A market is a set institution for exchanging property rights among economic agents. Markets can be physical (e.g., market square) or digital platforms. The efficiency of a market is judged by how well property rights are allocated—ideally, to those who value them the most.
Financial markets specifically deal with financial assets. The primary reasons for studying these markets are:
Reallocation of wealth over time (e.g. through riskless assets)
Insurance against future contingencies (e.g., investing in renewable energy if working in coal)
An important feature of financial markets is asymmetric information, where different agents possess varying knowledge about market prospects, impacting market efficiency. This information disparity can lead to inefficiencies if not regulated or managed.
Financial markets can be categorized into:
These involve the initial allocation of funds, such as:
Initial Public Offerings (IPOs)
Treasury auctions
In these markets, existing assets are traded among investors, allowing liquidity and efficient price discovery. Types include:
Stock markets
Bond markets
Derivative markets
Market efficiency implies that prices in the market reflect all available information. Factors affecting efficiency include the existence of a bid-ask spread, which can create friction and hinder trades.
Bid and ask prices reflect market dynamics where:
Bid Price < Ask Price
The spread can influence trading and market liquidity.
Liquidity is defined as the market’s ability to facilitate trading rapidly without significant price changes. Critical aspects related to liquidity include:
Market depth
Trading volume
Market stability
Limit Orders: Specify a price and quantity to be traded.
Market Orders: Specify a quantity to be traded at the best available price immediately.
In these markets, trades happen through a limit order book, either continuously or in call auctions.
Here, trades are facilitated by market makers or dealers who set prices and manage inventory.
Dealers profit through the bid-ask spread.
Risk management is critical, involving transaction costs and adverse selection due to asymmetric information.
Regulatory objectives include ensuring market efficiency, protecting against insider trading, and stabilizing market fluctuations.
Encouraging competition among exchanges
Imposing transparency requirements
Regulating price stability and trading structures
This document has covered the fundamental aspects of financial markets microstructure. As you embark on this course, consider exploring share prices and market differences among major stocks.
Find share prices, bid and ask prices for popular stocks.
Research the London Metal Exchange and discuss the pros and cons of maintaining physical trading floors.
In this lecture, we discussed the concept of liquidity within financial markets, focusing on its definition, measurement, and the challenges associated with measuring liquidity.
There are three primary types of liquidity to consider:
Market liquidity refers to a market’s ability to quickly facilitate the buying and selling of assets without significantly affecting their prices.
Definition: A market is liquid if an asset can be sold quickly at a stable price.
Monetary liquidity describes the ability of an asset to be converted into cash or goods.
Example: Cars vs. ice cream; cars are less liquid compared to ice cream.
Funding liquidity pertains to an economic agent’s ability to obtain cash or credit under acceptable conditions without substantial loss.
Example: A bank experiencing a liquidity shock when too many depositors withdraw funds simultaneously.
Liquidity is crucial for achieving market efficiency, which ensures assets are allocated to those who value them the most.
When markets are illiquid, they typically present two prices—higher prices for buyers and lower prices for sellers—leading to inefficiencies in trading.
Market depth quantifies the amount that must be traded to move the price of an asset by a specific amount.
Various measures can be used to assess liquidity, including spreads, price measures, price impact, and trading volumes.
The quoted spread St is the difference between the ask price Pa and the bid price Pb:
St = Pa − Pb
The normalized quoted spread st is calculated as:
$$s_t = \frac{S_t}{\frac{P_a + P_b}{2}} = \frac{S_t}{M_t}$$
The effective spread considers the price at which a transaction occurred compared to the mid-quote:
Ste = |Pt − Mt| ⋅ dt
where dt = 1 if the trade was initiated by the buyer, and dt = − 1 if initiated by the seller.
The realized spread represents the cost associated with holding an asset for a number of periods and is given by:
Str = Mt + Δ − Pt
In cases where one cannot access all necessary data, measures such as Lee and Ready’s algorithm can be applied to infer the direction of trades based on the proximity to bid or ask prices, particularly when trades occur within the quoted spread.
If only transaction prices are available, Roll’s measure can estimate the bid-ask spread by assuming that:
All trades have the same size.
Orders arrive randomly.
Price follows a random walk.
The spread can then be estimated using:
Spread = − 4 ⋅ Cov(Pt, Pt − 1)
This metric evaluates the average price an asset has traded throughout the day, weighted by volume.
$$\text{VWAP} = \frac{\sum P_t \cdot V_t}{\sum V_t}$$
This metric assesses the cost of not executing a perfect order:
Implementation Shortfall = (mT − p̄t) ⋅ κ − Opportunity Cost
We summarized the key concepts regarding liquidity in financial markets and emphasized the importance of choosing appropriate measures based on data availability. Next week, we will explore the primary factors driving spreads in markets.
9 Foucault, T., Pagano, M., & Roell, A. (2005). Market Liquidities.
We discussed liquidity—what it is, how to measure it, and its significance. The subsequent lectures will cover specific issues in financial markets, divided into three broad parts:
Model Setup: This part covers the mathematical models we will use to analyze financial markets, spanning approximately five lectures and culminating in a problem set.
Application of Models: Over three and a half lectures, we will apply our models to specific issues such as market fragmentation, transparency, liquidity, and corporate policy interconnections, concluding with another problem set.
Advanced Topics: We will explore modern financial topics not covered in the textbook, including digital markets, algorithmic and high-frequency trading, public information issues, market bubbles, and herding behavior. An additional class will focus on auction models.
Today, we will begin with the relationship between information and prices. We will also introduce our first model, Glosten and Milgrom’s model of information-based trading, which will help us understand how the bid-ask spread arises endogenously in the market.
Stock prices frequently fluctuate, often seemingly driven by news reports. However, it is challenging to ascertain the fundamental drivers behind these price movements. In financial markets, we approach understanding prices through the lens of expected future cash flows, which would be the market’s agreed valuation of an asset.
Traders generally engage in buying and selling for three primary reasons:
Risk Management: Traders may modify their portfolios to adjust their risk exposure. For example, if one works in a counter-cyclical industry, they may invest in pro-cyclical assets.
Funding Liquidity: Traders often need to convert assets into cash for various needs, which may influence their trading behavior.
Information Asymmetry: Traders possess different levels of information about the fundamentals affecting the asset’s value. If one trader believes they have valuable information that another does not, this may create a reason for trading.
It is essential to distinguish between public and private information:
Public Information: Information available to all market participants. If all participants simultaneously learn of a public update, prices may adjust without trading.
Private Information: Information known to some traders but not others. This generates asymmetry in the market and can lead to varying valuations of an asset.
The Efficient Market Hypothesis (EMH) posits that prices reflect all available information. EMH has three forms:
Weak Form: Prices reflect all past price information.
Semi-Strong Form: Prices reflect all available public information.
Strong Form: Prices reflect all available information, both public and private.
Challenges with the strong form of EMH include:
The No Trade Theorem: If all trades signal private information, traders would refrain from trading.
Incentives for Information Acquisition: If markets are perfectly efficient, traders lose motive to gather information since prices already reflect it.
Price Volatility: Markets exhibit fluctuations that cannot solely be attributed to public news.
Let us denote the information set at time t as Ωt, where this set is cumulative across periods.
The market valuation of an asset can be expressed as:
$$V_t = \mathbb{E}[C_s | \Omega_t] = \sum_{s=t}^{\infty} C_s \cdot \delta^{s-t}$$
where Cs denotes future cash flows and δ a discount factor.
The informational efficiency criterion indicates that:
Pt = 𝔼[V|Ωt]
Defining innovation in market valuation from t to t + 1 gives:
ϵt + 1 = Vt + 1 − Vt
The expected change in market valuation using the law of iterated expectations leads to:
𝔼[ϵt + 1|Ωt] = 𝔼[V|Ωt + 1] − Market Valuationt
Concluding that:
𝔼[ϵt + 1|Ωt] = 0
In essence, if market prices reflect complete information, they can be viewed as martingales, where future price performance is expected to align with the current price level.
This session introduced the crucial relationship between information and prices in financial markets, dissected the efficient markets hypothesis, and explored the mathematical foundations underlying these theories. Going forward, we shall delve into Glosten and Milgrom’s model, illustrating the nature of trading under asymmetric information.
Models are simplifications of reality that focus on specific aspects to provide insights into complex systems. The famous quote by George Box states:
"All models are wrong, but some models are useful."
The Glosten-Milgrom model, in particular, simplifies the interactions in markets to study the effect of asymmetric information on pricing and liquidity.
The Glosten-Milgrom model characterizes a market where two types of traders interact with a dealer or market maker:
The model operates in a dynamic setting, albeit separable across periods.
Each period involves a long-lived dealer and a trader (either informed or uninformed).
The model considers a unique trader interacting with the dealer in each period.
Traders are classified based on their information:
Informed Traders (Speculators): They possess private information about the fundamental value of the asset.
Uninformed Traders (Noise Traders): They trade for exogenous reasons such as liquidity needs or hedging, without knowledge of the asset’s fundamental value.
With probability π, a trader is a speculator, and with probability 1 − π, they are a noise trader.
Each trader can submit a market order to:
Buy one unit of the asset.
Sell one unit of the asset.
Abstain from trading.
Noise traders buy with probability βb and sell with probability βs.
The dealer is risk-neutral and quotes an ask price at and a bid price bt for one unit of the asset.
The dealer does not know if the incoming trader is a speculator or a noise trader.
The dealer operates in a competitive environment, leading to zero expected profits.
The bid and ask prices are defined as:
$$\begin{aligned}
a_t &= E[V \mid \text{buy order}] \\
b_t &= E[V \mid \text{sell order}]\end{aligned}$$
Where V is the fundamental value of the asset.
The profit of a speculator, given their action dt, is defined as:
$$\begin{aligned}
\pi(d_t) =
\begin{cases}
1 \cdot V - a_t & \text{if } d_t = 1 \text{ (buy)} \\
b_t - 1 \cdot V & \text{if } d_t = -1 \text{ (sell)} \\
0 & \text{if } d_t = 0 \text{ (abstain)}
\end{cases}\end{aligned}$$
For an equilibrium, the following must hold:
The dealer’s quotes must yield zero profit.
The speculator’s trading strategies must maximize expected profits based on the bid and ask prices.
The bid-ask spread is influenced by the level of information asymmetry and the presence of noise traders.
Increased informed trading leads to a larger spread, emphasizing the role of information in price setting.
In the long run, the prices converge to the fundamental value due to the accumulation of information:
pt → V as time t → ∞
This implies that while the bid and ask prices may initially deviate, over time they reflect the asset’s true value.
The spread increases with π, the probability of informed trading.
As β = βb + βs increases, the spread decreases.
The spread is maximized when uncertainty about V is highest (θ = 1/2).
The model assumes a dealer-centric framework, lacking market-clearing mechanisms.
Only fundamental values matter; speculation and resale opportunities are not considered.
The Glosten-Milgrom model serves as a fundamental framework to analyze the effects of asymmetric information on market pricing and liquidity. While it abstracts away many real-world complexities, its insights into the functioning of markets and the role of noise trading remain relevant.
Previous classes focused on the relationship between information and prices in markets, particularly price efficiency.
Introduction to the Glosten-Milgrom model, which explores the impact of asymmetric information on the bid-ask spread.
Key observations were made about the trade-off between market liquidity and price discovery.
Today’s lecture covers additional factors generating bid-ask spread and empirical methodologies to distinguish these factors.
In a dealer market, the market maker incurs various order processing costs, such as trading fees, clearing, and settlement fees as well as overhead expenses (e.g., office rent, salaries).
If dealers are not perfectly competitive, they can earn excess profits or rents, affecting the spreads traders face.
The model assumes one asset with a fundamental value v known to informed traders but unknown to dealers or uninformed traders.
Market valuation, denoted as μt, is based on public information and updated each period.
Half spreads are denoted as sat for ask and sbt for bid prices, where:
Ask price = μt − 1 + sat
Bid price = μt − 1 − sbt
Introducing transaction cost γ for each transaction shifts the spreads, leading to:
New Ask price = μt − 1 + sat + γ
New Bid price = μt − 1 − sbt − γ
The new spread St becomes:
St = (sbt + sat) + 2γ
The primary challenge: interpreting observed spreads to separate transaction costs from adverse selection costs.
The dynamics of these costs will differ over time—while transaction costs disappear as trading activity normalizes, the adverse selection costs will persist.
Short-term deviations in prices arise from both adverse selection and transaction costs.
Dealers’ inventory influences prices; short-run deviations depend on the recent trades.
Introduction to the Stoll (1978) model, positing that illiquidity arises due to dealer inventory costs.
Dealers must hold inventory and face risk in evaluating the assets they buy and sell, introducing a risk premium for their positions.
The model assumes risk-averse dealers who are price takers.
Dealers maximize utility defined over their wealth Wt + 1, which includes asset positions and cash flows.
Supply functions adjust according to expected future valuation of the assets and perceived risks:
Wt + 1 = (Asset position) + (Cash holdings)
Utility can be expressed as:
U = E[Wt + 1] − r ⋅ Var(Wt + 1)
The expression generates a linear supply function based on price, incorporating risk aversion and volatility:
$$y_t = z_t + \frac{p_t - \mu_t}{\rho \sigma^2}$$
In contrast to earlier models, the Stoll model shows that prices deviate from expected values, driven by risk and inventory costs:
Price deviation ∝ Inventory position
Traders may benefit from price inefficiencies if they encounter dealers with favorable inventory positions.
Future readings will focus on empirically assessing contributions of spread components: adverse selection, transaction costs, and inventory risk.
Homework exercises from Chapter 3 of the textbook will reinforce today’s concepts, specifically those around inventory risk.
In this lecture, we explore the determinants of market depth, specifically focusing on how trade size affects market prices. The key themes include:
Interaction between adverse selection, inventory risk, and order processing costs in determining the spread.
Market depth and its relation to trade size.
Review of the Kyle model, an extension of the Glosten-Milgrom model.
Market depth refers to the market’s ability to absorb large trades without significantly impacting the price.
A deeper market can accommodate larger trades with smaller price changes.
Adverse Selection: Refers to situations where informed traders have information about asset values that uninformed traders do not.
Inventory Risk: Occurs when dealers face costs due to holding non-neutral inventory positions.
Order Processing Costs: Costs incurred by traders when placing orders which may vary based on the size of the trades.
Larger trades typically signal stronger information regarding asset values.
In the context of the Glosten-Milgrom model, larger buy orders indicate higher fundamental values, and larger sell orders indicate lower values.
Dealers prefer to maintain neutral inventory positions. As trade size increases, the risk associated with holding larger positions rises, resulting in wider spreads for larger trades.
The increase in inventory risk leads to a demand for larger trading premiums.
These costs can vary depending on whether they are fixed per order or proportional to trade size. For example:
Fixed costs per trade: Costs decrease per unit as trade size increases.
Percentage-based costs: Remain constant regardless of trade size.
The Kyle model incorporates informed and uninformed traders and specifies a trading mechanism where trades are processed in batches (call auction):
Let v be the fundamental value, which is distributed normally.
Informed trader (speculator) submits a market order of size x.
Uninformed traders exhibit random demand u which is also normal with mean zero.
The net profit π of the informed trader is given by:
π = x ⋅ (v − p)
Where p is the market price established after all orders are processed.
The market maker provides a supply schedule that links price p and aggregated order flow q:
p = μ + λq
Where λ represents the price impact coefficient, estimating how much the price changes for a given order size.
The coefficient λ can be derived as follows:
$$\lambda = \frac{\text{Cov}(v, q)}{\text{Var}(q)}$$
In equilibrium, both the dealer’s pricing strategy and the speculator’s trading strategy are consistent with each other, leading to optimal values for λ and the speculator’s aggressiveness β.
Market depth D can be expressed as:
$$D = \frac{1}{\lambda} = \frac{2\sigma_u}{\sigma_v}$$
From this equation, we infer that market depth is greater when there is more noise trading and lower fundamental volatility.
Through empirical studies, we can estimate how various factors contribute to liquidity by measuring price impacts, particularly evaluating how depth and spread react to trade sizes in real market conditions.
The lecture outlines the theoretical foundations of market depth determinants through the Kyle model, emphasizes the importance of trade size in relation to adverse selection and inventory risks, and highlights the necessity for empirical validation in understanding these concepts in practice.
The lecture builds on the discussion of liquidity covered in previous lectures, particularly Lecture 2, where various empirical measures of liquidity were explored. Different theories explaining the existence of liquidity, including diverse selection, order costs, and inventory risk, have been proposed.
We focus on determining the relative importance of the factors affecting market liquidity. The objective is to estimate the contributions of:
Adverse Selection
Order Processing Costs
Inventory Risk
This investigation follows insights from Lecture 4, where dynamic impacts of these factors were discussed.
The effects of these factors on liquidity have differing time horizons:
Order Processing Costs: Short-lived effects that can be reversed by the next transaction.
Adverse Selection: Long-lasting impact on prices.
Inventory Risk: Medium-term effects that diminish over time.
To facilitate our analysis, the following notation is established:
λ: Price impact associated with adverse selection.
β: Price impact coefficient related to market maker’s risk conversion (inventory risk).
γ: Costs associated with order processing.
The data used consists of:
Transaction prices (Pt)
Net market order flow (Qt)
Order sign (Dt)
From the Glosten-Milgrom model, the transaction price Pt can be described as:
Pt = Ut + γDt
where Ut represents the market valuation and Dt is the directional trade.
Considering the first difference in prices:
ΔPt = Pt − Pt − 1 = ΔUt + γΔDt
This equation captures the changes in market valuation (ΔUt) modified by order processing costs depending on trade directions.
The authors estimate:
$$\begin{aligned}
\text{Assumption 1:} & \quad \gamma_1 = 0 \\
\text{Assumption 2:} & \quad \lambda_0 = 0\end{aligned}$$
indicating that order processing costs are independent of quantity traded, and the direction of trade conveys no information without volume.
In the second stage, they estimate parameters by setting γ0 and λ1 to values γ0 = 0.04 and λ1 = 0.1. These results reveal significant contributions from both costs.
The early empirical models often had limited data, with some using transactions from the New York Stock Exchange from the early 1980s. Concerns with the estimation procedures, particularly the two-stage regression model, raise questions about robustness.
The research by Hasbrouck in1988 yielded findings about autocorrelation in the order flow, revealing that order flows were negatively autocorrelated, likely due to inventory management practices by dealers.
The discussion leads to the understanding of how various components contribute differently to the spread. Empirical estimates suggest:
Order costs are the largest contributor (over 60% of the spread).
Inventory concerns account for approximately 30%.
Adverse selection contributes around 10%.
Lou et al. (1997) found that adverse selection significantly influences markets during market openings when new information is incorporated. In contrast, during market closures, order processing costs become more prominent.
Researchers have developed models to estimate the Probability of Informed Trading (PIN):
$$PIN = \frac{P(\text{Information Event}) \cdot P(\text{Trade from Informed})}{P(\text{Any Trade})}$$
Using comprehensive datasets, it has been identified that approximately 19% of trades come from informed traders, with variances depending heavily on the trading environment and anonymity of markets.
We learn that factors such as adverse selection, inventory costs, and order processing are key determinants of market liquidity.
The limitations of classical models signal the need for more advanced methodological approaches to capture the complexities of market dynamics.
For further reading, reference the ongoing studies and the working papers mentioned during the lecture, particularly regarding the varying components affecting liquidity across market conditions.
In this series of lectures on financial market microstructure, we will explore the transition from dealer-driven markets to order-driven markets. This lecture focuses on the characteristics, risks, and models associated with order-driven markets.
In our previous lectures, we discussed:
Determinants of market depth and liquidity.
Adverse selection and inventory risk.
The Kyle model which examines price impact due to adverse selection with non-trivial trades.
Empirical estimation of spread determinants and their implications on liquidity.
We found that order costs are significant contributors to illiquidity, primarily due to their role as a catch-all term.
Order-driven markets allow market participants to submit orders (limit and market orders) directly, eliminating the need for a dedicated dealer.
Limit Orders: Orders to buy or sell at a specified price or better.
Market Orders: Orders to buy or sell at the current best available price.
In dealer markets, the information structure is dominated by the dealer. In contrast, order-driven markets allow all participants to act as liquidity providers.
Market traders submit market orders based on a price schedule.
Limit traders set their orders and face non-execution risk and delay risk.
Non-Execution Risk: The risk that a limit order may never be executed.
Delay Risk: The uncertainty regarding when a limit order might be executed.
In order-driven markets, prices are determined by:
The marginal price for units traded, denoted as p′(q) where q is the order size.
The total amount paid for buying quantity q can be expressed as:
P(q) = ∫0qp′(q′) dq′
where P(q) is the total payment made for quantity q.
Market traders optimize the quantity to buy by equating their marginal valuation θi(q) to the marginal price:
θi(q) = p′(q)
This model analyzes limit order markets, allowing for an understanding of price efficiency and order book depth.
A single asset with unknown value v distributed according to G(v).
Competitive limit traders establish a limit order book with symmetries in order size.
Limit traders’ prices are determined by the expected fundamental value conditioned on the order size:
p′(q) = E[v|Q ≥ q]
Referring to the gap between ask and bid:
S = p* − pb
Where p* is the ask price and pb is the bid price. The distinction arises because the highest ask conditions on the buyer’s willingness, while the lowest bid conditions on the seller’s willingness.
The presence of adverse selection is crucial, as it affects how limit traders set prices based on their information about asset values.
Order-driven markets generate price schedules that can differ significantly from dealer markets due to the lack of centralized information asymmetry.
Different market structures (e.g., tick sizes, priority rules) have significant implications for trading dynamics and market outcomes.
The lecture emphasizes how traders navigate order-driven markets and how their behavior influences price dynamics. Future lectures will delve further into market dynamics and explore practical aspects such as market design and strategies.
Discuss the advantages and disadvantages of limit vs. market orders.
Explain how adverse selection affects liquidity in order-driven markets.
Analyze the implications of tick sizes on market efficiency.
In today’s session, we will cover two main topics:
Problems based on Kyle’s Model from last week.
Problems based on Glosten’s Model presented in the last session.
Feel free to ask questions in the chat as we progress.
Kyle’s Model explores competition among speculators in a market setting. The assumptions and structure of the model are as follows:
Agents:
One informed trader (speculator).
One dealer (market maker).
Implicit noise traders (not depicted in slides).
Asset Characteristics: An asset with a fundamental value v, which follows a normal distribution: v ∼ 𝒩(μ, σv2).
The informed trader (speculator) determines her volume of order x based on her information regarding v:
xi = β(v − μ)
where β is the speculator’s aggressiveness.
The market maker sets the price p according to:
p = 𝔼[v ∣ q] = μ + λq
where q is the total order size comprising orders from speculators x and noise traders u. - The price impact coefficient λ is given by:
$$\lambda = \frac{\text{Cov}(v, q)}{\text{Var}(q)}$$
In the current exercise, we extend the model to n informed traders. Each trader i has the same linear trading strategy:
xi = β(v − μ)
To find the equilibrium aggressiveness β, we need to maximize each speculator’s expected profit:
Πi = 𝔼[xi ⋅ (v − p)]
Substituting the price and evaluating the expected profit leads to the first order condition:
v − μ − λ(Nxi + (N − 1)β(v − μ)) = 0
Solving for β:
$$\beta = \frac{1}{\lambda(n + 1)}$$
This indicates that as n increases, β decreases because each trader’s impact diminishes with increased competition.
To derive λ: - Set the dealer’s total order:
q = nβ(v − μ) + u
Using covariance and variance assumptions, we find:
$$\lambda = \frac{n \beta \sigma_v^2}{n^2 \beta^2 \sigma_v^2 + \sigma_u^2}$$
Market depth is defined as:
$$\text{Depth} = \frac{1}{\lambda}$$
From the derived expressions, we see that market depth is increasing in n:
$$\text{Depth} = \frac{n+1}{\sqrt{n}(\sigma^2_v/\sigma^2_u)}$$
This implies that as the number of informed traders increases, the market depth increases, as each traders’ impact on the price becomes comparatively less significant.
For informed traders, expected profits will decrease as n increases due to competitive behavior reducing overall profitability:
Π = n ⋅ 𝔼[xi(v − p)]
Taking the expected value before traders know v, results in a generalized profit equation reducing with n.
In today’s session, we explored the implications of competition among informed traders using Kyle’s model and derived important economic insights regarding trading strategies, market depth, and the impact of noise traders.
This document serves as a comprehensive guide for understanding key concepts related to Glosten’s model within the context of financial markets, particularly focusing on limit order books, discrimination in pricing, informed and uninformed trading, and the implications of these market mechanics.
Informed Trader: A trader who possesses more information than the average market participant.
Uninformed Trader: A trader who does not possess specific market information.
Limit Trader: A trader participating in the market by submitting limit orders, contrasting with a dealer who can see the entire order book.
While Kyle’s model incorporates dealers who can observe the entire trade size queue and set prices accordingly, Glosten’s model introduces limit traders who rely on partial information (only that their limit order was executed) and cannot discern the entire market dynamics.
In Glosten’s model, the marginal price for the qth unit of the asset can be determined through the conditional expectation of the asset value given that the total trade size exceeds a certain threshold q:
P′(q) = 𝔼[V ∣ Trade size > q]
Where:
P′(q) is the marginal price for the qth unit.
V is the fundamental value of the asset.
The expectation is conditional on the condition that the total trade size exceeds q.
In limit order book markets, prices are discriminatory, meaning that the price paid for different units will vary as the trader "climbs" the book, contrasting with a dealer who offers a single price based on the whole order size. This concept is crucial when considering order execution strategies.
The limit trader’s behavior can be analyzed using an example wherein the trade size q is assumed to follow an exponential distribution. If we denote μ as the mean size and λ as the pricing impact parameter, we define the expected value based on this distribution as follows:
𝔼[V ∣ Trade size = q] = μ + λq
To derive the asset’s marginal price, we utilize iterated expectations:
𝔼[V ∣ Trade size > q] = 𝔼[μ + λQ ∣ Q > q]
Where the conditional distribution is represented using the law of total expectation.
To find the conditional expectation of an exponential distribution, we identify the conditional probability density function:
$$f_{Q|Q>y_k}(q) = \frac{f_Q(q)}{\mathbb{P}(Q>y_k)}$$
Where yk is a position in the limit order book.
As a result, after substituting in the relevant exponential probabilities, we derive the cumulative depth of the limit order book considering price a:
$$y(a) = \frac{(1 - F(a))}{\mathbb{P}(Q > y_k)}$$
Informed traders maximize their utility by choosing their order size optimally based on the known value V. Given the cumulative depth y(v), the informed trader trades up until they reach a price where the marginal cost meets the marginal benefit.
For a competitive limit trader in equilibrium, the expected profit must equal zero:
p′(q) = 𝔼[V ∣ Q > q]
Where p′(q) is the marginal cost determined by the limit order book depth.
It is observed that as the proportion of informed traders π increases or as the fundamental value’s volatility σ increases, the market depth y(a) decreases, leading to a thinner limit order book.
When π increases, traders become more cautious, making them less willing to post limit orders, thus reducing liquidity.
Increased σ heightens the uncertainty in valuation, leading to adverse selection and further discouraging limit orders.
The exploration of Glosten’s model deepens the understanding of how market participants behave under uncertainty and informs strategies surrounding trade execution. The relationship between informed and uninformed trading demonstrates the complex dynamics in limit order books that impact not only pricing but also overall market liquidity.
In this lecture, we will explore limit order book markets, also known as order-driven markets. We will build upon Claussen’s model and delve into important aspects of market design and the dynamics of trader orders.
Limit Orders: Orders to buy or sell at a specified price or better.
Market Orders: Orders to buy or sell at the best available current price.
Limit traders provide liquidity in a manner distinct from dealers in dealer-driven markets. The informational environment differs for limit traders, leading to varying market outcomes, including price inefficiencies.
Several dimensions influence the regulation and operation of order-driven markets:
Tick Size
Priority Rules
Dealers’ Presence
Tick size refers to the minimum price movement of a security. Tick sizes can affect:
Market Depth
Spread
Trader Participation
Let:
E(V|Q) = Expected Price Given Trade Size Q
Where Q represents the total order size.
When analyzing a limit trader who offers the 15th best bid, we can represent the supply curve generated by the limit order book graphically.
When tick sizes are reduced:
Total profit of limit traders decreases.
Depth may be reduced as limit traders exit the market.
The spread can decrease due to smaller rounding errors.
We will consider how traders decide between market and limit orders under dynamic conditions.
Regulations aimed at improving liquidity or market depth may have unintended adverse consequences by distorting traders’ incentives.
In alternative market structures, we can replace time priority with pro rata allocation where limit orders are executed proportionally:
$$\text{If Market Order Size is } X, \text{ then each Limit Order gets } \frac{(X - \text{filled amount})}{\text{total amount available}}.$$
This can increase market depth but may also lead to reduced profits for limit traders.
Incorporating dealers into order-driven markets can initially seem beneficial but may lead to the crowding out of limit traders:
Dealers can observe and provide price improvements for market orders.
Limit traders may become less active as the dealer picks off profitable orders.
Adding dealers can enhance liquidity during volatile periods but may reduce it overall during stable market conditions. They serve as a liquidity insurance mechanism during periods of low depth.
The study of limit order book markets reveals intricacies in market design that warrant careful consideration, particularly regarding regulatory approaches and trader dynamics.
Market design involves creating rules and frameworks that enhance market liquidity. However, regulations intended to improve liquidity may backfire by distorting agents’ incentives and ultimately reducing market depth.
Decreasing tick size can improve liquidity for small orders.
However, it may not have the same positive effect for large orders.
Adding a dealer can have unforeseen effects on market dynamics.
The focus here is on how traders make decisions regarding liquidity—whether to take (market orders) or make (limit orders) liquidity.
Market Orders: Executed immediately at the prevailing market price, but may have a worse price due to the bid-ask spread.
Limit Orders: Can provide better prices than market orders but involve non-execution risk and possible delay.
Traders face risks associated with limit orders not being executed (equivalent to infinite delay).
Adverse selection is a concern: limit orders can be "picked off" by better-informed traders.
Two primary models illustrate different aspects of the choice between market and limit orders:
In this model, there’s no adverse selection; the focus is on non-execution risk and delay.
Traders evaluate whether to place limit or market orders based on their own valuations and expected market behavior.
Incorporates both non-execution risk and adverse selection.
Concludes that patient traders generally place limit orders, while impatient traders opt for market orders.
We construct a simplified framework to analyze trader behavior:
Assume prices for bids (B) and asks (A) are exogenous.
Fundamental value (V) is uniformly unknown but agreed upon.
Each trader has their own valuation adjusted by an idiosyncratic component y.
Valuation for each trader is given by V + y, where y is uniformly distributed.
The expected payoff for different order types is modeled as follows:
$$\begin{aligned}
\text{Market Order Sell:} & \quad \text{Payoff} = B - (V + y) \\
\text{Limit Order Sell:} & \quad \text{Payoff} = p_{BM} \cdot (A - (V + y)) \\
\text{Market Order Buy:} & \quad \text{Payoff} = (V + y) - (A) \\
\text{Limit Order Buy:} & \quad \text{Payoff} = p_{SM} \cdot ((V + y) - B)\end{aligned}$$
Where:
pBM is the probability a market order to buy arrives.
pSM is the probability a market order to sell arrives.
The trader’s choice depends on their idiosyncratic valuation y and can be visualized through profit lines:
Different segments of the valuation spectrum corresponding to market and limit orders.
Traders with a high y will tend to submit market orders to obtain immediate execution.
Traders with a lower y will submit limit orders in hopes of obtaining better prices.
Probabilities pBM and pSM are determined based on the distribution of y as follows:
For a market order to sell, pBM reflects the proportion of traders with a sufficiently negative ŷ.
Likewise, for a market order to buy, pSM represents the probability that y is sufficiently positive.
Trader indifference between strategies leads to equilibrium cut-offs:
$$\begin{aligned}
V + y_{\text{hat}} &: \text{Indifference between Selling Limit and Buying Limit} \\
V + y_{\underline{\text{bar}}} &: \text{Above this point, traders will mainly submit market orders} \\
V + y_{\text{bar}} &: \text{Below this point, traders will mainly submit market orders to sell}\end{aligned}$$
In summary:
We explored how the choice between market and limit orders is influenced by traders’ individual valuations.
Market design regulations must be approached with caution—they can alter incentives and liquidity.
Future discussions will delve into market fragmentation and its implications for trading in multiple platforms.
Today, we will be discussing market fragmentation. We will explore its implications, especially in the context of financial markets and how it affects trading costs, price discovery, and liquidity.
In the past few weeks, we have examined:
Order Driven Markets - Focusing on Glosten’s Model and the behavior of limit traders as liquidity providers similar to dealers.
Parlor’s Model - Addressing trader decisions between limit orders and market orders, emphasizing the resiliency of limit order books.
Market Design - Exploring how market structures influence trading activities and outcomes.
Market fragmentation occurs when multiple markets trade the same asset. The consequences include variations in trading costs and price discovery.
Historically, assets were often traded on a single exchange. For example, stocks listed on the New York Stock Exchange were not traded elsewhere.
Over recent decades, cross-listing and admitted for trading phenomena have emerged, allowing the same stock to be traded on various exchanges without formal listing.
Priority Rule Violations: Fragmentation can lead to violations of price, time, and visibility priority rules.
Search Costs: Increased number of markets complicates the search for the best price, leading to higher transaction costs.
Worse Price Discovery: Information about fundamental values becomes dispersed, hampering effective price discovery.
Total Liquidity: Total liquidity may be less than in a consolidated market resulting in less overall trader participation.
Lower Trading Costs: Competition among exchanges can drive down order processing costs.
Better Price Discovery: Multiple prices may provide more data points for accurate inference.
Increased Total Liquidity: Fragmented markets can enhance total liquidity as liquidity providers may be incentivized to compete in multiple venues.
Consider the case of Dutch stocks before and after 2003:
Before 2003, stocks of Dutch companies were only traded on Euronext.
Following the introduction of competing platforms, Euronext dramatically reduced its order entry fees, showcasing how competition influences trading costs.
The misalignment of incentives between brokers and traders creates search cost issues, leading to inefficiencies in price finding. Effective regulations (e.g., order protection rules in the US) aim to mitigate these issues.
In Kyle’s model, we explore how information asymmetry affects market behavior:
Let V be the fundamental value of the asset, distributed normally.
Agents include informed traders, liquidity traders, and market makers, where the market maker sets prices based on aggregate order flow Q.
The order size X of the informed trader is given by:
X = β(V − μ)
where μ is the pre-trade expected value.
The market price P as a function of total order size Q is defined as:
P = λQ
Where λ is derived from the covariance of order flow with asset value.
The equilibrium parameters β and λ can be solved simultaneously:
Cov(X, V) = σV and σU
Trading costs can be viewed as the expected loss for uninformed traders:
Average Trading Cost = σV × σU
As we delve deeper into market fragmentation, we will assess its implications on liquidity and price discovery using our established models to better understand the intricacies of fragmented markets.
In this document, we explore a chiral model with fragmented markets. We analyze how market segmentation impacts traders, price discovery, market depth, and overall market welfare.
Consider two independent markets.
Each market has:
A dealer who is competitive.
One insider who knows the exact asset value D.
Noise traders whose order flow is split into two independent flows:
u1 and u2
The goal is to compare outcomes in fragmented markets versus consolidated markets.
Let λi represent price impact in market i.
The price Pi in market i is given by:
Pi = λiQi + εi
where Qi is the total order flow in market i and εi represents random noise.
The expected prices in both markets are equal:
𝔼[Pi] = Expected Value in Consolidated Market
Variance for prices:
Var(Pi) = Var(ui) cancels with noise variance
Markets operate simultaneously:
Informed and uninformed traders place orders at the same time.
Price realization happens afterward, preventing arbitrage between markets.
As the dealers observe trade flows from both markets, prices converge post-trade.
Linear strategies for informed traders yield:
$$\beta_i = \frac{\sigma_{u_i}}{\sigma_{v_i}}$$
where σui is volatility of noise traders and σvi is the volatility of informed traders.
Aggregate trading volume comparison:
V − μ ⋅ ∑σui (fragmented markets)
versus
V − μ ⋅ σu (consolidated market)
Resulting outcome:
∑σui > σu ⇒ Higher total order sizes in fragmented markets.
Computed profits of informed traders:
Profits = σv ⋅ ∑σui/2
Incomplete order flows and higher profits lead to increased loss for uninformed traders compared to consolidated markets.
Depth in each fragmented market is lower than in consolidated markets.
Aggregate depth can still be higher in fragmented contexts due to additional trader segmentation creating more opportunities for liquidity.
Informed trading reveals information and serves as a proxy for price discovery:
σv2/3 (fragmented)
versus
σv2/2 (consolidated)
If informed traders can choose between markets:
They may split across markets to avoid competition, increasing their overall profit.
Noise traders tend to prefer deeper markets and might migrate towards liquidity-rich exchanges.
Market fragmentation presents both risks and benefits:
Higher liquidity depth.
Potential adverse effects on uninformed traders’ profits.
Price discovery benefits from increased amounts of information but traders must evaluate actual engagement across markets.
In this lecture, we explore the concept of market transparency within the realm of financial markets microstructure. Before diving in, let us briefly review the previous lecture, which focused on market fragmentation and its implications.
Market fragmentation occurs when the same asset is traded on multiple platforms, leading to both costs and benefits.
Costs include greater adverse selection due to weakened competition among informed traders and potentially reduced risk-sharing.
Benefits of fragmentation include lower order processing costs due to competition among market platforms, which can ultimately enhance market efficiency.
Market depth can also improve in fragmented markets as various platforms may contribute to increased aggregate liquidity.
Market transparency refers to the availability of information on prices, costs, and other crucial data that influences trading decisions in financial markets. Financial markets are generally more transparent compared to other markets.
Three primary categories of transparency are identified:
Pre-trade Transparency: Information available before a trade is executed, such as quotes and limit order books.
During-trade Transparency: Information about who one is trading against can affect trading behavior significantly.
Post-trade Transparency: Information about past trades, prices, and other historical data that can be utilized for strategic trading decisions.
Exchanges have access to all types of information and make decisions regarding data release. A conflict exists between the need to provide information for market efficiency and the desire to protect proprietary data.
Considerations about transparency arise when evaluating information sharing between fragmented markets. If two platforms trade the same asset, the extent of transparency can mitigate the fragmentation issues if participants can observe the same market data.
Pre-trade transparency can substantially influence trader behavior. It affects decision-making processes based on:
Availability of quotes from various dealers.
The degree to which traders can benefit from price improvements.
In illiquid markets, quotes may not be readily available, prompting traders to actively solicit quotes from dealers.
This paradox illustrates the inefficiencies resulting from market power when prices are not publicly observable:
Consumers incur search costs C when seeking prices across stores.
In equilibrium, if search costs exist, stores will set the monopoly price since consumers cannot efficiently shop around.
This suggests that market transparency allows for increased competition, which lowers prices and improves trader welfare.
Market depth λ significantly affects trading decisions. When the depth is unknown, traders are hesitant, impacting overall trading volume and liquidity. Formally, the relationship between trader profit maximization and market depth can be derived as follows:
The profit maximization problem for an informed trader is given by:
maxx𝔼[x(V − P)] = 𝔼[x(V − μ − λx + q)]
where V is the true valuation, P is the market price, μ is the average signal, x is the order size, and q represents noise from uninformed traders.
The first-order condition is:
$$V - \mu - 2 \lambda x = 0 \implies x = \frac{V - \mu}{2 \lambda}$$
Under uncertainty about market depth, expected order size becomes:
$$x_{\text{expected}} = \frac{V - \mu}{2 \mathbb{E}[\lambda]}$$
Using Jensen’s inequality, the expected volume in a transparent market is higher than in an opaque market, establishing that higher uncertainty reduces trading volume, thereby hampering market efficiency.
Market transparency plays a vital role in the functioning of financial markets, as evidenced by the effects of pre-trade and depth transparency on trading behavior and market liquidity. As markets evolve, ongoing regulatory measures seek to balance transparency with the need for exchanges to maintain a competitive edge.
In financial markets, the implications of market transparency on order flow dynamics are complex and multifaceted. This document outlines the consequences of both pre-trade and post-trade transparency, particularly focusing on the behavior of informed and uninformed traders concerning their order flow and the resulting market pricing mechanisms.
We consider a market with one asset that has a fundamental value V which can take one of two possible states: high (VH) or low (VL), each with equal probability. Thus, the mean market valuation of the asset is given by:
$$\text{Market Valuation} = \frac{V_H + V_L}{2}$$
Dealers: Participants who quote prices for the asset. They are assumed to be competitive and risk-neutral. Traders: Individuals submitting market orders. For simplification, we consider two types of traders:
Informed traders: Traders with knowledge of the true value of the asset.
Uninformed (liquidity) traders: Traders without knowledge of the asset’s true value.
The correlation of order flow between informed and uninformed traders significantly affects market behavior. When both orders come from informed traders:
If VH, both orders will be buy orders.
If VL, both orders will be sell orders.
Conversely, when orders arise from uninformed traders, buy and sell orders are less correlated.
1. Opaque Market:
Dealers quote prices without access to the total order flow (i.e., they are blind to other orders).
Quoting behavior is based on a standard Kyle model approach, where dealers assess the probability of facing an informed or uninformed order.
2. Transparent Market:
Dealers can condition their quotes on total market order flow.
If both orders are buys, they infer that traders are informed and set ask prices to VH.
If both orders are sells, set bid prices at VL.
If the orders are mixed (one buy, one sell), they will assume both are uninformed and require a profit margin.
Generally, visibility of the order flow leads to improved price discovery, with prices being more closely aligned with the fundamental values VH and VL.
Post-trade transparency involves the visibility of historical trading data and outcomes. Key implications include:
Improved Market Efficiency: With historical data, traders can make better-informed decisions.
Informed Traders Suffer: They may face wider spreads and adverse selection, resulting in reduced profitability.
We introduce a model where orders appear in sequence:
First Order (Observed): All dealers see this order; thus, expectations adjust based on the order flow.
Second Order: Dealers can identify the type of trader (informed or uninformed) based on the first order’s behavior, resulting in differentiated pricing strategies.
The bid price from the uninformed dealer when the first order is a buy can be denoted as:
Bid Price (First Order Buy) = VH if second order is buy also.
If the second order is a sell, the uninformed trader adjusts expectations:
Expected Value = Market Price View
This illustrates how the order sequence impacts dealer pricing and market outcomes.
Identifying the trader’s identity has considerable implications:
Uninformed traders may receive better prices due to a lack of adverse selection.
Informed traders often face higher costs as their reputation precedes them.
This analysis leads to the conclusion that enhanced transparency benefits uninformed traders at the cost of informed traders, impacting liquidity and market dynamics.
Market transparency significantly influences trading behavior, impacting prices, spreads, and information asymmetry:
Uninformed Traders: Benefit from better terms of trade.
Informed Traders: Face adverse selection and wider spreads.
Regulation often aims to enhance transparency to protect uninformed traders, though such policies may inadvertently reduce overall market efficiency. Understanding these dynamics is crucial for trading strategies and market design.
These notes summarize the lecture on the value of liquidity in financial markets, focusing on how liquidity affects asset prices, informed vs. uninformed trading, and models of liquidity premium.
Transparency generally benefits uninformed traders and can harm informed traders.
Regulators advocate for transparency, which may lead to tensions with market makers who prefer confidentiality.
In financial markets, liquidity is defined as the ability to buy or sell assets without causing a significant impact on their prices.
We posit that an asset has some fundamental value and that its market price deviates from this value due to liquidity issues. The central question is:
How does limited liquidity affect the asset value?
U.S. Treasury securities are issued either as notes (long-term, 2-10 years) or bills (short-term, < 1 year). They can be traded freely in secondary markets.
Investors may face different prices for a note and a bill with the same face value nearing maturity.
Notes might trade at a discount due to their perceived lower liquidity compared to bills.
Investors typically require a return to compensate for potential liquidity costs when trading. The liquidity risk premium is the added return required on less liquid assets.
This model highlights how to evaluate assets based on their resale values and liquidity considerations:
$$P_t = \frac{\mathbb{E}[P_{t+h}]}{(1 + R)^h}$$
where Pt is the initial price, R is the required rate of return, and h is the holding period.
The returns from trading can be expressed as follows:
$$\begin{aligned}
P_t &= M_t \left(1 + \frac{s}{2}\right) \\
P_{t+h} &= M_{t+h} \left(1 - \frac{s}{2}\right)\end{aligned}$$
Here, Mt and Mt + h represent market price at times t and t + h, respectively, and s is the spread.
This leads to:
$$1 + R = \frac{1 + r + \frac{s}{2}}{1 - \frac{s}{2}} \tag{Liquidity Premium}$$
If we apply logarithmic approximations and small R and s assumptions, we get:
$$R \approx r + \frac{s}{H}$$
where H is the holding period.
Investors have varying holding periods which affect their choices:
Short-term investors (frequent traders) opt for lower spread assets.
Long-term investors may accept higher spreads for potentially higher returns.
Using a budget set analogy from consumer choice theories, different indifference curves illustrate the preferences of investors based on risk and return.
Investors maximize utility by choosing assets that align with their holding period strategy.
Pension funds may favor less liquid assets due to their long-term nature.
This lecture illustrated the critical role liquidity plays in asset pricing and market dynamics. Different types of investors trade differently based on their liquidity preferences and holding periods, indicating a complex relationship between liquidity, return, and market behavior.
Liquidity risk refers to the uncertainty regarding the ease with which an asset can be bought or sold in the market without affecting its price. It is characterized by:
Imperfect and fluctuating nature of liquidity over time.
Arbitrary correlations between liquidity of different assets across the market.
Inspired by the Capital Asset Pricing Model (CAPM), the Liquidity CAPM was proposed by Acharya and Pedersen in 2005.
In CAPM, the expected return on an asset J is given by:
E[RJ] = Rf + βJ(E[RM] − Rf)
where:
E[RJ]: expected return on asset J.
Rf: risk-free interest rate.
βJ: measure of the asset’s risk relative to the market.
E[RM]: expected market return.
The CAPM asserts that only systematic risk, denoted by β, is relevant since unsystematic risk can be diversified away.
In the Liquidity CAPM, the real returns r that investors care about are defined as:
rJ = RJ − SJ
where RJ is the nominal return and SJ is the liquidity spread (difference between bid and ask prices).
Thus, the expected excess return can be modified to:
E[RJ − SJ] = Rf + βJ(E[RM − SM] − Rf)
The liquidity risk premium λM is related to:
λM = E[RM] − SM − Rf
The aggregate beta is influenced by four components:
β1: Covariance of nominal returns of asset J and market returns.
β2: Covariance between the liquidity spreads of asset J and the market (low beta indicates good liquidity).
β3: Covariance with nominal returns during illiquid market conditions (high beta indicates a good asset).
β4: Covariance of asset’s liquidity with market returns (high beta helps in hedging).
The empirical results suggest that all components contribute significantly to asset pricing.
The fundamental law in finance states that there are no free lunches or arbitrage opportunities:
Assets generating identical cash flows must price similarly.
If arbitrage exists, traders will exploit it until the opportunities disappear.
Barriers to arbitrage include:
Costs of executing trades (e.g., short selling costs).
Market frictions prevent optimal execution of arbitrage strategies.
Consider the scenario where Treasury Bills and Bonds generate the same cash flows but are priced differently. This can be explained by:
Costs of arbitrage that prevent traders from exploiting price discrepancies.
Market participants facing similar trading costs which eliminate profit from arbitrage.
In Over-The-Counter (OTC) markets, trading is decentralized, and prices emerge from the negotiation between buyers and sellers:
Traders have heterogeneous valuations for assets.
Search costs lead to market power for dealers and create spreads between bid and ask prices.
Let:
Q: Fraction of population that supplies the asset.
C: Discount on dividend valuation by low-value traders (C ∈ (0, 1)).
Search costs result in a maximum price the trader is willing to pay (Ā) and minimum bid price (B̄).
The equilibrium conditions in OTC markets ensure that:
A = Ā and B ≤ B̄
Where the market clears, and both buyers and sellers remain indifferent to trading at these prices.
Liquidity affects asset pricing and the required return for holding assets. Illiquid assets often trade at a discount, while assets with volatile liquidity demand a liquidity premium. This framework is essential for understanding asset pricing and risk in the financial markets.
Recommended exercise:
Explore effects of dividends in the context of a no-arbitrage model, consider how liquidity affects pricing.
Today we are covering two distinct topics in the field of financial markets microstructure:
Liquidity and Corporate Policy
Digital Markets
We will begin by recalling some of the key concepts we discussed in last week’s lecture regarding liquidity and its effect on market prices.
Last week, we analyzed how liquidity affects the market valuation of securities. The relationship differs from traditional perspectives which focused on how liquidity causes deviations from fundamental values. This shift in focus highlights the importance of liquidity in determining the price investors expect when buying and selling stocks.
Investors may not plan to hold a security indefinitely; rather, they may intend to sell it under varying future circumstances. This behavior leads to trading costs which can cause selling prices to deviate below the fundamental value of the asset.
We explored several models that illustrate this relationship:
Fixed Spread Model: A simplification incorporating a fixed trading spread.
Capital Asset Pricing Model (CAPM): An analysis of how liquidity enters this traditional asset pricing model.
OTC Trading Model by Galeano and Patterson: A search model accounting for limited liquidity due to market makers’ power.
In today’s session, we will examine the intersection of liquidity and corporate policy. The main focus will be on understanding how liquidity influences a firm’s ability to raise capital and its implications for corporate governance.
Firms require financing to invest in profitable projects. The relationship between market liquidity and the cost of capital is critical because:
$$\text{Cost of Capital} \propto \frac{1}{\text{Market Liquidity}}$$
Higher liquidity implies lower costs of capital, facilitating more efficient investments and capital raising.
The lifecycle of firm funding changes as the business grows:
Business Angels: Initial funding from individuals willing to invest in startups.
Venture Capital: Expansion phase funding from specialized investment firms.
Initial Public Offering (IPO): Access to public equity markets for wider investor participation.
The graphical presentation of this lifecycle highlights sources of funding and changes in ownership structures.
Corporate governance often requires the alignment of interests between shareholders and management. Key points include:
Separation of ownership and control can lead to misaligned interests.
Shareholder activism may be necessary for efficient corporate oversight, particularly when market liquidity is limited.
The concept of the Wall Street Walk describes the option for dissatisfied shareholders to sell their shares if they disagree with management.
The ability of large shareholders to intervene often depends on the market liquidity: - In illiquid markets, the cost of exiting (through selling shares) is higher, which may encourage longer-term investment. - Shareholder interventions may improve governance but depend on the market’s liquidity state.
The distinction between information possessed by the market versus that held by firm management can influence decision-making. Firms may gauge market reactions post-announcement and adjust strategies accordingly. Relevant points:
The interaction can act as a feedback mechanism for management decisions.
The example of Hewlett-Packard and the Challenger incident illustrates the varying efficiency of market information compared to firm-level knowledge.
In conclusion, liquidity plays a crucial role in shaping corporate policies, financing opportunities, and governance structures. Understanding these relationships aids in comprehending the broader dynamics within financial markets.
Market liquidity affects capital costs and investment opportunities.
Corporate governance requires alignment of interests between shareholders and management.
Market reactions provide insights for managerial decision-making processes.
In the latter part of the lecture, we will transition to discussing the impact of digital markets, including blockchain and cryptocurrencies, and their ramifications on financial markets.
In this lecture, we explore the relationship between managerial compensation, incentive problems within firms, and the impact of digital markets and cryptocurrencies on financial systems. Below, we summarize the key points, concepts, and equations.
Principal-Agent Problem
Concept: The conflict of interest that arises when the goals of the principal (shareholders) differ from those of the agent (managers).
Incentive alignment: Important to ensure managers act in the best interest of the shareholders.
Objective: Design a salary structure that encourages managers to exert effort in maximizing firm value.
Pay-for-performance: Compensation linked to measurable performance indicators.
Key metrics for performance evaluation: Company profits Stock price
Fundamental value of the firm, denoted as V, can be high or low.
Probability of high outcome θH depends on manager’s effort (denote as e = 1 for high effort and e = 0 for no effort).
Low outcome probability: θL
Effort incurs a cost C.
The ideal contract would pay managers C + 1 for high effort and zero otherwise, yielding an indifference point:
WH − WL ≥ C
The compensation must ensure that the incentive constraint is met, meaning:
The difference in compensation for successful outcomes must exceed the cost of effort.
If the contract is based on realized profits V:
Incentive constraint:
Δθ(WH − WL) ≥ C
where Δθ = θH − θL. - Aim to minimize expected wage payments to the manager, 𝔼[W], while meeting the constraint.
Here, stock price serves as an indicator of managerial effort.
If the stock price is a direct function of the manager’s effort, it reflects the manager’s performance more transparently:
Similar incentive structure as the contract based on company value, but easier to enforce due to public availability of stock prices.
Expected payments can be lower than fixed contracts, which can result in significant savings for shareholders.
Efforts of managers are typically hard to quantify. Thus, performance indicators based on outcomes are necessary.
Non-negativity constraint applies: Wages must be non-negative due to limited liability.
Managers may skew their decisions based on reputation concerns, affecting risk-taking behavior:
Risk averse managers may avoid beneficial investments fearing reputational damage.
Conversely, risk-seeking behavior can arise from the desire to gain notoriety.
Trading Costs: Reduced due to electronic platforms.
Market Structure: Shift from dealer-based systems to order-driven markets through algorithmic trading.
Market Fragmentation: Geographical boundaries have diminished in importance; platforms have consolidated.
Transparency: Improvement in access to information, though anonymity has its limitations.
Latency: The time delay for transactions has decreased drastically, enabling rapid trades and arbitrage.
Algorithmic trading allows for enhanced execution of trades but may introduce liquidity challenges when market conditions change abruptly.
High-frequency trading (HFT) highlights issues of access to millisecond execution times and may exacerbate market crashes under certain conditions.
Definition: A distributed ledger technology facilitating secure and transparent transactions without intermediaries.
Use Case: Offers potential for decentralized exchanges where transactions can occur without traditional financial institutions.
Pros:
Lower transaction costs by eliminating intermediaries.
Transparency and record-keeping of transactions.
Ability to implement smart contracts, reducing counterparty risk.
Cons:
Limited processing capacity can cause delays (e.g., Bitcoin’s typical processing time).
Counterparty risks persist without a central clearinghouse.
Market fragmentation and execution risk can lead to inefficiencies.
The transformation of financial markets through digitalization has substantial implications for corporate governance, compensation structures, and overall market efficiency. The advent of blockchain and cryptocurrencies presents both exciting opportunities and challenges, requiring careful consideration of their effects on existing financial systems.
For students interested in delving deeper into these topics, pursuing courses in contract theory, corporate finance, and digital markets is highly recommended.
In this lecture, we explore the topic of High-Frequency Trading (HFT) within the framework of financial market microstructure. HFT represents a sophisticated form of algorithmic trading characterized by high speeds and high turnover rates. Before delving into HFT, some concepts from previous discussions were revisited.
Last week, we discussed:
Liquidity and its interaction with corporate governance.
Inter-firm decisions and their implications in financial markets.
The effects of globalization and digitalization on market transactions, specifically improvements in transparency, liquidity, and reduced trading costs.
The current state of cryptocurrencies and blockchain technologies.
Event: On a recent Monday, crude oil futures recorded negative prices, trading at $-37 per barrel.
Question: How could negative prices occur?
Explanation:
Mass cancellations of positions by speculators ahead of contract rollovers.
Selling pressure overwhelmed demand due to strategic traders needing to exit positions in light of anticipated delivery.
This event exemplifies how algorithmic trading dynamics can lead to unexpected market outcomes.
Algorithmic trading encompasses various trading strategies, including:
High-Frequency Trading: Utilized by arbitrageurs for profit maximizing.
Order Execution Optimization: Algorithms split large orders into smaller child orders to minimize market impact.
A recent paper presented at a seminar unveiled that:
Uninformed traders can utilize algorithms effectively, investing time and resources into sophisticated trading strategies.
Terminology:
Parent Orders: Large orders submitted by institutional investors.
Child Orders: Smaller orders generated from parent orders by algorithms.
From the empirical study, key points include:
Average parent order attempts to trade approximately $300,000 over 84 minutes, equating to about 5% of market volume during the order’s lifespan.
On average, a parent order spawns 63 runs, producing about 539 child orders.
Market orders represent less than 0.4% of all child orders, in contrast to over 50% for retail investors. Most child orders are limit orders (approx. 80%).
The median time to order execution is roughly 5 milliseconds.
HFT involves adapting trading strategies to exploit small discrepancies in prices, requiring significant investments in technology to reduce latency. HFT accounts for over 50% of all trading volume in the US.
Speed Advantage: Traders are willing to spend substantial investments to improve execution speed.
Market Impact: Both filled and unfilled orders can have discernible impacts on price, revealing trader beliefs and expectations.
The model presents a continuum of profit-maximizing institutions who can choose to become high-frequency traders:
Ui = V + Yi
where Ui is the overall value for trader i, V is the fundamental value, and Yi is the idiosyncratic component of the trader’s value.
HFT traders learn the fundamental value V before others—this intrinsic speed provides two advantages:
They learn V prior to executing trades.
They encounter trading opportunities with a higher probability than slower traders.
This lecture highlighted the complexity and strategic considerations associated with high-frequency trading. As markets evolve rapidly towards algorithm-driven frameworks, the understanding of market dynamics, trader behaviors, and price formation becomes increasingly intricate.
In the next sections, we will explore specific research papers related to HFT, aiming to dissect the theoretical foundations and empirical investigations to broaden our understanding further.
In this lecture, we explore the equilibrium properties of a high-frequency version of the Gloucester-Milgram model, focusing on the behavior of fast and slow traders in the market. We identify how the expectations of the dealer influence the spread and equilibrium prices.
Fast Traders: Traders with high expected valuations who are always willing to buy the asset.
Slow Traders: Traders with lower valuations who decide to trade based on their private valuations.
Dealer: The intermediary responsible for setting the ask and bid prices depending on the market conditions.
The valuations of traders can be categorized as follows:
Fast traders with good news have the highest valuation.
Fast traders with bad news have the lowest valuation.
Slow traders with high private valuations can value the asset more than fast traders with conflicting signals.
We assume parameters of the model such that:
$$\epsilon > \delta > \frac{\epsilon}{2}$$
where ϵ is the standard deviation of the fundamental value V and δ is the standard deviation of the private valuation y. This implies that fundamental values are more significant than private valuations, yet both remain important in the trading decisions.
We identify three types of equilibria, denoted as P1, P2, and P3:
In this equilibrium, the ask price and bid price are closely spaced (narrow spread). All types of traders (both fast and slow) actively buy and sell the asset.
In this scenario, the spread widens.
Fast traders with good news and high private valuations choose to buy.
Slow traders continue to buy if their valuations are high.
Fast traders with low valuations or bad news are driven out of the market.
This is characterized by a wider spread where:
Only fast traders with extreme valuations (good news and high private valuation) trade.
Slow traders are excluded from market participation.
A theoretical case where no trade occurs, termed as the "poor equilibrium." This equilibrium is not expected to occur under normal conditions.
The P1 equilibrium is Pareto dominant over P3, providing better trading outcomes for all participants.
As the proportion of informed traders α increases, the welfare of others decreases due to adverse selection, leading to market inefficiencies.
For fast traders (let’s denote their valuations as V and Y), the profit can be expressed as:
Profitfast = 𝔼[V|buy] − ask price
In equilibrium P1, their expected valuation as they trade under good news is:
𝔼[V|Trade] = μ + ϵ
For slow traders, the profit evaluation changes to:
Profitslow = 𝔼[Y|buy] − bid price
where their expected valuation conditional on trading is affected by their private information.
Through period zero, investors choose to become fast traders or remain slow based on their costs. This leads to a competitive dynamic where larger institutions can absorb the cost of being informed and capitalize on multiple market interactions.
Overall, determined by the expectations, market parameters, and institutional capacities, we wind up with equilibria that create adverse selection effects diminishing welfare as the number of informed traders in the market increases. The insights derived from the Gloucester-Milgram model extend into understanding market functioning under high-frequency trading regimes.
In financial markets, traders can submit either market orders (which execute immediately at the current market price) or limit orders (which set a maximum purchase price or minimum sale price). The choice affects liquidity and transaction costs.
Trading costs can include:
Make-and-take fees charged by exchanges
Bid-ask spreads, which represent the difference between the buying (ask) and selling (bid) prices
The model compares market orders (MO) and limit orders (LO), each with associated fees:
FMO: Fee per share for market orders
FLO: Fee per share for limit orders (only paid on execution)
The total revenue for the exchange per trade (F) is given by:
F = FMO + FLO
Let V be the fundamental value of the asset, and let Yi represent individual trader valuations which are uniformly distributed.
The trader’s valuation can be defined as:
Valuation = V + Yi
To compute bid (a) and ask (b) prices in equilibrium, we apply the condition of indifference between market and limit order submission:
For BUY orders (high valuation traders):
Profit from MO = (V + Yi) − (a + FMO)
Profit from LO = (V + L) − (b + FL0) ⋅ P(S|LO)
Where P(S|LO) is the probability a limit order executes, leading to the condition of indifference:
ProfitMO = ProfitLO
This gives us two equilibrium conditions which can be solved for bid and ask prices.
The determined spread depends on the order fees as follows. For a given spread S, the half spread is:
$$\text{half-spread} = \frac{b - a}{2}$$
Increase in FLO (limit order fee):
Increases the spread as limit orders become less appealing, thus widening the bid-ask spread.
Increase in FMO (market order fee):
Decreases the spread as market orders become costlier, encouraging the use of limit orders.
The concept of payment for order flow involves brokers receiving payments from dealers to direct order flow, which impacts the posted bid-ask spreads and overall market efficiency.
With payment for order flow, some dealers may gain exclusive access to certain types of orders, and their quotes will reflect this information asymmetry—differing from the classic competitive equilibrium obtained without such arrangements.
The exercises undertaken provide insights into the interaction between trading costs, trader behavior, and the characteristics and inefficiencies that arise in markets under various conditions, including payments for order flow and asymmetric fee structures.
This document summarizes the complexities of Kyle’s model combined with the Stoll model, focusing particularly on the behavior of risk-averse dealers.
In Kyle’s model, we consider the following elements:
One asset with a fundamental value denoted by D, which is normally distributed.
A single informed trader who selects the order size ξ.
A number of noise traders submitting random orders, which are normally distributed with mean zero.
Competitive dealers that set a price schedule based on total order flow Q.
The total order flow Q is defined as:
Q = ξ + ν
where ν is the order flow from noise traders, and the total order influences the price P, given by:
P = P0 + λQ
where P0 is an intercept and λ is a slope parameter determined by market dynamics.
A notable deviation in this iteration of Kyle’s model is the introduction of risk-averse dealers, modeled by a utility function:
$$U(W) = \mathbb{E}[W] - \frac{1}{2} \gamma \text{Var}(W)$$
where γ represents the risk aversion coefficient, W denotes future wealth, and dealers hold initial asset z0 and cash c0 (the latter of which turns out to have no impact).
Dealers in this model are assumed to take the price P as given, which generates an inconsistency since prices are determined by the order flow. This leads to the following decision rule:
If profits are positive per trade, dealers might want to buy up to a limit because of their risk aversion.
In contrast, under risk neutrality, dealers would buy/sell infinitely if profits exist strictly on either side.
The goal here is to find 𝔼[V|Q] and Var(V|Q). From econometrics, we know:
$$\mathbb{E}[Y | X] = \mathbb{E}[Y] + \frac{\text{Cov}(Y,X)}{\text{Var}(X)} (X - \mathbb{E}[X])$$
Applying this to 𝔼[V|Q]:
$$\mathbb{E}[V | Q] = \mathbb{E}[V] + \frac{\text{Cov}(V,Q)}{\text{Var}(Q)}(Q - \mathbb{E}[Q])$$
The important distinction here is that in standard models, 𝔼[Q] = 0. In our risk-averse model, it becomes necessary to account for 𝔼[Q] being centered around x0.
To find Var(V|Q):
$$\text{Var}(V | Q) = \text{Var}(V) - \frac{\text{Cov}(V,Q)^2}{\text{Var}(Q)}$$
In Part C, we analyze the dealer’s willingness to supply at price P. The supply curve equation follows:
Y(P) = f(P)
where Y is the quantity of the risky asset and P is the price.
The supply decisions depend on the expected profit:
In the case of risk neutrality, dealers may freely trade due to perceived profits.
Under risk aversion, the risk impacts their willingness to engage in trades.
To determine quantity Y:
Dealers maximize:
$$U(W) = \mathbb{E}[W] - \frac{1}{2} \gamma \text{Var}(W)$$
This involves differentiating the utility function concerning Q.
This exploration of Kyle’s model integrated with risk-averse dealer behavior emphasizes the importance of understanding conditional expectations and the resulting implications on supply curves in financial markets. Use of econometric principles allows us to quantify decisions based on the observations of order flow.
We shall focus on the topic of high-frequency trading (HFT) and its impact on market dynamics and public information.
HFT introduces endogenously informational asymmetries among traders.
Increased competition from HFT can harm liquidity by widening bid-ask spreads, primarily due to concerns over informed trading.
While HFT allows for quicker price discovery, it does not significantly improve efficiency in price formation.
The concept of HFT is compared to an arms race where traders invest unnecessarily to gain speed advantages over each other.
Investment in speed results in an overall reduction in liquidity without enhancing price discovery.
A solution proposed to counter this issue is replacing continuous auctions with frequent batch auctions (e.g., auctions every 100 milliseconds).
HFT traders can exploit arbitrary price differences between correlated assets like ETFs and futures contracts over short time intervals.
The degree of correlation between asset prices increases with the observation time. However, it remains zero or very low at microsecond intervals, indicating persistent arbitrage opportunities for the fastest traders.
The relationship between the spread s and trader profits can be formalized as:
$$\begin{aligned}
\text{Expected Profit}_{market\ maker} = \lambda_{invest} \cdot \frac{s}{2} - P(\text{jump}) \cdot \text{loss from sniping}\end{aligned}$$
Where:
λinvest: frequency of uninformed traders entering the market.
P(jump): probability of market value jumping due to new information.
The equilibrium condition requiring market makers to be indifferent whether to act as market makers or snipers can be expressed as:
$$\begin{aligned}
\text{E}[\text{Profit}_{market\ maker}] = \text{E}[\text{Profit}_{sniper}]\end{aligned}$$
This equilibrium condition dictates the market spread and demonstrates that irrespective of the number of HFT participants, the liquidity does not necessarily improve, as represented by:
$$\begin{aligned}
s^* = f(\lambda, J, \text{other parameters}) \text{ and does not depend on } n.\end{aligned}$$
Higher-order beliefs influence trader actions based on conjectures about others’ information and strategies.
Public announcements often lead to increased trading volumes — a phenomenon that needs explanation beyond traditional models.
Consider a setup where an asset’s value θ comprises two independent components, θ1 and θ2, observed through various signals.
The impact of public signals on prices typically results in changes in both ask and bid prices, representing shifts in expected asset valuation:
$$\begin{aligned}
\text{Ask Price} = \mathbb{E}[\theta | \text{signal}]
\end{aligned}$$
The spread, however, remains unaffected by the public signals and does not directly correlate with trading volumes.
Despite the saturation of HFT traders, their presence does not inherently enhance market efficiency or liquidity.
Transitioning from continuous auctions to batch auctions may mitigate asymmetric information effects and slow down the arms race in trader speed.
The interaction between public information and trading volumes remains an area of active investigation, with implications for understanding trader behavior and market dynamics.
We begin our exploration of the contour model by examining a simple example that illustrates the divergence of second-order beliefs. We will illustrate how this disagreement impacts trading strategies and results in increased trading volumes.
Consider two groups of traders, denoted as Group I and Group J. The fundamental value of an asset, Θ, can be represented as a sum of two components:
Θ = ΘI + ΘJ
Traders in Group I receive an informative signal about ΘI plus some noise, while traders in Group J receive a signal about ΘJ plus noise. Additionally, there exists a public signal about the total Θ. For simplification, we assume that these random variables are independent and normally distributed with zero mean.
Let’s first analyze the situation without a public signal. Trader I will form an expectation of Θ based on the information available to them:
E[Θ|information of trader I] = E[ΘI|XI] + E[ΘJ|lack of information]
As trader I only has information about ΘI, we have:
E[Θ|information of trader I] = E[ΘI|XI] + 0 = E[XI]
Thus, trader I’s second-order belief regarding trader J’s valuation of Θ is zero:
E[ΘJ|XI] = 0
Similarly, trader J will also have no insight into trader I’s valuation. Notably, trader I’s opinion about trader J’s valuation does not depend on XI.
Now, consider the case where both traders observe a common signal Y that provides information about the total Θ. The conditional expectation that trader I has regarding Θ becomes:
E[ΘI|XI, Y]
In this case, trader I can expect:
E[ΘI|Y] + E[ΘJ|XI, Y]
Given the normality of the distribution, we express this as:
E[ΘJ|Y] = βXI + cY
where β and c are constants reflecting the relationship between the signals and the expectations.
The second-order belief, concerning trader I’s expectations about trader J’s valuation, will decrease in XI indicating that a higher private signal leads trader I to assume that trader J will value the asset less:
E[ΘJ|XI] ↓ as XI↑
The intuition arises from the assumption that if trader I receives a strong signal concerning ΘI, then they may infer that ΘJ must be lower:
ΘJ = Θ − ΘI
The model consists of three periods:
In Period 1, traders in Group I trade based on their signals.
In Period 2, Group J traders enter and trade.
In Period 3, the true value Θ is realized, and traders consume the assets.
Each trader is allowed to condition their demand Di on the price P. The expected wealth for trader I is represented as:
WI = DI ⋅ (P2 − P1)
with exponential utility given by:
$$U(W_I) = E[W_I] - \frac{\gamma}{2}Var(W_I)$$
In addition, we will assume a normal distribution for aggregate supply. The market clearing condition in Period 1 requires the total supply U1 to match the demand DI. Similarly, in Period 2, the aggregate supply U2 needs to match the total demands from Group J.
To find the equilibria, we consider the linearity of price with respect to various signals. The prices are expressed as:
P1 = a1ΘI + bY + cU1
P2 = a2ΘJ + b′Y + c′U2
where ai, bi, ci are coefficients to be determined.
The optimal demand and price equations depend on first and second-order beliefs. Notably, second-order beliefs significantly influence the expected utility of traders when making their decisions, showing that these higher-order beliefs play a crucial role in trading outcomes.
Our analysis shows that the divergence of second-order beliefs necessitates trading and influences trading volume, particularly around public announcements that introduce additional information. Consequently, public signals lead to increased trading activity due to discrepancies in individual valuations among traders.
In future studies, it will be important to further investigate the implications of such models on trading dynamics and market behavior, especially the incorporation of varying trader horizons and the distribution of information among participants.
In this exercise class, we revisit the topics of post-trade transparency from Lecture 9 and the value of liquidity from Lecture 10. The fundamental question is to show that price discovery is more efficient under transparent markets compared to opaque markets.
Price discovery refers to the process through which the market determines the price of an asset based on supply and demand dynamics. The efficiency of price discovery can be measured using the variance of the difference between the market price and the true value of the asset.
Let k denote the regime (transparent or opaque).
Let subscripts 1 and 2 denote time periods.
The fundamental value of the asset is denoted as v.
The mean of the asset values is denoted as μ.
The standard deviation is denoted as σ.
The probability that both traders are informed is given by π.
The buyers and sellers of the asset may be informed or uninformed traders.
Informed Traders: Traders with knowledge about the true value of an asset.
Liquidity Traders: Uninformed traders whose order flow is not based on information about the fundamental value.
The model incorporates two periods of trading with one asset:
Informed traders have a probability π of being present.
If both traders are informed, they make correlated trades leading to better price discovery.
If both are liquidity traders, order flow is random, thus lowering price discovery efficiency.
1. Transparent Market: All dealers observe the first order. 2. Opaque Market: Only the dealer executing the first order observes it.
In a transparent market, during period one, prices can be represented as follows:
$$\begin{aligned}
\text{Ask Price: } P_{1}^{\text{ask}} &= \mu + \pi \sigma \\
\text{Bid Price: } P_{1}^{\text{bid}} &= \mu - \pi \sigma\end{aligned}$$
Calculating the average expected square deviation (residual variance) over the two periods:
$$E[(P_{t} - v)^2] = \frac{1}{2} \left( (1 - \pi)^2 \sigma^2 + (1 - \pi) \sigma^2 \right)$$
In an opaque market, in the first period, the expected price variance can be shown as:
E[(P1 − v)2] = (2π − 1)(σ2)
The prices for the second period are constant and equal to the fundamental value v, hence the variance in period two is always zero.
After calculating the residual variances in both types of markets, we see:
$$\begin{aligned}
\text{Residual Variance (Transparency)} &= (1 - \pi)(1 + \pi)\sigma^2 \\
\text{Residual Variance (Opacity)} &= (1 - \pi)(2\sigma^2)\end{aligned}$$
To compare the efficiency of price discovery:
$$\frac{(1 + \pi)}{2} \sigma^2 < 2\sigma^2 \quad \text{if } \frac{1}{2} < \pi < 1$$
Thus, price discovery is more effective in a transparent market.
We demonstrated that post-trade transparency enables better price discovery relative to opaque market structures. This understanding is crucial for evaluating market efficiencies and the role of transparency in financial systems.
In this note, we delve into the concepts of liquidity and its valuation within financial markets using various models. Notably, we will cover the Gordon model, liquidity capital asset pricing model (CAPM), and the Patterson model.
The Gordon growth model, or dividend discount model, focuses on determining the price of a stock based on its future dividend payments. More formally, it is represented as:
$$P = \frac{D}{r - g}$$
where P is the price of the stock, D is the expected dividend, r is the required rate of return, and g is the growth rate of the dividends.
In this model, we investigate how dividends affect the liquidity premium. The liquidity premium is defined as:
Liquidity Premium = r − R
where r is the required rate of return, and R is the observed rate of return accounting for transaction costs due to illiquidity.
We assume:
Investors buy and hold the stock for one period,
The constant spread s is relative to the asset price,
The fundamental value at time t is denoted by μt while the mid price is represented as Mt = μt.
The rate of return 1 + R, can be defined considering both price appreciation and dividends received. The return can be expressed as follows:
$$1 + R = \frac{M_{t+1} + D}{M_t}$$
Substituting for the buy and sell prices accounting for spreads, we can rearrange and find the relationship between these variables:
$$1 + R = \frac{\left(1 - \frac{s}{2}\right) \mu_{t+1} + \mu_t D}{\left(1 + \frac{s}{2}\right) \mu_t}$$
From the derived expressions, the liquidity premium decreases with increased dividend yield D:
R = R − D ⇒ Liquidity Premium is decreasing in D.
The intuition is that dividends are unaffected by liquidity costs, and as a higher portion of returns comes from dividends, the effect of liquidity decreases.
The Patterson model examines how the spread is affected by the probability of meeting a dealer, denoted as ϕ.
The generated spread s in the model is represented as:
s = f(ϕ, other parameters),
where the specific functional form of f will depend on parameters like bargaining power and the probability of value switching ψ.
If $\psi > \frac{1}{2}$: An increase in ϕ raises the spread due to higher valuation leading to more willing buyers.
If $\psi < \frac{1}{2}$: An increase in ϕ lowers the spread as it makes dealers more competitive due to increased chances of finding buyers.
The non-monotonic effect arises because:
When ϕ increases, the likelihood of trading increases, pushing buyers to offer more for the asset, which tends to increase the spread.
Conversely, an increase in trading opportunities also decreases the dealers’ market power, leading to a decrease in the spread.
By analyzing these models, we gain deeper insights into how liquidity impacts the financial markets, specifically how dividends influence the liquidity premium and how spreads react to changing probabilities in trading environments. Understanding these relations helps us grasp more complex market dynamics effectively.
Financial markets microstructure studies the processes and outcomes of exchanging assets under specific trading rules. Key focus areas include:
High-frequency trading (HFT) and algorithmic trading (Algo trading)
Price discovery
Market liquidity and efficiency
Behavioral aspects of trading
High-frequency trading likely creates informational asymmetries and can impair market liquidity.
Algo traders do provide liquidity in the market but can also lead to issues with adverse selection.
The Condor model shows how public announcements can generate high trading volumes, contradicting standard economic models that predict price adjustments without actual trading.
The model explores higher-order beliefs where a trader’s expectation about other traders’ beliefs impacts their own trading decisions.
Bubbles are often difficult to characterize formally. Here are several definitions:
Webster Dictionary: A bubble indicates trading in high volumes at inflated prices.
Wikipedia: Trade in high volumes at prices significantly deviating from intrinsic values, including negative bubbles.
Investopedia: A surge in equity prices, often unjustified by fundamentals, followed by a drastic drop during a sell-off.
Chicago Fed: Existence of bubbles when market prices significantly exceed fundamental valuations for a prolonged timeframe.
Enron: An energy company that saw massive growth in the 90s, followed by a collapse in the early 2000s.
US Housing Bubble: A significant increase in housing prices leading up to the 2008 financial crisis, not supported by fundamentals like building costs or population growth.
Cryptocurrency Bubble: Sudden surges in Bitcoin prices, followed by sharp declines.
Uranium Bubble (2006): A price surge following the flooding of a Canadian uranium mine, leading to speculation about supply shortages.
Herding occurs when traders ignore their private signals in favor of public information, leading to similar trading actions:
Different forms of herding exist based on rational responses to new information.
The model demonstrates why trading decisions can cascade through the market, often leading to incorrect outcomes where poor private signals dominate.
Let V denote the fundamental value, which is either low or high:
Decision Dt: Binary, where D = 1 indicates buying, and D = 0 indicates not buying.
Payoff for buying is defined as:
Payoff = V − M
where M is the market price.
Traders’ beliefs regarding the fundamental values evolve over time:
Let Pt: Probability that V is high upon trader arrival.
Public belief Qt: An updated belief based on previous actions.
Posterior belief Rt: Combined information from Pt and observed actions.
The decision to invest is determined as follows:
$$D_t = \begin{cases}
1 & \text{if } R_t \geq R^\ast \\
0 & \text{if } R_t < R^\ast
\end{cases}$$
Herding arises when:
Agents act sequentially and become influenced by observable actions of previous traders.
Initially high levels of private information become less visible due to public consensus.
The existence of a threshold Pt* where investment occurs based on perceived public value rather than private signals.
Cascading occurs when early trading decisions disproportionately influence later traders, leading everyone to either buy or sell despite fundamentals.
Pt* = f(Qt)
Where f is a function that shows the dependence of private belief on public information.
Adding noise traders introduces randomness into decision-making, which may prevent herding.
In the context of price flexibility, herding can still occur if mispricing influences decision-making.
Understanding financial market microstructure and the implications of herding behavior can help in grasping how and why bubbles arise. Various models illustrate the complexity of trader behavior and market dynamics, highlighting the importance of information both private and public.
This learning session focuses on the concepts of speculative and non-speculative bubbles, herding behavior in financial markets, and a specific model by Abreu and Brunnermeier (2003). We will explore how information asymmetry and the aggregation of beliefs influence trading behaviors and price dynamics.
A bubble can be described as a deviation of the market price from its fundamental value. There are different types of bubbles:
Speculative Bubbles: These occur when traders drive up the price of an asset beyond its intrinsic value based on expectations of future price increases, not supported by fundamentals.
Non-Speculative Bubbles: Occur when fundamentals do not support the price movement, often resulting from collective trading behaviors driven by information asymmetries.
In some scenarios, all traders possess information about the true value V of an asset. However, market makers do not know how many of the traders are informed, leading to a slow adjustment of prices. This can result in a situation where:
Price > Fundamental Value (Low Asset Value) (Bubble)
Conversely, if many informed traders sell while the price remains high, it can also suggest a bubble in the opposite direction.
Herding can arise due to:
Information asymmetry among traders, leading to failures in aggregating information.
Reputation concerns among managers in an investment context.
Consider managers tasked with making investments for their clients. They can either be classified as smart or dumb, receiving signals about asset values. If both receive the same signal, they will act similarly. However, if a manager thinks their reputation may depend on following the herd, they might choose to invest in line with other managers rather than solely based on signals.
If Manager 1 decides to invest, Manager 2, despite having a different signal, may also invest to avoid being labeled as dumb. This can lead to collective investments that affect price dynamics, even if the signals differ.
The model introduces information aggregation failures and factors in the importance of higher-order beliefs among traders. It describes the price pt of an asset over time:
At time t = 0, the asset value is V0 = 1.
The value grows at rate G.
At T0, this growth rate slows down to R < G.
The price might continue to rise at the former rate G until it adjusts.
The price continues to increase until either:
A certain time τ is reached at which the price adjusts to the new fundamental value.
Enough rational traders (κ) decide to sell, triggering a price adjustment.
The traders in the model are classified as:
Rational Traders: These traders are progressively informed about the asset’s mispricing and will make decisions to sell based on their beliefs and conditions.
Behavioral Traders: These traders do not realize a slowdown. They assume that growth G will persist indefinitely.
Behavioral traders will continue to buy the asset, believing incorrectly that its price will climb indefinitely. This results in significant mispricing that can lead to bubbles persisting until the market corrects.
The presence of coordination is necessary for bubbles to persist or burst. Traders face a dilemma between selling early to secure profits and waiting to maximize returns. When traders coordinate, it can lead to sudden price adjustments or crashes.
Exogenous Crash: Occurs when some external event triggers a price adjustment.
Endogenous Crash: Triggered by the collective behavior of traders once a critical mass of informed rational traders decides to sell.
The study of bubbles, herding behavior, and information asymmetry offers insights into financial market dynamics. Understanding these concepts helps to clarify how expectations and collective actions can lead to significant price movements, often contrary to fundamental values.
This set of notes covers key concepts from financial market microstructure, focusing on herding models, bubbles, and auction mechanisms.
In the previous lecture, we explored the idea of herding in financial markets. Key points include:
Public vs. Private Signals: Public information can outweigh private signals. If traders perceive public information as informative enough, they may ignore their private signals.
Implications of Herding: When all agents rely on the same public information, decisions may lead to incorrect outcomes regarding the underlying state of the market.
Key Models:
Sniffing-Sorenson Model: This model illustrates how herding can lead to price bubbles.
Bruin-Bruno-Meyer Model: Focuses explicitly on theorizing bubbles and includes higher-order expectations and lack of common knowledge regarding mispricing.
Today’s lecture will focus on auction models and their relevance in financial markets. The models explored are applicable beyond financial domains, with common uses in:
Contextual ad auctions (e.g., Google ads).
Spectrum auctions for mobile service providers.
Private value first-price auctions; model definitions discussed below.
Auction Types: Sealed bids, open bids, first-price, second-price, single-sided, double-sided auctions, etc.
Information Symmetry: Auctions can have symmetric or asymmetric information structures.
Private vs. Common Valuations: Traders may have different private valuations for the asset or agree on a common value with different information.
The setting of the model is:
Setup: One item for sale with n potential buyers.
Each buyer i has a private valuation vi, where vi is independently and identically distributed (iid).
Bids are made simultaneously and the highest bid wins.
Agents are assumed to be rational and risk-neutral:
Profiti = vi − bi
where:
bi: bid made by agent i,
vi: private valuation of agent i.
The probability of winning if bid bi is:
P(Winner) = P(bi > bj for all j ≠ i)
Given symmetric increasing bidding strategies β(x), the winning probability translates into:
P(bi > β(y1)) where y1 = max (b1, b2, …, bn − 1).
To maximize expected profits:
$$\frac{d \text{E}[\text{Profit}_i]}{db_i} = 0$$
Leading to finding equilibrium strategies β.
The desired equilibrium can be realized by solving a differential equation for the equilibrium bidding strategy:
β(x) given G(β − 1(bi)).
The insights gained from this model allow for analogies to different auction types. The next steps will include examining:
Common value first-price auction.
Second-price auctions.
Double auctions.
These models provide a powerful framework for understanding trading mechanisms and market dynamics in financial contexts.
If there are any questions about the material covered, please feel free to raise them during class or after the lecture.
In a common value first price auction, there exists one item for sale that has a fundamental value V which is the same for all bidders. The value V is unknown, but all bidders receive informative private signals denoted as Xi. Each Xi is a noisy estimate of the fundamental value V with zero noise assumed for simplicity.
The auction format is still a first price sealed bid auction: all bidders submit their bids simultaneously.
The bidder with the highest bid wins the item, pays their own bid, and the others pay nothing.
The expected profit for a winning bidder can be expressed as:
πi = 𝔼[V|win] − Bi
where Bi is the bid submitted by bidder i. The winner’s curse refers to the idea that winning the auction implies that the winning bid was higher than that of all other bidders, which suggests that the asset may be less valuable than initially thought.
The expected value conditionally on the private signal Xi and winning can be denoted as follows:
𝔼[V|Xi, win] < 𝔼[V|Xi]
indicating that the expectation of the value of the item (given the win) is lower than the expectation based on the private signal alone.
Assimize all agents follow a symmetric bidding strategy represented as β(X). Let Y1 be the highest signal among competitors. The conditioning on the highest signal involves a relationship that allows us to define expectations based on the signals received:
𝔼[V|Xi, Y1]
An individual bidder’s optimal strategy can be characterized by choosing whom to mimic — denoting this type Z — which leads to a modified expected profit from bidding:
π(Z) = ∫ − ∞Y1(𝔼[V|Y1]−β(Z))f(Y1)dY1
Here, f(Y1) is the distribution of signals.
Taking the first-order condition to find optimal Z leads to:
$$\frac{d\pi(Z)}{dZ} = 0$$
Resulting in a differential equation which can be solved under certain boundary conditions.
Bid shading occurs for two main reasons in common value auctions:
Bidders shade their bids below their valuations to increase the likelihood of a positive profit.
The winner’s curse implies that winning suggests a lower true value of the asset than expected.
The framework of double auctions differs from single auctions, where two agents, one buyer and one seller, interact:
The buyer submits a bid BB and the seller submits an asking price BS.
The trade occurs if BB > BS.
The expected profit for the buyer is given by:
πB = 𝔼[XB] − BB
The expected profit for the seller simplifies to:
πS = BS − 𝔼[XS]
This scenario can be interpreted as a variation of second-price auctions, with the role of "winning" effectively switching based on the bid comparisons.
Meyerson-Satterthwaite theorem highlights the inefficiencies in double auctions, stating there is no mechanism ensuring efficient trades occur without incurring a deficit or requiring external compensation. The double auction may lead to trade ceasing even when the seller’s valuation is lower than the buyer’s due to bid shading.
In conclusion, understanding common value first price auctions and their implications (like the winner’s curse and bid shading) is essential for analyzing auction dynamics in economic scenarios. The exploration of double auctions extends these concepts into more complex interactions between buyers and sellers, revealing inherent inefficiencies in the mechanisms employed.