V. ADVERSE SELECTION,
TRADING AND SPREADS
A. Information and Trading
•
•
•
•
•
•
•
The economics of information is concerned with how information along with the
quality and value of this information affect an economy and economic decisions.
Information can be inexpensively created, can be reliable and, when reliable, is
valuable.
The simplest microeconomics models assume that information is costless and all
agents have equal access to relevant information.
But, such assumptions do not hold in reality, and costly and asymmetric access to
information very much affects how traders interact with each other.
Investors and traders look to the trading behavior of other investors and traders for
information, which affects the trading behavior of informed investors who seek to
limit the information that they reveal.
Here, we discuss the market mechanisms causing prices to react to the information
content of trades (market impact or slippage), and how traders and dealers can react
to this information content to maximize their own profits (or minimize losses).
This chapter is concerned primarily with problems that arise when traders and other
market participants have inadequate, different (asymmetric information availability)
and costly access to information.
Adverse Selection
•
Adverse selection refers to pre-contractual opportunism where one contracting
party uses his private information to the other counterparty’s disadvantage.
– For example, the adverse selection problem can arise when a pyromaniac purchases fire
insurance.
– The agent (insured or customer) has private information with respect to the higher anticipated
costs of the insurance coverage or lease, but pays a “pooling” premium for the incident or
casualty coverage.
– This private information affects the behavior or insurers and other insured clients, in what
might otherwise be taken to be a sub-optimal manner, referred to as the adverse selection
problem.
•
•
•
In a financial trading context, adverse selection occurs when one trader with secret
or special information uses that information to her advantage at the expense of her
counterparty in trade.
Trade counterparties realize that they might fall victim to adverse selection, so they
carefully monitor trading activity in an effort to discern which trades are likely to
reflect special information.
For example, large or numerous buy (sell) orders originating from the same trader
are likely to be perceived as being motivated by special information. Trade
counterparties are likely to react by adjusting their offers (bids) upwards
(downwards), resulting in slippage.
B. Noise Traders
• Noise traders trade on the basis of what they falsely believe to be special
information or misinterpret useful information.
• Noise traders make investment and trading decisions based on incorrect
perceptions or analyses.
• Do noise traders distort prices? Maybe yes if:
– noise traders trade in large numbers
– their trading behavior is correlated
– their effects cannot be mitigated by informed and rational traders
• Milton Friedman suggested that traders who produce positive profits do
so by trading against less rational or poorly informed investors who tend
to move prices away from their fundamental or correct values.
• Fama argued that when irrational trading does occur, security prices
will not be significantly affected because sophisticated traders will react
quickly to exploit and eliminate deviations from fundamental economic
values.
• Figlewski [1979] suggested that it might take irrational investors a long
time to lose their money and for prices to reflect security intrinsic
values, but they are doomed in the long run.
Impact of Noise Traders
•
Noise traders might be useful, perhaps even necessary for markets to function.
– Without noise traders, markets would be informationally efficient and maybe no one would
want to trade.
– Even with asymmetric information access, informed traders can fully reveal their superior
information through their trading activities, and prices would reflect this information and
ultimately eliminate the motivation for information-based trading.
– That is, as Black [1986] argued, without noise traders, dealers would widen their spreads to
avoid losing profits to informed traders such that no trades would ever be executed.
•
However, noise traders do impose on other traders the risk that prices might move
irrationally.
– This risk imposed by noise traders might discourage arbitrageurs from acting to exploit price
deviations from fundamental values.
– Prices can deviate significantly from rational valuations, and can remain different for long
periods.
– Arbitrageurs might ask themselves the following question: “Does my ability to remain solvent
exceed the asset price’s ability to remain irrational?”
C. Adverse Selection in Dealer Markets
• Bagehot [1971] described a market where dealers or market makers stand
by to provide liquidity to every trader who wishes to trade, losing on trades
with informed traders but recovering these losses by trading with
uninformed, noise or liquidity motivated traders.
• The market maker sets prices and trades to ensure this outcome, on
average.
• The market maker merely recovers his operating costs along with a "normal
return.“
• In this framework, trading is a zero sum or neutral game; uninformed
investors will lose more than they make to informed traders.
• Market makers observe buying and selling pressure on prices, set prices
accordingly, often making surprisingly little use of fundamental analysis
when making their pricing decisions.
• The theoretical model of Kyle [1985] describes the trading behavior of
informed traders and uninformed market makers in an environment with
noise traders.
Kyle [1985]: Informed Traders, Market Makers and Noise Traders
•
•
•
•
•
•
•
•
•
Suppose two rational traders have access to the same information and are otherwise
identical. They have no motivation to trade.
Now, suppose they have different information. Will they trade? Not if one trader
believes that the other will trade only if the other has information that will enable
him to profit in the trade at the first trader’s expense.
Rational traders will not trade against other rational traders even if their information
differs. This is a variation of the Akerlof Lemon Problem.
So, why do we observe so much trading in the marketplace? Most of us believe that
others are not as informed or rational as we are or that others do not have the same
ability to access and process information that we do.
Kyle examines trading and price setting in a market where some traders are
informed and others (noise or liquidity traders) are not.
Dealers or market makers serve as intermediaries between informed an uninformed
traders,
Dealers set security prices that enable them to survive even without the special
information enjoyed by informed traders.
Kyle models how informed traders use their information to maximize their trading
profits given that their trades yield useful information to market makers.
Furthermore, market makers will learn from the informed trader’s trading, and the
informed trader's trading activity will seek to disguise his special information from
the dealer and noise traders.
The Kyle Framework
•
Consider a one-time period single auction model involving an asset that will pay in one time period v~N(p0; Σ0)
–
–
•
There are three risk-neutral trader types:
–
–
–
•
•
•
•
•
•
•
a single informed trader with perfect information
many uninformed noise traders
and a single dealer or market maker who acts as an intermediary between all trades.
There is no spread and money has no time value.
Market makers and noise traders seek to learn from informed trader actions who seeks to disguise his information in
a batch market (markets accumulate orders before clearing them).
The informed trader determines x, an appropriate share transaction volume that maximizes trading profits: π = E[(v p) x│v]where p is the market price of the asset.
Noise traders and the market maker will observe total share purchases X = x + u where u reflects noise trader
transactions, bidding up the price of shares p as X increases.
The market maker cannot distinguish between and u, but does correctly observe total demand X.
Neither the dealer nor the noise traders know which trades or traders are informed, but they try to discern informed
demand x from noisy signal X.
Noise traders submit market orders for u shares randomly.
–
–
•
•
•
p0 is the unconditional expected value of the asset.
Variance, Σ0, can be interpreted to be the amount of uncertainty that the informed trader’s perfect information resolves
Noise traders demand u shares, where u is distributed normally with mean E[u] = 0 and variance σu2: u~N (0; σu2).
Informed traders do not know how many shares uninformed traders will trade, but does know the parameters of the distribution of the
demand.
Informed and noise traders submit their order quantities x and u to the market maker in a batch market.
The market maker observes the net market imbalance X sets p such that total order flow X = x + u clears.
The market maker observes X then sets the price as a function of the sum of x and u: p = E[v│x + u].
The Informed Trader’s Problem: Profit Maximization
•
•
•
•
Kyle's Bayesian Learning model assumes that informed investor demand x can be expressed as a simple linear function of v:
x = α + βv where α and β are simple coefficients.
Similarly, the security’s price p, set by the market maker or dealer, is also assumed to be a simple linear function of demand:
p = μ + λ(x + u), where μ and λ are also simple coefficients.
Thus, informed investor demand x is a linear function of true security value v and the security price p is a linear function of
the sum of informed and uninformed investor demand X = (x + u).
The informed trader’s problem is to determine the optimal purchase (or sale) quantity x:
(1)
𝑚𝑎𝑥 𝐸 𝜋 = 𝐸 𝑣 − 𝑝 𝑥 𝑣 = 𝐸 𝑣 – 𝜇 − 𝜆 𝑥 + 𝑢 𝑥 𝑣
𝑥
= 𝐸 𝑣𝑥 – 𝜇𝑥 − 𝜆𝑥 2 − 𝜆𝑢𝑥
•
To maximize the informed trader’s profits,:
(2)
•
𝜕𝐸[𝜋]
= 𝑣 – 𝜇 − 2𝜆𝑥 − 𝜆𝑢 = 0
𝜕𝑥
Note that λ must be positive for the second order condition (the second derivative must be negative to hold for
maximization. We rearrange terms to obtain:
2𝜆𝑥 = 𝑣 – 𝜇
(3)
(4)
•
𝑥 =–
which is linear in v as Kyle proposed it would be. Now, we see that our coefficients α and β are simply:
α=–
•
𝜇
1
+ 𝑣
2𝜆 2𝜆
μ
,
2λ
β=
1
2λ
Thus, the informed trader linear demand function x = α + βv is set based on the dealer pricing function.
Dealer Price Setting
• We now explore our coefficients α and β.
• The dealer observes total order flow X = x + u and sets a single market clearing
price p = E[v│x + u] where x = α + βv.
• Since v and X are normally distributed, we apply the Projection Theorem to p:
(5)
𝐶𝑂𝑉 [𝑣, 𝑥 + 𝑢]
𝑝=𝐸 𝑣 +
𝑥 + 𝑢 − 𝐸[𝑥 + 𝑢]
𝑉𝐴𝑅[𝑥 + 𝑢]
• This dealer pricing function has a straightforward interpretation.
• The sensitivity of the dealer price to total share demand is a function of the
covariance between the stock’s value and total demand for the shares.
– Thus, if the dealer believes that total demand for the stock increases dramatically with
its intrinsic value v (unknown to him, but known to the informed trader), the price that
the dealer sets for shares will be very sensitive to total demand.
– If the informed trader dominates trading, the dealer will set the price of the security
mostly or entirely as a function of total demand for the security.
– Sensitivity to total demand will diminish as uninformed demand volatility increases.
– As total demand deviates more from expected demand, share prices will increase.
Informed Trader Demand and Dealer Price Adjustment
•
Since x = α + βv, and Σ0 is the variance of asset payoffs v, the variance of informed trader demand VAR[x] will
equal 20. This means that VAR[x+u] = Σ0 + u2, which means that the dealer pricing equation is:
(6)
•
𝑝=𝐸 𝑣 +
β is the slope of a line plotting a random dependent variable X or (x + u) with respect random variable v:
(7)
•
𝐶𝑂𝑉 [𝑣, 𝑥 + 𝑢]
𝑥 + 𝑢 − 𝛼 − 𝛽𝐸[𝑣]
𝛽 2 Σ0 + 𝜎𝑢2
𝑝=𝐸 𝑣 +
𝜆=
𝜇 = 𝐸 𝑣 + 𝜆 −𝛼 − 𝛽𝐸[𝑣]
Next, we will use α and β coefficients from above to demonstrate that μ = E[v]:
(10)
•
𝛽Σ0
+ 𝜎𝑢2
𝛽 2 Σ0
which implies μ = p + λ(-x - u) and:
(9)
•
𝑥 + 𝑢 − 𝛼 − 𝛽𝐸[𝑣]
Kyle suggested a linear relationship between the security price and its demand: p = μ + λ(x + u). This implies a
slope λ equal to:
(8)
•
𝛽Σ0
+ 𝜎𝑢2
𝛽 2 Σ0
𝜇=𝐸 𝑣 + 𝜆
μ
1
μ 1
1
μ
−
𝐸[𝑣] = 𝐸 𝑣 +
− 𝐸[𝑣] = 𝐸 𝑣 + = 𝐸 𝑣
2λ 2λ
2 2
2
2
Now, we will rewrite λ, substituting in for β:
(11)
𝛽Σ0
𝜆= 2
=
𝛽 Σ0 + 𝜎𝑢2
1
Σ
2λ 0
1
2λ
2
Σ0 + 𝜎𝑢2
Solving for Demand and Pricing Coefficients
•
We for λ, starting by multiplying λ and the right hand side of equation 11 by the denominator of the
right hand side of the equation:
𝜆
(12)
•
2
Σ0 + 𝜎𝑢2 =
1
Σ
2λ 0
Simplify the left hand side, then multiply both sides by λ and simplify further by subtracting 1/4Σ0
from both sides:
1
1
Σ0 + 𝜆𝜎𝑢2 =
Σ
4λ
2λ 0
(13)
(14)
(15)
(16)
•
•
•
•
1
2λ
1
4
1
Σ0 + 𝜆2 𝜎𝑢2 = Σ0
2
1
𝜆2 𝜎𝑢2 = Σ0
4
𝜆 =
1 Σ0
1 Σ0
=
2
4 𝜎𝑢
2 𝜎𝑢2
λ is positive and that our second order condition for profit maximization has been fulfilled.
 is the dealer price adjustment for total stock demand or the illiquidity adjustment.
Σ0/u2 is the ratio of informed trader private information resolution to the level of noise trading.
The dealer price adjustment is proportional to the square root of this ratio, increasing as private
information Σ0 is increasing and decreasing as noise trading increases. This means that if the dealer
determines that the informed trader resolves a substantial level of risk relative to the amount of noise
trading, the level of dealer price adjustment will be large.
Informed Trader Demand
• Informed investor demand coefficients α and β are:
𝛼 =–
E[v]
,
β=
Σ0
𝜎𝑢2
1
Σ0
𝜎𝑢2
The informed trader demand function is:
𝑥 = α + βv = –
(17)
E[v]
Σ0
𝜎𝑢2
+
𝑣
Σ0
𝜎𝑢2
=
(𝑣 − 𝐸[𝑣])𝜎𝑢
Σ0
• Informed trader transaction sizes will increase as the variance of
uninformed noise demand for shares σu2 increases.
• This increased noise volume will better enable informed traders to
camouflage their information advantage over the dealer.
• As the informed trader’s information improves relative to the dealer, the
informed trader will seek to camouflage his advantage by reducing his
trade volume. The informed trader will earn his profits by maintaining
more profit on a per share basis rather than on transaction volume.
Dealer Price Setting
•
Recall from equation (7) that the market maker sets the price at p,
which we will rewrite using the result of equation 16:
(18)
(19)
𝑝=𝐸 𝑣 +
𝛽 Σ0
𝛽 2 Σ 0 +𝜎𝑢2
𝑥 + 𝑢 − 𝛼 − 𝛽𝐸[𝑣] = 𝐸 v x + u .
p = μ + λ(x + u) = E[v] + (x+u)
• Notice that the dealer price p is the security’s expected value E[v]
conditioned on total demand [x+u] for the security.
• Higher noise or uninformed trader uncertainty reduces the security price
unless total demand (x+u) is negative.
• The opposite is true for value or cash flow uncertainty, which increases the
informed trader's informational advantage.
• The market maker sets the price at p, such that the informed trader buys
(sells) whenever v > E[v] (v < E[v]), and buys (sells) more aggressively as
this difference increases.
Informed Trader Profits
•
Notice that some of these implications might be clarified with the following
informed trader profit function:
(20)
E π = 𝐸 [v − E v ]x = 𝐸 [v − E v ]
𝑣
Σ0
𝜎2
𝑢
•
•
•
•
•
•
–
E v
Σ0
𝜎2
𝑢
=
Σ0
Σ0
𝜎2
𝑢
= Σ0 𝜎𝑢2 =
Σ0 𝜎𝑢
Informed trader profits are linear and increasing in the quality of their information
advantage and the demand uncertainty σu of noise traders.
A larger value for Σ0 implies a greater deviation in the security value v (known by
the informed trader) from its expected value E[v] (estimated by the dealer).
A larger value for Σ0 implies a larger information advantage to the informed trader.
Greater dealer price uncertainty increases informed trader trading profits.
Increased uninformed trader uncertainty and its associated increase in transactions
mean that the informed trader is better able to disguise from the market maker his
information advantage and trading activity through the increased noise trader
volume σu. This ability to camouflage means that the informed trader can trade
more aggressively, taking larger positions (x or –x) and profits in the stock without
accurately revealing his transactions to the market maker.
The market maker sets a price p such that informed traders earn their profits
indirectly from noise traders. The market maker loses on trades with informed
traders and earns profits on trades with noise traders, earning a competitive profit as
long as informed traders successfully camouflage their intentions at the ultimate
expense of noise traders.
Illustration: Kyle's Adverse Selection Model
• Suppose that the unconditional value of a stock in a Kyle framework is
normally distributed with an expected value equal to E[v] = $50 and a
variance equal to Σ0 = 30.
• An informed trader has private information that the value of the stock is
actually v = $45 per share.
• Uninformed investor trading is random and normally distributed with an
expected net share demand of zero σu2 equal to 5,000.
• The dealer can observe the total level of order volume X = x + u where u
reflects noise trader transactions and x reflects informed demand, but the
dealer cannot distinguish between x and u.
• The ability of the informed trader to camouflage his activity is directly
related to σu2 and inversely related to x.
• What would be the level of informed trader demand for the stock? We solve
for x as follows, using equation 17:
•
𝑥 =–
E[v]
Σ0
𝜎𝑢2
+
𝑣
Σ0
𝜎𝑢2
=–
50
30
5,000
+
45
30
5,000
=
−5
= −64.5497
. 07746
Informed Demand and Dealer Pricing Coefficients
• The informed trader would wish to sell an infinite number of shares to earn
a $5 profit on each, but cannot because the dealer would correctly infer that
his share sales convey meaningful information, and the dealer's price
revisions would lead to slippage.
• Thus, at what level does the dealer set his price, given the total demand X =
[x + u] = -64.5497 + 0 that he observes? First, we solve for parameters in
the dealer pricing equation:
α=−
𝐸[𝑣]
=−
Σ0
𝜎𝑢2
β=
1
Σ0
𝜎𝑢2
.
𝜆 =
=
50
30
5,000
1
30
5,000
= −645.49
= 12.91
1 Σ0
1
60
=
= .0387
2 𝜎𝑢2
2 10,000
The Dealer Price
• The Dealer sets her price as follows:
𝛽Σ0
𝑝=𝐸 𝑣 + 2
𝑥 + 𝑢 − 𝛼 − 𝛽𝐸 𝑣
𝛽 Σ0 + 𝜎𝑢2
= 50 +
12.91∙30
12.912 ∙30 +5,000
−64.5497 + 0 + 645.49 − 12.91 ∙ 50
= 50−.0387 ∙ 64.5497 = 47.50
•
•
•
D. Adverse Selection and the Spread
Walrasian markets assume perfect and frictionless competition and symmetric information
availability.
In security markets, imperfect competition, bid-offer imbalances and frictions often reveal
themselves in bid-offer spreads. Here, we are concerned with the determinants of the bid-offer
spread.
The evolution of prices through time should provide insight as to what affects the spread.
–
–
–
–
–
If market frictions were the only factors affecting the spread, we should expect that, in the absence of new
information, execution prices would tend to bounce between bid and ask prices. Thus, frictions such as
transactions costs will tend to be either leave execution prices unchanged or to change in the opposite
direction. Thus, transactions costs tend to induce negative serial correlation in asset prices.
Asymmetric information produces positive serial correlation in asset prices. Suppose that asymmetric
information is the only source of the spread, such that transaction prices reflect information communicated
by transactions. Transactions executed at bid prices would cause permanent drops in prices to reflect
negative information and transactions executed at offer prices would cause permanent increases in prices
to reflect positive information. If price changes are solely a function of random news arrival, price changes
will be random. If the distribution of information is asymmetric, prices will exhibit positive serial
correlation as informed traders communicate their information through their trading activity (recall that
this is what informed traders try to avoid in the Kyle model). Thus, the extent to which information
distribution is asymmetric will affect the serial correlation of asset prices.
Inventory costs (such as unsystematic risk from the dealer’s inability to diversify) will tend to cause
negative serial correlations in price quotes. Transactions at the bid will tend to cause risk averse dealers to
reduce their bid quotes as they become more reluctant not to over-diversify their inventories. Similarly,
transactions at the ask will tend to cause dealers to raise their quotes as they become more reluctant not to
under-diversify their inventories.
Inventory costs and transactions costs will tend to lead towards negative serial correlation in security
prices.
Asymmetric information availability will lead towards positive serial correlation in security prices as
transactions communicate new information.
The Demsetz [1968] Immediacy Argument
• Order imbalances impose waits on impatient traders requiring immediacy.
• The costs of providing immediacy to liquidity traders include order
processing costs (transactions costs), information and adverse selection
costs, inventory holding costs, costs of absorbing inventory risks and costs
of providing trading options.
• The bid-offer spread provides the dealer compensation for assuming these
costs on behalf of the market.
• In the Demsetz [1968] analysis, buyers and sellers of a security are each of
two types, one of which who wants an immediate transaction and a second
who wants a transaction, but can wait.
• Buy and sell orders arrive to the market in a non-synchronous fashion,
causing order imbalances.
• An imbalance of traders demanding an immediate trade forces the price to
move against themselves, causing less patient traders to pay for immediacy.
• The greater the costs of trading, and the greater the desire for immediacy,
the greater will be the market spread.
Glosten and Milgrom [1985] Information Asymmetry Model
•
•
•
•
•
•
The Glosten and Milgrom [1985] adverse selection model assumes that dealer spreads are based on
the likelihood π that an informed trader will trade.
Trades arrive to the market maker, each with some random chance of originating from either an
informed or uninformed trader.
The asset can take on one of two prices, a high price PH and a low price PL, each with probability ½.
Uninformed traders will not pay more than PH for the asset and they will not sell for less than PL.
Risk neutral liquidity traders value the asset at (PH + PL)/2.
When setting his quotes, the dealer needs to account for the probability that an informed trader will
transact at his quote, and will set his bid price Pb and ask price Pa as follows:
Pb = πPL + (1 – π)(PH + PL)/2
Pa = πPH + (1 – π)(PH + PL)/2
•
The spread is simply the difference between the ask and bid prices:
Pa - Pb = π(PH - PL)
•
•
Thus, in the single-period Glosten and Milgrom model, the spread is a function of the likelihood that
there exists an informed trader in the market and the uncertainty in the value of the traded asset.
The greater the uncertainty in the value of the traded asset as reflected by (PH + PL)/2, and the
greater the probability π that a trade has originated with an informed trader, the greater will be the
spread.
The Stoll [1978] Inventory Model
•
•
•
•
A dealer needs to maintain inventories in assets in which he makes a market to sell to investors as
well as cash to purchase assets from investors.
Suppose that a dealer currently without inventory in a particular asset trades so as to maximize his
expected utility level.
Assume that the dealer's wealth level could be subject to some uncertain normally distributed
security return r whose expected value is zero and variance σ2.
The dealer is willing to quote a bid price Pb to purchase this security whose consensus value or price
is P. Our problem here is to determine the maximum price that a risk averse dealer would be willing
to bid:
E[U(W – Pb + (1+r)P)] = U(W)
E[U(W + (P - Pb) + rP)] = U(W)
•
•
The dealer’s expected utility after the security purchase is a function of his current level of wealth W,
his bid price Pb and his uncertain return. Our problem is to solve this equality for Pb.
We start by performing a Taylor series expansion around the left side of the equality:
E[U(W) + (P - Pb + rP)(U’(W)) + ½(P - Pb + rP) 2(U”(W)) + …] = U(W)
•
Since E[r] = 0, E[rP]U’(W) can be dropped from the equality and σ2 = E[(P - Pb + rP)2]:
E[U(W) + (P - Pb)(U’(W)) + ½(σ2)U”(W) + …] = U(W)
Simplifying the Stoll Pricing Model for the Bid
•
Drop left-hand side higher order terms not explicitly stated in the above equality:
(P - Pb)(U’(W))  - ½(σ2)U”(W)
(P - Pb)  - ½(σ2)U”(W)/ U’(W)
(P - Pb)  ½(σ2)ARA
•
•
•
where -U”(W)/U’(W) is the dealer’s Arrow-Pratt Absolute Risk Aversion Coefficient (ARA).
The greater the dealer’s risk aversion, or the greater the uncertainty associated with the asset, the greater will be the
discount associated with his bid.
The dealer maintains an initial inventory Wp of securities Wp = W0. The dealer borrows and lends at the riskless rate r,
assumed to be zero. This initial inventory of securities Wp is an optimal portfolio that maximizes the dealer's utility U(W)
subject to his initial wealth constraint:
E[U(Wp(1+rp) + (P - Pb) + rP)] = U(W)
where ri is the uncertain return on the tradable asset i and rp is the uncertain return on the optimal portfolio Wp. Expand
the left side of this equation around E[U(Wp)], assume that E[r] = E[rp] = 0, and drop higher order terms:
E[U(Wp) + (P - Pb)(U’(Wp)) + ½(σ2)U”(Wp) + 2×½×E[rP×rpWp]×U"(Wp)…] = U(W)
•
The covariance of cash flows between profits on the optimal portfolio and the tradable asset i is 2×½×E[rP×rpWp] =
σi,P. Simplifying and solving for (P - Pb), we find that the dealer’s bidding discount from the consensus price is:
(P - Pb) = σI,PARA + ½(σ2)ARA
•
The dealer will decrease his bid as his inventory of securities increases and as the covariance between inventory returns
and asset returns increases.
Obtaining the Stoll Ask Price
• Similarly, the premium based the dealer’s ask or offer price Pa is obtained
as follows:
(Pa - P) = -σI,PARA + ½(σ2)ARA
• Thus, the dealer will increase his offer premium as his inventory of
securities decreases and as the covariance between his inventory returns
and the asset returns decreases. The bid-offer spread is simply the sum of
the bid discount and offer premium:
(Pa - Pb) = (σ2)ARA
• The greater the dealer’s risk aversion, or the greater the uncertainty
associated with the asset, the greater will be the dealer’s spread. The
dealer’s inventory level does not affect the size of the spread.
The Copeland and Galai [1983] Options Model
• A bid provides prospective sellers a put on the asset, with the
exercise price of the put equal to the bid.
• An offer provides a call to other traders.
• Thus, when the dealer posts both bid and offer quotes, the spread is,
in effect, a short strangle provided to the market. Both legs of this
strangle are more valuable when the risk of the underlying security
is higher.
• An options pricing model can be used to value this dealer spread.
• Ultimately, this "implied premium" associated with this dealer
spread is paid by liquidity traders who trade without information.
• Informed traders make money at the expense of the dealer, who
ultimately earns it back at the expense of the uninformed trader.
• The Copeland and Galai model, as presented here, does not have a
mechanism to allow trading activity to convey information from
informed traders to dealers and uninformed traders. Nonetheless, in
the Copeland and Galai option-based model, the spread widens as
the uncertainty with respect to the security price increases.