ABSTRACT Title of dissertation: Essays on Asset Pricing Su Li, Doctor of Philosophy, 2012 Dissertation directed by: Professor Albert \Pete" Kyle Department of Finance This dissertation consists of three essays. The rst essay is titled \Speculative Dynamics I: Imperfect Competition, and the Implications for High Frequency Trading". In this essay, I analyze the nature of imperfect competition among informed traders who continuously generate and exploit private information about a risky asset?s liquidation value which follows either a mean reverting process or random walk. I nd the following results: (i) The combined trading of multiple informed traders is much more aggressive than the monopolistic trader in Chau and Vayanos (2008). (ii) The equilibrium price is even more revealing of the informed trader?s private information. (iii) Market depth improves as the number of informed traders increases. (iv) In the limit of continuous trading, market is strong form e cient while aggregate pro ts of the informed traders remain bounded away from zero, in sharp contrast to the corresponding results in Holden and Subrahmanyam (1992), and Foster and Viswanathan (1993). (vi) Informed traders? inventories follows a Brownian motion, therefore enabling them to contribute signi cantly to total trading volume and price variance. These results shed light on empirical ndings regarding high frequency traders by helping explain why they remain pro table despite aggressive competition with each other, why their trading volume is very high, to what extent they improve e ciency, and through what mechanism they improve liquidity. The second essay is titled \Speculative Dynamics II: Asymmetric Informed Traders". In this essay, I study the strategic interaction between hierarchical duopolistic informed traders who continuously generate and exploit private information about a risky asset?s liquidation value, which follows either a mean reverting process or random walk. I nd the following results: (i) Both traders duopolize the private information they both observe and the more informed trader monopolizes the additional exclusive private information. (ii) The common private information is incorporated into prices more e ciently than the monopolistic private information. (iii) In the limit of continuous trading, both traders? inventories based on their shared information follow Brownian motions. (iv) The trader with less superior information has more contribution to the trading volume and price volatility when the frequency of trading is su ciently high. (v) As trading becomes more frequent, the less informed trader?s expected pro ts may fall but converges to a strictly positive constant in the limit. The third essay is titled \Real Options and Product Di erentiation". In this essay, I de- velop a continuous time real investment model in an oligopoly industry where the products are heterogenous. Although the heterogenous products assumption can lower each rm?s incentive to exercise the growth options prematurely, the preemption strategy is still prof- itable. Essays on Asset Pricing by Su Li Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial ful llment of the requirements for the degree of Doctor of Philosophy 2012 Advisory Committee: Professor Albert \Pete" Kyle, Chair/Advisor Professor Steven L. Heston Professor Mark Loewenstein Professor Dilip Madan Professor Daniel Vincent c Copyright by Su Li 2012 Dedication To my family. ii Acknowledgments This dissertation would not have been possible without the the guidance of my committee members, help from my friends, and support from my wife and two children. Firstly, I would like to express my deepest and most sincere gratitude to my advisor Professor Albert \Pete" Kyle. Ever since I entered the program, I am impressed not only by his wisdom and wide knowledge, but also by his willingness to spend time talking with students. I am thankful for his considerable patience and con dence on me, and his guidance and encouragement during the last three years of research. I am deeply grateful to Professor Mark Loewenstein. He has a very high standard of research and encouraged me to tackle di cult problems from the beginning. He was always there when I had questions and helped me through many di culties. Professor Steven L. Heston taught me many things about research including the GARCH model and option games. He even taught me how to make better presentations and how to write referee reports. I am very grateful to Professor Dilip Madan for chatting with me about research for the last few years. Special thanks goes to Professor Daniel Vincent for willing to participate in my dissertation committee. I would like to thank Wei Li, who as a good friend, has always been willing to help and give his best suggestions. His discussions with me have lead to many interesting ndings. I would also like to thank all the other Ph.D. students in the program especially Katie Moon, Danmo Lin, Mina Lee, and Onur Tosun for their kindness, friendship and support. On the personal side, I am deeply grateful to my wife Jianzhu Li for her support and iii for \tolerating" my pursuit of a second Ph.D. degree. I appreciate everything she has been doing. I also would like to thank my daughter Yifan Li and son Yixuan Li for bringing joys to my life. Their love has given me strength to complete graduate study after going through many years of work. iv Table of Contents 1 Speculative Dynamics I: Imperfect Competition, and the Implications for High Fre- quency Trading 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Equilibrium Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 The Market Maker?s Inference Problem . . . . . . . . . . . . . . . . . 13 1.2.3 The Informed Traders? Optimization Problem . . . . . . . . . . . . . 15 1.2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Asymptotic Properties of Equilibrium in the Limit of Continuous Trading . . 17 1.3.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.3.2 Asymptotic Properties of Equilibrium . . . . . . . . . . . . . . . . . . 20 1.3.3 Properties of the Perfectly Competitive Equilibrium . . . . . . . . . . 24 1.4 Numerical Illustrations and Comparative Statics . . . . . . . . . . . . . . . . 25 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2 Speculative Dynamics II: Asymmetric Traders 44 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.1 Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.2 Equilibrium Strategies and Value Functions . . . . . . . . . . . . . . 54 2.3 Inference Problems, Optimizations and The Equilibrium . . . . . . . . . . . 56 2.3.1 The Market Maker?s Inference Problem . . . . . . . . . . . . . . . . . 56 2.3.2 The Less Informed Trader?s Inference Problem . . . . . . . . . . . . . 57 2.3.3 The More Informed Trader?s Optimization Problem . . . . . . . . . . 59 2.3.4 The Less Informed Trader?s Optimization Problem . . . . . . . . . . 62 2.3.5 Equilibrium and Trading in the Continuous Time Limit . . . . . . . . 63 2.4 Properties of Equilibriums and Comparative Statics . . . . . . . . . . . . . . 66 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3 Real Options and Product Di erentiation 79 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2.1 Demand Function and Capacity Process . . . . . . . . . . . . . . . . 81 3.2.2 Risk Neutral Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3 Symmetric Open Loop Equilibrium for CES Demand Function . . . . . . . . 83 3.3.1 The Open Loop Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 83 3.3.2 A Preemption Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4 Linear Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.4.1 The Open Loop Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 93 3.4.2 A Preemption Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 v A Appendix 104 B Appendix 114 vi Chapter 1 Speculative Dynamics I: Imperfect Competition, and the Implications for High Frequency Trading 1.1 Introduction Fama (1970) suggests that in a strong form e cient market, price re ects all public and private information. Will it be possible for traders with superior private information to earn strictly positive expected pro ts in a strong-form e cient market? Holden and Subrah- manyam (1992) and Foster and Viswanathan (1993) nd that this is not possible and instead they reach a Bertrand-like result1. The intuition is that if there are two or more strategic traders receiving the same private information, each trader tries to preempt the others with the result that information is revealed almost instantaneously and each trader?s expected pro ts quickly dissipate to zero in the limit of continuous trading. Surprisingly, Chau and Vayanos (2008) nd that positive expected pro ts are possible while the market is strong form e cient. A monopolistic informed trader privately observes the ow of information and chooses to trade aggressively on her information to push the price towards her valuation of the asset2. In the limit, the information asymmetry disappears but the insider?s pro ts 1The informed traders in Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993) submit market orders to the market maker instead of a demand schedule. The intuition is similar to that of Back and Paulsen (2009), in which each rm tries to preempt other rms by investing earlier with the result that the value of growth options equal to zero and the outcome is competitive. 2According to Chau and Vayanos (2008), the monopolist is \impatient" for three reasons: (1) time discounting, (2) public revelation of information, (3) mean-reversion of pro tability. 1 converge to a positive constant. A similar result can also be found in Hellwig (1982) in which competitive informed traders submit market order conditioning on past prices. Even as the time interval between trades converges to zero, insiders can make positive returns while pushing the price to an arbitrarily closed to the e cient value. The di erence among these models depends on the arrival of private information. Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993) follow the assumptions in Kyle (1985) in which private information is one-shot and the asset?s value is xed. The private information is received when the trading session starts and there is a predetermined date when information is publicly announced and the asset is liquidated. In Chau and Vayanos (2008), the informed trader receives new information repeatedly, the fundamental value of the asset is stochastic, and trading takes place over an in nite horizon. It is reasonable to believe that in actual nancial markets, there are several information- aly large strategic investors consistently generating new information and trading to pro t from the information. In this paper, I study the nature of imperfect competition among those traders. One of the purposes is to examine how imperfect competition and repeated arrivals of private information a ect market e ciency, market liquidity, trading volume, price volatility, and the pro tability of informed traders, especially relative to the monopolistic case in Chau and Vayanos? model and the oligoplistic case in Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993). The other purpose of this paper is to shed light on high frequency trading by examining the properties of equilibrium in the limit of continuous trading. The paper can explain why high frequency traders3 remain pro table 3According to Chlistalla (2011), there are three types of high frequency traders: (i) electronic market 2 despite aggressive competition with one another, why they contribute a signi cant portion to trading volume and price volatility, and to what extent do those traders improve market e ciency and add to market liquidity. In the model, there is a riskless bond and a risky asset. The liquidation value of the risky asset which follows a stochastic process can only be observed by informed traders. Like the \market order" model in Kyle (1985), multiple identically informed strategic traders and exogenous liquidity traders execute batched market orders against competitive risk neutral market makers. The informed traders receive new information each period and trading takes place until the asset is liquidated at a random date. I prove that there exists a unique linear equilibrium, obtain the closed form solution up to a set of nonlinear equations, and derive analytical forms when trading becomes continuous. Not surprisingly, the combined trading of multiple informed traders is more aggressive than the monopolistic trader in Chau and Vayanos, the equilibrium price is even more revealing of the informed trader?s private information, and market depth improves as the number of informed traders increases. Oligopolistic imperfect competition makes the informed traders trade more aggressively than a monopolist, thus improving market e ciency and increasing aggregate trading volume. The e ects of imperfect competition on market depth is slightly more di cult to interpret since it has two opposite e ects. One the one hand, with increasing competition, initially the net order ow will contain more information relative to the noise trading, and therefore the adverse selection is more severe and market depth is worse. On the other hand, as market becomes more e cient, there is less private information contained in the order ow, thus making; (ii) statistical arbitrage strategies; (iii) liquidity detection. 3 improving market depth. In the stationary state, I show that the second e ect dominates and imperfect competition among informed traders makes market deeper, thus improving liquidity. Surprisingly, the model uncovers some important but unexpected results in the limit as the time interval between trades goes to zero. The rst result is that the variance of the private information held by informed traders goes to zero at a rate proportional to the time interval between rounds of trading. This is much faster than the corresponding strong from e ciency result in the Chau and Vayanos? model, where the convergence rate is proportional to the square root of the time interval between rounds of trading. The second result is that the aggregate pro ts of the informed traders remain bounded away from zero. As the number of informed traders increases, their aggregate pro ts fall, tending to zero only as the number of informed traders becomes large. To be more speci c, the aggregate pro ts near continuous trading is inversely proportional to the square root of the number of informed traders. The result has the avor of Cournot competition, not the avor of Bertrand competition found in the one-shot private information model of Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993). The third result concerns volume and volatility. The trading volume4 of informed traders over t is of order p t, the same magnitude as the trading volume of liquidity traders. This implies that in the limit informed traders make a non-negligible contribution to total trading volume and thus price volatility, and the fraction converges to one as the number of informed traders becomes large. The result is novel since in almost all 4Trading volume is not well de ned in continuous time Kyle?s model since total variation of Brownian motion over any nite time is in nite. 4 the continuous time market microstructure model with price impact (Kyle?s model and many extensions), the contributions to trading volume and price volatility by informed traders is negligible compared to the contributions of liquidity traders. The results near continuous trading can help explain some empirical ndings regard- ing high frequency trading. In recent years, nancial markets have witnessed rapid growth in high frequency trading5, made possible by the evolution of technology. High frequency traders are a subset of algorithmic traders. Those traders apply mathematical algorithms to either public or private statistical information, and they use fast computers to implement the algorithms, transmitting orders in a few milliseconds or less. High frequency traders contribute signi cantly to trading volume6. Despite aggressive competition with one an- other, high frequency traders remain pro table7. Kirilenko et al. (2011) document that high frequency traders are consistently pro table. They even turned a pro t on the day on May 6, 2010 Flash Crash. I focus on the type of high frequency traders who are pursuing low latency statistical arbitrage strategies. According to Chlistalla (2011), these traders \seek to correlate be- tween assets and try to pro t from the imbalance in these correlations". It might not be appropriate to label those traders as \informed" if one de nes information as corporate news on \merger and acquisition decisions" or \content of earnings announcements". However, 5As pointed out by Duhigg in Stock Traders Find Speed Pays, in Milliseconds (New York Times, July 23, 2009), \Average daily volume has soared by 164 percent since 2005, according to data from NYSE. ..., stock exchanges say that a handful of high-frequency traders now account for more than half of all trades." 6In \The Real Story of Trading Software Espionage" (AdvancedTrading.com, July 10, 2009), Iati mentions that \High-frequency trading rms, which represent approximately 2% of the trading rms operating in the U.S. markets today, account for 73% of all U.S. equity trading volume." 7Iati?s article states that \TABB group estimates that annual aggregate pro ts of low-latency arbitrage strategies exceed $21 billion, spread out among the few hundred rms that deploy them." 5 if those traders are faster and better than average market participants in gathering and processing information on order ows and price movements on the security and any other correlated securities to generate private signals which they can pro t from, it is reasonable to call them \informed"8. At rst glance, it appears that Chau and Vayanos (2008) can be a good model for those high frequency statistical arbitrageurs. Although their model has good insights on market e ciency and pro tability, it does not explain trading volume and market liquidity. The fraction of trading volume of the monopolistic informed trader in Chau and Vayanos is essentially zero in the limit of continuous trading, and the market becomes less liquid as the frequency of trading increases. Their model does not address the question of what happens to high frequency traders? pro ts if they compete very aggressively with each other. By focusing on imperfect competition in a non-cooperative duopolistic setting, I am able to show that high frequency traders incorporate private signals into prices much faster than in Chau and Vayanos? model, high frequency traders add to market liquidity by competing aggressively with each other, they contribute signi cantly to trading volume, and most important of all, those traders remain pro table despite exploiting from the same information set and implementing the same algorithms. After the ash crash of May 6, 2010, a recent policy proposal suggests that batching orders less frequently can reduce the participation rate and pro ts of high frequency traders and improve market depth. My model suggests that such a regulation would have the opposite e ect of reducing liquidity. If each high frequency trader has to pay an entry cost 8Hendershott and Riordan (2011b) nd that the market orders by high frequency traders have information advantage. 6 and operating cost, the number of high frequency traders might decrease with less frequent order batching, with the result that less competition will lead to market being less liquid. This paper belongs to the literature on strategic trading with asymmetric information. In the pioneering work of Kyle (1985), a monopolistic insider uses liquidity traders as cam- ou age, reveals her private information gradually, and exploits her monopoly power over time when facing a competitive risk neutral market maker. In the subsequent extensions by Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993), due to the imperfect competition among identically informed traders, almost all private information is revealed only after a few trading rounds. Foster and Viswanathan (1994) replace homogenous private information with a hierarchical information structure to study learning among strategic in- formed traders. Foster and Viswanathan (1996) relax the assumption even further to allow for a more general correlation structure among the signals received by multiple informed traders. They show the initial correlation among the signals has a strong e ect on the trad- ing strategies and informativeness of prices. Traders initially compete aggressively on the common part of the private information and later play a \waiting" game by making smaller bets and trying to infer private information exclusive to others. Back et al. (2000) solve a similar problem in continuous time and derive a closed-form solution. The traders in my model exploit their private signals via market orders. Rosu (2009) directly models the limit order book. He also nds a similar prediction that higher com- petition causes smaller price impact. However, competition in his model is measured by how fast traders arrive in the market whereas in my model, competition is measured by the 7 number of informed traders. In his model, traders trade for liquidity reasons and there is no asymmetric information, whereas the motive for trade in my model comes from private information. Martinez and Rosu (2011) study a very similar problem. They tackle the problem directly in setup similar to that of Back (1992) and focus on non-stationary equilibrium. In order to generate linear equilibrium in continuous time, they assume an informed trader to have uncertainty aversion regarding the level of the asset value (Informed traders care more about the change in the value of asset than the level) and impose a technological constraint on the market maker. My paper, on the other hand, does not require such assumptions and focuses on stationary equilibrium. In my model, high frequency traders are risk neutral. Therefore, the model cannot explain the phenomenons that high frequency traders reverse their inventories frequently, and move in and out of short-term positions very quickly. Future work may explain pattern of mean-reverting inventories by making the traders risk averse instead of risk neutral. The paper is organized as follows. In Section II, I describe the model, solve the linear equilibrium, prove its uniqueness. Section III characterizes the equilibrium near continuous trading. Section IV shows some comparative statics results and derives empirical implica- tions. Section V concludes. 1.2 The Model Assumption 1: Securities 8 I consider an economy with a single consumption good. There are a riskless bond with zero interest rate and a non-dividend paying risky asset with a liquidation value vn which evolves stochastically. Trading takes place from t = 0 to t = +1 at the discrete points tn (tn = nh), until the risky asset is liquidated where h is the time interval between the auctions. At the end of each period, there is a probability 1 exp ( rh) = rh + o(h2) that the risky asset is liquidated. I further assume the riskless bonds are in perfectly elastic supply. The liquidation value vn follows a mean-reverting process or random walk: vn vn 1 = ( v vn 1)h+ v;n: (1.1) In the above speci cation, determines the adjustment speed of the liquidation value vn to its long run xed target v. is assumed to be greater than or equal to zero such that the prices are stationary. If = 0, then vn follows a random walk. The innovation v;n is independently and normally distributed with mean zero and variance 2vh. Assumption 2: Market Participants and Information Structure The risk neutral market participants consist of a competitive market maker, M (M is a positive integer) informed strategic traders, and a number of liquidity traders. The informed traders are each assumed to be able to perfectly observe the liquidation value vn at each period. Conditional on that the asset has not yet been liquidated at the beginning of the nth period, I in fp ; v j0 ng is each informed trader?s information set at t = nh, and Imn = fp j ng is the market makers? information set. At each period, both the informed traders and liquidity traders submit market orders 9 to the market maker. The liquidity traders? order is denoted by un, which is normally distributed with mean zero and variance 2uh. I further assume un is uncorrelated with v;n. I denote the market order submitted by the jth informed trader at the nth period (t = nh) by xj;n. In equilibrium, I should have x1;n = ::: = xM;n = xn because of symmetry argument. Assumption 3: Timing of events I assume at the nth period, the informed traders and the noise traders submit their demands before new information arrives. After submitting their market orders, the informed traders observe n and thus vn. The market maker observes the total order imbalance yn = PM j=1 xj;n +un, then sets the price pn equal to the expected value of the asset based on the history of order ows, and takes the other side of the trade. At the end of the period, there is a probability (1 exp ( rh)) that the liquidation value vn is public announced, the risky asset is liquidated and investors pro ts are realized. Pricing Since the market maker is assumed to be competitive and risk neutral, therefore, at period n she sets the price pn equal to the expected value of the asset after she receives the total batched market order yn = x1;n + :::+ xM;n + un. Therefore, pn = E[ +1X n0=n (1 exp ( rh)) exp ( r(n0 n)h)vn0jI m n 1; yn]; (1.2) where (1 exp ( rh)) exp ( r(n0 n)h) is the probability that the asset is liquidated at the end of the n0th period. Lemma 1.2.1: The price pn is a linear function of the market maker?s expectation of 10 the current liquidation value of the risky asset E(vnjImn ): pn = (1 exp ( rh)) 1 exp ( rh)(1 h) E(vnjI m n ) + h v 1 exp( rh)(1 h) : (1.3) Proof : See Appendix A. Optimization Suppose the risky asset were liquidated at a random future date h. Given that the asset has not been liquidated at nh < h, the jth informed trader?s pro ts that accrue to her from period n should equal to the di erence between the value of her position ( P n v xj; ) and the cost of this position ( P n p xj; ): j;n = X n (v p )xj; (1.4) = +1X n0=n (1 exp( rh)) exp( r(n0 n)h)( n0X =n xj; (vn0 p )) Since informed traders are risk neutral, at the nth period, the jth informed trader tries to maximize her expected trading pro ts: max xj;n0 n E[ +1X n0=n (1 exp( rh)) exp( r(n0 n)h)( n0X =n xj; (vn0 p ))jI i n 1]: (1.5) 11 Lemma 1.2.2: The jth informed trader?s objective function can be written as: max xj;n0 n E[ +1X n0=n exp( r(n0 n)h)xj;n0( (1 exp ( rh)) 1 exp ( rh)(1 h) vn0+ h v 1 exp( rh)(1 h) pn0)jI i n 1]: (1.6) Proof : See Appendix A. 1.2.1 Equilibrium Concept The equilibrium concept in this paper is similar to the previous literature. I follow Foster and Viswanathan (1996) closely here and let Xj = (xj;1; :::; xj; ) (for each j) and P = (p1; :::; p ) represent the strategy functions were the asset liquidated at h. A Bayesian Nash equilibrium of the trading game is a M + 1 vector of strategies (X1; :::; XM ; P ) such that: 1. For any j = 1; :::;M and all n = 1; :::; , I have for X 0j = (x 0 j;1; :::; x 0 j; ) E[ j;n(X1; :::; Xj; :::; XM ; P )jI i n 1] E[ j;n(X1; :::; X 0 j; :::; XM ; P )jI i n 1] (1.7) 2. For all n = 1; :::; , I have pn = (1 exp ( rh)) 1 exp ( rh)(1 h) E(vnjI m n ) + h v 1 exp( rh)(1 h) : (1.8) Therefore, the market maker sets the price equal to the expected value of the risky asset conditional on her information set inferred from the order ow. Each risk neutral informed trader, taking as given the price process set by the market maker and the strategies of 12 other informed traders, submits market orders to maximize the expected pro ts taking into account the e ect on the price. I restrict attention to stationary linear Markov eqilibrium. In order to set the price pn which takes a linear form in Equation (1.3), the market maker has to solve the inference problem about vn. I then conjecture that informed trader j?s optimal strategy at period n is to submit market orders which depend linearly on the pricing error de ned as the di erence between vn 1 and the market maker?s conditional estimate v^n 1 = E(vn 1jImn 1), i.e., xj;n = j(vn 1 v^n 1); (1.9) to maximize her expected pro ts. 1.2.2 The Market Maker?s Inference Problem To solve the market maker?s inference problem, I use Kalman ltering. Conjecture that at the end of the (n 1)th period, the market maker believe vn 1 to be normally distributed with mean v^n 1 and variance v9. Then, at the nth period, after observing the order imbalance yn, the market maker updates her belief about vn 1 in the form of vn 1 = E(vn 1jIn 1) + 1 h yn + n (1.10) = v^n 1 + 1 h (Xn + un) + n; 9 v is strictly greater than 2vh since the informed traders observe v;n 1 after they submit the market order at the (n 1)th period. 13 where is the inference parameter for the market maker to be derived next. Since vn follows a mean-reverting process (random walk if = 0) in Equation (1.1), vn has the following expression: vn = (1 h)v^n 1 + h v + yn + (1 h) n + v;n: (1.11) Therefore, the market maker?s posterior belief about vn is normally distributed with mean E(vnjI m n ) = v^n = (1 h)v^n 1 + h v + yn (1.12) and variance var(vnjI m n ) = var((1 h) n + v;n) = (1 h) 2var( n) + 2 vh: (1.13) Stationary condition requires that var(vnjImn ) = v. Lemma 1.2.3: Given the trading strategy of the informed traders de ned in equation (1.9), the market maker?s inference parameter is given by = (1 h) vM M2 2 v + 2uh ; (1.14) and the variance of the market maker?s belief on vn satis es the equation (1 h)2 v 2uh M2 2 v + 2uh + 2vh = v: (1.15) 14 Proof : See Appendix A. 1.2.3 The Informed Traders? Optimization Problem At the beginning of each period, each informed trader submits market orders given the price process generated by the market maker to maximize the present value of the expected pro ts scaled by (1 exp ( rh))1 exp ( rh)(1 h) . The j th informed trader?s optimization problem becomes E[ +1X n0=n xj;n0(vn0 v^n0)e r(n0 n)hjI in 1]: (1.16) In the optimization, the informed trader takes account of how his trading and his estimate of the trading by other market makers in uence the market price, which equal to market makers estimate v^n. I conjecture that the value function for the jth informed trader is quadratic with respect to vn 1 v^n 1 and takes the form of Vj;n = B(vn 1 v^n 1) 2 + C: (1.17) Later, I will prove the quadratic form is sustained and valid in Lemma 1.2.4. The value function must satisfy the following Bellman equation: V (vn 1; v^n 1) = maxxnE[xn(vn v^n) + e rhV (vn; v^n)jI i n 1]: (1.18) The solution of the above Bellman equation is provided in the following theorem: 15 Lemma 1.2.4: Given the price process set by the market maker, each informed trader?s strategy (equation (1.9)) is characterized by a trading intensity parameter , given by = (1 2e rhB )(1 h) (M(1 2e rhB ) + 1) : (1.19) Equation (1.18) has a quadratic solution of the form V (vn 1; v^n 1) = B(vn 1 v^n 1)2 + C where B and C satisfy the following set of equations: B = (1 h)2(1 e rhB ) (1 +M 2Me rhB )2 (1.20) and C = e rhB( 2 2u + 2 v)h 1 e rh : (1.21) Proof : See Appendix A. 1.2.4 Equilibrium Proposition 1.2.1: There exists a unique linear Markovian equilibrium characterized by ve parameters , v, , B and C which satisfy the system of ve nonlinear equations: (1.14), (1.15), (1.19), (1.20) and (1.21). The expressions for v, and are given by v = 2vh 1 (1 h) 2 M(1 2q1)+1 ; (1.22) 16 = s (1 2q1)(1 (1 h)2 M(1 2q1)+1 ) M u v ; (1.23) and = (1 h) p M v q (1 2q1(1+M(1 2q1))) 1+M(1 2q1) (1 h)2 M u(1 2q1) + u ; (1.24) where q1 = M + 1 3M (1.25) 1 6M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 ((M + 1)2 3Z)3 1 6M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 + ((M + 1)2 3Z)3 and Z = e rh(1 h)2 < 1. Proof : See Appendix A. 1.3 Asymptotic Properties of Equilibrium in the Limit of Continuous Trad- ing 1.3.1 Research Questions It is important to study how imperfect competition a ects the properties of equilibrium, especially when the frequency of trading becomes very high approaching to the limit of con- tinuous trading. Firstly, it is intuitively to believe that increasing competition will make the already \impatient" monopolistic trader in Chau and Vayanos (2008) even more \impatient". 17 The aggregate trading of multiple informed traders competing strategically should be more aggressive than in the monopolist case. Chau and Vayanos nd that when the time interval between rounds of trading is small, the variance of private information not incorporated into price v is proportional to p h. How does the more aggressive trading by informed traders improve the convergent rate? Can the rate be of the same or higher order than p h? Secondly, does more competition improve market depth or not? As shown in the non- stationary setup in Holden and Subrahmanyam (1992), imperfect competition has two op- posite e ects. The market is less liquid in the beginning of the trading sessions since the net order ow contains more private information relative to the noise. After most information is revealed in the remaining of trading sessions, the market becomes very deep since price is already very e cient and there is less information asymmetry between the informed traders and market maker. It would be interesting to know which e ect dominates in the stationary state. Thirdly, Foster and Viswanathan (1993) show that identically informed traders? pro ts converge to zero in a continuous trading limit since by trading more frequently the traders have more opportunities to preempt each other. In their model and the model of Holden and Subrahmanyam (1992), private information is one-shot and the liquidation value of the risky asset is xed. It is important to examine whether this Bertrand-like result still holds in the model where private information arrives repeatedly and the value of the asset is stochastic. In the steady state of this model, the informed traders earn zero pro ts if and only if the price impact is zero10. The market maker cannot set price impact to zero because there is 10This is because the market maker sets price e ciently and informed traders bene t when the liquidity 18 always new information coming in and the market maker has to learn from the net order ow. As long as the price impact is strictly positive in the limit, informed traders? pro ts remain bounded away from zero. Lastly, in almost all dynamic market microstructure models with price impact, the trading volume contributed by the informed traders is negligible compared to the trading volume of liquidity traders when trading is continuous. For the same reason, the fraction of price volatility contributed by informed traders is also zero. Even though the monopolistic trader in Chau and Vayanos trades very aggressively, his trading volume over a short interval h is of order h 3 4 11, which is much smaller than p h contributed by the liquidity trader over the same short period. In the presence of imperfect competition, the aggregate trading of informed traders should be more aggressive relative to the monopolistic case. But it remains to be shown whether the results will change qualitatively such that the trading volume of the informed traders is comparable to the trading volume of liquidity traders. If so, informed traders will also contribute signi cantly to price volatility as well. To answer the above questions, I derive the asymptotic properties of the equilibrium near continuous trading in the following theorem. trader moves price away from its e cient value. 11Since trading intensity parameter p h and pricing error is of order h 1 4 , the trading volume over one period is of order h 3 4 . 19 1.3.2 Asymptotic Properties of Equilibrium Proposition 1.2.2: In the limit of continuous trading (h ! 0), the asymptotic behaviors of v, , , B and C are given by: lim h!0 v h = 2v(1 +M(1 2q0)) M(1 2q0) (1.26) lim h!0 = u(1 2q0) v p 1 +M(1 2q0) (1.27) lim h!0 = v u s 1 1 +M(1 2q0) (1.28) lim h!0 B = q0 u v p 1 +M(1 2q0) (1.29) lim h!0 C = q0 u v r ( p 1 +M(1 2q0) + s 1 1 +M(1 2q0) ): (1.30) where q0 = M + 1 3M (1.31) 1 6M 3 q (M + 1)3 18M 9 + p ((M + 1)3 18M 9)2 (M2 + 2M 2)3 1 6M 3 q (M + 1)3 18M 9 p ((M + 1)3 18M 9)2 (M2 + 2M 2)3: Proof : See Appendix A. From the above theorem, the parameter which measures the uncertainty of the market maker about the underlying pro tability of the risky asset, v, converges to 0 when h goes to 0. The value v is also the variance of private information not incorporated into price at 20 each period. The notation v h means that all private information is re ected in the price, information asymmetry disappears, and the market is strong form e cient when trading is continuous. Although we reach the same conclusion regarding market e ciency as in Chau and Vayanos? model, there is still some di erence in how e cient the price becomes or how di erent the convergence rate is for small h. Since v p h for the case of a monopolistic informed trader and v h for our imperfectly competitive case, we have v(M=1) v(M>1) 1p h with the ratio converging to in nity when h ! 0. Therefore, the equilibrium price with M 2 informed traders is even more revealing of the informed traders? private information than the monopolist case. We should also expect that trading intensity is qualitatively di erent. In Chau and Vayanos, when there is only one monopolistic informed trader, p h. However, is of order 1 when the market is populated with multiple informed traders. This implies that imperfect competition makes traders trade much more aggressively and bring information into the price much more quickly. Therefore, market makers learn more from the order ows and set more e cient price. Next, I examine the trading volume contributed by informed traders to check whether it is comparable to the trading volume of liquidity traders in continuous trading. One can tell that is of order 1 and jvn 1 v^n 1j p h. Then over one trading period, the absolute aggregate trading volume of an informed trader jxnj = jvn 1 v^n 1j is of order p h. Since the trading volume contributed by liquidity trader is of the same order junj = u p h, it follows that the informed traders generate a non-negligible fraction of total trading volume because the ratio jXnjjunj converges to a positive constant bounded away from zero. The next 21 theorem derives the fraction of trading volume contributed by informed traders. Proposition 1.2.3: In the continuous trading limit (h! 0), de ne M to be the fraction of trading volume contributed by the informed traders. The value of M can be expressed as M = M p 1 2q0 M p 1 2q0 + 1 (1.32) which depends only on the number of informed traders. In most dynamic models with price impact, the informed traders? trade does not have a di usion component which contributes to volatility in the limit of continuous trading since their fraction of trading volume is zero. In my model, as illustrated above, informed traders contribute signi cantly to trading volume, and thus they should contribute signi cantly to price volatility as well. It is trivial to write pn as pn = pn pn 1 = (1 exp ( rh)) 1 exp ( rh)(1 h) v^n (1.33) = (1 exp ( rh)) 1 exp ( rh)(1 h) (M (vn 1 v^n 1) + un): The price variance can therefore be written as V ar( pn) h = ( (1 exp ( rh)) 1 exp ( rh)(1 h) )2 2(M2 2 v + 2uh) h : (1.34) The next theorem illustrates the contribution of price variance by informed traders and liquidity traders. 22 Proposition 1.2.4: In the continuous trading limit (h! 0), the price variance limh!0 V ar( pn) h can be decomposed into two components: (i) a contribution from informed traders given by ( rr+ ) 2 2 2 2 v(1+M(1 2q0)) M(1 2q0) ; (ii) a contribution from liquidity traders given by ( rr+ ) 2 2 2u. The total price variance which is the sum of these two components, is ( rr+ ) 2 2v , independent of the number of traders. Proof : (i) and (ii) are trivial to prove. To prove the last point regarding total price variance, observe that 2(M2 2 v + 2 u) = 2v 2u 1 1 +M(1 2q0) ( M2(1 2q0)2 1 +M(1 2q0) 2u 2v 2v(1 +M(1 2q0))) M(1 2q0) (1.35) + 2u) = 2v 2u 1 1 +M(1 2q0) (M(1 2q0) 2 u + 2 u) = 2v : Finally, I examine each informed trader?s pro tability. The expected pro ts can be written as (1 exp ( rh)) 1 exp ( rh)(1 h) V (vn 1; v^n 1) = (1 exp ( rh)) 1 exp ( rh)(1 h) (B(vn 1 v^n 1) 2+C): (1.36) The term (vn 1 v^n 1)2 converges to 0 when h ! 0. But (1 exp ( rh)) 1 exp ( rh)(1 h)C converges to a positive constant from Proposition 1.2.2. Hence, competition makes the aggregate pro ts fall, but it does not drive pro ts to zero. The results are in sharp contrast with the ones found in Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993) although 23 sharing a similar result in terms of market e ciency. 1.3.3 Properties of the Perfectly Competitive Equilibrium I examine another class of asymptotic results by taking the limit as M goes to in nity. It is easy to verify that when M ! +1, limM!1M2q0 = 1. Then substituting 1M2 for q0 in Proposition 1.2.2, I can derive the properties of the perfectly competitive equilibrium in the limit of continuous trading in the next theorem. Proposition 1.2.5: In the perfectly competitive case (i.e., when the number of traders goes to in nity), the asymptotic properties of the equilibrium now becomes: lim h!0;M!+1 v h = 2v (1.37) lim h!0;M!+1 p M = u v (1.38) lim h!0;M!+1 p M = v u (1.39) lim h!0;M!+1 M 3 2B = u v (1.40) lim h!0;M!+1 M 3 2C = u v r (1.41) lim M!+1 M = 1 (1.42) . Since 2vh is the variance of new private information the informed traders learn at each period, limh!0;M!+1 vh = 2 v implies that in the perfectly competitive case, there is no 24 information left on the table. The result on suggests that although each individual trader?s trading intensity can be in nitesimally small, the aggregate trading intensity can be very large as the number of traders increases. The results on and C suggest that as the number of informed traders increases, market depth improves, but aggregate pro ts fall, tending to zero only as the number of informed traders becomes large. The result on M , the fraction of trading volume from informed traders, can be arbitrarily close to 1 as the number of traders is large enough. 1.4 Numerical Illustrations and Comparative Statics In what follows, I numerically illustrate how information structure and imperfect compe- tition among informed traders a ect market e ciency, market liquidity, trading volume, price volatility, and expected pro ts of the informed traders. I also provide some empirical implications. Since M is the number of informed traders in the market, the model reduces to the monopolist case if M is set to be 1. Most of the comparative static analysis is concerned with the e ect of changing M . One issue concerns the comparison between the duopolist case M = 2 and monopolist case M = 1. I nd that adding just one more informed trader to the monopolist case will change the asymptotic properties of equilibrium in Chau and Vayanos? model qualitatively. Another issue concerns within the oligopolistic situation. As we increase M from M = 2 to large values, the competition among informed traders increases. I study how changing the intensity of competition a ects the properties of equilibrium. 25 Market E ciency I set the parameters such that v = u = = 1 and r = 0:05. v measures the market maker?s uncertainty about the liquidation value of the risk asset. It is therefore a measure of the e ciency of price, with a smaller value corresponding to a more e cient price. v = 0 corresponds to the scenario where information asymmetry vanishes and the market is strong-form e cient. If increasing the number of imperfect competitors makes traders willing to incorporate more private information into the price, the price should become more informative and we therefore expect a smaller v as we increase M . As trading becomes more frequent (h is smaller), the noncooperative setting results in a more aggressive competition, making the already \impatient" informed traders even more \impatient". To illustrate this intuition, I show how v varies with h and M in Figure 1.1(A). As shown in the gure, v monotonically decreases with h for the monopolist case and for the oligopolist cases when M = 2, 3 and 10. The value of v declines more rapidly for M 2 than for M = 1. If we x h and vary only the number of informed traders, v is found to be inversely related with the number of informed traders M . This con rms the previous intuition that increasing competition makes the market more e cient. Next I examine the asymptotic properties of v. According to Chau and Vayanos (2008), v p h as h! 0 in the monopolist case M = 1. I prove in Proposition 1.2.2 that v h in the oligopolist case M 2. In Figure 1.1(B), I show how the scaled value of v varies with h for di erent M . I scale v by p h for M = 1 and h for M 1. From the gure, hp h approaches to a constant for M = 1, con rming the asymptotic result obtained for the 26 monopolistic trader. When M 2, vh converges to a positive constant con rming the asymptotic property of v obtained in Proposition 1.2.2. Since the ratio v(M=1) v(M 2) p h h !1 as h converges to zero, private information is revealed much more quickly and price becomes more e cient when there are multiple traders in the market. Figure 1.2 illustrates how the degree of competition a ects the asymptotic properties of v. When M is large enough, limh!0 v 2vh can be very close to 1. Since v is always greater than 2vh, in the limit as M ! 1, we have the result of strong-form e ciency, consistent with intuition based on perfect competition. 27 Figure 1.1: (A) v as a function of h for M = 1, M = 2, M = 3 and M = 10. (B) Scaled v as a function of h. v is scaled by p h for M = 1, and by h for M 2. Parameter values: v = u = = 1 and r = 0:05. 28 Figure 1.2: limh!0 v 2vh as a function of the number of informed traders M . Trading Intensity and Expected Quantity of Informed Trading I have shown that, with increasing competition, informed traders reveal more information through their trading. Intuitively, one should expect that the aggregate trading intensity should be higher, and the fraction of trading volume contributed by informed traders should be higher when M increases. I demonstrate numerically how each trader?s trading intensity and aggregate trading volume generated by the informed traders per period E(jXnj) vary with h, respectively, for the cases when M = 1, 2 ,3 and 10 in Figure 1.3. As can be seen from the gure, when we compare monopoly M = 1 with duopoly M = 2, the increased competition between the two traders induces each duopolist to choose a higher 29 Figure 1.3: (A) The trading intensity parameter as a function of h for M = 1, M = 2, M = 3 and M = 10. (B) The aggregate expected trading volume per period E(Xnj) as a function of h for M = 1, M = 2, M = 3 and M = 10. Parameter values: v = u = = 1 and r = 0:05. 30 trading intensity as shown in Figure 1.3(A) when h is small. But the result reverses sign as we continue adding more informed traders when h is small. For M 2, the more the informed traders, the less trading intensity from each individual trader. Each trader?s optimal strategy is to exploit less the investment opportunity determined by the di erence between the true valuation of the asset and the price set by the market maker. Consequently, each trader may actually trades less intensely as more traders become informed. In aggregate, competition does make the traders behave more aggressively since the aggregate trading intensity M increases as competition becomes more intensive. In other words, when h is small, for M 2, the value of is decreasing in M while the value of M is increasing in M . Figure 1.4 illustrates how the asymptotic trading intensity limh!0 and aggregate trading intensity M limh!0 vary with the number of informed traders. Although each individual trader trades less intensely, the aggregate trading intensity monotonically increases with M . As shown in Figure 1.3(B), the expected aggregate quantity of informed trading per pe- riod E(jXnj) monotonically increases with the number of traders M . Therefore, although each individual trader tends to submit lower demand when the market becomes more compet- itive, the aggregate trading volume which contains private information increases, conveying more information to the market maker. The asymptotic properties of and E(jXnj) can be inferred from Figure 1.3. In the monopolist case, is of order p h and E(jXnj) = p v is of order h 3 4 . In the imperfectly competitive case, converge to a positive constant and E(jXnj) / h 1 2 . Intuitively, as trading becomes more frequent, there is less liquidity trading at each period to provide camou age. 31 Figure 1.4: limh!0 and limh!0M as functions of the number of informed traders. 32 Figure 1.5: Fraction of trading volume of informed traders as a function of h for M = 1; 2; 3 and 10. Parameter values: v = u = = 1 and r = 0:05.. Therefore, the informed traders trade less intensely and scale back the trading volume at each period. It can also be noted that in the monopolist case, the insider generates a negligible fraction of total trading volume, whereas in the imperfectly competitive case, the total aggregate volume submitted by the informed traders is comparable to the volume by the liquidity traders. The results are illustrated in Figure 1.5. When M = 1, the ratio quickly converges to zero when h is small. But the ratios converge to positive constants when M > 1 and become 1 when M ! +1. 33 Price variance Since the aggregate trading volume is comparable to the trading volume of the liquidity traders, following similar argument and Proposition 1.2.3, the informed traders? contribution to the total price variance is also non-negligible. Figure 1.6 illustrates how the total price variance (blue lines) and its contribution by the informed traders (red lines) varies with the time interval between rounds of trading and the number of informed traders. In the monopolist case, although total price variance increases when h is small, the contribution by the monopolistic trader converges to zero. The liquidity trader therefore contributes almost all of the price volatility near continuous trading. In the imperfectly competitive case, not only do informed traders contribute signi cantly to the total price variance near continuous trading, but the ratio increases as the number of informed traders increases. My numerical calculation make it reasonable to believe that the ratio converges to 1 when M !1. Market Liquidity and Pro tability Stationarity requires that the price impact is a time independent constant. Figure 1.7 illustrates the e ect of competition on . If we x h and increase the number of informed traders, declines accordingly. This is because in a steady state, more competition decreases information asymmetry between informed traders and market maker, there is less adverse selection. When the number of informed traders is xed and trading frequency increases, numerical calculations show that increases and converges to a positive constant. Two opposite e ects occur as trading becomes more frequent. On the one hand, there is less 34 Figure 1.6: Total price variance (blue lines) and the contribution by the informed traders (red lines) as a function of h for M = 1; 2; 3 and 10. Parameter values: v = u = = 1 and r = 0:05.. 35 Figure 1.7: The market liquidity parameter as a function of h for M = 1; 2; 3 and 10. Parameter values: v = u = = 1 and r = 0:05. liquidity trading over each period and the adverse selection problem is more severe. On the other hand, as market becomes more e cient, there is less information asymmetry between the informed traders and the market maker, and the adverse selection problem is less severe. It can be shown from the gure that the rst e ect dominates with the adverse selection parameter a decreasing function of h. The fact that remains strictly positive in the continuous trading limit ensures that informed traders make strictly positive expected pro ts. This is because informed traders 36 bene t from the liquidity traders. Their pro ts are higher when price impact is higher and when liquidity traders are able to move the price further away from the e cient value giving informed traders opportunities to trade. To give a more rigorous explanation, remember that pro t margin per share is v which is of the order p h when h is small. The demand submit in period n in absolute term by each informed trader is proportional to p h. Then at each period, each informed trader earns a small expected pro ts in the order of h. The present value of the aggregate pro ts at any period n is proportional to P1 k=n e r(k n)hh, which is nite when h ! 0. Hence, our model predicts that each informed trader can still earn positive expected pro ts in the continuous trading limit. But imperfect competition does make each informed trader worse o . To demonstrate the e ect of competition on expected pro ts, we plot in Figure 1.8 the aggregate expected pro ts of informed traders of informed traders as a function of h for di erent M . When M 2, the aggregate pro ts monotonically decrease with M for xed h and converge to positive constants as h! 0. Figure 1.9 illustrates how the asymptotic price impact limh!0 varies with the number of informed traders. limh!0 monotonically decreases with M . When the number of traders is large enough, the market can be in nitely deep with limh!0 very close to zero. Therefore, the aggregate pro ts of informed traders also converges to zero as M !1. Empirical Implications Hendershott and Riordan (2011b) nd that high frequency traders? marketable orders have information advantage. Those traders trade in the direction of permanent price changes and in the opposite direction of transitory pricing errors. Therefore, it is reasonable to believe 37 Figure 1.8: Aggregate expected pro ts as a function of h for M = 1; 2; 3 and 10. Parameter values: v = u = = 1 and r = 0:05. 38 Figure 1.9: limh!0 as a function of the number of informed traders. 39 that the high frequency traders who demand liquidity are \informed" traders. They may not possess corporate inside information but are better and quicker than the average market participants in gathering and processing information from massive market wide data to generate private signals. The informed traders in my model can be treated as high frequency traders when they are able to trade their information quickly (i.e, the frequency of trading is very high). The unique results from my model in terms of e ciency, liquidity, trading volume, price volatility, and expected pro ts can help us better understand the e ects of high frequency traders on nancial markets than previous literature on strategic trading. Hendershott et al. (2011), Hendershott and Riordan (2011a), Hendershott and Riordan (2011b) and Brogaard (2010) provide evidences showing that the existence of high frequency traders are bene cial to the market. Those traders provide more e cient quotes and the growth of high frequency traders accompanies improvements in market liquidity, no matter whether the traders are liquidity providers or liquidity demanders. In addition, high fre- quency traders contribute signi cantly to trading volume. For example, Hendershott and Riordan (2011b) estimate that high frequency traders initiating trades are responsible for roughly 43% of trading volume in large stocks from a unique dataset from Nasdaq. High fre- quency traders are found to implement highly correlated strategies but remain to be highly pro table despite aggressive competition with one another. At rst glance, Chau and Vayanos (2008), Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993) seem to be good candidates to be used to understand high frequency trading. All three models can predict that high frequency trading make price more e cient, 40 although traders in those models are \impatient" to reveal their superior information very quickly for di erent reasons. However, all three models cannot explain trading volume. The fraction of trading volume contributed by informed traders is essentially zero in continu- ous trading limit. Chau and Vayanos (2008) shows that market is thinner when trading is more frequent, and Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993) show that market can be in nitely illiquid when trading starts. Their predictions cannot perfectly explain the empirical fact that market liquidity increases with more participation of high frequency traders. In addition, Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993) predict that pro ts of informed traders quickly converge to zero when there is imperfect competition. Zero expected pro ts are inconsistent with the pro tability of the imperfectly competitive high frequency traders. Therefore, both models cannot ex- plain why high frequency traders can survive in the long run in an imperfect competitive environment. My model combines the two important features: (i) imperfect competition and (ii) con- tinuous arrival of new private information. As the number of high frequency traders increases due to ease of entry: (i) information is incorporated into prices more quickly and therefore speeds up price discovery; (ii) high frequency traders participate in an even larger fraction of total trading volume and price volatility, (iii) high frequency traders add to market liquidity by competing aggressively with one another, and (iv) high frequency traders should make pro ts which are bounded away from zero (like Cournot and not like Bertrand competition) unless the number of high-frequency traders is very large. 41 1.5 Conclusion In this paper, I analyze how imperfect competition among informed traders a ects market e ciency, liquidity, trading volume and the pro tability of informed traders. The combined trading of multiple informed traders is more aggressive than the monopolistic trader, the equilibrium price is even more revealing of the informed traders? private information, and market depth improves as the number of informed traders increases. In the continuous trading limit, the variance of private information held by informed traders goes to zero at a rate proportional to the time interval between rounds of trading. This is much faster than the corresponding strong from e ciency result in the Chau and Vayanos model, where the convergence rate is proportional to the square root of the time interval. In addition, in the limit as the time interval between rounds of trading goes to zero, the aggregate pro ts of the informed traders remain bounded away from zero and they contribute signi cantly to the total trading volume and price volatility. If high frequency traders are \informed" in a sense that they are able to generate prof- itable private signals consistently by processing information from order ows and price move- ments of securities across market, then this model provides a reasonable characterization of those traders. My results suggest that the entry of more high frequency traders improves market e ciency by incorporating information more quickly into price, improves market liquidity by lowering price impact, and increases the fraction of trading volume from high frequency traders. But those traders remain pro table despite exploiting the same informa- tion set and implementing similar algorithms. 42 Future research can extend the results of this paper in two directions. First, explain why high frequency traders quickly reverse their inventories, we may add risk aversion. Second, the assumption that traders are identically informed is too strong. The assumption does not allow the more realistic scenario in which the informed traders learn from each other. Li (2012) extends this paper by introducing a hierarchical information structure in which there is one strictly better informed trader and one less informed trader. Kyle et al. (2012) further relax the assumption in this paper even further to allow for a more general correlation among streams of private information in which each trader has to forecast the forecasts of other traders. 43 Chapter 2 Speculative Dynamics II: Asymmetric Traders 2.1 Introduction In a nancial market, there are a few large institutional traders who actively spend resources generating information and competing aggressively with one another. These informed traders are found to follow positively correlated strategies. Li (2011) provides a theoretical frame- work that characterize the equilibrium behavior of those traders. If their information sets and strategies are perfectly correlated, adding more informed traders can speed up price discovery and improve market depth. They contribute signi cantly to trading volume and price variance. In the seminal paper of Kyle (1985) and many extensions, the informed trader?s contributions to the total trading volume and price variance are negligible to the contributions by the liquidity trader in continuous trading. This is because that the informed trader?s strategy is continuous (of order dt) and liquidity trading is a Brownian motion (of order p dt). and manage to earn a pro t strictly bounded away from zero even when the fre- quency trading is very high near continuous trading. This is in sharp contrast to the strategic trading models with one-shot private information. In the one shot private information setup of Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993), multiple informed traders? pro ts vanish due to imperfect competition in continuous trading. In this paper, I relax the assumption in Li (2011) of identically informed traders. This 44 assumption ignores the possibility that informed traders produce di erent information. If their signals are imperfectly correlated, some traders have information that others don?t possess. The inference problem becomes more complicated relative to the scenarios of one monopolistic informed trader or multiple identically informed traders. On the one hand, informed traders who do not know all the private information will infer any additional information they do not have from past order ows and price movements, like uninformed market makers. On the other hand, informed traders will treat their available information di erently: they trade cautiously on any exclusive information they may possess to exploit their monopoly power, and they trade aggressively on information they share. By allowing the signals observed by the informed traders to di er, especially when new information arrives repeatedly, makes the problem very di cult to solve analytically. One has to overcome the in nite regress problem arising in a dynamic model when traders try to infer each other?s information. In this paper, we bypass the problem by focusing on a hierarchical duopolistic setting. To be more speci c, we consider an economy with a risky asset which is going to be liquidated at a random date. The liquidation value is not xed, but rather evolves in a stochastic way and is only perfectly observed by an informed trader. The other trader, however, observes a stream of noisy signals of the liquidation value. These noisy signals are also observed by the rst trader. Therefore, the second trader is less informed and the rst trader is strictly better informed. Both traders submit marketable orders to the market maker along with liquidity traders at each trading period. The risk neutral market maker cannot see each individual trader?s order but rather observes the batched order. She 45 then sets the price equal to the expected value of the asset by solving an inference problem, and takes the other side of the trade. The history of order imbalances provides information not only to the market maker, but to the less informed trader as well. Like the market maker, the less informed trader imperfectly infers the more informed trader?s additional information from the history of order ows by subtracting her own contribution. In the meanwhile, she has to compete with the more informed trader concerning any private information they both observe. There is another problem which is associated with the existence of linear equilibrium in continuous trading when traders have perfectly correlated signals. Holden and Subrah- manyam (1992) and Back et al. (2000) nd that linear equilibrium does not exist when informed traders are identically informed in the limit of continuous trading. This is because risk neutral informed traders try to preempt each other causing the private information to be revealed instantaneously, with trading intensity and price impact becoming in nite large at the start of the trades (at t = 0+). Through the remaining of the trading sessions (t > 0), there is no information asymmetry between the informed traders and therefore market is in nitely deep. It is reasonable to believe that in a hierarchical private information setting of Foster and Viswanathan (1994) where both asymmetric informed traders share a com- mon noisy signal about the value of a risky asset to be liquidated at a predetermined date, there is no linear equilibrium when trading is continuous1. In an economy where an risky asset?s liquidation value is stochastic and private information arrives repeatedly, Li (2011) 1What should happen when trading is continuous is that both traders would trade on the common private information so aggressively such that the common private information incorporated into price almost instantaneously when trading starts and the more informed trader is able to monopolize the additional exclusive information through the remaining trading session like the informed speculator in Kyle (1985). 46 has found that stationary linear equilibrium does exist in the limit of continuous trading with identically informed traders. In this paper, we study the equilibrium in an economy with duopolistic traders endowed with hierarchical information sets and a risk neutral market maker. Our model di ers from Foster and Viswanathan (1994) in several aspects. In Foster and Viswanathan?s model, all the private information is known at time zero, the liquidation value of the asset is xed before trading starts, and there is a predetermined date when the asset is liquidated. In this paper we have more realistic assumptions: both traders receive continuous ows of new information, the fundamental value of the risky asset is stochastic, and the time for the risky asset to be liquidated is random, following a geometric distribution. In their paper, Foster and Viswanathan characterize the equilibria numerically and focus on time series properties of equilibrium in an economy with only four trading periods. Because of the above argument on the existence of equilibria in continuous trading, their paper does not address the economic properties of equilibrium when trading periods are shortened to continuous limit. In this paper, by making di erent assumptions and focusing on stationary equilibrium, we are able to derive the equilibria in a closed form, able to derive economic propertis like to market e ciency, market liquidity, and pro tability. By analyzing the properties of the equilibrium, the overall private information held by both traders can be decomposed into two orthogonal components: a \common" component of the private information known to both traders and \monopolistic" component known only to the more informed trader. Consistent with the results found in Chau and Vayanos 47 (2008) and Li (2011), the variances of both pieces of information converge to zero in the continuous trading limit. However, the rates of how the two components of information are incorporated into prices are quite di erent. When h is small (denote h to be the time interval between trades), the variance of common component of private information not incorporated into prices is proportional to h, while the variance of the monopolistic component of private information not incorporated into prices is proportional to the square root of h. This suggests that the common private information is incorporated more e ciently than the additional monopolistic information. The degree of competition is measured by the precision of the information stream that is shared by both traders. Increasing the precision can have two e ects on market depth. On the one hand, there will be more common private information and less monopolistic information at each period and therefore more information will be incorporated into prices conditional on the total private information. Since the market orders carry more informa- tion, market maker should set a higher price impact. On the other hand, over time price should become more e cient, there should be less information asymmetry among the mar- ket participants, and the market should be more liquid. It can be shown in the stationary state, the second e ect dominates, thus the more precise the shared private information, the more liquid the market becomes. Following the same logic, the inference parameter which measures the sensitivity of belief to order ow for the less informed trader monotonically also decreases with . In almost all dynamics models with price impact, the contribution to the trading volume 48 or price variance from informed trading is negligible. This is because in the continuous time limit as the interval between trades h goes to zero, the value of informed trading is of order h, whereas the value of liquidity or noise trading strategy is a Brownian motion of order p h. Li (2011) shows that with imperfect competition and continuous arrival of new private information, the informed traders? strategy can be a Brownian motion in the limit as h! 0. Therefore, informed traders can contribute signi cantly to the total trading volume and add a volatility component to the price formation process. In this model, since imperfectly competitive informed traders share a stream of signals, their strategies on the common private information also become Brownian motions as h ! 0 and therefore the contributions to the total trading volume and price variance by the two traders should be non-negligible in the limit. A somewhat surprising result is that, contrary to conventional wisdom, when the fre- quency of trading is su ciently high, the less informed trader may contribute a more sig- ni cant fraction to total trading volume and price variance than the more informed trader. A trader may rationally trade more because her information is \worse". The intuition for why this happens is that the less informed trader trades more aggressively on the common private information, and the di erence in trading on the common information may dominate the trading volume in the additional private information by the more informed trader. The more informed trader?s strategy on her additional private information is of order h 3 4 which is negligible compared to p h when h is small enough. Although not formally stated in Foster and Viswanathan (1994), by following the same 49 spirit of Holden and Subrahmanyam (1992) and Foster and Viswanathan (1993), when the time interval between trades goes to zero, the informed traders compete so aggressively based on shared information that this information is incorporated into prices almost instan- taneously and the pro ts of the less-informed trader go to zero. This paper generates an opposite result in terms of the pro tability of the less informed trader. When the interval between rounds of trading is small, both the pro ts of the more informed and less informed traders remain bounded away from zero, with the pro ts of the more informed trader much greater than the pro ts of the less informed when the signals are not highly correlated. The plan of the paper is as follows. In Section II, I describe the model. The equilibrium and its asymptotic properties are presented in Section III. Section IV shows some comparative statics and empirical implications. Section V concludes. 2.2 The Model 2.2.1 Economy Assumption 1: Securities Trading takes place from t = 0 to t = +1 at the discrete points tn (tn = nh) through time where h is the time interval between the auctions. There are a riskless bond with zero interest rate and a non-dividend paying risky asset with a stochastic liquidation value vn. At the end of each period, there is a probability 1 exp ( rh) that the risky asset is liquidated. I further assume the riskless bond is in perfect elastic supply. The liquidation value vn follows a mean-reverting process or random walk: 50 vn vn 1 = ( v vn 1)h+ v;n: (2.1) In the above speci cation, determines the adjustment speed of the liquidation value vn to its long run xed target v; is assumed to be greater than or equal to zero implying that the prices are stationary2; and the innovation v;n is independently and normally distributed with mean zero and variance 2vh. Assumption 2: Market Participants and Information Structure The risk neutral market participants consist of a competitive market maker, two informed strategic traders, and a number of liquidity traders. The two strategic traders, however, are asymmetrically informed with a hierarchical information structure such that one trader is less informed and the other is better informed. Therefore, their information sets are hierarchical. At each period, the less informed trader observes a noisy signal sn about v;n (the innovation term in equation (2.1)) in the form sn = v;n + p 1 2en (2.2) where 0 1 and en N(0; 2vh) is normally distributed and uncorrelated with other shocks in the economy. The better informed trader observes vn perfectly but also observes the noisy signal sn each trading period. The variable measures the level of precision of the noisy signal received by both traders. When = 0, the less informed trader is completely uninformed. When = 1, both traders are identically and perfectly informed. 2If = 0, then vn follows a random walk. 51 I in = fp ; v ; s j ng is the more informed trader?s information set at t = nh, I l n = fp ; s j ng is the less informed trader?s information set, and Imn = fp j ng is the market maker?s information set. Clearly, we have I in k I l n k I m n and the information sets are nested. Assumption 3: Timing of Events At the beginning of the nth period, the informed traders and the liquidity traders submit their demands before new information arrives. Let zn denote the less informed trader?s market order and xn denote the more informed trader?s market order. After the market orders are submitted, both informed traders observe sn but only the more informed perfectly observes vn. The market maker observes the total order ow yn = xn + zn + un, then sets the price pn equal to the expected value of the asset based on the history of orders ows, and then clears the market. Pricing Since the market maker is assumed to be competitive and risk neutral, at period n she sets the price pn equal to the expected value of the asset after she receives the total batched market order yn = xn + zn + un. Therefore, pn = E[ +1X n0=n (1 exp ( rh)) exp ( r(n0 n)h)vn0 jI m n ]; (2.3) where (1 exp ( rh)) exp ( r(n0 n)h) is the probability that the asset is liquidated at the end of the n0th period. Lemma 2.2.1: The price pn is the following linear function of the market maker?s 52 expectation of the current liquidation value of the risky asset E(vnjImn ): pn = (1 exp ( rh)) 1 exp ( rh)(1 h) E(vnjI m n ) + h v 1 exp( rh)(1 h) : (2.4) Proof : See Appendix B. Although the market maker observes the total order ow, she cannot distinguish the contributions made by the more informed trader, less informed trader and the liquidity trader. The less informed trader, however, by observing the price set by the market maker, can infer the sum of the orders by the more informed trader and liquidity trader, xn + un, therefore enabling her to maker a sharper inference about vn than the market maker. Similar to the pricing function of Lemma 2.2.1, the less informed trader?s valuation of the risky asset at the nth period equals to the expected value conditional on her information which is denoted by ln: ln = E[ +1X n0=n (1 exp( rh)) exp( r(n0 n)h)vn0jI l n] (2.5) = (1 exp ( rh)) 1 exp ( rh)(1 h) E(vnjI l n) + h v 1 exp( rh)(1 h) : Optimization Since informed traders are risk neutral, then at the nth period, the better informed trader tries to maximize her expected trading pro ts: max xn0 n E[ +1X =n (1 exp( rh)) exp( r(n0 n)h)( n0X =n x (vn0 p ))jI i n 1]: (2.6) 53 Lemma 2.2.2: The better informed trader?s objective function can be written as: max xn0 n E[ +1X n0=n exp( r(n0 n)h)xn0( (1 exp ( rh)) 1 exp ( rh)(1 h) vn0+ h v 1 exp( rh)(1 h) pn0)jI i n 1]: (2.7) Proof : See Appendix B. Following a similar argument, the objective function of the less informed trader can be expressed as: max zn0 n E[ +1X n0=n exp( r(n0 n)h)zn0( (1 exp ( rh)) 1 exp ( rh)(1 h) vn0 (2.8) + h v 1 exp( rh)(1 h) pn0)jI l n 1]: 2.2.2 Equilibrium Strategies and Value Functions Suppose that conditional on information up to period n 1, the market maker believes that vn 1 is normally distributed with mean v^n 1 and variance v;n 1, while the less informed trader believes the mean to be v n 1 and variance to be v;n 1. Then, v;n 1 measures the total variance of private information not incorporated into prices at the (n 1)th period. The value of v;n 1 measures the variance of private information withheld exclusively by the better informed trader, and therefore v;n 1 = v;n 1 v;n 1 measures the variance of private information withheld by both informed traders. 54 I hypothesize that the more informed insider submits demand xn = (vn 1 v n 1) + (v n 1 v^n 1) (2.9) to maximize the present value of her expected pro ts scaled by (1 exp ( rh))(1 exp ( rh)(1 h)) which is quadratic with respect to vn 1 v n 1 and v n 1 v^n 1: V i(vn 1 v n 1; v n 1 v^n 1) = A(vn 1 v n 1) 2+B(v n 1 v^n 1) 2+C(vn 1 v n 1)(v n 1 v^n 1)+E: (2.10) The strategy and value function not only depend on vn 1 but also depend on v n 1 since the more informed trader understands that her action will be closely followed by the less informed trader as well as the market maker. The less informed insider submits demand zn = (v n 1 v^n 1) (2.11) to maximize the present value of her expected pro ts scaled by (1 exp ( rh))(1 exp ( rh)(1 h)) , which is quadratic with respect to v n 1 v^n 1: V l(v n 1 v^n 1) = F (v n 1 v^n 1) 2 +G: (2.12) Note that the order submission strategy of the more informed trader (Equation 2.9) can be decomposed into two parts. The rst term is proportional to the di erence between the perfect signal she receives and the information she shares with the other trader with intensity 55 . The second term is proportional to the di erence between the information she shares and the estimation of vn 1 by the market maker with intensity . The less informed trader, bases her trade on the di erence between the information she shares and market maker?s estimation on vn 1 with intensity . 2.3 Inference Problems, Optimizations and The Equilibrium 2.3.1 The Market Maker?s Inference Problem The market maker uses the total order ow yn, together with her prior belief on vn 1 to form the posterior belief about vn 1: vn 1 = E(vn 1jI m n 1) + 1 h (xn + zn + un) + n (2.13) = v^n 1 + 1 h (xn + zn + un) + n: She believes that vn 1 is normally distributed with mean E(vn 1jImn ) in the form E(vn 1jI m n ) = v^n 1 + 1 h yn (2.14) and variance var( n). Then according to the stochastic process of vn (equation (2.1)), she believes that vn N(v^n; var((1 h) n + v;n)) is normally distributed with conditional mean E(vnjI m n ) = v^n = (1 h)v^n 1 + yn + h v (2.15) 56 and conditional variance var(vnjI m n ) = (1 h) 2var( n) + 2 vh: (2.16) In the next theorem, we derive the explicit form of the inference parameter and set up nonlinear equation which can be solved for v. Lemma 2.3.1: Given the trading strategies of the informed traders de ned in equations (2.9) and (2.11), the market maker?s inference parameter is given by = (1 h)( v + ( + ) v) 2 v + ( + )2 v + 2uh : (2.17) The uncertainty of the market maker?s belief on the risky asset?s liquidation value v satis es the following nonlinear equation: v = (1 h) 2( v ( v + ( + ) v)2 2 v + ( + )2 v + 2uh ) + 2vh: (2.18) Proof : See Appendix B. 2.3.2 The Less Informed Trader?s Inference Problem In this subsection, we solve the inference problem of the less informed trader. At period n, the less informed insider can infer the realization of xn + un since the trader needs only to subtract her contribution zn from the total order ow yn. Because xn + un contains private 57 information which is exclusive to the more informed trader about vn 1, the less informed trader can update her belief about vn 1 after observing pn: vn 1 = v n 1 + 1 h (xn + un (v n 1 v^n 1)) + "n (2.19) = v n 1 + 1 h (xn + un (v n 1 v^n 1)) + "n with the conditional mean E(vn 1jI l n) = v n 1 + 1 h (xn + un (v n 1 v^n 1)) (2.20) and conditional variance var("n). When forming the posterior belief about vn, the trader needs to take into account of the signal sn. Hence, the conditional mean and conditional variance of vn are given by v n = E(vnjI l n) = (1 h)v n 1 + h v + (xn + un (v n 1 v^n 1)) + sn: (2.21) and var(vnjI l n) = (1 h) 2var("n) + (1 2) 2vh (2.22) respectively. In the following theorem, we derive the explicit form of the less informed trader?s inference parameter and setup the nonlinear equation for v. Lemma 2.3.2: Given the more informed trader?s strategy de ned in equation (2.9) 58 and the price process determined by the market maker in equation (2.15), the inference parameters of the less informed trader is given by 1 h = v 2 v + 2uh : (2.23) The trader?s uncertainty about the liquidation value of the risky asset v satis es the fol- lowing nonlinear equation: = (1 h)2 2v 2 uh 2 v + 2uh + (1 2) 2vh: (2.24) Proof : see Appendix B. 2.3.3 The More Informed Trader?s Optimization Problem At period n, the more informed trader submits demand xn to maximize the the expected pro ts scaled by 1 exp( rh)1 exp( rh)(1 h) E[ +1X n0=n xn0(vn0 v^n0)e r(n0 n)hjI in 1]: (2.25) We have conjectured that the trader?s value function V1(vn 1; v n 1; v^n 1) in period n, evaluated at after submitting the market order xn, is a quadratic function of vn 1 v n 1 and v n 1 v^n 1, as shown in equation (2.10). In equilibrium, V i(vn 1; v n 1; v^n 1) must satisfy 59 the following Bellman equation V i(vn 1; v n 1; v^n 1) = maxxnE[(vn v^n)xn + e rhV i(vn; v n; v^n)jI i n 1]: (2.26) Lemma 2.3.3: Given the price process determined by the market maker and the strategy of the less informed trader, the trading strategy of the more informed trader is characterized by: = (2 ) 1[1 h 2e rhA (1 h) + e rhC( )(1 h)] (2.27) and = (2 ) 1[1 h 2e rhA (2.28) + 2e rhB( )(1 h ) + e rhC( ) e rhC (1 h x )] where = e rh(A 2 +B( )2 C ( )) (2.29) The Bellman equation (2.26) has the solution of quadratic form A(vn 1 v n 1) 2 +B(v n 1 v^n 1)2 +C(vn 1 v n 1)(v n 1 v^n 1)+E, with A, B, C and E characterized by the following 60 set of nonlinear equations: A = (1 h ) + e rhA(1 h )2 + ( ) ]2 (2.30) + e rhC(1 h )( ) ; B = (1 h ( + )) + e rhB[1 h ]2; (2.31) C = (1 h ( + )) + (1 h ) (2.32) + 2e rhB( ) [1 h ] + e rhC(1 h )[1 h ]; and E = e rh[A( 2 2uh+ (1 2) 2vh) (2.33) + B(( )2 2uh+ 2 2vh) C( ( ) 2uh) + E]: Proof : See Appendix B. 61 2.3.4 The Less Informed Trader?s Optimization Problem Having solved the dynamic programming of the more informed trader, we then focus on the less informed trader?s optimization problem. Conditional on her most recent information set, she submits market order zn at period n to maximize the present value of expected pro ts scaled by 1 exp( rh)1 (1 exp( rh))(1 h) , E[ +1X n0=n zn0(vn0 v^n0)e r(n0 n)hjI ln 1]: (2.34) The value function of the less informed trader is assumed to be quadratic in v n 1 v^n 1 and solves the following Bellman equation in equilibrium: V l(v n 1; v^n 1) = maxznE[zn(vn v^n) + e rhV l(v n v^n)jI l n 1]: (2.35) The following theorem characterizes the trading strategy of the less informed trader and the solution for the Bellman equation. Lemma 2.3.4: Given the trading strategy of the more informed trader and the pricing rule determined by the market maker, the optimal strategy of the less informed trader is characterized by = (1 2e rh F )(1 h ) 2 (1 e rh F ) : (2.36) The Bellman equation (2.35) has the solution in a quadratic form F (v n 1 v^g 1) 2 +G, with 62 F and G characterized by the following set of nonlinear equations: F = (1 h ) + e rhF (1 h )2 (2.37) and G = e rhF [( )2 2 v + ( ) 2 2uh (2.38) + 2 2vh] + e rhG: Proof : See Appendix B. 2.3.5 Equilibrium and Trading in the Continuous Time Limit Having solved the inference problems for the market maker and the less informed trader and the optimization problems for both informed traders, in the next theorem we establish the necessary and su cient conditions for the stationary equilibrium. Proposition 1: The linear stationary equilibrium of the model with hierarchical infor- mation structure is characterized by 13 parameters ( , , v, v, , , , A, B, C, E, F and G) which are solutions to the system of equations (2.17 - 2.18), (2.23 - 2.24), (2.27 - 2.34) and (2.36 - 2.39). We are mostly concerned with the asymptotic properties of the equilibria in the limit of continuous trading, i.e., h! 0. When there is only one monopolistic informed insider, Chau and Vayanos prove that the variance of information not incorporated into prices at each 63 trading period is proportional to p h and the trading intensity is of order p h. By contrast, in an economy with multiple identical informed traders, the variance of private information left at each period is proportional to h and the trading intensity converges to a strictly positive constant. In this paper, since the total private information can be decomposed into two orthogonal components, we should expect that the variance of common private information not incorporated into prices v at each period is proportional to h and the variance of additional private information v left at each period converges to zero at rate of order of p h. The intensity at which the more informed trader trades, based on her additional private information, , should converge to zero at the order p h. The intensities at which both traders trade, based on their common information, and , should converge to positive constants. The next theorem establishes the asymptotic behavior of the equilibrium. Proposition 2: When the time interval between rounds of trading converges to zero, we have lim h!0 = 0 a p h (2.39) lim h!0 = 0 a p h (2.40) lim h!0 p h = b (2.41) lim h!0 = 0 (2.42) lim h!0 = 0 (2.43) lim h!0 v = Oh+ L p h (2.44) 64 lim h!0 v = L p h (2.45) and lim h!0 A = A0 limh!0B = B0 lim h!0 C = C0 (2.46) lim h!0 E = E0 limh!0 F = F0 lim h!0 G = G0; where 0, 0, a, b, 0, 0, L, O, A0, B0, C0, E0, F0 and G0 satisfy the following set of nonlinear equations: 0 = ( 0 + 0)O + bL ( 0 + 0)2O + 2u ; (2.47) 0 = bL 2u ; (2.48) b A0a 0 0 + A0 20 B0( 0 0)2 ; (2.49) 0 = 1 0 2A0 0 + 2B( )(1 0 ) + C( ) 0 C (1 0 0) 2 0 ; (2.50) 0 = (1 2 F0)(1 0) 2 (1 F ) ; (2.51) F = 0(1 ( 0 + 0)) + F0(1 ( 0 + 0)) 2; (2.52) B0 = 0(1 ( 0 + 0)) +B0(1 ( + )) 2; (2.53) 1 2A0 0 C0( 0 0) = 0; (2.54) b2 (r + 2 )A0 + 2A0ab+ A 2b2 b2C0( ) = 0; (2.55) 65 E = A( 2 2u + (1 2) 2v) +B(( ) 2 2u + 2 2v) C( ( ) 2 u) r ; (2.56) G = F ( )2 2u + 2 2v r (2.57) 2.4 Properties of Equilibriums and Comparative Statics In what follows, we illustrate how information structure, the strategic interaction between the two informed traders, and increasing trading frequency a ect market e ciency, liquidity, trading strategies, trading volume, and expected pro ts of the informed traders. The variable measures the precision of the noisy signal. Intuitively, it also measures the degree of information asymmetry and hence the degree of imperfect competition between the two informed traders. The higher the value of , the more precise the stream of noisy signals observed by both traders, the more information shared by both traders and less private information exclusively held by the more informed trader. To illustrate the e ect of imperfect competition and the strategic interaction between the two traders, I compare the results of when 0 1 with two benchmark cases. When = 1, both traders are equally informed and there is no information asymmetry between them; when = 0, the less informed trader is completely uninformed and the more informed trader is a monopolist on the perfect signal she receives. Market E ciency Since the information sets are hierarchical, I can decompose the overall private infor- mation held by the informed traders vn v^n into two orthogonal components: the private information known to both traders v n v^n and the information exclusive to the more in- 66 formed trader vn v n. It is natural to expect that both traders act like duopolists on the common component of the information, and the more informed trader is able to monopolize the information exclusive to herself. Following the intuitions, we should expect that v, which measures the market maker?s uncertainty about the rst component, is of order h when h is small and v, which measures the variance about the second component, is of order p h3. Therefore, the shared private information should be incorporated into prices in a much higher rate than the monopolistic private information. Because v = v+ v and the p h dominates h for small h, the variance of total private information not incorporated into price should be of order p h. Since p h! 0 as h ! 0, the market approaches to strong-form e ciency in the continuous trading limit with no information asymmetry among the informed traders and market maker. To illustrate the properties of market participants beliefs about the liquidation value vn, we plot the variance v, v and v against h respectively, for the cases when = 0:7, = 1 and = 0 in Figure 2.1. When = 0:7, v is always greater than v implying that the less informed trader is making a sharper inference about the liquidation value of the asset than the market maker. The di erence between v and v is actually v which measures the variance of v n v^n. As h becomes smaller, v and v also decrease converging to zero and so does their di erence, v. We then compare the results when = 0:7 to the benchmark cases when = 1 and = 0. The stream of \noisy" signal received by the less informed trader becomes perfect 3In the monopolist case, Chau and Vayanos (2008) prove that the private information not incorporated into price is of order p h when h is small. In the oligopolist case in Li (2011), the author proves that the convergence rate is proportional to h. 67 Figure 2.1: v, v and v as functions of h for the cases when = 0, 0:7 and 1. Parameter values: v = u = = 1 and r = 0:05. 68 and she is as informed as the other trader if = 1. As illustrated in Figure 2.1, v is always zero and v = v. If = 0, the fact that the less informed trader is completely uninformed and she can make an inference no better than the market maker leads to v = v. Trading Intensity Parameters: , and We have demonstrated that the more informed trader monopolizes any additional private information. Remember that is the intensity with which the more informed trader trades on this information, should be of order p h when h is small. and are the intensities with which the traders trade on the common private information. From the asymptotic properties derived in Proposition 2, we expect and converge to positive constants when h is closed to zero. We rst examine how the trading intensity parameters vary with the time interval between trades and the precision of the stream of noisy signals. In Figure 2.2, we plot , and against h for the cases when = 0:3, = 0:9 and = 1. As h becomes smaller, so do , and because the variance of noise trading is getting smaller at each trading period. The value of converges to zero except at = 1 when the \more" informed trader has no additional private information. The values of and converge to strictly positive values. In addition, is always greater than except at = 1 implying that the less informed trader always acts more aggressively on their common private information than the more informed trader. We next study the asymptotic properties of , and . We let b = limh!0 p h , 0 = limh!0 and 0 = limh!0 . Figure 2.3 illustrates how b, 0 and 0 vary with . The value of 69 Figure 2.2: The trading intensity parameters ( , and ) as functions of h for cases when = 0:3, 0:9 and 1. Parameter values: v = u = = 1 and r = 0:05. 70 b rst increases with slowly when is small, but increases sharply with when it is close to 1. Since limh!0 is found to be strictly positive at = 1, we should expect b! +1 as ! 1. The result is consistent with the counterpart found in Chau and Vayanos (2008) in which b is in nity if there is no adverse selection between the monopolistic insider and the market maker. Although the two traders act like competing duopolists, the asymptotic properties of trading intensities are quite di erent from the counterparts found in Li (2011). Li nds that the trading intensity can be su cient large when the information asymmetry between the identically informed traders and the market maker is small enough in the limit of continuous trading, and that the trading intensity is inversely related to information asymmetry. One can tell from Figure 2.3(B) and (C) that 0 and 0 remain bounded at = 0. 0 is found to be a non-monotonic function of . 0 rst monotonically decreases with and later monotonically increases with when is close to 1. 0 is a monotonically increasing function for all values of . Inference Parameters: and Li (2011) nds that more competition leads to a smaller price impact parameter and hence a more liquid market. The degree of competition is measured by the number of the traders in the market. In this paper, the number of traders is xed to be two and the degree of competition is measured by the precision of the noisy signal both traders receive. We expect that the same conclusion still holds: as increases, the market should become more liquid in the stationary state where the market maker is facing a less severe adverse selection. 71 Figure 2.3: The asymptotic properties of trading intensity parameters (b, 0 and 0) as functions of h for cases when = 0:3, 0:9 and 1. Parameter values: v = u = = 1 and r = 0:05. 72 Figure 2.4(A) and (B) con rm the conjecture. For a xed h, the price impact or the inference parameter for the market maker (inverse measure of market depth) monotonically decreases with . The inference parameter for the less informed trader is also found to be negatively related to since the trader becomes more informative about the value of the asset as the stream of signals she observes becomes more precise. When = 1, strictly equals to zero since the \less" informed trader is perfectly informed and learns nothing from the history of order ows. In Figure 2.4(C), we plot 0 = limh!0 and 0 = limh!0 against . We nds that 0 is always greater than 0 except at = 0. This is because the less informed trader is more informed than the market maker and learns less from the order ow. Expected Quantity of Informed Trading Intuition might suggest (incorrectly) that the better informed trader trade more than the less informed trader on average because she has more private information. However, since the less informed trader trades more intensely on the shared private information, the di erence between how they trade the common information may dominate the trading volume by the more informed trader on her exclusive private information. We con rm the intuition in Figure 2.5. In Figure 2.5(A), we x = 0:9 and = 1. When h is relatively large, the contribution to the total trading volume by the more informed trader is greater than that by the less informed trader. The relation switches sign when h becomes smaller or the frequency of trading is higher. In Figure 2.5(B), we let = 0:6 and = 1; in (C), we let = 0:6 and = 0. We observe similar phenomena as in (A). But the frequency at which 73 Figure 2.4: (A) and (B) The inference parameters , as functions of h for = 0, 0:3, 0:9 and 1. Parameter values: v = u = = 1 and r = 0:05. (C) 0 and 0 as functions of . 74 the less informed trader starts to dominate the better informed trader in terms of trading volume becomes higher if we decrease or . Expected Pro ts Although Foster and Viswanathan (1994) do not address the issue of each informed trader?s expected pro ts in the limit of continuous trading, it is reasonable to believe that as the time interval between trades converges to zero, the informed traders compete so aggressively on shared private information that the information is re ected into prices in- stantaneously and the pro ts of the less informed trader go to zero. The better informed trader however, can earn strictly positive pro ts by trading the remaining monopolized in- formation slowly. Li (2011) has shown that identically informed traders can earn strictly positive pro ts when trading is continuous if the information arrives repeatedly. It is reason- able to believe that in this model, despite both informed traders compete very aggressively on their shared stream of noisy signals, in the limit, the less informed trader can still earn a non-zero expected pro ts. Figure 2.6 illustrates how each informed trader?s expected pro ts vary with the time interval between trading and the precision of the ow of the noisy signals observed by both traders. We nd that the less informed trader?s expected pro ts increase when becomes higher. Intuitively, the better the information received by the less informed trader, the more market power and hence the better investment opportunity she has. A more interesting result is that as h converges to zero, both the pro ts of the more and less informed traders remain strictly positive, with the pro ts of the more informed trader greater than the pro ts 75 Figure 2.5: Contributions to the total trading volume. Parameter values: v = u = 1 and r = 0:05. 76 of the less informed trader. When = 1, we reach to the benchmark case where both traders are equally informed. We should expect them to earn the same expected pro ts and it is con rmed in the Figure. 2.5 Conclusion In this paper, we examine how the correlation among ows of information received by in- formed traders a ects the market e ciency, market liquidity, trading volume and their ex- pected pro ts, especially in the limit of continuous trading. We consider a very special case in which duopolistic informed traders? information sets are nested. We nd the total private information can be decomposed into two components with each component incorporated into prices in a qualitatively di erent manner in the limit. The shared private information is incorporated into prices much more quickly than the information held exclusively by the more informed trader. One can nd that the less informed trader acts more aggressively than the better informed trader. When the frequency of trading becomes high enough, the less informed trader who has worse information contributes more trading volume contrary to conventional wisdom. The pro ts for the less informed trader may fall as trading becomes more frequent, but converge to a positive constant in the limit as the time interval between trades goes to zero. 77 Figure 2.6: (A) and (B) The expected pro ts for each informed trader as functions of h for cases when = 0:4, 0:8 and 1. (C) Expected pro ts for each trader as functions of in the limit of continuous trading. Parameter values: v = u = = 1 and r = 0:05. 78 Chapter 3 Real Options and Product Di erentiation 3.1 Introduction Ever since the pioneering work of McDonald and Siegel (1986), the real options literature often either assumes the rm to be a monopolist (Pindyck (1988), He and Pindyck (1992)) or assumes the rm to be perfectly competitive (Dixit (1989) ). Very little literature has modeled the imperfect competition among rms in exercising real options (option games). Smets (1993) provides the rst approach to model real options game in a duopoly industry. In his model, the capital stock choice is discrete and there is a upper bound in the total industry capital stock. He also randomly picks a rm to be the leader by investing rst (the other rm which invests later is called the follower) when time comes to make investment. The subsequent work that follow Smets?s approach include Grenadier (1996) and Williams (1993) . In the situation where the capital stock choice is continuous, implying that each rm can make arbitrarily small investment, the rms face a pre-emption problem. Each rm has incentive to preempt its competitors to prematurely invest to extract more rents. Grenadier (2002) claims to solve the equilibrium in an oligopoly industry by using a \myopic rm" approach. When a \myopic" rm is evaluating the optimal time to invest, it assumes that all other rms? capital stock is xed forever. The equilibrium derived in his approach has 79 lower growth options value relative to the monopolist case but is still positive. Such an equilibrium, however is not subgame perfect. It is an \open loop" equilibrium instead of a \closed loop" equilibrium, as pointed out by Back and Paulsen (2009). In an \open loop" equilibrium, each player in the game cannot observe the other players? actions. While in a \closed loop" equilibrium, all past play is common knowledge. In Grenadier (2002)?s setup, each rm can infer its opponent?s capital stock from the price of the product. It can be shown that, by knowing its opponent?s capital stock, each rm has incentive to preempt its opponent by investing earlier than the conjectured optimal time in Grenadier?s equilibrium. As long as the value of the real option is positive, each rm will have incentive to invest till the real option value drops to zero. This means that the preemption never disappears as long as the growth options is positive. Back and Paulsen further conjecture that the equilibrium with pre-emption should be competitive in an oligopoly industry with elastic demand such that the real option value remains zero all the time . In Grenadier (2002) and Back and Paulsen (2009), the products are homogenous (per- fect substitute). This paper examines whether the incentive to preempt could diminish if we assume heterogenous products instead. By making such an assumption, each rm has some limited monopoly power over its product. I examine whether such monopoly power, although limited, could decrease the competition of exercising options to such an extent that preemption is no longer pro table and the value of growth options is still positive. However, we nd that under the assumption of non- exibility of capital stock, preemption is always pro table even though the products are imperfect substitute. 80 The plan of the paper is as follows. In Section II, we describe the model. We solve the equilibria using the \myopic" approach for the CES demand case in Section III and for the linear demand case in Section IV. We nd the equilibrium is not subgame perfect in either case. Section V concludes. 3.2 Model Setup 3.2.1 Demand Function and Capacity Process I make the following assumptions: (1) There exits an industry composed of n rms each producing a single non-storable di erentiated product. In section III, I use the CES (constant elasticity of substitution) demand function. The inverse demand function is given by Pi = Y (t)( nX j=1 q j ) 1q 1i (3.1) where dY = Y dt+ Y dz is an exogenous shock process to demand. measures the degree of di erentiation, ranging from 0 for independent goods to 1 for perfect substitutes. The demand function is derived from a n-good industry with representative consumer with utility function given by U(q) = ( Pn i=1 q i ) (Xavier Vives 2000). Another speci cation can be found in Dixit and Stiglitz (1977): U(q) = ( Pn i=1 q i ) 1= . In section 3.4 , I use the linear inverse demand function: Pi = Yt qi P j 6=i qj. (2) At time t, each rm produces qi(t) Ki(t) units of output where Ki is the capital 81 stock of rm i. For simplicity, zero variable cost of production is assumed. The operating pro t ow is denoted i(Y (t); Ki(t); K i(t)) = Piqi. (3) At any point in time, each rm can invest in additional capacity to increase its output. Each unit of capacity costs I. I assume that Ki is a nondecreasing process implying completely irreversibility of capital stock. I also assume that there exists an investment strategy which is characterized by the trigger function Y (Ki; K i). Whenever Y (t) rises to the trigger function, rm i increases its capacity level. (4) The rm i is solving the following problem: V i[Y;K i (t); K i(t)] = max Ki E( Z 1 0 e rt i[Y (t); Ki(t); K i(t)]dt Z 1 0 e rtIdKi) (3.2) 3.2.2 Risk Neutral Measure Following Carlson et al. (2004), let Bt denote the price of the riskless asset with dynamics dBt = rBtdt, where r is the risk-free interest rate, and let St be the risky asst with dynamics dSt = Stdt+ Stdzt: (3.3) We use assets B and S to de ne a new probability measure (Q) under which z^t = zt + r t is a standard Brownian motion. Under this new measure, the demand dynamics becomes dYt = (r )Ytdt+ Ytdz^t (3.4) 82 where = > 0. 3.3 Symmetric Open Loop Equilibrium for CES Demand Function 3.3.1 The Open Loop Equilibrium I consider the case when there are two rms in the industry. Proposition 3.1 1. In a symmetric open-loop equilibrium, the trigger strategy for rm i is given by Y = I 2 2(1 1 1 )( + )K 1 = K1 (3.5) with = I 2 2(1 1 1 )( + ) and K = Ki = Kj to be each rm?s capacity. 2. The rm i?s value is given by Vi(Y;K) = I(1 2 1 ( 1 1)( + 1) ) 1( 1 1( 1) + 1 )K 1( 1)+1Y 1 + Y 2 1K : (3.6) The value of assets in place is Fi(Y;K) = Y 2 1K + I 1( 1 1( 1) + 1 )K 1( 1)+1Y 1 = J(K)Y + E(K)Y 1 (3.7) and the value of growth option is Gi(Y;K) = I 2 1 ( 1 1)( + 1) 1( 1 1( 1) + 1 )K 1( 1)+1Y 1 = C(K)Y 1 : (3.8) 83 Proof: From standard arguments in the literature on investment with uncertainty, V i(Y;Ki; K i) and Y (Ki; K i) are solutions to the following di erential equation: 1 2 2Y 2V iY Y + (r )Y V i Y rV i + i(Y;Ki; K i) = 0; (3.9) with the following boundary conditions @V i @Ki (Y (Ki; K i); Ki; K i) = I; (3.10) @2V i @Ki@Y (Y (Ki; K i; Ki; K i) = 0; (3.11) and @V i @K i (Y (Ki; K i); Ki; K i) = 0: (3.12) Grenadier provides a simple approach to solve the equilibrium strategies without involving xed point problem. He considers a myopic rm i that ignores all potential competitive exercise. The value of the myopic rm is denoted as M i(Y;Ki; K i). Then, I will denote the myopic rm?s marginal output by mi(Ki; K i), with mi(Y;Ki; K i) = @M i @Ki . It can be shown that mi and Y (Ki; K i) satisfy the following di erential equation: 1 2 2mY Y + (r )mY rm+ @ @Ki = 0 (3.13) 84 subject to m(Y (Ki; K i); Ki; K i) = I (3.14) and @m @Y (Y (Ki; K i); Ki; K i) = 0: (3.15) The function m(Y;Ki; K i) represents the value of a perpetual American call option, where the option has an exercise payo of I, and a zero exercise price. It can be easily veri ed that @ @Ki = Y (t)(K i +K i) 2( K2 1i + K 1 i K i): (3.16) The solution for Equation 3.9 is m = BY 1 + 2 2Y ( + )K 1 : (3.17) From the boundary conditions, we have BY 1 + 2 2Y ( + )K 1 = I (3.18) and B 1Y 1 1 + 2 2( + )K 1 = 0: (3.19) Since rms are symmetric, we should have Ki = Kj = K, and the above equation 85 becomes @ @Ki = Y (t)2 2 ( + 1)K 1: (3.20) Equation 3.19 can be reduced to B = 2 2( + )K 1 1Y 1 1 : (3.21) Then, substituting the expression for B into 3.18, we have 2 2( + )K 1Y 1 + 2 2Y ( + )K 1 = I; (3.22) which leads to the solution for Y , Y (K) = I 2 2(1 1 1 )( + )K 1 = K1 ; (3.23) where = I 2 2(1 1 1 )( + ) : From a similar argument, we can also calculate the myopic trigger strategy when two rms have di erent capital stock in the beginning (Ki 6= K i). Y (Ki; K i) = I (1 1= 1) (K i +K i) 2( K 2 1 i +K 1 i K i) (3.24) The value of the rm: assets in place and growth options. Our next goal is to solve Equation 3.9 with the boundary conditions. Given rm i?s 86 instantaneous pro t function i = Y (K i +K j ) 1K i , the solution can be written as follows Vi = AY 1 + Y (K i +K j ) 1K i = A(K)Y 1 + Y 2 1K : (3.25) The subscript is ignored since in equilibrium K1 = K2 = ::: = Kn. From the boundary conditions, we have @V @K (Y ; K) = I (3.26) which is equivalent to A0(K) 1Y 1 + Y 2 1 K 1 = I: (3.27) Solving Equation 3.27, we have A0(K) = (I Y 2 1 K 1 )Y 1 (3.28) = (I 2 1 ) 1K 1( 1) = I(1 2 1 ( 1 1)( + 1) ) 1K 1( 1): By integrating A0(K), we nd that A(K) = Z 1 K A0(k)dk = I(1 2 1 ( 1 1)( + 1) ) 1 1 1( 1) + 1 K 1( 1)+1: (3.29) 87 Therefor, rm i?s value can be expressed as Vi = I(1 2 1 ( 1 1)( + 1) ) 1 1 1( 1) + 1 K 1( 1)+1Y 1 + Y 2 1K : (3.30) The value of assets in place is given by Fi(K;Y ) = Y 2 1K +I 1( 1 1( 1) + 1 )K 1( 1)+1Y 1 = J(K)Y +E(K)Y 1 ; (3.31) and the value of growth option is given by Gi(K;Y ) = I 2 1 ( 1 1)( + 1) 1( 1 1( 1) + 1 )K 1( 1)+1Y 1 = C(K)Y 1 : (3.32) 3.3.2 A Preemption Strategy To determine whether there exists a preemption strategy, I follow the approach in Paulsen (2006). Suppose K10 = K20 = K0. If both rms were to play the myopic strategy de ned in Proposition 2, then both rms would always hold the same capacity and the expected present value of the future cash ow would be given by Vi = I(1 2 1 ( 1 1)( + 1) ) 1 1 1( 1) + 1 K 1( 1)+10 Y 1 + Y 2 1K 0 : (3.33) Now consider the strategy for rm 1: invest K > K0 and keep this capacity constant until time when K2 = K, then play the symmetric trigger. We assume rm 2 plays the 88 myopic trigger and time is the rst passage time of Y (K;K). From time on, both rms will hold the same capacities, thus rm 1?s expected payo is given by the value function V (Y ; K;K) discounted by e r . Up to time , rm 1?s capacity is xed and only K2 increases. Firm 1?s expected pro t is thus W (Y;K;K2) = EY;K2 [ Z 0 e rt( (Yt; K;K2) rIK)dt+ e r V (Y ; K;K)]: (3.34) From the Feynman-Kac theorem, W (Y;K;K2) satis es the ODE 1 2 2Y 2WY Y + (r )WY rW + ( (Y;K;K2) rIK) = 0; (3.35) subject to the boundary conditions WK2(Y (K2; K); K;K2) = 0; (3.36) W (0; K;K2) = IK; (3.37) and W (Y (K;K); K;K) = V (Y (K;K); K;K): (3.38) The solution of the ODE is in the form of W (Y; K;K2) = Y ( K +K 2 ) 1 K I K + A( K;K2)Y 1 . From the boundary condition, @W @K2 = ( 1)Y (K2; K)( K +K 2 ) 2 K K 12 + AK2( K;K2)Y 2(K2; K) = 0; (3.39) 89 we have AK2( K;K2) = ( 1)( K +K 2 ) 2 K K 12 Y 1 1(K2; K) : (3.40) Integrating AK2( K;K2), we can nd A( K;K2) = A( K) Z 1 K2 AK2( K; s)ds: (3.41) To nd A( K), we have to use the last boundary condition. First, we need to calculate V (Y ( K; K); K). From the result we have in section 3.3.1, Vi = I(1 2 1 ( 1 1)( + 1) ) 1 1 1( 1) + 1 K 1( 1)+1Y 1( K)+ Y ( K)2 1 K : (3.42) From the last boundary condition, A( K) = Z 1 K AK2( K; s)ds+ I(1 2 1 ( 1 1)( + 1) ) 1 1 1( 1) + 1 K 1( 1)+1: (3.43) Since the integrals cannot be derived in the closed forms except for a few parameter values, I evaluate the integrals numerically to nd the expected payo from the preemption strategy. Some numerical examples. First, we set = 0:2, = 0:12, r = 0:12, I = 1, = 1 and = 0:5. One can tell from simple calculation (not shown here), 1 = 3. This is actually the set of parameters in the counter-example provided by Paulsen (2006), in which the products are perfect substitute. 90 We then plot W (Y (1; 1); K; 1) vs. K. It can be shown from Figure 3.1 that W (Y (1; 1); K; 1) is increasing in K up to a value greater than 1 and then decreasing. Following the argument in Paulsen (2006), the symmetric trigger Y (Ki; K i) is not sub-game perfect, and it is pro table for the rm to exercise the growth option early to preempt the second rm. We then keep xed and decrease from 1 to 0.5. Figure 3.2 shows the preempting value function W (Y (1; 1); K; 1) as a function of K. In addition, the preemption strategy still exists but the local maxima is less than the case when = 1. I also tried other values of and calculate W (Y (1; 1); K; 1) numerically. As long as is positive and less than 1, it is always pro table for one rm to deviate the conjectured strategy in Proposition 1 and preempt if the other rm plays the conjectured strategy. If = 0, then the utility of the representative customer is not well de ned. However, we can de ne the form of logarithm as U(q1; q2) log (q 1 + q 2). Then the price can be written as Pi(qi; q i) = (q i + q i) 1q 1i . Fig. 3.3 shows the expected pro t W (Y (1; 1); K; 1) from the preemption strategy as a function of K. Clearly, W (Y (1; 1); K; 1) is monotonically decreasing when K > 1. The myopic strategies form a Nash equilibrium. The result is not surprising since Heston and Loewenstein have already proved in the perfect substitute case that if the elasticity is one (when the representative customer has a utility in the form of log (qi + q i)), each rm is indi erent between waiting and preempting. 91 Figure 3.1: = 1, = 0:5 Figure 3.2: = 0:5, = 0:5 92 Figure 3.3: = 0:5, = 0 (equivalent to the case in which the representative customer has a log utility function.) 3.4 Linear Demand 3.4.1 The Open Loop Equilibrium Now we study the case in which the inverse demand function is linear. The inverse demand function is given by Pi = Yt Ki X j 6=i Kj: (3.44) The value of ,0 = = 1 measures the degree of product di erentiation. When = 1, the goods are perfect substitute, when = 0, the goods are independent, and when < 0, the goods are complements. Baldursson (1998), Grenadier (2002) and Back and Paulsen (2009) all have studied the open loop equilibrium in an oligopoly setting with the 93 assumption of linear demand. Our assumption is the same except that the products are not homogenous. Again, we assume that there exist two rms in the industry. The instantaneous pro t for rm i (i = 1; 2) can be expressed as i = Ki(Y Ki K i): (3.45) In the symmetric equilibrium, Ki = K i and = Ki(Y ( + )Ki). Proposition 3.2: 1. The trigger strategy is given by Y (Ki; K i) = 1 (2 Ki + K i + rI) ( 1 1)r : (3.46) At equilibrium, Ki = K i and we have Y (Ki) = 1 [(2 + )Ki + rI] ( 1 1)r (3.47) 2. The value of marginal investment is given by m(Yt; Ki; K i) = Yt 2 Ki + K i r I + 2 Ki+ Kjr ( 1 1)Y 1 Y 1t : (3.48) 3. In the symmetric equilibrium, the value of rm i is given by V (Y;Ki) = C(Ki)Y 1 + Y Ki (2 + )K2i r (3.49) 94 where C(Ki) = Z +1 Ki I + 2(2 + )Kr Y (K) Y 1 dK = r 1(2 + ) Y 1+1(Ki)+ r( 1 1) 1(2 + ) 2 1 1 2 1 Y 1+2(Ki): (3.50) Proof : We rst calculate the marginal bene t @ i @Ki = Yt 2 Ki K i: (3.51) Let m(Yt; Ki; K i) denote the value of the marginal investment. m(Yt; Ki; K i) satis es the di erential equation, 1 2 2Y 2mY Y + (r )Y mY rm+ @ i @Ki = 0 (3.52) and is subject to the boundary conditions: m(Y ; Ki; Kj) = I; (3.53) @m @Y = 0: (3.54) The solution of m(Yt; Ki; K i) takes the form of m(Yt; Ki; K i) = Yt 2 Ki + Kj r + AY 1t : (3.55) 95 From the boundary conditions (value matching and smooth-pasting), we have Y t 2 Ki + Kj r + AY 1t = I (3.56) and 1 + 1AY 1 1 t = 0: (3.57) By solving Equation 3.56 and 3.57, we nd the trigger strategy Y (Ki; K i) andm(Yt; Ki; K i) as follows Y (Ki; K i) = 1 1 1 2 Ki + K i + rI r (3.58) and m(Yt; Ki; K i) = Yt 2 Ki + K i r 1 1 Y 1 1 Y 1t : (3.59) We then need to nd the value function of the rm Vi(Yt; Ki; K i). The function Vi satis es the following di erential equation 1 2 2 @2Vi @Y 2 + (r ) @Vi @Y rVi + = 0; (3.60) subject to @Vi @Ki (Y ; Ki) = I; (3.61) with = Ki(Y ( + )Ki) = Y Ki ( + )K2i . 96 The solution of Vi can be expressed as Vi = C Y 1 + Y Ki ( + )K2i r : (3.62) From the boundary condition, we have C 0 (Ki)Y 1 + Y 2( + )Ki r = I: (3.63) Solving the above equation, we have C 0 (Ki) = I + 2( + )Kir Y Y 1 : (3.64) From the fact that 2( + )Ki r = 2( + ) 2 + ( Y ( 1 1) 1 I); (3.65) we can write C 0 (Ki) as C 0 (Ki) = I2 + + 1 2( + ) (2 + ) 1 Y Y 1 : (3.66) One can verify that Z 1 Ki 1 Y 1 dKi = r 1 (2 + ) Y 1+1 (3.67) and Z 1 Ki 1 Y 1 1 dKi = r( 1 1) 1 ( 1 2)(2 + ) Y 1+2: (3.68) 97 We then can evaluate C(Ki) by integrating C 0 (Ki) C(Ki) = Z 1 Ki C 0 (K)dK = + r I 1(2 + )2 Y 1+1(Ki) r( 1 1)( 1 2( + )) ( 1 2) 2 21(2 + )2 Y 1+2(Ki): (3.69) The growth option is then given by G(Y;Ki) = ( 2r( + )I 1(2 + )2 Y 1+1(Ki) + r( 1 1)(2( + ) 1) ( 1 2) 2 21(2 + )2 Y 1+2(Ki))Y 1 : (3.70) 3.4.2 A Preemption Strategy To determine whether there exists a preemption strategy, I again follow the approach in Paulsen (2006). Suppose K10 = K20 = K0. If both rms were to play the myopic strategy de ned in Proposition 2, then both rms would always hold the same capacity and the expected present value of the future cash ow would be given by V (Y;K0; K0) = C(K0)Y 1 + Y K0 ( + )K20 r : (3.71) Now consider the strategy for rm 1: invest K > K0 and keep this capacity constant until time when K2 = K, then play the symmetric trigger. We assume rm 2 plays the myopic trigger and time is the rst passage time of Y (K;K). From time on, both rms will hold the same capacities, thus rm 1?s expected payo is given by the value function V (Y ; K;K) discounted by e r . Up to time , rm 1?s capacity 98 is xed and only K2 increases. Firm 1?s expected pro t is thus W (Y;K;K2) = EY;K2 [ Z 0 e rt( (Yt; K;K2) rIK)dt+ e r V (Y ; K;K)]: (3.72) From the Feynman Kac theorem, W (Y;K;K2) satis es the ODE 1 2 2Y 2WY Y + (r )WY rW + ( (Y;K;K2) rIK) = 0; (3.73) subject to the boundary conditions WK2(Y (K2; K); K;K2) = 0; (3.74) W (0; K;K2) = IK; (3.75) and W (Y (K;K); K;K) = V (Y (K;K); K;K): (3.76) The general the solution is in the form W (Y;K;K2) = KY K( K + K2) r IK + A(K;K2)Y 1 (3.77) since (Y;K;K2) = K(Y K K2). 99 From the value-matching condition, we have K r + AK2(K;K2)Y 1(K2; K) = 0: (3.78) Solving this equation, we have AK2(K;K2) = K r Y 1(K2; K) : (3.79) Integrating AK2(K;K2), we can nd the expression for A(K;K2), A(K;K2) = A(K) Z 1 K2 AK2(K;K)dK = A(K)+ K r 2 1 ( 1 1)r 1 ( 1 1)(2 K2 + K + rI) 1 1 ; (3.80) which can be simpli ed into A(K;K2) = A(K) + K 2 1 1 (2 K2 + K + rI) 1 1 : (3.81) From the last boundary condition, we can determine A(K): A(K;K) = A(K) + K 2 1 1 (2 K + K + rI) 1 1 = C(K): (3.82) Solving the equation, we nd that A(K) = C(K) K 2 1 1 (2 K + K + rI) 1 1 : (3.83) 100 Some numerical examples I set r = 0:12, = 0, = 0:2 and it is easy to nd that 1 = 3 and 2 = 1. I rst study the case in which = 2, = 1:5 and the goods are imperfect substitute. In g. 3.4, I plot W (Y (1; 1); K; 1) as a function of K. It is increasing in K up to a value around 1:2 and then decreasing. Hence the symmetric trigger strategy Y (Ki; K i) is not rm i?s best response to rm i?s strategy. I also tried smaller value of (not shown here). No matter how small is, as long as is positive, the function W (Y (1; 1); K; 1) has a local maxima at some K which is greater than 1. This means that although product di erentiation can decrease competition among rms, the open-loop equilibrium is still not sub-game perfect since pre-emption strategy exists for arbitrary positive values. I then keep = 2 and set = 0. This is the case in which goods are independent and rms are monopolists. As we can see from g. 3.5, W (Y (1; 1); K; 1) has a slope of zero at K = 1 and is decreasing when K > 1. This is expected since rms are monopolists and the equilibrium strategies do not depend on other rms? capacities. Finally, I examine the case in which < 0 and the goods are complements. I set = 2 and = 1. The value function W (Y (1; 1); K; 1) of rm 1 preempting at Y (1; 1) is shown in g. 3.6. Again, we nd that W (Y (1; 1); K; 1) has a local maxima at K > 1. This illustrates that even for the case of complimentary goods, each rm has an incentive to preempt. 101 Figure 3.4: = 2 and = 1:5 Figure 3.5: = 2 and = 0 102 Figure 3.6: = 2 and = 1 3.5 Conclusion I develop a continuous time real option model in an oligopoly industry with heterogenous products. I nd that although the heterogenous products assumption lowers the incentive for each rm to prematurely exercise the growth options, the preemption strategy is still pro table for each rm when the reverse demand function is either CES or linear with non- operating exibility. 103 Chapter A Appendix Proof of Lemma 1.2.1: It is easy to verify that vn0 = (1 h) n0 nvn + h v n0X =n (1 h)n 0 n + n0X =n (1 h)n 0 n v; : (A.1) Taking expectations in Equation A.1, we have E(vn0 jI m n ) = (1 h) n0 nE(vnjI m n ) + h v(1 (1 h) n0 n+1): (A.2) Substituting into Equation 1.2, the price is equal to pn = +1X n0=n (1 exp ( rh)) exp ( r(n0 n)h)((1 h)n 0 nE(vnjI m n ) (A.3) + h v(1 (1 h)n 0 n+1)) = (1 exp ( rh)) 1 exp ( rh)(1 h) E(vnjI m n ) + h v 1 exp( rh)(1 h) : 104 Proof of Lemma 1.2.2: First, one can nd that +1X n0=n exp( r(n0 n)h)( n0X =n xj; (vn0 p )) = (A.4) xj;n +1X =n (exp( r( n)h)(v pn)) + xj;n+1 +1X =n+1 (exp( r( n)h)(v pn+1)) + xj;n+2 +1X =n+1 (exp( r( n)h)(v pn+2)) + ::: = xj;n ( +1X =n exp( r( n)h)v pn 1 exp( rh) ) + xj;n+1 ( +1X =n+1 exp( r( n)h)v exp( rh)pn 1 exp( rh) ) + xj;n+2 ( +1X =n+2 exp( r( n)h)v exp( 2rh)pn 1 exp( rh) ) + ::: (A.5) Substituting Equation A.5 into Equation 1.5 and using the result in Equation A.1, we 105 have E[ +1X =n (1 exp( rh)) exp( r(n0 n)h)( n0X =n xj; (vn0 p ))jI i n 1] = (A.6) E[xj;n( (1 exp ( rh)) 1 exp ( rh)(1 h) vn + h v 1 exp( rh)(1 h) pn) + exp( rh)xj;n+1( (1 exp ( rh)) 1 exp ( rh)(1 h) vn+1 + h v 1 exp( rh)(1 h) pn+1) + exp( 2rh)xj;n+1( (1 exp ( rh)) 1 exp ( rh)(1 h) vn+1 + h v 1 exp( rh)(1 h) pn+2) :::jI in 1] = E[ +1X n0=n exp( r(n0 n)h)( (1 exp ( rh)) 1 exp ( rh)(1 h) vn0 + h v 1 exp( rh)(1 h) pn0)jI i n 1]: Proof of Lemma 1.2.3: We rst compute each component of the covariance matrix of the vector (vn 1; yn) conditional on the market maker?s information set Imn 1: cov(yn; vn 1jIn 1) = cov(M (vn 1 g^n 1); vn 1jIn 1) = M var(vn 1jIn 1) = M v (A.7) var(ynjI m n 1) = M 2 2 v + 2 uh: (A.8) 106 Then under the market maker?s belief, we have the joint distribution of (vn 1; yn)0 0 B B @ vn 1 yn 1 C C A N( 0 B B @ v^n 1 0 1 C C A ; (A.9) 0 B B B B B B @ v M v v h M v h M v M2 2 v + 2uh 1 C C C C C C A ): Then applying the projection theorem, we have 1 h = (M2 2 v + 2 uh) 1M v; (A.10) which can be reduced to equation (1.14). Applying the projection theorem again, we can derive the variance of n as var( n) = var(vn 1jIn 1) 1 h M v (A.11) = v 1 h M v = v 2uh vM2 2 + 2uh : The uncertainty of market maker?s posterior belief about gn is given by var(vnjI m n ) = var((1 h) n + v;n) = (1 h) 2var( n) + 2 vh: (A.12) 107 By stationary condition, we must have var(vnjImn ) = v which leads to equation (1.15). Proof of Lemma 1.2.4: From equations (1.1) and (1.12), market maker?s estimation error on vn is vn v^n = (1 h)(vn 1 v^n 1) yn + v;n: (A.13) Substituting for vn v^n in equation (1.18), we nd V (vn 1; v^n 1) = maxxi;n(xi;n((1 h)(vn 1 v^n 1) (xi;n +Xi ;n)) (A.14) + e rh(B((1 h)(vn 1 v^n 1) (xi;n +Xi ;n)) 2 + 2 2uh+ 2 vh) + C)): The rst order condition yields xi;n = (1 2e rhB )(1 h)(vn 1 v^n 1) + (2e rhB 1)Xi ;n 2 (1 e rhB ) : (A.15) The second order condition requires that e rhB 1 < 0: (A.16) Because of symmetry argument, the only possible equilibrium is one in which their strate- gies are identical. We should have Xi ;n = (M 1)xi;n, which leads xi;n = (1 2e rhB )(1 h) (M(1 2e rhB ) + 1) (vn 1 v^n 1) = (vn 1 v^n 1); (A.17) 108 where = (1 2e rhB )(1 h) (M(1 2e rhB ) + 1) : (A.18) Substituting for xi;n back in the Bellman equation and matching the (vn 1 v^n 1)2 term and constant term, we nd B = [1 h M ] + e rhB(1 h M )2 (A.19) which can be reduced to B = (1 h)2(1 e rhB ) (1 +M 2Me rhB )2 ; (A.20) and C = e rhB( 2 2u + 2 v)h 1 e rh : (A.21) Proof of Proposition 1.2.1: First, we de ne q = e rh B and Z = e rh(1 h)2. From equation (1.20), we have f(q) = 4M2q3 4M(M + 1)q2 + ((M + 1)2 + Z)q Z = 0: (A.22) The cubic equation f(q) has three real roots: q1 = M + 1 3M (A.23) 1 6M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 ((M + 1)2 3Z)3 1 6M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 + ((M + 1)2 3Z)3 109 q2 = M + 1 3M (A.24) 1 + i p 3 12M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 ((M + 1)2 3Z)3 1 i p 3 12M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 + ((M + 1)2 3Z)3 and q3 = M + 1 3M (A.25) 1 i p 3 12M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 ((M + 1)2 3Z)3 1 + i p 3 12M 3 q (M + 1)3 + (9 18M)Z + p ((M + 1)3 + (9 18M)Z)2 + ((M + 1)2 3Z)3: It can be easily veri ed that for M > 1, we have f(0) = Z < 0; f(1=2) => 0; f(1) = (M 1)2 > 0: (A.26) In addition, the quadratic function f 0(q) = 0 has the two solutions q4;5 = 2(M + 1) p 4(M + 1)2 2((M + 1)2 + Z) 2M : (A.27) The solution with the positive sign q5 2 ( 1 2 ; 1): (A.28) Equations (A.26) and (A.28) imply that 0 < q1 < 12 and 1 2 < q3 < q4 < 1. From the second order condition (equation (A.16)) we have q < 1. In addition, from 110 equation (1.19) any root that makes economically feasible must lie in the range q 2 (0; 12). The only possible solution is q1. From equation (1.14) and equation (1.14), equation (1.19) can be rewritten as 1 h = vM 2 vM2 2 + 2uh = 1 2q1 1 +M(1 2q1) : (A.29) From Equation A.29, one can nd that = s 2u(1 2q1)h vM : (A.30) Substituting Equation A.30 into Equation 1.15, we can nd the expression for v v = 2vh 1 (1 h) 2 M(1 2q1)+1 : (A.31) Then, from Equations (A.30, 1.14), (A.24), (1.21) and the expression for q1, we can derive the remaining parameters , , B and C, respectively with the expressions for and given by = s (1 2q1)(1 (1 h)2 M(1 2q1)+1 ) M ; (A.32) = (1 h) p M v q (1 2q1(1+M(1 2q1))) 1+M(1 2q1) (1 h)2 M u(1 2q1) + u : (A.33) 111 Proof of Proposition 1.2.2: We rst de ne Sv = v=h; (A.34) and q0 = M + 1 3M 1 6M 3 q (M + 1)3 18M 9 + p ((M + 1)3 18M 9)2 (M2 + 2M 2)3(A.35) 1 6M 3 q (M + 1)3 18M 9 p ((M + 1)3 18M 9)2 (M2 + 2M 2)3: Then, as h approaches 0, equations (A.29) and (1.15) become M2Sv 2 M2Sv 2 + 2u = M(1 2q0) 1 +M(1 2q0) (A.36) Sv 2u M2 2Sv + 2u + 2v = Sv: (A.37) The set of the above nonlinear equations has the solution Sv = 2v(1 +M(1 2q0)) M(1 2q0) (A.38) and = u(1 2q0) v p 1 +M(1 2q0) : (A.39) 112 By the continuity argument, the limiting results of v and become lim h!0 v h = Sv = 2v(1 +M(1 2q0)) M(1 2q0) (A.40) and lim h!0 = u(1 2q0) v p 1 +M(1 2q0) : (A.41) Then, from equations (1.14), (A.28) and (1.21), we obtain the following asymptotic results for , B and C: lim h!0 = v u s 1 1 +M(1 2q0) ; (A.42) lim h!0 B = q0 u v p 1 +M(1 2q0); (A.43) lim h!0 C = q0 u v( p 1 +M(1 2q0) + s 1 1 +M(1 2q0) ): (A.44) 113 Chapter B Appendix Proof of Lemma 2.2.1: It is easy to verify that vn0 = (1 h) n0 nvn + h v n0X =n (1 h)n 0 n + n0X =n (1 h)n 0 n v; : (B.1) Taking expectations in Equation B.1, we have E(vn0jI m n ) = (1 h) n0 nE(vnjI m n ) + v(1 (1 h) n0 n+1): (B.2) Substituting into Equation 2.3, the price is equal to pn = +1X n0=n (1 exp ( rh)) exp ( r(n0 n)h)((1 h)n 0 nE(vnjI m n ) (B.3) + h v(1 (1 h)n 0 n+1)) = (1 exp ( rh)) 1 exp ( rh)(1 h) E(vnjI m n ) + h v 1 exp( rh)(1 h) : 114 Proof of Lemma 2.2.2: First, one can nd that +1X n0=n exp( r(n0 n)h)( n0X =n x (vn0 p )) = (B.4) xn +1X =n (exp( r( n)h)(v pn)) + xn+1 +1X =n+1 (exp( r( n)h)(v pn+1)) + xn+2 +1X =n+1 (exp( r( n)h)(v pn+2)) + ::: = xn ( +1X =n exp( r( n)h)v pn 1 exp( rh) ) + xn+1 ( +1X =n+1 exp( r( n)h)v exp( rh)pn 1 exp( rh) ) + xn+2 ( +1X =n+2 exp( r( n)h)v exp( 2rh)pn 1 exp( rh) ) + ::: (B.5) Substituting Equation B.5 into Equation 2.6 and using the result in Equation B.1, we 115 have E[ +1X =n (1 exp( rh)) exp( r(n0 n)h)( n0X =n x (vn0 p ))jI i n 1] = (B.6) E[xn( (1 exp ( rh)) 1 exp ( rh)(1 h) vn + h v 1 exp( rh)(1 h) pn) + exp( rh)xn+1( (1 exp ( rh)) 1 exp ( rh)(1 h) vn+1 + h v 1 exp( rh)(1 h) pn+1) + exp( 2rh)xn+1( (1 exp ( rh)) 1 exp ( rh)(1 h) vn+1 + h v 1 exp( rh)(1 h) pn+2) :::jI in 1] = E[ +1X n0=n exp( r(n0 n)h)( (1 exp ( rh)) 1 exp ( rh)(1 h) vn0 + h v 1 exp( rh)(1 h) pn0)jI i n 1]: Proof of Lemma 3.1: At period n, under the market maker?s belief, we have the joint distribution of (vn 1; xn + zn + un)0 conditional on her information set Imn 1 0 B B @ vn 1 xn + zn + un 1 C C A N( 0 B B @ v^n 1 0 1 C C A ; (B.7) 0 B B @ v v + ( + ) v v + ( + ) v 2 v + ( + )2 v + 2uh 1 C C A): Then applying the projection theorem on equation (2.1), we have 1 h = v + ( + ) v 2 v + ( + )2 v + 2uh : (B.8) 116 Applying the projection theorem to nd the conditional variance, we have var( n) = v ( v + ( + ) v)2 2 v + ( + )2 v + 2uh : (B.9) From equations (2.1), market maker?s posterior belief about vn is vn = (1 h)v^n 1 + h v + (xn + zn + un) + (1 h) n + v;n: (B.10) The stationary condition requires that v = (1 h) 2var( n) + 2 vh; (B.11) which leads to Equation (2.18). Proof of Lemma 2.3.2: Under the less informed trader?s belief, we have the joint distribution of (vn 1; xn + un)0 conditional on her information set I ln 0 B B @ vn 1 xn + un 1 C C A N( 0 B B @ v n 1 (v n 1 v^n 1) 1 C C A ; 0 B B @ v v v 2 v + 2uh 1 C C A): (B.12) Applying the projection theorem on equation (2.20), we have x 1 h = v 2 v + 2uh (B.13) 117 and var("n) = v 2 2v 2 v + 2uh : (B.14) Since the less informed insider observes a signal in the form sn = v;n+ p 1 2en, then under the less informed insider?s belief, v;n = sn + s;t (B.15) and var( s;t) = (1 2) 2vh: (B.16) Thus, the less less informed trader?s posterior belief about vn is vn = (1 h)v n 1 + h v + (xn + un (v n 1 v^n 1)) + (1 h)"n + sn + s;n: (B.17) The steady state condition requires that v = (1 h) 2( v 2 2v 2 v + 2uh ) + (1 2) 2vh: (B.18) Proof of Lemma 2.3.3: Using the results in section 2.3.2 and 2.3.3, we rst compute 118 the less informed trader?s estimation error of vn at period n vn v n = (1 h)vn 1 + h v + v;n [(1 h)v n 1 + h v (B.19) + (xn + un (v n 1 v^n 1)) + sn] = [1 h](vn 1 v n 1) (xn + un) + (v n 1 v^n 1) + v;n sn; then compute the market maker?s estimation error relative to the less informed trader?s, v n v^n = (1 h)v n 1 + h v + (xn + un (v n 1 v^n 1)) + sn (B.20) [(1 h)v^n 1 + h v + (xn + zn + un)] = (1 h )(v n 1 v^n 1) + ( )(xn + un) zn + sn; and nally the market maker?s estimation error vn v^n = (1 h)(vn 1 v^n 1) (xn + zn + un) + v;n: (B.21) Substituting for the above three equations into the Bellman equation (2.26), the more 119 informed trader solves the following problem: V (vn; v n; v^n) = maxxnfxn[(1 h)(vn 1 v^n 1) (xn + zn)] (B.22) + e rh[A((1 h)(vn 1 v n 1) xn + x (v n 1 v^n 1)) 2 + A( 2 2uh+ (1 2) 2vh) + B((1 h )(v n 1 v^n 1) + ( )xn zn) 2 + B(( )2 2uh+ 2 2vh) + C((1 h)(vn 1 v n 1) xn + (v n 1 v^n 1)) ((1 h )(v n 1 v^n 1) + ( )xn zn) C ( ) 2uh+ E]g: After simpli cation, the rst order condition of the better informed trader?s value function with respect to xn is 2 xn = [1 h 2e rhA (1 h) (B.23) + e rhC( )(1 h)](vn 1 v n 1) + [1 h 2e rhA + 2e rhB( )(1 h ) + e rhC( ) e rhC (1 h )](v n 1 v^n 1) 120 and the second order condition is = e rh(A 2 +B( )2 C ( )) > 0: (B.24) Rewriting the rst order condition leads to the expression xn = (vn v n 1) + (v n 1 v^n 1) with the and de ned in equations (2.27) and (2.29). Substituting xn back into the Bellman equation, and by matching the coe cient of (vn v n 1) term, the (vn v n 1)(v n 1 v^n 1), the (v n 1 v^n 1) 2 term and the constant term, we are able to setup the nonlinear equations (2.31 to 2.34) that determine A, B, C and E. Proof of Lemma 2.3.4: We rst substitute for expressions of v n v^n and vn v^n found in previous proof into the Bellman equation (2.35). The less informed trader then solves the following optimization problem: V2(v n 1; v^n 1) = maxznfzn[(1 h )(v n 1 v^n 1) zn] (B.25) + e rhF [(1 h )(v n 1 v^n 1) zn] 2 + e rhF [( )2 2 g + ( x ) 2 2uh + 2 2gh] + e rhGg: After simpli cation, the rst order condition of the less informed trader?s value function with respect to zn is zn = (1 2e rh F )(1 h ) 2 (1 e rh F ) (v n 1 v^n 1) (B.26) 121 and the second order condition is (1 e rh F ) > 0: (B.27) Since the optimal strategy is assumed to be zn = (v n v^n) which is proportional to v n v^n, we have = (1 2e rh F )(1 h ) 2 (1 e rh F ) : (B.28) Substituting for zn = (v n v^n) back into the objective function and matching the coe cients of the quadratic term (v n 1 v^n 1) 2 and the constant term, we are able to setup the nonlinear equations (2.37) and (2.39) which determine F and G. Proof of Proposition 2.3.2: To study the asymptotic properties of the equilibrium for small h, we make the following assumptions: ? 0 a p h; (B.29) x ? 0 a p h; (B.30) v ? Oh; (B.31) v ? L p h; (B.32) v ? L p h+Oh; (B.33) ? b p h; (B.34) 122 ? 0 (B.35) ? 0 (B.36) Substituting the asymptotic forms of , , v v, v, and into equation 2.17, we have = ( 0 + 0)O + bL ( 0 + 0)2O + 2u ; (B.37) into Equation 2.23, we have = bL 2u ; (B.38) into Equation 2.36, we have 0 = (1 2 F0)(1 0) 2 (1 F ) ; (B.39) into Equation 2.37, we have F = 0(1 ( 0 + 0)) + F0(1 ( 0 + 0)) 2; (B.40) B0 = 0(1 ( 0 + 0)) +B0(1 ( + )) 2; (B.41) and into Equation 2.29, we have 0 = 1 0 2A0 0 + 2B( )(1 0 ) + C( ) 0 C (1 0 0) 2 0 ; (B.42) where 0 = + A0 2 B0( )2. 123 Substituting the asymptotic forms into Equation 2.27, the constant term should equal to zero which leads to 1 2A0 0 C0( 0 0) = 0; (B.43) and the p h term should equal to b p h, which leads to b A0a 0 0 + A0 20 B0( 0 0)2 = 0: (B.44) We then substitute the asymptotic expressions of , , x into equation 2.31, match the coe cient of the p h term and reach the following equation: b2 (r + 2 )A0 + 2A0ab+ A 2b2 b2C0( ) = 0: (B.45) 124 Bibliography Back, Kerry, \Insider Trading in Continuous Time," The Review of Financial Studies, 1992, 5 (3), pp. 387{409. and Dirk Paulsen, \Open-Loop Equilibria and Perfect Competition in Option Exercise Games," Review of Financial Studies, 2009, 22 (11), 4531{4552. , C. 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