Quantitative trading - page 29
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Lecture 3, part 1: Information and Prices (Financial Markets Microstructure)
Lecture 3, part 1: Information and Prices (Financial Markets Microstructure)
The professor outlines three broad parts of the class. The first part focuses on setting up mathematical models to represent relevant aspects of financial markets, which will be applied to specific issues. This part aims to provide a theoretical foundation for understanding market dynamics. The second part applies these models to investigate specific issues, such as the costs and benefits of fragmentation in financial markets and the interconnections between liquidity and corporate policy. By applying the models to real-world scenarios, students can gain practical insights into the functioning of financial markets. Finally, the third part of the course covers specific topics that are highly relevant to modern financial markets but not covered in the textbook. These topics include digital markets, algorithmic and high-frequency trading, public information, and issues dealing with bubbles and herding in financial markets. This part of the course aims to explore cutting-edge developments and challenges in the field.
The professor then begins discussing the relationship between information and prices. They introduce the first model that explores these issues, starting with the bid-ask spread that arises as a result of conscious decisions made by traders. The lecture delves into the reasons why traders buy or sell assets in the market and how these reasons may or may not affect the valuation of the asset.
The first reason discussed is to form a risk profile unique to the individual trader. Traders may buy or sell assets based on their personal risk preferences, which may not directly impact the valuation of the asset. The second reason is for funding liquidity, where individuals need access to liquid funds or want to invest excess funds. This type of trading also does not affect the profitability or future cash flows of the asset and therefore does not change its valuation. Finally, the lecturer highlights that trading based on the fundamentals of the market, such as the state of the economy and external factors, does affect the profitability and future cash flows of the asset and thus changes its valuation.
The lecture then focuses on the different types of information, distinguishing between public and private information, and their impact on asset valuation. Public information is information that is available to all market participants and can be understood and evaluated in the same way by everyone. It can lead to a reevaluation of an asset's value without generating new disagreements among traders. On the other hand, private information creates an information asymmetry among traders, as individuals may have access to specific information that others do not possess. This private information can lead to differences in traders' perceptions of an asset's value based on their knowledge of specific factors.
The speaker discusses how public information, within the confines of the classical economic paradigm, cannot generate new trade but can lead to the reevaluation of asset values by incorporating widely available information. However, the speaker emphasizes that models with asymmetric information are meant to demonstrate why insider trading is illegal and show what would happen if it did take place in markets.
The academic interpretation of private information is explained, stating that all information, whether public or private, is in principle available to everyone. However, some traders may be better at analyzing the information and have superior knowledge of the stocks' fundamentals. The lecture explores the connection between information and prices, highlighting how prices coordinate information from different individuals to enable efficient allocation in society. The lecturer also touches upon the different forms of market efficiency, including weak efficiency, which implies that prices incorporate historical information.
The lecture moves on to discuss different levels of market efficiency and how the efficient markets hypothesis implies that prices must be efficient in some ways. The strongest form of efficiency, known as the "strong form," requires prices to reflect all public and private information immediately. However, this notion creates challenges, such as the "no trade theorem," as traders with private information would refrain from trading since any desire to trade would become too informative and act as a public signal, rendering any potential trade unprofitable. Furthermore, if prices were perfectly efficient in the strong form, there would be no incentives for traders to acquire information, as any piece of knowledge obtained would immediately be incorporated into prices, making the information not worth acquiring.
The lecture discusses the paradoxes in the Efficient Market Hypothesis, highlighting the lack of explanation for price volatility. The hypothesis does not provide a clear understanding of how information is incorporated into prices, and it fails to explain three important phenomena: the equity premium, the volatility of prices, and the size of market movements. To address these issues, the video explains the concept of the market value of an asset and introduces two approaches to examining it, including the discounted cash flows approach, which considers the expectation of future cash flows.
The speaker introduces the concept of market valuation, which can be calculated by taking the expectation of an asset's future cash flows and discounting them by a discount factor that accounts for uncertainty. While there is an objective fundamental value for an asset, public information may not fully incorporate all private information, and traders may not have complete knowledge of the true fundamental value. The efficiency of asset prices in terms of reflecting market valuation is determined by the extent to which the price aligns with the market valuation. If the price equals the market valuation, then the price is considered semi-strong efficient. The lecture concludes by defining innovation in market valuation as a random variable from the perspective of an investor.
The presenter proceeds to describe how to calculate the expectation of the innovation in market valuation. The formula involves taking the expectation of the innovation in the next period, which can be expressed as the expected value of the expected value of the market valuation in the next period. By applying the law of iterated expectations, the nested expectation can be simplified, resulting in the expectation of the innovation in the next period being equal to the current expectation of the market valuation minus the expected value of the market valuation given the current information. The presenter notes that if prices are efficient, the expectation of a change in prices is zero, indicating that the best estimate of the future price of the asset is the current price.
The lecturer explains that in the semi-strong form of efficiency, the stock price is considered a martingale from the viewpoint of market participants. This means that although the price can go up or down, the best estimate of the future price is simply the current price. This implication of informational efficiency is a popular assumption made in asset pricing literature. The next section of the course will introduce a particular model of trading under asymmetric information, building upon these foundational concepts.
In the next section of the course, the presenter delves into a specific model of trading under asymmetric information. This model aims to capture the dynamics and outcomes that arise when traders possess varying levels of information about the market and its assets.
The lecturer begins by highlighting the importance of understanding how information asymmetry affects trading behavior and market outcomes. In real-world financial markets, it is common for some traders to have access to private information that others do not possess. This imbalance in information can significantly influence the decisions and strategies adopted by market participants.
The presenter introduces the concept of adverse selection, which occurs when traders with private information selectively participate in the market, leading to adverse effects on market efficiency. Traders with superior information may choose to trade only when they believe they have an advantage, while traders without such information may become hesitant to engage in transactions due to the increased risk of adverse outcomes.
To illustrate the impact of adverse selection, the lecturer provides an example of a market with two types of assets: high-quality assets and low-quality assets. Traders possess private information about the quality of the assets, but this information is not observable to other market participants. Consequently, traders with knowledge of high-quality assets are more likely to engage in trading, while those without such information may opt out or demand higher prices for their assets. This leads to a situation where the market becomes dominated by low-quality assets, as informed traders are unwilling to trade high-quality assets due to the adverse selection problem.
The lecturer proceeds to discuss the consequences of adverse selection on bid-ask spreads. As traders with private information selectively trade, they create a wider spread between the bid and ask prices. The bid-ask spread reflects the cost of trading and serves as compensation for the risk associated with adverse selection. In an asymmetric information environment, the bid-ask spread widens to account for the increased uncertainty faced by traders who lack complete information about the assets.
Furthermore, the lecturer introduces the notion of signaling in markets. Signaling refers to the strategic actions taken by traders to reveal their private information to others. By engaging in certain behaviors or transactions, traders attempt to convey information about the quality or value of their assets. This signaling process helps mitigate adverse selection and improve market efficiency by reducing information asymmetry.
The lecturer provides an example of signaling through price choice. In a market with two types of assets, a seller with a high-quality asset may set a higher price to signal its quality, while a seller with a low-quality asset may set a lower price. Observing the prices set by sellers, potential buyers can infer information about the underlying asset quality. This signaling mechanism allows for a better allocation of assets and reduces the adverse effects of information asymmetry.
To further explore the dynamics of trading under asymmetric information, the presenter introduces the concept of moral hazard. Moral hazard arises when individuals take greater risks or engage in undesirable behavior due to a lack of full accountability for the consequences. In financial markets, moral hazard can manifest when traders possess private information that influences their trading decisions.
The lecturer highlights that the presence of moral hazard can affect market outcomes and efficiency. Traders with private information may be inclined to take riskier positions or engage in activities that exploit their informational advantage, potentially leading to market distortions or inefficiencies. Understanding the implications of moral hazard is crucial for designing effective regulations and market mechanisms that promote transparency and fair trading practices.
This section of the course focuses on trading under asymmetric information, adverse selection, signaling, and moral hazard. By studying these concepts, students gain insights into the complexities of real-world financial markets, where varying levels of information among traders can significantly impact trading behavior and market outcomes.
Lecture 3, part 2: Glosten-Milgrom Model (Financial Markets Microstructure)
Lecture 3, part 2: Glosten-Milgrom Model (Financial Markets Microstructure)
In this section of the course, the lecturer emphasizes the significance of models in capturing specific issues present in real-world financial markets. The focus is on the Glosten-Milgrom model, which provides valuable insights into how informational issues impact pricing and market liquidity.
The Glosten-Milgrom model revolves around a long-lived dealer who interacts with two types of traders: privately informed speculators and uninformed noise traders. The model assumes that the speculators aim to maximize their expected profits while concealing their private information, using noise traders as a cover. On the other hand, noise traders trade with fixed probabilities due to liquidity or risk needs.
The lecturer discusses the assumptions and equilibrium concept of the Glosten-Milgrom model. It is a static game with asymmetric information, and the equilibrium concept is a base Nash equilibrium. The dealer sets bid and ask prices to maximize profits while ensuring zero profit for themselves. Speculators make decisions based on their expected gains. The lecture poses thought-provoking questions regarding the absence of uninformed speculators, the necessity of a dealer in the model, and the willingness of the dealer to trade with better-informed speculators.
Further analysis of the Glosten-Milgrom model reveals that the dealer's zero-profit prices should be equal to the expected valuations of the asset, considering all relevant trades and orders. The lecturer explains how buy and sell orders from noise traders and speculators, who possess information about the true asset value, affect the expected profit of speculators. The optimal strategy for speculators is described, considering different asset values in relation to the ask and bid prices.
The lecturer also emphasizes how orders reveal information about the asset's valuation. The probability of a buy order from a noise trader versus a speculator is explained, with speculators providing more information about the asset's true value. The conditional expectation of the asset's value is expanded using the law of total probability, enabling the calculation of ask and bid prices with greater accuracy.
Using the Glosten-Milgrom model, it is confirmed that the ask price will be higher than the bid price, leading to efficient market prices in the semi-strong form. However, this efficiency relies on dealers being competitive and obtaining no trading margin. If dealers have market power, prices will deviate from the fair market valuation of the asset. The lecture includes a simple example of the Glosten-Milgrom model with a binary fundamental asset value.
The lecture delves into deriving the ask and bid prices in the Glosten-Milgrom model's equilibrium with trade. The calculation involves explicit consideration of the asset value distribution, taking into account purchase and sale orders. The ask and bid prices are expressed as functions of the model's parameters, leading to an equilibrium in the model.
Illiquidity and the computation of the quoted spread in the market are discussed, with the spread increasing with the probability of informed trading and decreasing with the presence of noise traders. Comparative statics demonstrate how the spread is influenced by the degree of initial uncertainty about the asset value. The lecture also touches on a multi-period setting, where persistent asset value and informative trade flow contribute to price discovery over time.
The Glosten-Milgrom model is further examined, highlighting that the long-term price of an asset will converge to its true fundamental value, indicating strong-form efficiency within this model. However, the speed of price discovery depends on the proportion of informed traders, creating a trade-off between price discovery and liquidity in the market. Balancing these aspects in market design can be challenging. The lecture acknowledges limitations such as the centralized dealer model and the absence of market clearing, which may not fully capture real-world market dynamics.
Finally, the lecture concludes by discussing the final drawback of the Glosten-Milgrom model, which solely considers the fundamental value without incorporating speculation or resale. This limitation means that traders who perceive a gain in buying an asset at value v do not consider the potential illiquidity when selling the asset in the future. Nonetheless, the Glosten-Milgrom model remains a flexible and straightforward framework to understand the impact of adverse selection on prices and address specific questions in the market. The model also underscores the importance of noise trading in maintaining market liquidity. The chapter concludes with exercises on the Glosten-Milgrom model for interested readers to explore.
Lecture 4: Determinants of Liquidity (Financial Markets Microstructure)
Lecture 4: Determinants of Liquidity (Financial Markets Microstructure)
The video discusses two factors that can generate bid-ask spreads in dealer markets. The first factor considered is the impact of order processing costs on the spread. The lecturer explains that dealers in the market have various costs to manage, such as trading fees, clearing and settlement fees, office rent, and research and analysis expenses. These costs, along with any extra profits that dealers may receive, are ultimately borne by traders who trade with them, thereby affecting the spread. The lecturer emphasizes the importance of empirically disentangling these costs to understand their effects on market liquidity and price discovery.
The second factor discussed is the Glosten-Milgrom model and how a dealer's transaction cost affects bid and ask prices in the market. In this model, only speculators have information about the fundamental value of an asset, while other market participants have limited information represented by omega_t. The market values an asset at mu_t, which is the conditional expectation of the fundamental value given public information. The ask price is determined as the market valuation at the beginning of period t plus the half spread plus the dealer's transaction cost (gamma), while the bid price is calculated as the market valuation minus the half spread minus gamma. Therefore, the spread is extended by the transaction cost, and the overall spread comprises two components: transaction costs and adverse selection costs.
The video highlights the challenge of determining the portion of the quote spread that arises from order costs versus adverse selection costs when examining a single pair of quotes. However, by observing the dynamics of quotes over time, it becomes possible to infer this information. The effects of these costs on prices differ in terms of their dynamic impact. The lecturer presents an equation that demonstrates how the realized price paid by a trader is centered around the exact market valuation plus the direction of trade multiplied by the half spread, plus or minus the order costs. The dynamic impact of order costs differs from that of adverse selection costs, with the deviation of price from the ex-ante market valuation given by the realized spread plus the transaction order component. The video also discusses the calculation of expected future price.
The long-term impact of trade on prices is then examined by analyzing the adverse selection and order processing components of half spreads. The expectation of future prices is approximated as the ex-ante market valuation minus one. While the adverse selection term has a permanent effect and shifts prices in the respective direction, with future trades contributing more information and further shifting prices, the order processing costs only make the spread wider and do not have any long-term effects on shifting average prices. Future trades reverse the impact of order processing costs, while the adverse selection term remains more permanent, causing prices to shift in the respective direction permanently.
The video further explores the effect of adverse selection versus order processing costs on market valuation. Adverse selection has a permanent effect on future market valuations by shifting them based on the relevant adverse selection component. In contrast, order processing costs simply widen the spread without exerting any long-lasting impact on average prices. Moving on to Stahl's model from 1978, the video suggests that dealer inventory costs may be responsible for market illiquidity or bid-ask spreads. Dealers are required to hold inventory for some time, and if the asset price changes during that period, it can become costly for the dealers. Therefore, dealers may demand a premium for holding positive or negative positions in the asset. The model relies on the assumption of risk aversion among dealers.
The video then discusses how market makers or dealers submit their competitive demand schedules or supply schedules. These schedules provide separate prices for any given amount that a trader might want to buy or sell, essentially forming a limit order book filled exclusively by the dealer. The model focuses on deriving the supply and demand schedules, assuming no informed traders and only noise traders. Information arises in the market through public announcements, which can shift the valuation of all agents in the market, thereby introducing the risk of changing valuations. The lecturer notes that while it is natural to assume dealers are inherently risk-averse, risk aversion arises instrumentally due to regulation. If dealers are obligated by regulatory requirements to maintain positions or given margins, they cannot take large positions in any given asset, behaving as if they are risk-averse.
Next, the lecturer explains a model in which dealers in the market hold positions they acquired during a single period and can only unwind them in the next period when the fundamental value has changed. Dealers must buy or sell the asset today and then sell or buy it later, respectively. However, they do not face future illiquidity because they can unwind the position at the exact value at which they bought or sold it. Additionally, the model assumes the presence of a single competitive dealer or a very large number of dealers, which does not affect the argument. A representative dealer has an initial position called "endowment" in the asset.
The video further discusses the asset demand or supply by a dealer and how the dealer acts as a price taker in a competitive market. The dealer's decision to supply or demand is driven by the desire to maximize utility defined over their next period wealth, which is a random variable perceived by the dealer at time t. The dealer's wealth at the beginning of period t + 1 is determined by their future asset position and cash holdings, denoted as zt plus one and ct plus one, respectively. Utility is defined as a function of this wealth. The solution algorithm to this model is complex and may not be intuitive.
The assumption of competitive dealers is also discussed, which is more reasonable if there are a million dealers in the market. In this case, no single dealer can significantly impact prices, and they all act as price takers, optimizing based on the demand and supply schedules set by all other dealers in the market. The maximization problem for any given dealer involves deciding how much of the unit to supply given any fixed price. The mean-variance preferences of dealers are also considered in terms of the expectation of future wealth. The expectation of future wealth is determined by the agent's future position in the asset, while the variance of future wealth arises from the asset holding as the value of cash is a riskless asset in this model.
The video then presents the algorithm for deriving the supply schedule of a dealer given a fixed price. The dealer's utility function is used to obtain the asset supply function, which is proportional to the difference between the price and the mid-quote. The mid-quote represents the price at which the dealer is not willing to supply or buy any units of the asset and depends on the dealer's initial position in the asset. A larger initial position leads to a lower mid-quote, resulting in lower prices for buying and selling the asset.
The lecture delves into how liquidity is affected by a dealer's risk aversion and preference for mean standard deviation. The larger the dealer's risk aversion and the more volatile the asset valuation, the less willing the dealer is to accept larger positions. This results in a steeper price schedule and a larger price impact, ultimately leading to lower market depth. The speaker also explains how the dealer's preference for mean standard deviation affects their willingness to supply to the market, with the slope of the function determining the dealer's supply limit.
The lecturer describes the dealer's indifference to buying and selling an asset based on the yields. The only scenario where the dealer can achieve equilibrium is when the slope of the function is exactly zero, meaning that the dealer would be indifferent to buying or selling an asset at any price. This equilibrium point creates a single point of discontinuity determined by the dealer's positive or negative position. In contrast to mean-variance preferences, this standard deviation preference model generates bid-ask spreads as natural measures. These spreads represent the distance between the prices at which the dealer is willing to accept a positive position and the price at which the dealer is willing to accept a negative position in the asset. Additionally, the model depicts that incoming random noise trades can move the prices, causing the mid-price to deviate from the expected value.
Dealers aim to maintain a neutral position in the asset in the medium to long run. However, this is not always possible, and any non-zero inventory can cause prices to deviate from market valuation. As a trader, one can benefit from this price inefficiency when encountering a dealer with a favorable position. However, if the terms of trade are unfavorable or if there is an urgent need to buy a specific amount of the asset, the trader may face difficulties. Nevertheless, in the long run, prices will converge back to the efficient level as dealers eliminate their inventory.
The video concludes by presenting a graph that visually demonstrates the effects of adverse selection, order processing costs, and inventory costs on market valuation. The adverse selection component has a permanent effect, resulting in long-run price changes. On the other hand, the order processing cost component only affects current prices and has no impact on expected trade prices. The inventory control cost component has a medium-run effect on prices, gradually fading away. In the next lecture, the speaker plans to estimate the importance of each mechanism and their empirical contributions to the spread, providing relevant exercises on inventory risk from the reading list. From an economic perspective, order processing costs are considered trivial compared to other factors.
Lecture 5, part 1: Depth determinants, Kyle Model (Financial Markets Microstructure)
Lecture 5, part 1: Depth determinants, Kyle Model (Financial Markets Microstructure)
The lecture begins by discussing the determinants of market depth and how trade size affects market prices, building upon the previous lecture's discussion on spread determinants. The main question addressed is why traders have to pay more when trading in large amounts, with larger trades typically having a larger spread and moving the price further away from the efficient level, indicating limited market depth.
The lecture introduces the Kyle model, which is a widely used model in financial markets microstructure literature that allows for flexible trade sizes. It is mentioned that the second part of the lecture will cover the empirical estimation of factors contributing to liquidity, including estimating price impact, depth, and the proportion of informed trading.
The video explores potential factors that explain the phenomenon of paying more for larger trades, including adverse selection, inventory risk, and order processing costs. Adverse selection and inventory risk are considered valid explanations for limited market depth, as dealers are reluctant to take on large positions due to the associated risk and require larger premiums from traders. However, the lecture distinguishes between costs paid to dealers and costs paid to the exchange when discussing order processing costs.
The relationship between market power, limited market depth, and the costs arising from imperfectly competitive dealers is also discussed. Imperfectly competitive dealers may set wider spreads and extract surplus from traders, leading to larger spreads for any given trade size compared to perfectly competitive dealers. However, it is unclear whether this difference generated by imperfect competition would be larger or smaller for large trade sizes. Additionally, the lecture acknowledges that order processing costs may increase or decrease in the per-stock expression depending on whether traders pay a fixed fee or a percentage fee. The difficulty of arranging, clearing, or settling large trades may also result in higher transaction costs.
The lecture then focuses on the Kyle model, which explains the link between adverse selection and market depth. It discusses assumptions regarding traders' behavior, emphasizing that traders' actions and whether they are informed or noise traders can affect the fundamental value of the asset. The lecturer explains the key aspects of the Kyle model, particularly the role of a speculator who strategically chooses their order size to minimize price impact.
The assumptions made in the Kyle model are discussed, comparing them to Stahl's model from the previous class. In the Kyle model, market makers are assumed to be risk-neutral and competitive, making zero profits. Dealers observe only the aggregate market flow and cannot distinguish between speculative and noise orders. Orders are cleared in batches through a call auction, rather than a continuous auction.
The speculator in the Kyle model has access to the true value of the asset and strategically buys a certain number of units, aiming for a net gain. The price at which the trade will clear is not explicitly observed by the speculator when choosing the order price. The noise trader in the model has random demand, but no information is conveyed about the fundamental value.
The lecturer explains the market makers' role in the Kyle model, where they make zero profit and submit a pricing schedule. The market price is determined by the order size of traders, and the trade is executed at the market price. The model seeks equilibrium, where the speculator's strategy is based on the fundamental value of the asset. The speculator is assumed to use a linear strategy, with the coefficient beta determining their aggressiveness in reacting to market news. The market maker knows the speculator's strategy, estimates the asset's likely value from the total aggregate order flow, and extracts the relation between the order flow and asset value.
The lecture discusses the price impact coefficient, lambda, and its role in determining the market maker's strategy. The coefficient is estimated as the regression coefficient of the trade size (q) on the asset value (v), and it measures the linear relationship between trade size and price impact. The video explains the linear pricing equation and how the distance between the realized trade price and the ex-ante market valuation is linearly related to the trade size (q) with the coefficient lambda. The inverse of lambda serves as a measure of market depth, indicating how much trade can be conducted before the price changes by one dollar.
The measure of information conveyed by trade size (q) in the Kyle model is also discussed. The expectation of q represents the neutral information, while q minus the expectation of q indicates the information conveyed by trade size. This measure can be used to estimate the size of the fundamental value (v), with a larger trade size (q) suggesting a larger value (v). The speaker presents an alternative interpretation of the coefficient by normalizing all variables by their standard deviations, making it easier to understand the correlation between the fundamental value (v) and trade size (q). The process of obtaining the results from the slides using conditional probability density functions (PDFs) and applying Bayes' rule is briefly outlined.
The lecture covers the derivation of the probability density function (PDF) in the Kyle model using Bayes' rule. The PDF is obtained by multiplying the probability of a given value of the fundamental value (v) and the probability of a given value of the trade size (q) given v, divided by the probability of q. It is explained that all three PDFs (fundamental value, trade size given fundamental value, and trade size) are assumed to be normal distributions, and the lecture demonstrates how to express them in terms of mean and variance.
Finding the optimal speculator strategy in the Kyle model is discussed. The linear pricing equation is given to the speculator, and it is plugged into the expression for the speculator's profit, resulting in a quadratic expression in the number of units traded (x) with a unique maximum. The lecture concludes that the optimal trading strategy for the speculator is given by beta times the difference between the fundamental value (v) and the estimated value (mu), where beta is determined by 1 over 2 times lambda. It is noted that the model focuses on linear strategies, and other equilibria with non-linear pricing rules and trading strategies are not explored due to the computational complexity involved.
The lecture discusses the trading strategy of speculators in the Kyle model, highlighting that speculators expect to make a positive average profit. Competitive and risk-neutral intermediaries in the market make zero profits, while noise traders generate negative profits on average. However, the lecture mentions that the negative profit for noise traders may be offset by gains in risk portfolios or liquidity needs, which are not explicitly modeled. The model is considered complete after deriving the dealer's optimal pricing strategy given the speculators' trading strategy and solving the speculators' trading problem given the dealer's pricing strategy. The aggressiveness parameter (beta) and the price impact coefficient (lambda) are expressed in terms of known model parameters, with beta being higher when the fundamental value of the asset is less volatile.
The video delves into factors that affect a speculator's profit per unit traded and how this can lead to the need for more trading to compensate for low profits. When the profit per unit is not substantial, the marginal expense and loss of increasing trade size and trading at more adverse prices are relatively lower. Market depth, determined by 2 times the standard deviation of noise trader demand (sigma u) divided by the standard deviation of the fundamental value (sigma v), increases with less insider trading and more noise trading. The insider's profit improves as the number of noise traders in the market increases and as the asset's value becomes more volatile. The lecture also covers the computation of the residual variance and its dependence on trade size (q).
The lecture addresses the concept of residual variance, which represents the uncertainty remaining in the market regarding the fundamental value after all the information contained in the trade size has been considered. The conditional variance quantifies the amount of unknown information that remains after trading, which is expected to be lower than the initial uncertainty because trade sizes convey information about the fundamental value. In the Kyle model, the insider speculator reveals half of their information, and overall market depth is limited due to insider trading. The model is described as more comprehensive than the Bloom-Milgram model, as it allows for exploration of the effects of trade size. Unlike the Bloom-Milgram model, the speculator in the Kyle model is not a price taker.
The lecturer highlights the significance of the Kyle model in the context of batch auction markets with a dealer intermediary. This model enables traders to choose their desired position on a predetermined price schedule, which influences the price they will receive. The lecture mentions potential extensions of the model to incorporate dynamics, multiple trading rounds, infinite auctions, and additional insiders. These extensions introduce more competition, increased aggressiveness, and greater liquidity, leading to both price discovery and higher market liquidity.
The focus of the lecture turns to the extensions of the Kyle model, considering dynamic models with multiple insiders and the role of dealers as intermediaries. It is acknowledged that achieving both liquidity and price discovery can be challenging due to the fact that speculators prefer market power and seek to avoid competition. The lecture explores the consequences of different assumptions, such as imperfect competition among dealers, which can result in less liquid markets and pricing inefficiencies, as well as the impact of risk-averse market makers, introducing inventory concerns into the model. Ultimately, the Kyle model is presented as a more advanced and effective theoretical basis for market analysis compared to the Stahl model.
The lecturer concludes the discussion by emphasizing that the Kyle model provides a robust framework for understanding market dynamics and analyzing various factors that influence market liquidity and pricing. The model's ability to incorporate trade size and the behavior of different market participants, such as speculators, intermediaries, and noise traders, offers valuable insights into real-world market scenarios.
Moreover, the lecture highlights that the Kyle model can be further extended to address more complex dynamics and considerations. For instance, incorporating multiple insiders can capture the presence of various informed traders competing in the market. Additionally, introducing dynamics into the model can account for time-varying factors and the evolution of market conditions over multiple trading rounds. These extensions enhance the model's realism and provide a more comprehensive understanding of market outcomes.
The lecturer also acknowledges that achieving a balance between liquidity provision and price discovery is a challenging task. Speculators, who play a crucial role in liquidity provision, tend to prefer market power and may avoid competition. This inherent tension between liquidity and price discovery necessitates further exploration and analysis to identify optimal market structures and mechanisms that foster both objectives.
In conclusion, the Kyle model serves as a valuable tool for studying market microstructure and understanding the determinants of market depth, pricing, and liquidity. Its ability to capture the impact of trade size, the behavior of market participants, and the role of intermediaries contributes to a comprehensive analysis of financial markets. With the potential for extensions and refinements, the Kyle model continues to be a significant theoretical framework for market analysis and an avenue for future research in the field of financial economics.
Lecture 5, part 2: Empirics of Illiquidity (Financial Markets Microstructure)
Lecture 5, part 2: Empirics of Illiquidity (Financial Markets Microstructure)
During the lecture, the professor delves into the estimation of the empirics of illiquidity by untangling the contributions of various theoretical factors to market liquidity. This discussion builds upon previous talks on empirical measures of liquidity and the theories that elucidate how liquidity is influenced by factors like adverse selection, order costs, and inventory risk.
To facilitate the analysis, the professor introduces notation for three key factors: lambda, which represents the adverse selection component of the price impact coefficient; beta, which pertains to the price impact coefficient associated with inventory risk concerns; and gamma, which serves as a catch-all component for liquidity. The data utilized in the estimation process comprises transaction prices, net market order flow, and order sign. The objective is to assess the effect of a specific transaction's order size and sign on the overall market.
Next, the lecturer explores the estimation of the effect of a given transaction on future transaction prices in financial markets. They employ the Gloston Milgram model, which incorporates order processing costs. According to this model, the transaction price consists of the market valuation and the order processing cost component, which depends on the direction of trade. By taking the first difference, the change in market valuation can be obtained, reflecting the adverse selection component that incorporates the information from the previous transaction. The lecturer demonstrates how this information can be utilized to estimate future transaction prices, resulting in an expression that contains no unobservable terms except for the noise term.
Moving forward, the lecturer explains the process of estimating the adverse selection component and the order processing cost component separately. The estimation is carried out in two stages. The first stage involves finding that inventory or order processing costs are independent of the quantity traded. In the second stage, a regression is performed solely considering the direction of trade and the trade volume. The lecturer mentions a specific research paper that employs transaction data from the New York Stock Exchange in the early 1980s, cautioning that the number of observations in that paper is limited.
Furthermore, the limitations of a two-stage estimation procedure used in a particular paper are discussed. This paper solely estimated two factors contributing to illiquidity while neglecting inventory costs. The lecturer points out that disentangling inventory costs from the adverse selection components poses challenges, making it impossible to identify the coefficients separately. The lecturer then highlights that order flow exhibits autocorrelation, and the introduction of split orders adds a positive degree of autocorrelation to an otherwise uncorrelated order flow. Subsequent researchers attempted to estimate all three components of the spread using this specification, resulting in an autoregressive process of order one, which alters the expression in the estimated equation.
The speaker proceeds to discuss a study in which the authors estimated an equation for 20 major stocks at the New York Stock Exchange. The study revealed that the order flow's autocorrelation is, in fact, negative rather than positive. They attributed this finding to dealers' inclination to immediately unwind any inventory they may have accumulated, rather than order splitting. The study further determined that order costs account for over 60% of the spread, emphasizing the significance of order costs in driving illiquidity. Additionally, one-third of the spread is driven by dealers' inventory concerns, while 10% is attributed to the adverse selection component. The study also found that the adverse selection component is strongest in the morning.
The lecture then explores how adverse selection and inventory concerns are balanced during the opening and closing hours of trading. In the morning, the market incorporates all the accumulated information from off-market hours, including news and information generated by markets worldwide. This leads to a need for the market to rapidly assimilate a significant amount of information, affecting prices. In the evening, traders aim to unwind their positions before the end of the trading day, which significantly impacts closing prices. However, this inefficiency is swiftly resolved in the morning through after-hours trading that reverts prices back to the market valuation established before the closing auction.
Moreover, the lecturer discusses two papers that estimate the impact of trades on prices and the extent of adverse selection in financial markets. The first paper focuses on estimating the impulse response of prices to trades and finds a significant short-term effect, but a smaller long-term impact due to order processing costs. The study confirms the hypothesis of adverse selection, as the impact is more pronounced for less liquid stocks. The second paper utilizes a model to estimate the probability of informed trading, assuming an arrival process of traders that includes both informed and uninformed traders. The model identifies significant adverse selection in financial markets.
The speaker then delves into the estimation of the probability of informed trading (PIN) using data from the New York Stock Exchange spanning from 1983 to 1998. PIN represents the probability that any given trade originates from an informed trader. The estimation reveals that the median probability of informed trading across assets and stocks is approximately 19%, with 90% of stocks having a probability of informed trading between 10% and 30%. However, for the remaining 10% of stocks, particularly those with small capitalization and infrequent trading, the probability of informed trading can be much higher. Additionally, this probability is positively correlated with the spread and price volatility.
Furthermore, the speakers discuss how the probability of informed trading tends to be higher in more anonymous markets due to the difficulty of gaining a bad reputation as an informed trader. In such markets, informed traders are more inclined to engage in trading, which leads to increased illiquidity. The section concludes with a summary of the topics covered in the lecture, emphasizing the importance of order costs in determining liquidity costs. However, the authors remind viewers that order costs encompass various factors and that different components of the spread may have distinct explanations.
Lastly, the presenter mentions a blog post that recounts an incident in which the price of oil plummeted to negative values in the spring of 2020 due to constraints on physical inventories of oil storage. Additionally, the presenter recommends referring to Chapter 4 of a textbook, which provides variations of the Kyle model and exercises for further practice. Finally, a preview is given for the upcoming week's focus, which centers on the differences between dealer markets and limit order markets, as well as how traders and regulators can utilize heterogeneity to achieve their desired outcomes.
Lecture 6: Limit Order Book Markets (Financial Markets Microstructure)
Lecture 6: Limit Order Book Markets (Financial Markets Microstructure)
The lecture focuses on financial markets microstructure and delves into the distinction between dealer markets and order-driven markets. In dealer markets, an intermediary acts as a middleman and provides price quotes, essentially serving as a representative for all transactions. On the other hand, order-driven markets operate differently, where all participants submit their orders to a limit order book. Market orders can then directly access liquidity from this book without the need for a dedicated dealer. This technological shift has led to the emergence of online or electronic trading, where orders can be matched and routed automatically using electronic systems.
The video elaborates on the differences between dealer markets and order-driven markets. For market traders who submit market orders, the distinction between the two types of markets may not be significant. However, traders who choose to submit limit orders take on a role similar to dealers. By providing market liquidity, these limit order traders face non-execution risk and delay risk, which are not encountered by dealers. In order-driven markets, traders have the choice between market orders and limit orders, a choice that is not present in dealer markets. Despite the additional risks involved, limit orders are often preferred as they offer traders a better price for their transactions, even though they must accept the potential risks of non-execution and delay.
The video goes on to explain the concept and choice between market orders and limit orders in an order-driven market. Market orders are executed at the ask price, while limit orders are executed at the bid price. The lecturer emphasizes the self-balancing nature of markets and how the choice between market and limit orders can impact the depth and liquidity of the limit order book. The video discusses how the cost of submitting a limit order varies depending on market saturation. In a saturated market, the cost is higher, but the benefit becomes more significant when the market is thin. The lecture introduces a model by Glosten from 1994, which explores how prices are determined in an order-driven market and how limit traders establish their prices, ensuring the efficiency of prices. Additionally, the video touches on the determination of the depth of the limit order book and how traders make decisions about taking or providing liquidity.
Moving forward, the video delves into the composition of a limit order book with competitive traders for a single side of the market, specifically focusing on limit orders to sell and market orders to buy. Once the limit order book is constructed, a price schedule is created, and market traders face this schedule. Prices are adjusted based on volume, and the marginal price for buying a certain quantity of assets is defined as the price at which the last trade occurred. The total amount paid to purchase a volume q is obtained by integrating the marginal price over all trades taken, and the derivative of this total payment with respect to q yields the marginal price p prime of q.
A model of a market trader's decision-making process in a limit order book market is presented in the video. The model assumes the presence of one market trader per period, denoted as "i." The market trader determines the size of their buy order, denoted as "q," by equating their marginal valuation for an additional unit of the asset with the marginal price for that unit. The marginal valuation represents the trader's rate of substitution between money and assets, referred to as "theta i of q." The video explains that larger trades require more capital, resulting in additional costs to buy more assets. As a result, the effective willingness to pay for further units decreases.
The speaker discusses how the state of q relates to the fundamental value of the asset in financial markets microstructure. Although not explicitly outlined, the model assumes that a higher valuation for a trade of a given size suggests a higher asset value. The video explains how limit traders competitively post limit orders but are only executed if a market trader places an order equal to or greater in size. However, since limit traders do not have access to all the information, they know that if their order is executed, it is at least their specified size but may not be larger.
The video delves into the pricing of limit orders, explaining that the price set by a limit trader for the qth unit of an asset in a limit order book market is the conditional expectation of the asset's fundamental value, given that the order size is at least q. This creates an inside spread between the bid and ask prices as the order size approaches zero, resulting in a discontinuity due to the conditioning. The video highlights that limit traders always profit from the sale of the last units, as their prices yield zero average profit between different cases of large order sizes and optimistic news about the market trader's fundamental value. However, the marginal price for the first asset may or may not be lower than this.
The lecturer then discusses the conditions that the best ask and bid quotes in a limit order book market must satisfy. These quotes are conditioned on the willingness of market traders to buy and sell, but they cannot be conditioned on the specific amounts of buying or selling. This condition creates the inside spread, which represents the difference between the best ask and best bid prices. The lecturer also explains how the discreteness of prices comes into play, as prices often adhere to a tick size that limits the amount of undercutting between competitors.
To further illustrate the concepts, the lecturer introduces a model that is similar to the previous one but incorporates a discrete price grid with a constant tick size. The model assumes that limit orders are prioritized based on both time and price, with orders posted first being executed first, and limit orders with lower prices being executed ahead of those with higher prices. The lecturer introduces notation to represent the amount supplied at a certain price and the total amount that a market trader can obtain at prices no higher than a particular price. The model also takes into account that larger orders generally indicate higher valuations, which influences the calculation of expectations. The lecture concludes by explaining how competition functions within the model.
The lecture then explains the process of limit orders being supplied at each tick in a queue-like system. The marginal trader who submits the last order at each price level earns zero profit, while the next trader attempting to submit an order at the same price level no longer finds it profitable and moves on to the next tick. This process can be illustrated using a graph where the supply curve resembles a step function. The first trader typically earns an expected positive profit, while the last trader generally receives zero expected profit.
The lecturer proceeds to discuss the zero profit condition for marginal orders in a limit order book market. Infra-marginal orders can earn positive profit based on the expected profit of the limit trader multiplied by the probability of order execution, which is equal to the display cost "c." The expression connecting the price level "ak" and the cumulative depth "yk" consists of two components: the adverse selection term to the price and the execution risk component. The lecturer incorporates the display cost "c" into the graph and provides examples of binary and continuous random variables to model traders in the market.
The concept of equilibrium in financial market microstructure is then introduced. The model with continuous prices and a discrete equilibrium is utilized, where two prices, "a1" and "a2," are determined such that the depth at "a1" is equal to "qs" and the cumulative depth at "a2" is equal to "ql." The assumption is made that a noise trader employs one of four trades with equal probabilities: a small buy, a large buy, a small sell, or a large sell. The speculator also only trades in one of two units, either "qs" or "ql." Finally, the prices of "a1" and "a2" must satisfy two equations to determine the expected fundamental value for each order size on a particular side of the trade.
The speaker explains the concept of equilibrium in a simple limit order book market model. The equilibrium encompasses strategies from both groups of active players in the market: limit traders and market traders. Limit traders set prices, such as "a1" and "a2," based on the zero-profit condition, while market traders decide which orders to submit and trade optimally according to conditioned probabilities of being either informed or uninformed. The derived expressions demonstrate that the conditions for equilibrium are met, establishing this model as an equilibrium.
The video proceeds to discuss examples of limit order book markets. In one example, discrete price levels arise due to the discreteness of noise traders' strategies, resulting in only two possible order levels and effectively two possible groups of events. The price impact is positive due to the strategy of uninformed traders. Another example is introduced where market order sizes follow an exponential distribution. The price impact equation assumes lambda times x for some constant price impact factor lambda, which measures the informativeness of order flow. Although this example is significant, the video focuses primarily on how the limit order book is formed based on the behavior of market traders and presents a reduced-form analysis.
The speaker explains how to derive a conditional expectation using the conditional probability density function (pdf) of "q," which is based on the cumulative depth "yk" for limit traders. By applying Bayes' rule, the speaker demonstrates a straightforward method for calculating the expected fundamental value "v," which represents the price that the limit trader would set for the "ykth" unit of the asset. The final expression for the conditional pdf of trade sizes "q" incorporates the exponential distribution, and integration by parts is used to derive the linear price impact equation. The inclusion of the factor "1/theta" yields the inside spread of the equation.
The lecture concludes by summarizing the connection between the equation relating the tick "ak" and the cumulative depth at the tick "yk" in a market with predetermined ticks, taking into account display cost model parameters. The lecture emphasizes the impact of display cost and the need to invert the expression in a market with predetermined ticks. The lecturer indicates that the role of liquidity provision differs for limit traders and dealers due to their distinct informational environments, resulting in different market outcomes. The next lecture will explore how tick size and priority rules affect market outcomes through dynamic analysis that considers a parlor model where traders have the choice between limit orders and market orders. The instructor provides practice questions from the textbook for students to further reinforce their understanding.
The lecturer begins by introducing the concept of tick size, which refers to the minimum price increment at which securities can be quoted or traded. Tick size plays a crucial role in market microstructure as it affects the granularity of price levels and the potential profitability of limit traders. A smaller tick size allows for more price levels and finer price differentiation, which can lead to increased competition and tighter spreads in the market. On the other hand, a larger tick size may result in fewer price levels and wider spreads.
Next, the lecture explores the impact of tick size on the equilibrium outcome in a limit order book market. The model assumes that traders have the choice between submitting limit orders or market orders. Limit orders have priority over market orders, meaning they are executed first at a given price level. The lecturer explains that the tick size affects the number of limit orders that can be submitted and executed at each price level.
The speaker presents a parlor model to analyze the dynamic interaction between limit orders and market orders. In this model, traders alternate between submitting limit orders and market orders, based on the outcome of the previous round. The lecture focuses on the case where the tick size is small relative to the standard deviation of fundamental value changes. In this scenario, the equilibrium outcome is characterized by a stable price, where limit orders dominate market orders due to their priority.
The lecturer explains that the stability of the equilibrium price arises from a feedback mechanism. When a trader observes that the limit order book is thin, indicating low liquidity, they are more likely to submit a limit order. This increase in limit orders boosts liquidity in the market, attracting more market orders and reinforcing the dominance of limit orders. Conversely, when the limit order book is thick, indicating high liquidity, traders are more inclined to submit market orders, reducing the dominance of limit orders.
The lecture emphasizes that tick size plays a crucial role in this feedback mechanism. With a smaller tick size, there are more price levels, allowing for finer differentiation and a more effective feedback process. This leads to a more stable equilibrium price and tighter spreads. In contrast, a larger tick size limits the number of price levels, reducing the effectiveness of the feedback mechanism and potentially leading to a less stable equilibrium with wider spreads.
The lecturer also discusses the impact of priority rules on market outcomes. Priority rules determine the order in which orders are executed at a given price level. The lecture introduces two priority rules: price-time priority and pro-rata priority. Under price-time priority, the earliest submitted order at a given price level is executed first. Under pro-rata priority, orders at a given price level are executed proportionally based on their size.
The speaker explains that priority rules can affect market outcomes by influencing the behavior of traders. Price-time priority encourages traders to submit orders early to gain priority, which can lead to a higher level of displayed liquidity in the market. Pro-rata priority, on the other hand, incentivizes traders to submit larger orders to receive a larger share of the executed volume.
The lecture concludes by highlighting the interplay between tick size and priority rules in determining market outcomes. The choice of tick size affects the number of price levels and the effectiveness of the feedback mechanism, while priority rules influence the behavior of traders and the distribution of executed volume. Both factors play a significant role in shaping market dynamics and liquidity provision in a limit order book market.
Students are encouraged to further explore these topics through readings and practice exercises to deepen their understanding of market microstructure and its implications for trading strategies and market outcomes.
Exercise class 3, part 1 (Financial Markets Microstructure)
Exercise class 3, part 1 (Financial Markets Microstructure)
In the lecture covering financial markets microstructure, the speaker provides a detailed explanation of the differences between dealer markets and order-driven markets. In dealer markets, there is an intermediary who quotes prices and handles all transactions on behalf of market participants. On the other hand, in order-driven markets, participants submit their orders to a limit order book, and market orders are executed by taking liquidity from the book without the involvement of a dedicated dealer. The advent of electronic trading technology enabled the development of order-driven markets, where trading occurs online and orders are matched and routed automatically.
The video emphasizes that for market traders who submit market orders, the choice between dealer and order-driven markets doesn't make a significant difference. However, for traders who submit limit orders, they assume the role of dealers by providing market liquidity. These traders face non-execution risk and delay risk, which are not faced by dealers in dealer markets. In order-driven markets, traders have the choice between market orders and limit orders. Despite the risks involved, limit orders are preferred because they offer traders a better price on their transactions.
The concept of market orders and limit orders in an order-driven market is explained in the video. Market orders are executed at the ask price when buying or the bid price when selling. Limit orders, on the other hand, allow traders to specify the price at which they are willing to buy or sell the asset. The video highlights the self-balancing nature of markets and how the choice between market and limit orders affects the depth and liquidity of the limit order book. It also discusses how the cost and benefit of submitting a limit order vary depending on the market's saturation. In thin markets, the benefit of a better price outweighs the non-execution and delay risks.
The lecturer introduces a model proposed by Glosten in 1994, which examines how prices are determined in an order-driven market and how limit traders set their prices to ensure market efficiency. The video also touches upon how the depth of the limit order book is determined and how traders make choices between taking and making liquidity.
The composition of a limit order book with competitive traders for a single side of the market (e.g., sell orders) is explained. Once the book is composed, a price schedule is created, and market traders face this schedule. Prices are adjusted based on volume, and the marginal price for a desired quantity of assets to buy is determined. The total amount paid to buy a particular volume is calculated using the integral of the marginal price over all trades. The first derivative of this total amount paid represents the marginal price for the desired quantity.
The video presents a model of a market trader's decision-making process in a limit order book market. Each period, labeled as "i," is associated with a market trader who determines the buy order size "q" by equating their marginal valuation for the next unit of the asset with the marginal price for an additional unit. The marginal valuation represents the trader's marginal rate of substitution between money and assets. Larger trades require more capital, leading to decreasing effective willingness to pay for further units.
The speaker discusses the relationship between the state of quantity "q" and the fundamental value of the asset in financial markets microstructure. Although the video doesn't explicitly explain the exact connection between the state and the asset's value, the model assumes that higher valuations for a given trade size indicate a higher asset value. The speaker also explains how limit traders competitively post their limit orders, but the orders are only executed when a market trader places an order of equal or greater size. The limit trader may not know the exact size of the market order that executed their limit order.
The video delves into the pricing of limit orders in a limit order book market. The price set by a limit trader for the qth unit of an asset is given by the conditional expectation of the asset's fundamental value, given that the order size is at least q. This leads to the inside spread between the bid and ask prices as the order size approaches zero. However, the pricing equation creates a discontinuity at zero due to the conditioning. The video notes that limit traders always profit on the sale of the last units, as their price yields zero average profit between cases of large order sizes and optimistic news about the market trader's fundamental value. The marginal price for the first asset may or may not be lower than this.
The lecturer discusses the conditions that the best ask and bid quotes in a limit order book market must satisfy. These quotes are conditioned on the willingness of market traders to buy or sell but cannot be conditioned on the specific amount of buying or selling. This creates the inside spread, which represents the difference between the best ask and best bid prices. The lecturer also explains how the discreteness of prices comes into play, as prices are often subject to a tick size that limits the undercutting between competitors.
A model is introduced, which is similar to the previous one but incorporates a discrete price grid with a constant tick size. In this model, limit orders are prioritized based on time and price, with earlier orders executed first and lower-priced orders executed ahead of higher-priced ones. The lecturer introduces notation to denote the supplied amount at a certain price and the total amount that can be obtained by a market trader at prices no higher than that price. The model assumes that larger orders generally indicate higher valuations, which is taken into account when calculating the expectation. The lecture concludes with an explanation of how competition works in this model.
The video explains the process of limit orders being supplied at each tick in a queue-like system. The marginal trader who submits the last order at each price level earns zero profit, while the next trader who tries to submit an order at the same price level no longer finds it profitable. Consequently, they move on to the next tick. This process can be illustrated using a graph where the supply curve takes the form of a step function. The first trader typically achieves an expected positive profit, while the last trader generally receives zero expected profit.
The lecturer discusses the zero profit condition for marginal orders in a limit order book market. Infra-marginal orders can earn positive profit based on the expected profit of the limit trader multiplied by the probability of order execution, which is equal to the display cost "c." The expression connecting the price level "ak" and the cumulative depth "yk" comprises two terms: the adverse selection term and the execution risk component. The lecturer incorporates the display cost "c" in the graph and provides examples of binary and continuous random variables to model traders in the market.
The equilibrium in financial market microstructure is then discussed. The model with continuous prices and discrete equilibrium is used, involving two prices, "a1" and "a2." The depth at "a1" is equal to "qs," while the cumulative depth at "a2" is equal to "ql." The assumption is made that a noise trader employs one of four trades with equal probabilities: small buy, large buy, small sell, or large sell. The speculator also only trades in one of two units, either "qs" or "ql." Finally, the prices of "a1" and "a2" should satisfy two equations to determine the expected fundamental value for each order size on a specific side of the trade.
The speaker explains the concept of equilibrium in a simple limit order book market model. The equilibrium consists of strategies from both groups of active players in the market: limit traders and market traders. Limit traders set prices such as "a1" and "a2" based on zero-profit conditions, while market traders decide which orders to submit and trade optimally based on conditioned probabilities of being either informed or uninformed. The derived expressions show that the conditions for equilibrium are met, making this model an equilibrium.
The video discusses an example of limit order book markets where discrete price levels arise due to the discreteness of the noise trader's strategy, resulting in only two possible order levels and effectively two possible groups of events. The price impact is positive due to the strategy of uninformed traders. Another example is introduced where market order sizes are distributed according to an exponential distribution. The price impact equation assumes a constant price impact factor, lambda, which measures the informativeness of order flow. Although this example is significant, the video primarily focuses on how the limit order book is formed given the behavior of the market traders and provides a reduced-form analysis.
The speaker explains how to derive a conditional expectation using the conditional probability density function (pdf) of "q," which is based on the cumulative depth "yk" for limit traders. By applying Bayes' rule, the speaker demonstrates a simple way to calculate the expected fundamental value "v," which is the price that the limit trader would set for the "yk"-th unit of the asset. The final expression for the conditional pdf of trade sizes "q" makes use of the exponential distribution, and integration by parts is employed to derive the linear price impact equation. However, the factor of one over "theta" yields the inside spread of the equation.
The lecturer concludes the discussion on order-driven markets by focusing on the connection between the equation relating tick "ak" and the cumulative depth at the tick "yk," considering display cost model parameters. The lecture highlights the impact of display cost and the need to invert the expression in a market with predetermined ticks. The lecturer indicates that the liquidity provision role of a market differs for limit traders and dealers due to their distinct informational environments, resulting in different market outcomes. The next lecture will explore how tick size and priority rules affect market outcomes with dynamic analysis, considering a parlor model that gives traders the choice between limit and market orders. The instructor provides some practice questions from the textbook for students to work on.
represented as a function of the number of informed traders, demonstrating that as the number of informed traders increases, the aggregate profit of all speculators decreases while the profit of each individual speculator decreases as well.
Exercise class 3, part 2 (Financial Markets Microstructure)
Exercise class 3, part 2 (Financial Markets Microstructure)
The instructor introduces the glossing model, which is a market model similar to Kyle's model but with a limit trader instead of a dealer. In this model, the limit trader submits limit orders and does not have information about the total trade size queue. The limit trader can only condition upon the fact that their order has been executed. As a result, the price in this model will be discriminatory, meaning that the market trader who submits a market order will execute different parts of their order at different prices as they climb up the book.
To analyze this model, the instructor discusses the assumption of a distribution of trade sizes in the market and how informed traders behave to generate a linear price impact equation. They assume that limit traders do not affect this behavior. The instructor then delves into the expected value of the marginal price for the last unit traded and explains how it can be represented using the law of iterated expectations. They also express the expected value of the conditional trade size being larger than a given threshold.
Next, the instructor explains how to find the expected value of an asset given that the trade size is above a certain level. They derive a conditional probability density for trade sizes larger than a certain value and use it to compute the conditional expectation. The process involves taking the integral of the trade size with respect to the conditional density of trade sizes. They present two possible expressions for the final result.
The instructor further explains how to use the conditional density to find the expected value of the fundamental value given that the trade size is above a fixed level. They consider the total expected profit of the limit trader, taking into account the probability of trade, the profit from trade, and the display cost. It is assumed that limit traders are competitive. By considering the distribution of trade sizes and the distribution of the fundamental value conditional on trade size, an expression is derived that connects the price of a specific unit and the depth of the market.
The video then moves on from assuming tick sizes and looking for specific values to examining how informed traders would behave given market limits. The assumption is made that some traders are informed while others aren't, and informed traders optimize with a probability pi. Uninformed traders submit buy or sell orders with equal probability at an exponential distribution size. The scenario assumes a continuous limit order book with no tick size. The instructor provides a hint from the textbook that the conditional expectation for this setup can be found through the distribution parameter sigma.
The geometric intuition of the speculator's trading decision is discussed. The speculator aims to buy a certain portion of the asset when its value is above a minimal price denoted by a star. The market trader climbs up the supply curve, paying discriminatory prices for each unit bought. The optimal strategy for the informed trader is to submit an order size based on a proportion of the asset's value until the supply curve intersects with the value. The marginal cost of buying the first unit is given by the marginal price on the supply curve, while the marginal benefit is given by the value of the asset.
The instructor then discusses the relationship between marginal revenue and marginal cost in financial market microstructure. The trader will buy units as long as the marginal cost is below the value and the marginal price is below the marginal revenue. Part B of the video focuses on deriving the supply curve using the concepts discussed in Part A and the zero-profit condition. The zero-profit condition states that the marginal price of the qth unit should be equal to the expected value of the fundamental valuation. The probability of the market order coming from an informed trader can be determined using Bayes' rule.
The conditional probability of a trader being informed is discussed, given that the trade size is at least a certain value. The probability is calculated by multiplying the unconditional probability of a trader being informed by the probability of the informed trader submitting a buy order size of at least that value. Similar probabilities for uninformed traders are involved in the denominators, and upon simplification, an expression for the conditional probability is obtained with multiple exponential terms. This alpha value is necessary to compute the conditional expectation of the fundamental value, which helps in deriving the supply curve or the cumulative depth of the market.
The video discusses how the market book becomes thinner when there are more informed traders or when volatility increases. As more informed trading occurs, the cost of trading for the dealer increases, leading to less eagerness from limit traders to submit their orders. Similarly, the market depth decreases when volatility increases, making limit traders more reluctant to submit their orders. The math involved in these developments is relatively simple, and the intuition behind them aligns with what has been observed in many models.
Furthermore, the video explores the choice that informed traders face between trading at discriminatory prices in a limit order book or trading with a dealer while revealing their order size. The main difference lies in how prices are formed, as dealers condition prices on the total trade size, while limit traders condition prices on their order size being above a certain level. In general, traders should choose to trade small orders against a dealer to convey that they lack a strong information advantage. Conversely, they should choose to trade large orders using a limit order book to exploit the limited information of limit traders and obtain better prices than a dealer would offer.
Furthermore, the video explores the choice that informed traders face between trading at discriminatory prices in a limit order book or trading with a dealer while revealing their order size. The main difference lies in how prices are formed, as dealers condition prices on the total trade size, while limit traders condition prices on their order size being above a certain level. In general, traders should choose to trade small orders against a dealer to convey that they lack a strong information advantage. Conversely, they should choose to trade large orders using a limit order book to exploit the limited information of limit traders and obtain better prices than a dealer would offer.
Lastly, the instructor addresses the concern about tick sizes in the limit order book. In this context, tick sizes refer to certain fixed price levels that determine the allowable prices in the market. The larger the tick size, the more profit limit traders can make, potentially at the expense of market traders. As a result, submitting to a limit order book becomes less appealing compared to a dealer market where the dealer can quote any desired price.
Lecture 7, part 1: Market Design (Financial Markets Microstructure)
Lecture 7, part 1: Market Design (Financial Markets Microstructure)
In the previous lecture, the speaker provided a refresher on limit order book markets or order-driven markets, focusing on Claussen's model. This model highlighted that limit traders act as liquidity providers in the market, similar to dealers, but with a different approach due to facing an informational disadvantage. The lecture then introduced various dimensions of market design that can influence trading and the market environment in order-driven markets. These dimensions include tick sizes, priority rules, and the inclusion of dealers. The speaker emphasized that understanding these dimensions is crucial for effective market regulation, and their effects would be explored further in the lecture.
The main focus of the lecture was on the dynamic analysis of order-driven markets and the decision-making process for traders regarding whether to submit market orders or limit orders. This decision is a common one made by traders in real markets. The lecture delved into the regulation of tick sizes and its impact on market liquidity and depth. However, it also highlighted the potential unintended consequences of such regulations, as they may have the opposite effect and distort agents' incentives, leading to inefficient outcomes. Graphics were used to aid in explaining how the supply curve generated by the limit order book represents the information available to traders when a trade occurs.
The lecturer proceeded to discuss the concept of the zero-profit line for limit traders in a competitive market with continuous ticks. This line represents the price at which a limit trader sets their order to ensure no profit is made. However, with discrete ticks, the zero-profit line shifts as limit traders submit orders, potentially generating positive profits. In a market with time priority, the limit order book operates on a first-come, first-served basis, giving priority to earlier orders over later ones. Consequently, once the zero-profit point is reached at a specific price, no further limit orders will be present in the book.
The impact of reducing tick size in the market was then examined. Smaller tick sizes result in prices being set at finer increments, which geometrically translates to a decline in the profit potential for limit traders. The average profit for limit traders decreases, leading to a reduction in the number of limit traders participating in the market and subsequently less depth in the order book. While a decrease in the bid-ask spread may occur, it is typically minimal due to rounding errors, rather than a significant decrease.
Moving forward, the lecturer discussed the effects of tick size on financial market microstructure. Tick size refers to the minimum price increment at which a security can move. Decreasing the tick size leads to a narrower bid-ask spread and increased liquidity, but it can also drive some limit traders out of the market. As a consequence, market depth decreases, and the recovery of liquidity after trades becomes slower, impacting market resiliency. These conclusions were supported by tests conducted on the NYSE when the tick size moved from 1/8 to 1/16 of a dollar, aligning with the predicted effects. To prompt further discussion, the lecturer posed an open-ended question about the role of time priority in the market and included a quote by Man Winner on HFT 101.
The importance of tick size in market design and its impact on price priority versus time priority were then discussed. Smaller tick sizes give greater prominence to price priority relative to time priority. Lower tick sizes can be utilized to balance the two priorities, potentially driving out high-frequency traders while attracting slower traders. The lecture also introduced pro rata allocation as an alternative to time priority. Pro rata allocation allocates shares to all limit orders at a given price level proportionally based on their size when a market order is received.
The video then explored pro rata allocation in competitive markets. In such markets, the last limit trader who submitted an order at a given tick receives zero profit. However, all traders at that tick level are treated equally, resulting in zero profit for all traders collectively at that tick. Consequently, the aggregate supply curve in the market exhibits greater depth at any given tick. However, it does not necessarily imply that a larger quantity will be available at a specific price level.
The concept of pro-rata allocation was further examined, particularly in markets such as electronic futures for short-term interest rates and the market for two-year US Treasuries. While pro rata allocation can enhance depth at each price level, it may also lead to lower profits for limit traders, potentially driving them out of the market. The lecture also touched upon hybrid markets, where dealers are introduced into order-driven markets to provide additional liquidity. However, this inclusion may offset the benefits, as limit traders adapt their behavior to the presence of dealers.
Lastly, the lecturer discussed the actions of a dealer in a market with red ticks and price priority. Profitability for limit traders was reviewed, revealing that small orders yield profits while large orders result in losses. In such a scenario, the dealer observes the size of incoming market orders and must offer a price that surpasses the break-even point while still improving on the prices quoted in the limit order book. By doing so, the dealer can generate profits and potentially enhance the execution of the trade.
Moreover, the impact of dealers on the limit order book and overall market liquidity was addressed. Dealers can profit by quoting prices that are more favorable than those in the limit order book. However, this means they selectively take profitable limit orders while relaying only unprofitable ones back to the order book. As a result, limit traders are gradually driven out of the market, and the liquidity provided by dealers replaces the liquidity previously offered by limit traders. Consequently, the addition of dealers decreases market liquidity and depth in favorable market conditions. However, it can increase liquidity in unfavorable times by providing a form of liquidity insurance for the market.
In conclusion, the lecture shed light on various aspects of order-driven markets and their dynamics. It emphasized the significance of understanding market design dimensions such as tick sizes, priority rules, and the role of dealers in shaping market outcomes. The regulation of tick size was examined, revealing its impact on market liquidity, depth, and the behavior of limit traders.
The lecturer highlighted that reducing tick size may lead to a narrower bid-ask spread and increased liquidity but can also drive some limit traders out of the market. This reduction in market depth can result in slower recovery of liquidity after trades and affect the overall resiliency of the market. The effects of tick size were supported by empirical tests conducted on the NYSE, reinforcing the anticipated consequences.
The lecture also explored the interplay between price priority and time priority, emphasizing that smaller tick sizes elevate the importance of price priority over time priority. Lower tick sizes can be utilized to balance the two priorities, potentially attracting slower traders while discouraging high-frequency traders. Pro rata allocation was introduced as an alternative to time priority, which can enhance depth at each price level but may reduce profits for limit traders.
The role of dealers in order-driven markets was another focal point. It was revealed that dealers can profit by offering better prices than those in the limit order book, selectively picking off profitable limit orders while leaving unprofitable ones for the order book. Consequently, the liquidity provided by dealers replaces that of limit traders, potentially leading to decreased market liquidity and depth. However, the presence of dealers can offer liquidity insurance during unfavorable market conditions.
Throughout the lecture, graphics and examples were employed to illustrate key concepts and facilitate understanding. By delving into the intricacies of order-driven markets, the lecture provided valuable insights into market dynamics, the decision-making process of traders, and the potential consequences of market design choices.
The comprehensive analysis of limit order book markets, tick sizes, priority rules, and the role of dealers offered a deeper understanding of the intricacies and trade-offs involved in designing and regulating order-driven markets. The lecture served as a foundation for further exploration and discussion on the dynamic nature of these markets and the implications for market participants and regulators.
Lecture 7, part 2: LOB Markets - Dynamic Analysis (Financial Markets Microstructure)
Lecture 7, part 2: LOB Markets - Dynamic Analysis (Financial Markets Microstructure)
In this segment of the lecture, the focus shifts to the dynamic analysis of limit order book (LOB) markets, particularly the decision-making process of traders when it comes to choosing between taking or making liquidity and submitting market or limit orders. The lecturer delves into the trade-off involved in these choices, highlighting that market orders offer immediate execution but at the current market price, while limit orders have the potential for a better price but carry non-execution risk and are susceptible to adverse selection.
Two notable models are introduced for examining the dynamic analysis of the choice between market and limit orders: Christine Parlor's model and Foucault's model. These models differ in their considerations of adverse selection, non-execution risk, and delay, aiming to understand which types of orders are submitted by different traders. However, the lecturer acknowledges the complexity of performing a comprehensive dynamic analysis of LOB markets due to the multitude of factors at play.
The lecturer proceeds to discuss the dynamic dependence between future agents in financial markets microstructure. The attractiveness of submitting a limit order today is contingent upon the execution probability or the choices made by future agents who will trade against it. This creates a challenging dynamic loop where the submission of limit orders depends on future agents' choices, which, in turn, depend on execution probabilities. To illustrate this concept, a simple model is introduced where traders arrive and decide whether to submit a limit or market order for one unit of the asset. The choice is influenced by the probability of the limit order being executed, which is less than one, leading to a delay in the decision-making process.
The lecture further explores a model that considers non-execution risk, where traders are assigned a valuation as V plus y. While traders have different valuations for the asset, it is not due to possessing different information about the fundamental value V. V represents the fundamental value of the asset, known or unknown to all market participants. Traders assign different valuations for liquidity or risk management purposes. Each trader incorporates an idea of a credit component uniformly distributed on an interval, centered around zero, and independent across traders. The equilibrium probabilities of limit or market order executions are determined, taking into account the explicit delay, which serves as the unknowns sought in the model.
To facilitate understanding, the presenter introduces a graphical representation of four linear lines representing the profits associated with different types of orders in LOB markets. Rational traders are expected to choose the type of order that maximizes their expected profit based on their valuation Y. Traders with a high Y will opt for immediate buying through a market order, while those with high valuations but without a sense of urgency may risk a limit order to secure a better price. On the other hand, traders with low valuations will prefer to sell the asset. This decision-making process ensures that all traders have an opportunity to buy the asset at the best possible price.
The optimal trading strategy is discussed based on varying levels of urgency to sell. Traders with extremely low valuations of holding the asset are willing to sell at a lower price, while those with moderately low valuations may take the risk of a longer execution period in exchange for a slightly higher price. The probability of the next market orders to sell or buy can be computed based on the distribution of Y, the credit component of valuations, and the breakpoints in the graph. The lecture acknowledges that the equilibrium cut-offs and probabilities have not yet been determined.
The speaker then delves into the probabilities associated with the uniform distribution and the determination of cut-offs for y using indifference points. Indifference points represent the valuations at which traders are equally inclined to submit a limit order to sell or a limit order to buy, as the expected profits from both choices are equivalent. The speaker demonstrates how to solve the system and find the equilibrium using a simplified model. An example is provided, illustrating that traders with extreme valuations between -2 and -0.4 will submit market orders to sell, while those with high valuations between 1.4 and 2 will submit market orders to buy. Although the probability of execution for limit orders is low, traders are willing to take the risk due to the potential significant price improvement they can attain.
Additionally, the speaker mentions the inclusion of adverse selection in a model, along with non-execution risk. However, since these two frictions do not interact with each other significantly, the model does not offer substantial insights beyond what has already been explored in previous discussions of adverse selection (Clausten's model) and non-execution risk (Parlor's model). The speaker cautions that regulatory efforts aimed at enhancing market liquidity and depth may have unintended consequences, as evidenced by the various aspects of market design examined throughout the lecture.
As the lecture draws to a close, the speaker proposes an exercise for students to work on, exploring the effect of fees charged for limit orders and market orders within the framework of the Parlor model. This exercise encourages further exploration and analysis of the intricate dynamics and implications of different market mechanisms. Additionally, the lecture concludes by inviting viewers to consider enrolling in a forthcoming mechanism design course, indicating that there is more to be learned and discussed regarding the fascinating field of market dynamics and design.