 p 1 )= ( t4 1 ) (54) t4 ['(0)] = 0 : (55) 4
>< ^P DP t4 = It is straightforward to show that all strategies are optimal in equilibrium (N t4 = 3): ' 1 LT;b t4 = ' M (; p B ), ' LT;b t4 = '(0), and ' 3 LT;b t4 = ' M (; p 1 ). We refer to the proof of Proposition 1, ppendix, for the optimal strategies of small traders. Still as an example, and continuing to follow Fig. B4, we consider the opening LOB b t3 = [1]. traders pro ts are as follows: t3 [' M (1; p B 1 )] = (p B 1 t3 ) (56) [' L (1; p 1 )] = t3 ['(0)] = 0 (57) [' L (1; p B 1 )] = ( t3 p B 1 ) Pr (' M (; p B 1 ) j b t4 = []) (58) t3 [' M (1; p 1 )] = ( t3 p 1 ) : (59) traders strategies are similar, the only di erence being that j = for a market sell order, and that conditioning on b t3 = [1] now traders willing to buy can combine market and limit orders, and traders willing to sell can walk down the book ia a marketable order: [' ML (1; p 1 ; 1; p B 1 )] = ( t3 p 1 )+( t3 p B 1 ) Pr (' M (; p B 1 ) j b t4 = [1]) (60) t3 [' M (; p B )] = (p B 1 t3 ) + (p B t3 ) : (61) Pro ts for periods t 1 and t are similar and are omitted, the main di erence being that limit orders hae additional periods to get executed. B.3.. L&P framework n example of the extensie form of the game is presented in Fig. B5. The solution of the L&P framework follows the same methodology, but now the large trader soles Eq. (5). For breity, we proide again examples only for the last two periods of the trading game. The remaining two periods are soled in a similar way. Following Fig. B5, at we consider again the book b t4 = [0] with the following information set, t4 = [0; ' L (; p 1 ); ' L(1; p B 1 ); ' M(1; p B 1 )]. Notice that when at t 3 traders obsere ' M (1; p B 1 ), they don t know whether a small trader submitted a one-unit market order to sell, or whether a large trader submitted a market order combined with a dark pool order. Similarly, at t when traders obsere ' L (1; p B 1 ) it could be a one-unit limit order to buy or a combination of a limit and a dark pool order. s coming at Bayesian update their expectations on the state of the dark pool as follows (we omit that all probabilities are conditional to t4 ): 8 >: +1 with prob = 0 with prob = 1 with prob = [Pr t ' LD (1;p B 1 ;+1;ep Mid)+(1 (1 ) Pr t ' LD (1;p B 1 ;+1;ep Mid) Pr t 3 ' M (1;p B 1 ) ) Pr t ' L (1;p B 1 )][(1 ) Pr ' M (1;p B 1 )+ Pr ' MD (1;p B 1 ; t 3 t 3 Pr t ' LD (1;p B 1 ;+1;ep Mid) Pr t 3 ' MD (1;p B 1 ; 1;ep Mid)+(1 ) Pr t ' L (1;p B 1 ) Pr t 3 ' M (1;p B 1 ) [Pr t ' LD (1;p B 1 ;+1;ep Mid)+(1 [Pr t ' LD (1;p B 1 ;+1;ep Mid)+(1 ) Pr t ' L (1;p B 1 )][(1 (1 ) Pr t ' L (1;p B 1 ) Pr t 3 ' MD (1;p B 1 ; ) Pr t ' L (1;p B 1 )][(1 ) Pr ' M (1;p B 1 )+ Pr ' MD (1;p B 1 ; t 3 t 3 1;ep Mid) 1;ep Mid)] 1;ep Mid)] ) Pr ' M (1;p B 1 )+ Pr ' MD (1;p B 1 ; 1;ep Mid)] : t 3 t 3 (6) 5
We refer to the proof of Proposition 1 in ppendix for the pro ts of the small trader, and discuss the large trader s pro ts. We omit regular market orders that we hae already presented in the B framework, Eqs. (53) (55), and focus only on orders that inole the use of the P DP : e [' MD (1; p B ; 1; ep Mid )] = (p B t4 ) + ( p 1 +pb t4 ) Pr(^P DP t4 = +1) (63) e [' D ( 1; ep Mid )] = ( p 1 +pb t4 ) Pr(^P DP t4 = +1) (64) e [' D (+1; ep Mid )] = ( t4 e [' MD (1; p 1 ; +1; ep Mid )] = ( t4 p 1 ) + ( t4 p 1 +pb ) Pr(^P DP t4 = 1) (65) p 1 +pb ) Pr(^P DP t4 = 1) : (66) To determine the equilibrium strategies ' n a; t4 at for n N t4, the model has to be soled up to period t 1. We anticipate that because in equilibrium Pr t ' LD (1; p B 1 ; +1; ep Mid) = 0 and Pr t3 ' MD (1; p B 1 ; 1; ep Mid) > 0, N t4 = 5 and the strategies of the large trader are as follows: ' 1 LT; t4 = ' M (; p B ), ' LT; t4 = '(0), ' 3 LT; t4 = ' D (+1; ep Mid ), ' 4 LT; t4 = ' MD (1; p 1 ; +1; ep Mid), and ' 5 LT; t4 = ' M (; p 1 ). s for the B framework, we now consider the case b t3 = [1] with the following information set, t3 = [1; ' L (; p 1 ); ' L(1; p B 1 )]. The expected state of the PDP is: 8 Pr >< +1 with prob = t ' LD (1;p B 1 ;+1;ep Mid) Pr t ' LD (1;p ^P 1 DP t3 = Mid)+(1 ) Pr t ' L (1;p B 1 ) (1 )Pr >: 0 with prob = t ' L (1;p B 1 ) Pr t ' LD (1;p B 1 ;+1;ep Mid)+(1 ) Pr t ' L (1;p B 1 ) : (67) Compared to the B framework, the large trader has now additional strategies. market and dark orders: He can combine [' MD (1; p 1 ; +1; ep Mid)]= ( t3 p 1 ) + Pr(^P DP t3 = +1)( t3 Pr (' D ( p 1 +pb 1 ) ; ep Mid ) j t4 ) + Pr(^P DP t3 =0)[( t3 p 1 +pb 1 ) (68) Pr (' D ( ; ep Mid )j t4 ) + ( t3 p 1 +pb ) Pr (' MD (1; p B 1 ; 1; ep Mid)j t4 )] ; where t4 = [11; ' L (; p 1 ); ' L(1; p B 1 ); ' M(1; p 1 )]; [' MD (1; p B 1 ; 1; ep Mid)]= (p B 1 t3 ) + Pr(^P DP t3 = +1)f( p 1 +pb t3 ) [(1 ) + (1 Pr (' M (; p 1 )j ))] + ( p +pb t3 ) Pr (' M (; p 1 )j )g + Pr(^P DP t3 =0)( p 1 +pb t3 ) [Pr (' D (+1; ep Mid ) j t4 ) + Pr (' MD (1; p 1 ; +1; ep Mid) j t4 )] ; (69) where t4 = [0; ' L (; p 1 ); ' L(1; p B 1 ); ' M(1; p B 1 )]. 6
lternatiely, he can combine a limit and dark order on the bid side of the market: [' LD (1; p B 1 ; +1; ep Mid)]= ( t3 p B 1 ) Pr (' M (; p B 1 ) j ) + ( t3 [Pr (' MD (1; p B 1 ; 1; ep Mid)j t4 ) + Pr (' D ( ; ep Mid )j t4 )] +Pr(^P DP t3 = +1) Pr (' D ( ; ep Mid )j t4 )g ; p 1 +pb 1 )fpr(^p DP t3 =0) (70) where t4 = [; ' L (; p 1 ); ' L(1; p B 1 ); ' L(1; p B 1 )]. Finally, he can submit a pure dark pool order to sell or buy: [' D ( ; ep Mid )]= Pr(^P DP t3 = +1)f( p 1 +pb 1 t3 )[( Pr (' MD (1; p 1 ; +1; ep Mid)j t4 ) + Pr (' D (+; ep Mid )j t4 ) + Pr (' D ( ; ep Mid )j t4 ) + Pr ('(0)j t4 )) + (1 ) (Pr (' s (0)j t4 ) + Pr (' M (1; p 1 )j ))] + ( p 1 +pb t3 )[(1 ) Pr (' M (1; p B 1 )j ) + (Pr (' M (1; p B )j t4 ) + Pr (' MD (1; p B 1 ; 1; ep Mid)j t4 ))] (71) +( p +pb 1 t3 ) Pr (' M (; p 1 )j )g + Pr(^P DP t3 =0)( p 1 +pb 1 t3 ) [Pr (' MD (1; p 1 ; +1; ep Mid)j t4 ) + Pr (' D (+; ep Mid )j t4 )] [' D (+; ep Mid )]= Pr(^P DP t3 = +1)( t3 p 1 +pb 1 ) Pr (' D ( ; ep Mid )j t4 ) +Pr(^P DP t3 =0)[( t3 p 1 +pb ) Pr (' MD (1; p B 1 ; 1; ep Mid)j t4 ) (7) +( t3 p 1 +pb 1 ) Pr (' D ( ; ep Mid )j t4 )] ; where in both cases t4 = [1; ' L (; p 1 ); ' L(1; p B 1 ); '(0)]. B.3.3. L&C framework n example of the extensie form of the game is presented in Fig. B6. For consistency, we consider again the example of the book b t4 = [0] with the following information set, t4 = [0; ' L (; p 1 ); ' L (1; p B 1 ); ' M(1; p B 1 )]. In this case when at t 3 a trader obseres ' M (1; p B 1 ), he doesn t know whether a small trader submitted a one-unit market order, or whether a large trader submitted a market order combined with a dark pool order or a two-unit IOC dark pool order that was partially executed on the CDP. Therefore, traders coming at Bayesian update their expectations on the state of the dark pool by using an extended ersion of Eq. (6) which takes into account that ^CDP t4 = 0 with the additional probability Pr ' LD (1; p B 1 ; +1; ep Mid;t) Pr ' D ( ; p Mid;t3 ; p B 1 ). s coming at t 3 instead face the same t t 3 uncertainty as for the L&P framework. 7
Compared to the L&P framework, the large trader can now submit a IOC dark pool order with the following pro ts: e [' D (+; p Mid;t4 ; p 1 )]= t4 p 1 [1 + Pr(^CDP t4 = +1; 0)] p 1 +pb Pr(^CDP t4 = 1) (73) e [' D ( ; p Mid;t4 ; p B )]= p B [1 + Pr(^CDP t4 = 1; 0)] + p 1 +pb Pr(^CDP t4 = +1) t4 (74) To determine the equilibrium strategies ' n a; t4 at for n N t4, the model has again to be soled up to period t 1. We anticipate that because in equilibrium Pr t ' LD (1; p B 1 ; +1; ep Mid;t) = 0 and Pr t3 ' MD (1; p B 1 ; 1; ep Mid;t) > 0, N t4 = 4 and the strategies of the large trader are as follows: ' 1 LT; t4 = ' M (; p B ), ' LT; t4 = '(0), ' 3 LT; t4 = ' D (+1; p Mid;t4 ), ' 4 LT; t4 = ' D (+; p Mid;t4 ; p 1 ). s for the other two frameworks, we now consider the case b t3 = [1] with the following information set, t3 = [0; ' L (; p 1 ); ' L(1; p B 1 )]. We refer to Eq. (67) for the probabilities of the expected state of the CDP, ^CDP t3 = f0; +1g. The large trader can now submit a IOC dark pool order to sell (the execution probability on the CDP of a IOC dark order to buy is zero, so traders neer select this strategy): [' D ( ; p Mid;t3 ; p B )]= p B 1 + p B Pr(^CDP t3 = 0) + p 1 +pb 1 Pr(^CDP t3 = + 1) t3 (75) Notice that the pro ts of the combined market and dark orders, limit and dark orders and pure dark pool orders are similar to the ones presented for the L&P framework. The main two di erences are that rst in this framework the order is executed as soon as liquidity is aailable on the CDP and not at the end of the trading game; second that traders can use IOC dark pool orders. For example pro ts for a dark pool sell order are: [' D ( ; ep Mid;t )]= Pr(^CDP t3 = +1)( p 1 +pb 1 t3 )f1 + [Pr (' D (+; p Mid;t4 ; p 1 )j ) +Pr (' D (+; p Mid;t4 )j t4 )]g + Pr(^CDP t3 =0)( p 1 +pb 1 t3 ) [Pr (' D (+; p Mid;t4 ; p 1 )j ) + Pr (' D (+; p Mid;t4 )j t4 )]g ; (76) where, as for the L&P, t4 = [1; ' L (; p 1 ); ' L(1; p B 1 ); '(0)]. B.3.4. OM Results for OM presented in Fig. B1 are deried by straightforward comparison of the equilibrium strategies for the three frameworks: B, L&P and L&C. In Figs. B7 B1 we proide plots at t for the large trader s pro ts as a function of ; for both the L&P and L&C frameworks. Following the exposition in the main text, we focus on selling strategies. Each gure proides a graphical representation of the traders optimization problem. Fig. B7 shows how the introduction of a P DP changes the optimal order submission strategies of large traders by crowding out both market and limit orders, and generating OM. Consider rst OM in the L&P : compare Figs. B7 and B9 for the e ect of market depth, B7 and B11 for the e ect of spread. For the L&C, compare instead Figs. B8 and B10, and B8 and B1, respectiely. 8
B.3.5. TC and VC Results for T C and V C presented in Figs. B1 and B, respectiely, are obtained by comparing ll rates and olumes for the B, L&P and L&C frameworks, as shown in Eqs. (14) and (50) respectiely. s an example, we consider period t 1 of the B model and specify formulas for the estimated ll rate and olume in this period. Equilibrium strategies at t 1 for a large trader are as follows: ' 1 LT = ' M(;p B ), ' LT = ' ML(1; p B ; 1; p 1 ), '3 LT = ' L(; p 1 ), '4 LT = ' L(; p B 1 ), '5 LT = ' ML(1; p ; 1; pb 1 ) and '6 LT = ' M(;p 1 ). The ones for a small trader are: ' 1 ST = ' M(1;p B 1 ), ' ST = ' L(1; p 1 ), '3 ST = ' L(1; p B 1 ) and '4 ST = ' M(1;p 1 ). F R B t 1 ;[] = 1 (Pr t 1 V B t 1 ;[] = 1 (Pr ' 1 ST + Pr t 1 B.3.6. Spread, depth and welfare t 1 ' 4 ST )+ 1 (Pr t 1 ' 1 ST + Pr ' 4 ST )+ 1 t 1 ( Pr t 1 ' 1 LT + Pr t 1 ' 1 LT + Pr t 1 ' LT + Pr ' 5 LT + Pr ' 6 LT ) (77) t 1 t 1 ' LT + Pr ' 5 LT + Pr ' 6 LT ) : (78) t 1 t 1 Results for spread and depth (Fig. B3, Panel ), and for welfare (Fig. B3, Panel B) are obtained by comparing, respectiely, the two market quality measures and welfare alues for the B, L&P and L&C protocol. We refer to the proofs of Proposition and Proposition 3 for an example of how to compute these measures. 9
Table B1 Order submission strategies. This table reports the trading strategies, ', aailable to large (LT) and small traders (ST) for the three di erent frameworks considered: a benchmark model (B) with a limit order book (LOB), and either a continuous (L&C) or a periodic dark pool (L&P ) competing with a LOB. The LOB is characterized by a set of four prices, denoted by p z i, where z = f; Bg indicates the ask or bid side of the market, and i = f1; g the leel on the price grid. In the L&C, epmid;t indicates the spread mid-quote on the LOB preailing in period t. In the L&P, epmid indicates the spread mid-quote on the LOB preailing at the end of period t4, when the dark pool crosses orders. IOC indicates Immediate-or-Cancel orders. The LT trades up to two shares, j = f0; 1; g, while the ST trades up to one share. Strategies LT (B) Notation Strategies ST /(B L&C L&P ) Notation Market order of j shares ' M (j; p z i ) Market order of 1 share ' M(1; p z i ) Limit order of j shares ' L (j; p z 1 ) Limit order of 1 share ' L(1; p z 1 ) No trading '(0) No trading '(0) Market & limit order of 1 share each ' ML (1; p z i ; 1; pz 1 ) Marketable order of shares ' M (; p z ) dditional Strategies LT (L&C) Market & dark pool order of 1 share each ' MD (1; p z i ; 1; ep Mid;t) Limit & dark pool order of 1 share each ' LD (1; p z 1 ; 1; ep Mid;t) Dark pool order of j shares ' D (j; epmid;t) IOC on dark pool or market order of j shares ' DM (j; epmid;t; p z i ) dditional Strategies LT (L&P ) Market & dark pool order of 1 share each ' MD (1; p z i ; 1; ep Mid) Limit & dark pool order of 1 share each ' LD (1; p z 1 ; 1; ep Mid) Dark pool order of j shares ' D (j; epmid) 10
Table B Order submission probabilities at t1 and t: This table reports the submission probabilities of large (LT) and small traders (ST) for the orders listed in column 1 for the benchmark framework (B), for the model with a limit order book and a periodic dark pool (L&P ) and for the model with a limit order book and a continuous dark pool (L&C). We consider the t1 equilibrium strategies from the four-period model that opens with an empty book at t1, bt1 = [00], and the t equilibrium strategies from the three-period models that open at t according to the three equilibrium opening books (at t) from the four-period model, and that di er for the number of shares at the rst leel of the ask side of the book: bt = [00], b = [10], and b t t = [0]. Results are computed assuming that the tick size is equal to = 0:08 and the probability that a large trader arries is = 0:5. Panel - ST bt1 = [00] b = [00] b = [10] b = [0] t t t Trading Strategy B L&C L&P B L&C L&P B L&C L&P B L&C L&P ' M 1; p B 0.0399 0.049 0.0410 0.376 0.385 0.384 0.3706 0.374 0.3709 0.445 0.445 0.445 ' L 1; p 1 0.4601 0.4571 0.4590 0.64 0.615 0.616 0.150 0.131 0.147 0.068 0.068 0.068 ' L 1; p B 1 0.4601 0.4571 0.4590 0.64 0.615 0.616 0.1306 0.1305 0.1306 0.1406 0.1386 0.1377 ' M 1; p 1 0.3738 0.3741 0.3738 0.371 0.3741 0.3750 ' M 1; p 0.0399 0.049 0.0410 0.376 0.385 0.384 Panel B - LT bt1 = [00] b = [00] b = [10] b = [0] t t t Trading Strategy B L&C L&P B L&C L&P B L&C L&P B L&C L&P ' M (; p B ) 0.0399 0.049 0.0410 0.376 0.385 0.384 0.3706 0.374 0.3709 0.445 0.4067 0.4061 ' ML (1; p B ; 1; p 1 ) 0.686 0.780 0.778 0.1330 0.1354 0.134 0.0539 0.051 0.0536 0.0110 ' D ( ; epmid) 0.065 ' D ( ; epmid;t ) 0.0614 ' L (; p 1 ) 0.1915 0.1791 0.181 0.194 0.161 0.191 0.069 0.069 0.0693 0.0505 0.0181 0.0147 ' L (; p B 1 ) 0.1915 0.1791 0.181 0.194 0.161 0.191 0.0639 0.064 0.063 0.0714 0.0698 0.0713 ' ML (1; p 1 ; 1; pb 1 ) 0.048 0.054 0.054 0.0687 0.0700 0.0689 ' ML (1; p ; 1; pb 1 ) 0.686 0.780 0.778 0.1330 0.1354 0.134 ' M (; p 1 ) 0.3738 0.3741 0.3738 ' M (; p ) 0.376 0.385 0.384 ' M (; p ) 0.0399 0.049 0.0410 0.376 0.385 0.384 11
Panel : Order migration Panel B: Trade creation Fig. B1. Order migration and trade creation, small and large traders. This gure presents results for the two frameworks, L&P and L&C. The rst one combines a limit order book (LOB) and a periodic dark pool (P DP ); the second combines a LOB and a continuous dark pool (CDP ). For each framework we report in Panel order migration which is the aerage probability that an order migrates to the dark pool, and in Panel B trade creation that is the sum of two components. The rst one is the aerage ll rate in the dark pool. The second one is the di erence between the aerage LOB ll rate in the L&P or L&C, and the aerage LOB ll rate in the benchmark. We report results for both the four-period model that opens with an empty book at t1, bt1 = [00], and for the three-period models that open at t according to the three equilibrium opening books (at t) from the four-period model, and that di er for the number of shares at the rst leel of the ask side of the book: bt = [00], b = [10], and b t t = [0]. Results are computed assuming a tick size equal to = 0:08 and a probability that a large trader arries = 0:5. 1
Fig. B. Volume creation, small and large traders. This gure presents results for the two frameworks, L&P and L&C. The rst one combines a limit order book (LOB) and a periodic dark pool (P DP ); the second combines a LOB and a continuous dark pool (CDP ). For each framework we report olume creation (V C) that is the sum of two components. The rst one is the aerage olume in the dark pool, V (DRK P OOL). The second one is the di erence between the aerage LOB olume in the L&P or L&C, V (LOB), and the aerage LOB olume in the benchmark, V (B). We report results for both the four-period model that opens with an empty book at t1, bt1 = [00], and for the three-period models that open at t according to the three equilibrium opening books (at t) from the four-period model, and that di er for the number of shares at the rst leel of the ask side of the book: bt = [00], bt = [10], and b t = 0:5. = [0]. Results are computed assuming a tick size equal to = 0:08 and a probability that a large trader arries 13
Panel : Market quality Panel B: Welfare Fig. B3. Market quality and welfare, small and large traders. This gure presents results for spread, depth and welfare in the two frameworks, L&P and L&C. L&P combines a limit order book (LOB) and a periodic dark pool (P DP ); L&C combines a LOB and a continuous dark pool (CDP ). ll measures are computed as the aerage percentage di erence between their alue for the L&P or L&C framework and the benchmark framework. s spread and depth are exogenous in the initial period we do not include it in the aerage, while for welfare we consider three measures: the welfare of a small trader (ST ), the welfare of a large trader (LT ), and aggregate welfare.we report results for both the four-period model that opens with an empty book at t1, bt1 = [00], and for the three-period models that open at t according to the three equilibrium opening books (at t) from the four-period model, and that di er for the number of shares at the rst leel of the ask side of the book: bt = [00], b = [10], and b t t = [0]. Results are computed assuming a tick size equal to = 0:08 and a probability that a large trader arries = 0:5. 14
, j= b t3 = [00] Market & Limit Buy b t3 = [11], j = b t = [00] Limit Buy, j = b t3 = [] Market & Limit Buy Limit Buy, j = Limit Sell, j= b t = [01] b t = [0] b t = [0] Limit Sell, j= Market & Limit Sell, j= b t3 = [40] b t3 = [30] b t3 = [0], j= Market & Limit Buy Limit Buy, j =1 b t4 = [01] b t4 = [1] b t4 = [], j= No Trading, j= Market & Limit Sell b t = [10] 1, j=1 b t4 = [0] b t1 = [00] 1, j = Limit Buy Limit Sell b t = [00] b t = [00] b t = [01] b t = [10] Limit Buy Limit Sell b t3 = [10] b t3 = [1] b t3 = [30] b t3 = [0] 1 Marketable Sell Limit Buy b t4 = [0] b t4 = [11] b t4 = [] 1 No Trading b t = [00] b t4 = [0] Fig. B4. Benchmark model of limit order book (B), small and large traders. Example of the extensie form of the game for the benchmark model when the opening book at t1 is bt1 = [00]; where j indicates the number of shares traded by the large trader. We only present the equilibrium strategies. 15
, j= b t3 = [00] Market & Limit Buy b t3 = [11], j = Market & Limit Buy Limit Buy, j = Limit Sell, j= b t = [00] b t = [01] b t = [0] b t = [0] Limit Buy, j = Limit Sell, j= Dark Pool Sell, j=, j= b t3 = [] b t3 = [40] b t3 = [0] b t3 = [0], j= Market & Limit Buy Limit & Dark Pool Buy Dark Pool Sell b t4 = [01] b t4 = [1] b t4 = [] b t4 = [1], j= Market & Dark Pool Buy Dark Pool Buy, j=1 No Trading Market & Limit Sell b t = [10] 1 Market & Dark Pool Sell b t4 = [0], j= b t1 = [00] 1, j = Limit Buy Limit Sell b t = [00] b t = [00] b t = [01] b t = [10] Limit Buy Limit Sell b t3 = [10] b t3 = [1] b t3 = [30] b t3 = [0] 1 Marketable Sell Limit Buy b t4 = [0] b t4 = [11] b t4 = [] 1 No Trading b t = [00] b t4 = [0] Fig. B5. Limit Order Book and Periodic Dark Pool (L&P), small and large traders. Example of the extensie form of the game for the model with a periodic dark pool when the opening book at t1 is bt1 = [00], where j indicates the number of shares traded by the large trader. Books that belong to the same information set, and hence are undistinguishable, are in squared dashed boxes and hae the same line format. For example, bt4 = [] can be obsered when at t 3 either a small trader arries and submits a limit buy, or a large trader arries and submits a combined limit and dark order to buy. We only present the equilibrium strategies. 16
, j= b t3 = [00] Market & Limit Buy b t3 = [11], j= b t4 = [01], j = Market & Limit Buy Limit Buy, j = Limit Sell, j= b t = [00] b t = [01] b t = [0] b t = [0] Limit Buy, j = Limit Sell, j= Dark Pool Sell, j=, j= b t3 = [] b t3 = [40] b t3 = [0] b t3 = [0] Market & Limit Buy Dark Pool Buy Limit Buy, j=1 Dark Pool Sell b t4 = [1] b t4 = [1] b t4 = [] b t4 = [1] IOC &, j= Dark Pool Buy, j=1 No Trading Market & Limit Sell b t = [10] 1 Market & Dark Pool Sell b t4 = [0], j= b t1 = [00] 1, j = Limit Buy Limit Sell b t = [00] b t = [00] b t = [01] b t = [10] Limit Buy Limit Sell b t3 = [10] b t3 = [1] b t3 = [30] b t3 = [0] 1 Marketable Sell Limit Buy b t4 = [0] b t4 = [11] b t4 = [] 1 No Trading b t = [00] b t4 = [0] Fig. B6. Limit order book and continuous dark pool (L&C), small and large traders. Example of the extensie form of the game for the model with a continuous dark pool when the opening book at t1 is bt1 = [00], where j indicates the number of shares traded by the large trader. Books that belong to the same information set, and hence are undistinguishable, are in squared dashed boxes and hae the same line format. For example, bt4 = [1] can be obsered when a large trader arries at t 3 and submits either a dark pool order to sell or to buy. We only present the equilibrium strategies. 17
OrderType OrderType L&P and B: M,p B L&C and B: M,p B 0.8 t L&P: L&P: ML 1,p B ;1,p 1 L,p 1 0.8 t L&C: L&C: ML 1,p B ;1, p 1 L,p 1 0.6 L&P: D,p Mid B: ML 1, p B ;1, p 1 0.6 L&C: D,p Mid,t B: ML 1, p B ;1, p 1 0.4 B: L, p 1 0.4 B: L, p 1 0. 0. 0.5 0.6 0.7 0.8 0.9 1.0 t 0.5 0.6 0.7 0.8 0.9 1.0 t Fig. B7. Order migration on the L&P - b t =[0] Fig. B8. Order migration on the L&C - b t =[0] OrderType OrderType L&P and B: M,p B L&C and B: M,p B 0.8 t L&P: L&P: ML 1,p B ;1,p 1 L,p 1 0.8 t L&C: L&C: ML 1,p B ;1, p 1 L,p 1 0.6 L&P: D,p Mid B: ML 1, p B ;1, p 1 0.6 L&C: D,p Mid,t B: ML 1, p B ;1, p 1 0.4 B: L, p 1 0.4 B: L, p 1 0. 0. 0.5 0.6 0.7 0.8 0.9 1.0 t 0.5 0.6 0.7 0.8 0.9 1.0 t Fig. B9. Order migration on the L&P - b t =[10] Fig. B10. Order migration on the L&C - b t =[10] OrderType OrderType L&P and B: M,p B L&C and B: M,p B 0.8 t L&P: L&P: ML 1,p B ;1,p 1 L,p 1 0.8 t L&C: L&C: ML 1,p B ;1, p 1 L,p 1 0.6 L&P: D,p Mid B: ML 1, p B ;1, p 1 0.6 L&C: D,p Mid,t B: ML 1, p B ;1, p 1 0.4 B: L, p 1 0.4 B: L, p 1 0. 0. 0.5 0.6 0.7 0.8 0.9 1.0 t 0.5 0.6 0.7 0.8 0.9 1.0 t Fig. B11. Order migration on the L&P - b t =[00] Fig. B1. Order migration on the L&C - b t =[00] 18