Supplementary Appendix for Liquidity, Volume, and Price Behavior: The Impact of Order vs. Quote Based Trading not for publication

Transcription

1 Supplementary Appendix for Liquidity, Volume, and Price Behavior: The Impact of Order vs. Quote Based Trading not for publication Katya Malinova University of Toronto Andreas Park University of Toronto March 16, 2009 Abstract This document contains an extensive number of tables and graphs to support the numerical analysis in Sections 5 and 6 of the main text; these figures are numbered consecutively following those in the main text. This document also provides additional information about the institutional background of trading mechanisms, technical details of the information structure used in the main text and, a detailed description of the simulation process that we employed to obtain our results in Section 8 of the main text. katya.malinova@utoronto.ca; web: andreas.park@utoronto.ca; web: apark/.

2 C Appendix: The Institutional Background Our treatment of formation in the dealer market is stylized: effectively, people submit their market orders without knowing the and there are no standing quotes from dealers. Of course, in real markets dealers do post quotes, but they usually quote only a single bid and a single ask. Moreover, for most markets, dealers are commonly required to trade a guaranteed minimum number of units at this (for instance, on Nasdaq a quote must be good for 1000 shares for most stocks). Alternatively, on exchanges such as the TSX or Paris Bourse, the upstairs dealers are required to trade at the best bid or offer (BBO) that is currently on the book, unless the size of the trade is very large. Finally, trading systems or exchanges that include small-order routing (i.e. small orders are given to different dealers according to a pre-determined set of routing rules) require dealers to do improvement, that is they require dealers to give small size orders the best that is currently quoted. None of these institutional details contradict our setup. First, the defining feature of dealer markets is that the dealer will know the size of the trade when quoting the uniform for the order. Thus the dealer quotes cannot be hit in the same way as a standing limit order in a consolidated limit order book. Next, in the theoretical analysis of our paper we describe that the dealer charges different s for different quantities. The bid- and ask-s that she quotes would be for the minimum quantity that she must trade and this quantity may well be large ; in other words, the quoted ask may be ask 2 D, but when facing a small order, the dealer may offer ask 1 D. Third, in our model traders accurately anticipate the that they will be quoted. Consequently, quotes will be self-fulfilling. Finally, the BBO requirement in upstairs-downstairs markets is trivially satisfied in the hybrid market. D Appendix: Quality and Belief Distributions Financial market microstructure models with binary signals and states typically employ a constant signal quality q [1/2, 1], with Pr(S = v V = v) = q. Our framework has a continuum of possible qualities with a continuous density function and, as outlined above, we will map investors signals and their qualities into a continuous private belief on [0, 1]. The quality parametrization on [1/2, 1] is very natural, as a trader who receives a high signal h will update his prior in favor of the high liquidation value, V = 1, and a trader who receives a low signal l will update his prior in favor of V = 0. We thus use the conventional parametrization on [1/2, 1] in the main text. 1 Appendix for Trading Mechanisms & Market Dynamics

3 However, to characterize the map from investors signal and qualities into their private beliefs and to derive the distributions of the latter, it is mathematically convenient to normalize the signal quality so that its domain coincides with that of the private belief. We will denote the distribution function of this normalized quality on [0, 1] by G and its density by g, whereas the distribution and density functions of original qualities on [1/2, 1] will be denoted by G and g respectively. The normalization proceeds as follows. Without loss of generality, we will employ the density function g that is symmetric around 1/2. For q [0, 1/2], we then have g(q) = g(1 q)/2 and for q [1/2, 1], we have g(q) = g(q)/2. Under this specification, signal qualities q and 1 q are equally useful for the individual: if someone receives signal h and has quality 1/4, then this signal has the opposite meaning, i.e. it has the same meaning as receiving signal l with quality 3/4. Signal qualities are assumed to be independent across agents, and independent of the security s liquidation value V. Beliefs are derived by Bayes Rule, given signals and signal-qualities. Specifically, if a trader is told that his signal quality is q and receives a high signal h then his belief is q/[q + (1 q)] = q (respectively 1 q if he receives a low signal l), because the prior is 1/2. The belief π is thus held by people who receive signal h and quality q = π and by those who receive signal l and quality q = 1 π. Consequently, the density of individuals with belief π is given by f 1 (π) = π[g(π) + g(1 π)] in state V = 1 and analogously by f 0 (π) = (1 π)[g(π) + g(1 π)] in state V = 0. Smith and Sorensen (2008) prove the following property of private beliefs (Lemma 2 in their paper): Lemma 1 (Symmetric beliefs, Smith and Sorensen (2008)) With the above the signal quality structure, private belief distributions satisfy F 1 (π) = 1 F 0 (1 π) for all π (0, 1). Proof: Since f 1 (π) = π[g(π) + g(1 π)] and f 0 (π) = (1 π)[g(π) + g(1 π)], we have f 1 (π) = f 0 (1 π). Integrating, F 1 (π) = π f 0 1(x)dx = π f 0 0(1 x)dx = 1 f 1 π 0(x)dx = 1 F 0 (1 π). A direct implication of this lemma is that with symmetric thresholds, π b = 1 π s, a buy in state V = 1 is as likely as a sale in state V = 0, because β 1 = (1 µ)/2 + µ(1 F 1 (π b )) = (1 µ)/2 + µf 0 (1 π b b) = (1 µ)/2 + µf 0 (π s ) = σ 0. Similarly, β 0 = σ 1. Next, the belief densities satisfy the monotone likelihood ratio 2 Appendix for Trading Mechanisms & Market Dynamics

4 property because is increasing in π. f 1 (π) f 0 (π) = π[g(π) + g(1 π)] (1 π)[g(π) + g(1 π)] = π 1 π One can recover the distribution of qualities on [1/2, 1], denoted by G, from G by combining qualities that yield the same beliefs for opposing signals (e.g q = 1/4 and signal h is combined with q = 3/4 and signal l). With symmetric g, G(1/2) = 1/2, and G(q) = q 1 2 g(s)ds q q g(s)ds = 2 g(s)ds = 2G(q) 2G(1/2) = 2G(q) 1. (1) 1 2 E Simulation Procedure for Price Efficiency We employed the following data generation procedure for the simulations: We obtained 500,000 observations of trading days for each of the Poisson arrival rates ρ {5, 10, 15, 20, 25, 30, 35, 40, 45, 50} and levels of informed trading µ {.1,.2,.3,.4,.5,.6,.7,.8,.9}. Here, the Poisson arrival rate ρ implies that, on average, there are ρ traders (though some may choose not to trade). Fixing the true value to V = 1, s are closer to the true value if they are larger. To get a better sense of the effect of a small ρ for transparent vs. opaque hybrid markets, we also ran ρ {1, 2,...,14} for µ {.2,.5,.8}, as outlined in the main text. For each series, we first drew the number of traders for the session and performed the random allocation of traders into noise and informed. Thus overall there were, for instance, approximately 25,000,000 trades for ρ = 50. The informed traders were then equipped with a signal quality and a draw of the high or low signal for that quality, conditional on V = 1. Noise traders were assigned a random trading role. We then determined a random entry order and for the hybrid market performed an additional random draw to determine in which market the respective trader will be placing this order. Finally, we computed the end-s that would result for each trading sessions under each of the four trading regimes. One can think of these s at the s that would obtain at the end of a trading day. Our results are for the distribution of these final s. Our descriptions on properties of the empirical distributions are based on the case with ρ = 50. The rule of thumb is that the larger is ρ, the diverse the outcomes of trading rounds are and the closer are the distributions of end s to a smooth continuous function (for small values of ρ they resemble step functions). 3 Appendix for Trading Mechanisms & Market Dynamics

5 0.020 Ask(limit)-Ask(hybrid) Ask(hybrid)-Ask(dealer) mu mu Figure 4: Bid-Ask-Spreads in Limit Order, Dealer and Hybrid Markets. In each panel a line plots a difference of ask-s under specific market mechanisms as a function of the amount informed trading, µ, for a specific prior p. The left panel plots the difference of ask s for the first unit in the limit order and hybrid markets, the right plots the difference of ask s for small trades in the hybrid and dealer markets. The panels illustrate Proposition 3 (a). References Smith, L., and P. Sorensen (2008): Rational Social Learning with Random Sampling, mimeo, University of Michigan. 4 Appendix for Trading Mechanisms & Market Dynamics

6 m m K Cost(limit)-Cost(hybrid) K K K Cost(hybrid)-Cost(dealer) K0.01 K0.02 K K0.03 K Figure 5: Execution Costs for Large Orders in Limit Order, Dealer and Hybrid Markets. In each panel a line plots a difference of execution costs for larger orders under specific market mechanisms as a function of the amount informed trading, µ, for a specific prior p. The left panel plots the cost difference in limit order market and the hybrid market, the right panel plots the cost difference for the hybrid market and the dealer market. The panels illustrate Proposition 3 (b). m m K0.05 Impact(limit)-Impact(hybrid limit) K0.05 K0.10 K0.15 Impact(hybrid limit)-impact(dealer) K0.10 K0.15 K0.20 K0.25 Figure 6: Price Impacts of Small Trades in Limit Order, Dealer and Hybrid Markets. In each panel a line plots a difference of impacts of small orders under specific market mechanisms as a function of the amount informed trading, µ, for a specific prior p. The left panel plots the difference of impacts of small orders in the limit order market and the hybrid market, the right panel plots the difference of impacts of small orders in the limit order segment of the hybrid market and isolated dealer market. As ask 1 H > ask 1 D by part (a), the panels illustrate Proposition 3 (c). 5 Appendix for Trading Mechanisms & Market Dynamics

7 Impact(hybrid limit)-impact(limit), large trades Impact(dealer)-Impact(hybrid dealer), large trades m m Figure 7: Price Impacts of Large Trades in Limit Order, Dealer and Hybrid Markets. In each panel a line plots a difference of impacts of large orders under specific market mechanisms as a function of the amount informed trading, µ, for a specific prior p. The left panel plots the difference of impacts of large orders in the limit order segment of the hybrid market and the isolated limit order market market, the right panel plots the difference of impacts of large orders in the isolated dealer market and the dealer segment of the hybrid market. The panels illustrate Proposition 3 (c) Volume(limit)-Volume(hybrid) Volume(hybrid)-Volume(dealer) m m Figure 8: Volume in Limit Order, Dealer and Hybrid Markets. In each panel a line plots a difference of volume under specific market mechanisms as a function of the amount informed trading, µ, for a specific prior p. The first panel plots the difference of volume in limit order and hybrid markets, the second plots the difference of volume in hybrid and dealer markets. Both cleanly indicate the order expressed in Numerical Observation 1. 6 Appendix for Trading Mechanisms & Market Dynamics

8 Closing Price Limit Order Market - Dealer Market Table 1: Difference of Average Closing Prices: Limit Order vs. Dealer Market. This table is based upon the simulations described in the main text. Columns denote the entry rate ρ, rows the level of informed trading µ. Thus each entry denotes the difference of the average closing s in Limit Order and Dealer markets for a specific (ρ, µ)-combination. As the underlying true value is V = 1, the higher a is, the closer it is to the true value and thus the more efficient it is. Thus a positive difference of the average closing s indicates that the Limit Order Market is more efficient. As the table indicates, this is always the case but for the largest values of µ that we considered. This table relates to Numerical Observation 2. 7 Appendix for Trading Mechanisms & Market Dynamics

9 Closing Price Limit Order Market - Hybrid Market Table 2: Difference of Average Closing Prices: Limit Order vs. Hybrid Market. This table is based upon the simulations described in the main text. Columns denote the entry rate ρ, rows the level of informed trading µ. Thus each entry denotes the difference of the average closing s in Limit Order and hybrid markets for a specific (ρ, µ)-combination. As the underlying true value is V = 1, the higher a is, the closer it is to the true value and thus the more efficient it is. Thus a negative difference of the average closing s indicates that the hybrid market is more efficient. As the table indicates, this is always the case. This table relates to Numerical Observation 2. 8 Appendix for Trading Mechanisms & Market Dynamics

10 Closing Price Dealer Market - Hybrid Market Table 3: Difference of Average Closing Prices: Dealer vs. Hybrid Market. This table is based upon the simulations described in the main text. Columns denote the entry rate ρ, rows the level of informed trading µ. Thus each entry denotes the difference of the average closing s in dealer and hybrid markets for a specific (ρ, µ)-combination. As the underlying true value is V = 1, the higher a is, the closer it is to the true value and thus the more efficient it is. Thus a negative difference of the average closing s indicates that the hybrid market is more efficient. As the table indicates, this is always the case but for the largest values of µ that we considered. This table relates to Numerical Observation 2. 9 Appendix for Trading Mechanisms & Market Dynamics

11 µ =.1 µ =.2 µ = µ =.4 µ =.5 µ = µ =.7 µ =.8 µ =.9 Figure 9: First Order Stochastic Dominance of Closing Prices Dealer Market vs. Limit Order Book. The panels plot differences of empirical distributions as a functions of the F D (p) F L (p) and illustrates Numerical Observation 3 (a). 10 Appendix for Trading Mechanisms & Market Dynamics

12 FOSD dealer vs hybrid cdf(dealer) cdf(hybrid) cdf(dealer) cdf(hybrid) µ =.1 µ =.2 µ =.3 FOSD dealer vs hybrid FOSD dealer vs hybrid FOSD dealer vs hybrid cdf(dealer) cdf(hybrid) cdf(dealer) cdf(hybrid) cdf(dealer) cdf(hybrid) µ =.4 µ =.5 µ =.6 FOSD dealer vs hybrid FOSD dealer vs hybrid cdf(dealer) cdf(hybrid) cdf(dealer) cdf(hybrid) FOSD dealer vs hybrid µ =.7 µ =.8 µ =.9 Figure 10: First Order Stochastic Dominance of Closing Prices Dealer Market vs. Hybrid Market. The panels plot differences of empirical distributions as a functions of the F D (p) F H (p) and illustrates Numerical Observation 3 (b). 11 Appendix for Trading Mechanisms & Market Dynamics

13 FOSD LOB vs hybrid FOSD LOB vs hybrid FOSD LOB vs hybrid µ =.1 µ =.2 µ =.3 FOSD LOB vs hybrid FOSD LOB vs hybrid FOSD LOB vs hybrid µ =.4 µ =.5 µ =.6 FOSD LOB vs hybrid FOSD LOB vs hybrid FOSD LOB vs hybrid µ =.7 µ =.8 µ =.9 Figure 11: First Order Stochastic Dominance of Closing Prices Limit Order Book vs. Hybrid Market. The panels plot differences of empirical distributions as a functions of the F L (p) F H (p) and illustrates Numerical Observation 3 (c). 12 Appendix for Trading Mechanisms & Market Dynamics

14 ccdf(dealer) ccdf(lob) SOSD dealer vs LOB ccdf(dealer) ccdf(lob) SOSD dealer vs LOB ccdf(dealer) ccdf(lob) SOSD dealer vs LOB ccdf(dealer) ccdf(lob) µ =.1 µ =.2 µ =.3 SOSD dealer vs LOB SOSD dealer vs LOB ccdf(dealer) ccdf(lob) ccdf(dealer) ccdf(lob) SOSD dealer vs LOB ccdf(dealer) ccdf(lob) µ =.4 µ =.5 µ =.6 SOSD dealer vs LOB ccdf(dealer) ccdf(lob) SOSD dealer vs LOB µ =.7 µ =.8 µ =.9 Figure 12: Second Order Stochastic Dominance of Closing Prices Dealer Market vs. Limit Order Book. The panels plot differences of empirical distributions as a functions of the p 0 [F D(s) F L (s)]ds and illustrates Numerical Observation 4 (a). 13 Appendix for Trading Mechanisms & Market Dynamics

15 ccdf(dealer) ccdf(hybrid) SOSD dealer vs hybrid ccdf(dealer) ccdf(hybrid) SOSD dealer vs hybrid ccdf(dealer) ccdf(lob) SOSD dealer vs LOB ccdf(dealer) ccdf(hybrid) µ =.1 µ =.2 µ =.3 SOSD dealer vs hybrid SOSD dealer vs hybrid SOSD dealer vs hybrid ccdf(dealer) ccdf(hybrid) ccdf(dealer) ccdf(hybrid) µ =.4 µ =.5 µ =.6 SOSD dealer vs hybrid SOSD dealer vs hybrid SOSD dealer vs hybrid ccdf(dealer) ccdf(hybrid) ccdf(dealer) ccdf(hybrid) ccdf(dealer) ccdf(hybrid) µ =.7 µ =.8 µ =.9 Figure 13: Second Order Stochastic Dominance of Closing Prices Dealer Market vs. Hybrid Market. The panels plot differences of empirical distributions as a functions of the p 0 [F D(s) F H (s)]ds and illustrates Numerical Observation 4 (b). 14 Appendix for Trading Mechanisms & Market Dynamics

16 SOSD LOB vs. Hybrid SOSD LOB vs. hybrid SOSD LOB vs. Hybrid µ =.1 µ =.2 µ =.3 SOSD LOB vs. Hybrid SOSD LOB vs. Hybrid SOSD LOB vs. Hybrid µ =.4 µ =.5 µ =.6 SOSD LOB vs. Hybrid SOSD LOB vs. Hybrid SOSD LOB vs. Hybrid µ =.7 µ =.8 µ =.9 Figure 14: Second Order Stochastic Dominance of Closing Prices Limit Order Book vs. Hybrid Market. The panels plot differences of empirical distributions as a functions of the p 0 [F L(s) F H (s)]ds and illustrates Numerical Observation 4 (c). 15 Appendix for Trading Mechanisms & Market Dynamics

17 Closing Price Hybrid Transparent - Hybrid Opaque Table 4: Difference of Average Closing Prices: Transparent vs. Opaque Hybrid Market. This table is based upon the simulations described in the main text. Columns denote the entry rate ρ, rows the level of informed trading µ. Thus each entry denotes the difference of the average closing s in transparent and opaque hybrid markets for a specific (ρ, µ)-combination. As the underlying true value is V = 1, the higher a is, the closer it is to the true value and thus the more efficient it is. Thus a positive difference of the average closing s indicates that the transparent hybrid market is more efficient. As the table indicates, this is always the case but for small values of ρ that we considered; see also Table 5. This table support Numerical Observation 5 (a). 16 Appendix for Trading Mechanisms & Market Dynamics

18 Closing Price Hybrid Transparent - Hybrid Opaque for small rho Table 5: Difference of Average Closing Prices: Transparent vs. Opaque Hybrid Market small values of ρ. This table complements Table 5 and considers small values of ρ. is based upon the simulations described in the main text. Columns denote the entry rate ρ, rows the level of informed trading µ. Thus each entry denotes the difference of the average closing s in transparent and opaque hybrid markets for a specific (ρ, µ)-combination. As the underlying true value is V = 1, the higher a is, the closer it is to the true value and thus the more efficient it is. Thus a positive difference of the average closing s indicates that the transparent hybrid market is more efficient. As the table indicates, for small values of ρ, the opaque market may be more efficient. Further, for larger µ, the set of entry rates for which this applies is larger. This table support Numerical Observation 5 (a). 17 Appendix for Trading Mechanisms & Market Dynamics

19 FOSD Hybrid opaque vs transparent FOSD Hybrid opaque vs transparent FOSD Hybrid opaque vs transparent µ =.1 µ =.2 µ =.3 FOSD Hybrid opaque vs transparent FOSD Hybrid opaque vs transparent FOSD Hybrid opaque vs transparent µ =.4 µ =.5 µ =.6 FOSD Hybrid opaque vs transparent FOSD Hybrid opaque vs transparent FOSD Hybrid opaque vs transparent µ =.7 µ =.8 µ =.9 Figure 15: First Order Stochastic Dominance of Closing Prices Transparent vs. Opaque Hybrid Market. The panels plot differences of empirical distributions as a functions of the F D (p) F L (p). 18 Appendix for Trading Mechanisms & Market Dynamics

20 SOSD Hybrid opaque vs transparent SOSD Hybrid opaque vs transparent SOSD Hybrid opaque vs transparent µ =.1 µ =.2 µ =.3 SOSD Hybrid opaque vs transparent SOSD Hybrid opaque vs transparent SOSD Hybrid opaque vs transparent µ =.4 µ =.5 µ =.6 SOSD Hybrid opaque vs transparent SOSD Hybrid opaque vs transparent SOSD Hybrid opaque vs transparent µ =.7 µ =.8 µ =.9 Figure 16: Second Order Stochastic Dominance of Closing Prices Transparent vs. Opaque Hybrid Market. The panels plot differences of empirical distributions as a functions of the p 0 [F L(s) F H (s)]ds and supports Numerical Observation 5 (b). 19 Appendix for Trading Mechanisms & Market Dynamics