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Title: Financial Market
Description: A financial market is a market in which people trade financial securities, commodities, and other fungible items of value at low transaction costs and at prices that reflect supply and demand.
Description: A financial market is a market in which people trade financial securities, commodities, and other fungible items of value at low transaction costs and at prices that reflect supply and demand.
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Lecture notes on
FINANCIAL MARKETS
Marco Li Calzi
Dipartimento di Matematica Applicata
Universit` “Ca’ Foscari” di Venezia
a
February 2002
M
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Reflecting this heterogeneity, the course was dubbed “Topics in Economics” and I was given
a fair amount of leeway in its development
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These notes detail my choices after two years of teaching “Topics in Economics”
at the Master in Quantitative Finance of Bocconi University and a similar class more aptly
named “Microeconomics of financial markets” at the Master of Economics and Finance of
the Venice International University
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Each unit corresponds to a 90-minutes session
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Unit 7 requires less time than the standard session: I usually
take advantage of the time left to begin exploring Unit 8
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unive
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i
ii
Contents
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1 Introduction
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2 Decisions under risk
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3 Decisions under uncertainty
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1 Introduction
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2 Price uncertainty
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3 Real options
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4 Assessing your real option
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Optimal growth and repeated investments
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1 Risk attitude
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2 Risk attitude and expected utility
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3 Mean-variance preferences
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4 Risk attitude and wealth
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5 Risk bearing over contingent outcomes
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1 Introduction
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2 The Dutch book
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3 The red hats puzzle
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4 Different degrees of knowledge
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5 Can we agree to disagree?
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6 No trade under heterogenous priors
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Herding and informational cascades
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4 Excursions
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Normal-CARA markets
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1 Introduction
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2 An example
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3 Computing a rational expectations equilibrium
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4 An assessment of the rational expectations model
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Market microstructure: Kyle’s
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Noise trading: simulations
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14
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1 Introduction
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2 Judgement biases
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3 Distortions in deriving preferences
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4 Framing effects
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Market microstructure: Glosten and Milgrom’s model
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1 Introduction
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2 An example
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3 Competitive versus monopolistic market making
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12
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1 Introduction
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2 A model with noise trading
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3 Relative returns
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4 An appraisal
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5 Excursions
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15
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1 Introduction
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2 Myopic loss aversion
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3 A partial equilibrium model
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4 An equilibrium pricing model
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Expected utility and stochastic dominance
1
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When you plan to invest
your money in a long-term portfolio, you do not know how much will its price be at the
time of disinvesting it
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Decision theory is that branch of economic theory which works on models to help you sort
out this kind of decisions
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The first class is concerned with what is known as
decisions under risk and the second class with decisions under uncertainty
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2
Decisions under risk
Here is a typical decision under risk
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There is a family
of investment funds
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The return on each
fund is not known with certainty, but you know its distribution of past returns
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1
Let us model this situation
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There is a set A of alternatives (i
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, the funds)
out of which you must choose one
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For instance, assuming there are only three funds, your
choice problem may be summarized by the following table
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ty
-1%
20%
+2%
40%
+5%
40%
Fund β
return prob
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ty
2
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Def
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1 [Expected utility under risk] Define a real-valued utility function u over consequences
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Choose an alternative
which maximizes the expected utility
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Trusting the
distribution of past returns is a choice you make at your own peril
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The expected utility of Fund α is
U (α) = −1 · 0
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4 + 5 · 0
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6
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85 and U (γ) = 2
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According to the expected utility criterion, you should go for Fund β and rank α and γ
respectively second and third
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√
For instance, if u(r) = r + 3, we find U (α) ≈ 2
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62 and U (γ) ≈ 2
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The
best choice is now γ, which however was third under the previous utility function
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Before you can get her
to use this, there are a few questions that your CEO would certainly like you to answer
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Each one of us is free to develop his own way to reach a
decision
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Using expected utility is equivalent to taking decisions
that satisfy three criteria: 1) consistency; 2) continuity; 3) independence
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If you pick α over β
and β over γ, then you will pick α over γ as well
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Continuity means that your preferences do not change abruptly if you slightly change
the probabilities affecting your decision
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Independence is the most demanding criterion
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Choose
a third alternative γ
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If you’d pick α over β, then you should
also pick α over β
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On the other hand, if you adopt expected utility
as your decision making tool, you will be (knowingly or not) obeying these criteria
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Caveat emptor! There is plenty of examples where very reasonable people do not want to
fulfill one of the three criteria above
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Suppose the consequences
are given as payoffs in millions of Euro
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ty
0
1%
1
89%
5
10%
β
payoff
1
2
prob
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Between the two alternatives
γ
payoff
0
5
δ
prob
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ty
89%
11%
he would have picked γ
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Economists and financial economics, untroubled by this, assume that all agents abide
by expected utility
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To describe the choices of an expected utility maximizer, an economist
needs only to know the consequences, the probability distribution over consequences for
each alternative, how to compute the expected value and the utility function over the
consequences
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For the moment, however, let us go back to your CEO waiting for your hard-earned
wisdom to enlighten her
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In the example above, when the utility function was u(r) = r, the optimal
choice is√
Fund β which looks a lot like a risky stock fund
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It is the utility function which makes you prefer one over another
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This is a tricky issue, but I’ll say more about it in Lecture 4
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Suppose that consequences are monetary payoffs and assume
(as it is reasonable) that the utility function is increasing
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What is the sufficient condition such that
u(x) dF (x) ≥
u(x) dG(x)
for all increasing utility functions u?
Def
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2 [Stochastic dominance] Given two random variables α and β with respective
cumulative probability distributions F and G, we say that α stochastically dominates β if
F (x) ≤ G(x) for all x
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That is, α is less likely than β to be smaller than x
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3
If you happen to compare alternatives such that one stochastically dominates the other
ones and you believe in expected utility, you can safely pick the dominating one without
even worrying to find out what your “right” utility function should be
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Isn’t this “expected utility business” too artificial? Well, it might be
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Expected utility is what
economists use to model your behavior under risk
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We can put it down to a matter of decision procedures
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There
are other procedures which lead you to choices that are compatible with expected utility
maximization in a possibly more natural way
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Suppose that you are a fund manager
and that your compensation depends on a benchmark
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Each strategy will lead to a payoff at the end of the
year which is to be compared against the benchmark
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The performance of the benchmark is a random
variable B
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Moreover, since you are only one of many fund managers, you assume that the performance
of the benchmark is independent of which investing strategy you follow
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If your investing
strategy leads to a random performance α with c
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f
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While (naturally) trying to maximize your chances of getting your bonus, you will be behaving as if (artificially) trying to maximize a utility function u(x) = H(x)
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You might have read it already but, thank God (or free will), nobody can
answer this
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In
the example above, I mischievously assumed that you were willing to use the distribution
of past returns but this may sometimes be plainly wrong
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Their firstcut answer is to isolate the problem by assuming that the probabilities have already been
estimated by someone else and are known to the agent
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Whenever we do not assume that probabilities are
already known, we enter the realm of decisions under uncertainty
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See Bordley and LiCalzi (2000) for
the most recent summary
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3
Decisions under uncertainty
Here is a typical decision under uncertainty
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There
is a family of investment funds
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The
return on each fund is not known with certainty and you do not think that the distribution
of past returns is a good proxy
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Let us model this situation
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There is a list S of possible future scenarios
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e
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Each alternative
α in A is a function which tells you which consequence c you will be able to attain under
scenario s: that is, α(s) = c
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scenario
s1
s2
s3
s4
s5
Fund α
return
-1%
+2%
+2%
+2%
+5%
Fund β
return
-3%
-3%
-3%
+10%
+10%
Fund γ
return
2
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5%
2
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5%
2
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1
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Define a utility function over consequences
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Choose an alternative which maximizes the expected utility
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For instance, assessing P (s1 ) = 20%, P (s2 ) = 30%, P (s3 ) = P (s4 ) = 5%, and
P (s5 ) = 40% gets you back to the case studied above
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However, staying with the same utility function, if
you’d happen to assess P (s1 ) = P (s2 ) = P (s4 ) = P (s5 ) = 5%, and P (s3 ) = 80%, the
optimal choice would be γ
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What matters is not only your attitude
to risk (as embedded in your choice of u), but your beliefs as well (as embedded in your
probability assessment)
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Suppose the consequences are given as payoffs in millions of Euro
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scenario
s1
s2
s3
s4
s5
Fund α
payoff
0
1
2
3
4
5
Fund β
payoff
4
0
1
2
3
Suppose that you assess probabilities P (s1 ) = 1/3, and P (s2 ) = P (s3 ) = P (s4 ) = P (s5 ) =
1/6
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Any expected utility maximizer (if using an increasing utility function)
would pick β over α
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References
[1] R
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Li Calzi (2000), “Decision analysis using targets instead of utility
functions”, Decisions in Economics and Finance 23, 2000, 53–74
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Castagnoli (1984), “Some remarks on stochastic dominance”,Rivista di matematica
per le Scienze Economiche e Sociali 7, 15–28
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Irreversible investments and flexibility
2
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(In spite of contrary
advice from most academics, consultants and hence practitioners use the payback time and
the IRR as well
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If you have to pick one among many possible investments, pick the one
with the greatest NPV
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This
might work as a first rough cut, but it could easily led you astray
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Most of the material is drawn from Chapter 2
in Dixit and Pyndick (1994)
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2
Price uncertainty
Consider a firm that must decide whether to invest in a widget factory
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The firm
can be built at a cost of c = 1600 and will produce one widget per year forever, with zero
operating cost
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After
this, the price will not change anymore
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Presented with this problem, a naive CFO would compute the expected price p = 200
from the next year on
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(1
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A clever CFO would consider also the possibility of waiting one year
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The NPV for this investment policy is
NPV =
1 −1600
+
2
1
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1)t
≈ 773
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The
value of the flexibility to postpone the investment is 773 − 600 = 173
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2
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Ex
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2 Suppose that there exists a futures market for widgets, with the futures price for
delivery one year from now equal to the expected future spot price of 200
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To see why, check that you could
hedge away price risk by selling short futures for 11 widgets, ending up with a sure NPV
of 2200
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The futures market allows the firm to get rid of the risk but does
not improve the NPV of investing now
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3
Real options
We can view the decision to invest now or next year as the analog of an american option
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Here, we have the right to make an investment expenditure now or next
year and receive a random NPV
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This sort of situations, where the underlying security is a real investment, are known as
real options
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Let us compute the value of our investment opportunity using the real options approach
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Then F1 is a random
variable, which can take value
∞
t=0
300
− 1600 ≈ 1700
(1
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We want to find out what is F0
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The value of this portfolio
today is Π0 = F0 − nP0 = F0 − 200n
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Since P1 = 300 or 100, the possible values of Π1 are 1700 − 300n or −100n
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5
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The return from holding this portfolio is the capital gain Π1 − Π0 minus the cost of
shorting the widgets; that is, Π1 − Π0 − 170 = −850 − (F0 − 1700) − 170 = 680 − F0
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1)Π0 = 0
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This is of course the same value we have
already found
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4
Assessing your real option
Once we view an investment opportunity as a real option, we can compute its dependence on
various parameters and get a better understanding
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a) Cost of the investment
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5 − 0
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Hence, Π1 = F1 − nP1 = 0
...
Imposing a risk-free rate of
r = 10% yields
F0 = 1500 − 0
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We can use this relationship to find out for what values of c investing today is better
than investing next year
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Since the NPV of the payoffs
from investing today is 2200, we should invest today if 2200 > c+F0
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In the terminology of financial options, for low values of c the option is “deep in the
money” and immediate exercise is preferable, because the cost of waiting (the sacrifice of
the immediate profit) outweighs the benefit of waiting (the ability to decide optimally after
observing whether the price has gone up or down)
...
Fix again c = 1600 and let us now vary P0
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Suppose that we want to invest when the price goes up and we do not want
to invest if it goes down (we will consider other options momentarily)
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5P0 − 1600 − 1
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5nP0 is the price goes down
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5 − (1600/P0 ) is the number of widgets that we need to short to make the portfolio
risk-free, in which case Π1 = 800 − 8
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Recall that the short position requires a payment of 0
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65P0 − 160 and compute
the return on this portfolio
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65P0 −
160] = 0
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5P0 − 727
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However, if P0 is low enough we might never want
to invest, and if P0 is high enough it might be better to invest now rather than waiting
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From (2), we see that F0 = 0 when
P0 ≈ 97
...
Analogously, let us compute for which price we would always
9
invest today
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e
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The critical price P0 satisfies 11P0 − 1600 = F0 which,
ˆ0 = 249
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5P0 − 727
then F0 = 11P0 − 1600
Investment rule
and you never invest;
and you invest next year if price goes up;
and you invest today
...
Fix an arbitrary P0 and let us vary q
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5 and is independent
of q (yes, check it)
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5P0 ) + (1 −
q)(0
...
5P0 ); therefore the expected capital gain on widgets is [E(P1 ) − P0 ]/P0 =
q − 0
...
Since the long owner of a widget demands a riskless return of r = 10% but gets
already a capital gain of q − 0
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1 − (q − 0
...
6 − q)P0
per widget
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6 − q)nP0 = 0
...
5 we find (for P0 > 97) that
the value of the option is
F0 = (15P0 − 1455)q
...
What about the decision to invest? It is better to wait than to invest today as long as
F0 > V0 − c
...
5)P0 ]/(1
...
Note that P0 decreases as
q increases: a higher probability of a price increase makes the firm more willing to invest
today
...
Fix q = 0
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This leaves E(P1 ) = P0 but increases the variance of P1
...
The two possible values for Π1 are
19
...
75nP0 if the price goes up and −0
...
Equating these two values and solving for n gives n = 12
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21P0 irrespective of P1
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75P0 − 727
...
Why does an increase in uncertainty increase the value of
this option?
Ex
...
3 Show that the critical initial price sufficient to warrant investing now instead rather
ˆ
than waiting is P0 ≈ 388, much larger than the 249 found before
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K
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S
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10
3
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1
Introduction
Standard portfolio theory treats investments as a single-period problem
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The basic lesson from this static approach is that volatility is “bad”
and diversification is “good”
...
However, when we begin working over
multiperiod investment problems, some of the lessons of the static approach take a whole
new flavour
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Most of the material is drawn from Chapter 15 in Luenberger (1998)
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2
An example
At each period, you are offered three investment opportunities
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An identical but independently distributed
selection is offered each period, so that the payoffs to each investment are correlated within
each period, but not across time
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ty
1/2
1/3
1/6
You start with Euro 100 and can invest part or all of your money repeatedly, reinvesting
your winnings at later periods
...
What should you do?
Consider the static choice over a single period
...
More precisely, for
an investment of 100, the first lottery has an expected value of 150 and a 50% probability
of losing the capital
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6%
probability of losing the capital
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3%
probability of losing the capital
...
However, note that α dominates β
and γ under both respects
...
11
This intuition does not carry over to the case of a multiperiod investment
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Instead of maximizing your return, repeatedly
betting the whole capital on α guarantees your ruin
...
Each of
these portfolios leads to a series of (random) multiplicative factors that govern the growth
of capital
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With
probability 50%, you obtain a favorable outcome and double your capital; with probability
50%, you obtain an unfavorable outcome and your capital is halved
...
Over a long series
of investments following this strategy, the initial capital will be multiplied by a multiple of
the form
1
1
1
1
1
1
(2)
(2) (2)
...
The overall factor is likely to be about 1
...
Suppose now to invest using the (1/4, 0, 0) portfolio
...
Since the two outcomes are equally likely, the average multiplicative factor over two periods is (3/2) (3/4) = 9/8
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06066
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Ex
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4 Prove that this is the highest rate of growth that you can attain using a (k, 0, 0)
portfolio with k in [0, 1]
...
3
...
3
...
For each period t = 1, 2,
...
The capital evolves according to the equation
Xt = Rt Xt−1 ,
(5)
where Rt is the random return on the capital
...
In the general capital growth process, the capital at the end of n trials is
Xn = (Rn Rn−1
...
12
After a bit of manipulation, this gives
log
Xn
X0
1/n
=
1
n
n
log Rt
...
Since all Rt ’s are independent and identically distributed, the law of
large numbers states that the right-hand side of this expression converges to m as n → +∞
and therefore
Xn 1/n
log
→m
X0
as well
...
Roughly speaking, the
capital tends to grow exponentially at rate m
...
Thus, if we choose the
utility function U (x) = log x, the problem of maximizing the growth rate m is equivalent to
finding the strategy that maximizes the expected value of EU (X1 ) and applying this same
strategy in every trial
...
3
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Suppose that you have the opportunity to invest in a prospect
that will either double your investment or return nothing
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Suppose that you have an initial capital of X0 and that you can repeat
this investment many times
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If the outcome is favorable, the
capital grows by a factor 1 + α; if it is unfavorable, the factor is 1 − α
...
This situation resembles the game of blackjack, where a player who mentally keeps
track of the cards played can adjust his strategy to ensure (on average) a 50
...
With p =
...
5% and thus em ≈ 1
...
00125% gain each round
...
Suppose that there are only two assets available for investment
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The
other is a risk-free bond that just retains value — like putting money under the mattress
...
An investment left in the stock will have a
value that fluctuates a lot but has no overall growth rate
...
Nevertheless, by using these two investments in combination, growth can be achieved!
13
Suppose that we invest α of our capital in the stock and (1 − α) in the bond, with α in
[0, 1]
...
For this choice of α, em ≈ 1
...
The gain is achieved by using the volatility of the stock in a pumping action
...
When the stock goes up in certain period, some of its capital gains are
reinvested in the bond; when it goes down, additional capital is shifted from the bond to the
stock
...
Note that this strategy follows automatically the
dictum “buy low and sell high” by the process of rebalancing the investment in each period
...
3
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Assume that each asset moves independently of the other
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8%
...
Let us go back to the example of Section 3
...
Consider
all portfolio strategies (α1 , α2 , α3 ), with αi ≥ 0 for i = 1, 2, 3 and α1 + α2 + α3 ≤ 1
...
Under scenario s2 , the return
is R = 1 − α1 + α2 − α3
...
To find
an optimal portfolio, it suffices to maximize
m=
1
1
1
log (1 + 2α1 − α2 − α3 ) + log (1 − α1 + α2 − α3 ) + log (1 − α1 − α2 + 5α3 )
...
3
...
Prove that it is not unique, by
checking that (5/18, 0, 1/18) is also optimal
...
99%, higher than what was found
above
...
There is one stock that can be purchased at a price of Euro 100
per share
...
What is the log-optimal strategy over this period?
The log-optimal strategy is found by maximizing
2
m=
[log(1 − α + αr)]
...
We find that the log-optimal investment in the stock is α∗ ≈
...
82% per period
...
5
Excursions
An extensive list of several properties of the log-optimal strategy is given in MacLean et
alii (1992)
...
References
[1] Y
...
[2] D
...
Luenberger (1998), Investment Science, New York: Oxford University Press
...
C
...
T
...
Blazenko (1992), “Growth versus security in dynamic investment analysis”, Management Science 38, 1562–1585
...
M
...
O
...
15
4
...
1
Risk attitude
Consider a choice problem among the following three lotteries, whose expected values are
written by their name
...
ty
480
100%
β (525)
return prob
...
ty
1000
50%
0
50%
If we were to base our choices on the expected value, β would be our preferred choice
...
What can we say about the elusive notion of “risk”? While it is hard to define what
exactly “risk” means, certainly a sure outcome like α should be deemed riskless
...
) By contrast, let us call risky any
lottery which does not yield a sure outcome
...
However, avoiding risk cannot be the only reason driving a choice
...
An agent who chooses α is avoiding risk at the cost of
accepting a lower “return”
...
Def
...
4 An agent is risk neutral if he evaluates lotteries by their expected value
...
How do we tell which case is which?
There is a simple question we may ask the agent: suppose you possess lottery β; how much
sure money would you ask for selling it? We name this “price” c(β) called by the agent the
certainty equivalent of a lottery
...
That is, we should have c(β) = E(β)
...
If c(β) < E(β), the agent values β less than his expected
value: therefore, the risk in β reduces the value to him of holding it — we say that he is
risk averse
...
For a different way to say exactly the same thing, define the risk premium r(β) of a
lottery β as the difference between its expected value and its certainty equivalent
...
That is, he is willing to accept for sure a payment
which is less than the average payoff of β in order to get rid of the risk associated with β
...
2
Risk attitude and expected utility
All of this holds in general, even if the agent is not an expected utility maximizer
...
Thm
...
5 An expected utility maximizer is risk neutral (resp
...
, concave or convex)
...
Ex
...
8 Check that expected utility can rationalize any of the three choices in the example
above using different utility functions
...
This is evidence of the flexibility of the expected utility model
...
There are two assets
...
The other is a risky stock that has a random
return of R per euro invested; we assume that E(R) > 1 so that on average the stock is
more profitable than the riskless bond
...
The
agent must select a portfolio and invest a fraction α of his wealth in the risky asset and a
fraction 1 − α in the riskless bond
...
The maximization problem is maxα Eu(αR + 1 − α)
...
Therefore, the optimal portfolio
satisfies the first-order Kuhn-Tucker condition:
= 0 if 0 < α < 1
≤ 0 if α = 0
...
Therefore, we conclude
that the optimal portfolio has α∗ > 0
...
4
...
One that is very common
relies on the use of indices of location and dispersion, like mean and standard deviation
...
Risk, instead, is present if the standard deviation (or some other measure of dispersion) is positive
...
If offered several lotteries with the same standard deviation, a (greedy) agent prefers
the one with the highest expected value
...
Thus, a greedy and risk
17
averse agent has preferences represented by a functional V (µ, σ) which is increasing in µ
and decreasing in σ
...
It turns out that this is not consistent with the definition
of risk aversion given above
...
Ex
...
9 Suppose that a risk averse agent is an expected utility maximizer with utility
function u(x) = x for x ≤ 1 and u(x) = 1 + 0
...
Compare a lottery α
offering a payoff of 1 with probability 1 versus another lottery β offering a payoff of 1
...
While µ(α) = µ(β) and σ(α) = 0 < σ(β),
the agent strictly prefers β to α
...
Given an arbitrary value x∗ , consider the Taylor
expansion of u around x∗ :
u(x) = u(x∗ ) +
(x − x∗ )
(x − x∗ )2
(x − x∗ )3
u (x∗ ) +
u (x∗ ) +
u (x∗ ) +
...
Or, in more statistical terms, it looks only at the first two
moments of the probability distribution of the lotteries
...
ty
-1
0
...
001
β
return
1
-999
prob
...
999
0
...
However, most people is not indifferent between the
two
...
Skewness is zero for a symmetric distribution; it is positive if there
is a hump on the left and a long thin tail on the right; and negative in the opposite
case
...
Most commercial lotteries and
games of chance are positively skewed: if people like them because of this, the second-order
approximation cannot capture their preferences
...
18
4
...
To keep things simple, assume in the following that all risky decisions concern
monetary payoffs and that agents maximize expected utility
...
We consider one way in which wealth affects the risk attitude
...
If he
chooses α, his future wealth will be w + α; if he chooses b, w + b
...
If he is an expected
utility maximizer, this implies that Eu(w + α) > u(w + b)
...
4
...
Similar definitions holds for constant and increasing risk aversion
...
Thm
...
7 An agent has decreasing risk aversion (resp
...
, constant or increasing) in x
...
However, in applications it is very common to postulate that they
are risk neutral or that they have constant risk aversion because this greatly simplifies the
choice of their utility function
...
4
...
Ex
...
10 Suppose that the agent is an expected utility maximizer with a constant coefficient
of absolute risk aversion k > 0
...
Given a lottery X ∼ N (µ, σ), check that the preferences of the agent can be represented by
the functional V (µ, σ) = µ − (1/2)kσ 2
...
5
Risk bearing over contingent outcomes
Suppose that there is a finite number of states of the worlds (or scenarios)
...
, n) occurs with probability πi
...
The agent is endowed with the same initial income y in each scenario
and he derives a differentiable utility u(ci ) from consuming a quantity ci of the commodity
19
in the scenario si
...
t
...
i
Assuming an interior solution, the first-order condition requires [πi u (ci )] /pi to be constant
for all i
...
References
[1] J
...
G
...
[2] D
...
[3] A
...
D
...
R
...
20
5
...
1
Introduction
One traditional view about trading in financial markets is that this has two components:
liquidity and speculation
...
According to this view, high volume trading should
be explained mostly by differences in information among traders
...
It seems clear that the only way to explain the volume of trade is with a
model that is at one and at the same time appealingly rational and yet permits
divergent and changing opinions in a fashion that is other than ad hoc
...
In fact, there are even
cases where it might reduce the trading volume and lead to market breakdowns
...
5
...
In fact, even more is true:
when two or more risk-neutral agents have different beliefs about the probability of some
events and are willing to bet among them on these events, a bookie can arrange a set of
bets that each of the agents is willing to take and still guarantee himself a strictly positive
expected profit
...
Here is a simple illustration
...
Suppose that Ann
believes that in a month the Mib30 will be up with probability p while Bob believes that
this will happen with probability q < p
...
The bookie can make a sure profit by offering to each player a specific bet customized for
his beliefs
...
Given any x > 0, the bookie can cash a sure (and strictly
positive) profit of x − ε, where 0 < ε < x is the “sweteener” that induces agents to take the
bets
...
21
Bull
Ann’s bet
Bob’s bet
bookie’s bet
x
1−p
+ε
p−q
−x
1−q
p−q
x−ε
Bear
−x
x
p
p−q
q
+ε
p−q
x−ε
Ann’s uses her own beliefs to assess her bet
...
Bob, using his own beliefs, reaches a similar conclusion
...
Everybody
is happy (a priori) but the bookie is setting up an arbitrage to his advantage
...
Who is being exploited?
Well, if two agents hold different beliefs about the probability of an event, at least one
of the two must be “wrong”
...
Differing beliefs imply that there is an arbitrage
opportunity lurking somewhere, and agents who recognize this fact will forfeit trading and
revise their beliefs for fear of being exploited
...
It also motivates the wide-spread buzzword that there is only one “market
probability” to which agents’ beliefs should conform to; otherwise, so goes the argument,
they could be exploited
...
5
...
Consider three events:
in a month the Mib30 will go down of more than 5%, or it will be up of more than 5%, or
it will move less than 5%
...
Ann believes
that Ei occurs with probability pi , for i = 1, 2, 3
...
Assume 0 < p3 = q1 = r2 < p2 = q3 = r1 < p1 = q2 = r3 < 1
...
(Warning: solving this is longer than usual
...
3
The red hats puzzle
We should turn the intuition above into a formal argument
...
We illustrate this with a puzzle
...
Summoning the prisoners to his presence and commanding them
under penalty of death not to look upward, the King has a hat placed upon the head of
each one
...
Each prisoner can see the hat of his fellow
mates, but not his own
...
At least
one of you is wearing a red had
...
If you guess correctly, you will go
free; if you guess incorrectly, you will be instantly beheaded
...
Since they all know what the King has told them, none can
conclude anything on the colour of his own hat
...
On the second day, the white-hatted prisoners still see two red hats and the two redhatted prisoners still see one red hat
...
One day is passed and neither Ann nor Bob have
spoken and left
...
If he had not already left, it is because he
must have seen a red hat
...
“Hence, — Ann reasons, — since the only red hat I can see is on Bob’s head, that
which Bob must have seen on the first day must have been on mine
...
Both will infer
their hats to be red on the second day, and will go free
...
Consider the proposition “everyone sees a red hat”
...
Thus
on the first day the proposition is actually true but not everyone knows that it is true
...
The highest-level proposition is “everybody knows that everybody knows that everybody
knowns
...
Independently of which event you put within
brackets, this is an instance of what is called “common knowledge” of an event
...
The puzzle shows that sometimes the difference between staying in prison and going
free rests with understanding the difference
...
Since trading markets are public, both propositions are true; but the second
proposition conveys more information than the first one
...
5
...
Ex
...
13 Prove that, if the red hats had been k, then k prisoners would have gone free on
the k-th day
...
4
Different degrees of knowledge
There is a set Ω of states
...
When the true state is ω, the agent thinks that
all (and only all) the states in P (ω) may be true; in other words, the agent knows only that
the true state belongs to the set P (ω)
...
An agent for whom
P (ω) ⊆ E knows, in state ω, that some state in the event E has occurred or, more simply,
he knows that E must be true
...
There are two risk-neutral agents, Ann and Bob
...
In case of indifference, assume
that either agent prefers to turn down the bet
...
Trade is possible only if agents have different beliefs
...
, ω5 and the pairs (θ, ω) are drawn from the following prior
joint distribution, which is known to both agents:
ω1
ω2
ω3
ω4
ω5
θ=0
0
...
05
0
...
15
0
...
05
0
...
05
0
...
20
Ann and Bob have access to different information, which is represented by their information maps
...
A partitional information map may be specified
by the information partition it induces
...
5
...
Show that, if (ii) were false, in state ω the agent could
infer that the state is not ω : therefore, ω could not belong to P (ω)
...
Bob’s information partition is PB = ({ω1 }, {ω2 , ω3 }, {ω4 , ω5 })
...
For instance, consider state ω2
...
Using Bayes’ rule, she attributes probability 5/9 to θ = 0
...
Therefore, in state ω2 , agents have different beliefs and are willing to subscribe the bet
...
Consider now state ω5
...
In this state, no trade can take place
...
5
...
Using the results of Exercise 5
...
24
State
Ann
Bob
ω1
Bet
No bet
ω2
Bet
Bet
ω3
Bet
Bet
ω4
Bet
Bet
ω5
No bet
Bet
It seems that trade should occur in states ω2 , ω3 and ω4
...
To see
why, consider again state ω2
...
However, by computing
the table above, she knows only that Bob is not willing to bet in state ω1
...
But
then it must be exactly ω2 , in which case the probability that θ = 0 is 1/4 and therefore
Ann should not accept the bet
...
Ex
...
16 By a similar argument, prove that no trade can occur in state ω4
...
Trade is impossible when everybody behaves rationally, even if Ann and Bob have
different information maps! The public information revealed by their willingness of trade
realigns their probabilities and make them give up trade
...
If I know you are so smart, why
should I buy a piece of risky stock from you? If you are willing to sell it at a price p, you
must think that its value is less than p
...
As far as you are willing to sell, you must
know something that I do not know: then I better be careful and avoid being a sucker, so
I won’t buy
...
5
Can we agree to disagree?
In the example above, only two rounds of deductive thinking are needed to reach the
conclusion that no trade is preferable
...
Restricting for simplicity to the case of two individuals, here is how to formalize it
...
An event F is self-evident if for all ω
in F we have Pi (ω) ⊆ F for i = A, B
...
Ex
...
17 Let Ω = {ω1 , ω2 , ω3 , ω4 , ω5 , ω6 }
...
Prove that in any state the event E = {ω1 , ω2 , ω3 , ω4 } is not common knowledge
...
Ok, enough with abstract theory
...
But traders use probabilities
...
Suppose that an
25
agent has a prior probability distribution p(·) on Ω: for instance, he attaches probability
p(E) to an event E
...
If E ∩ P (ω) = ∅, it will be assigned a probability p(E|P (ω)); otherwise,
it will be deemed impossible
...
It turns out that a common prior and common
knowledge of the posteriors force people to have the same posterior probabilities for that
event
...
5
...
If the
information maps are partitional and in state ω ∗ it is common knowledge that each agent
i = 1, 2 assigns probability πi to some event E, then π1 = π2
...
The event F must be a union of
members of i’s information partition
...
Given two nonempty disjoint sets C and D with
p(E|C) = p(E|D) = πi , we have p(E|C ∪ D) = π
...
Hence π1 = π2
...
However, it cannot be common knowledge that the probability assigned
by Ann exceed that assigned by Bob
...
For instance, suppose that Ann and Bob are strictly risk averse
...
If Bob accepts to bet on “tails”, he reveals that he is
assigning “heads” a probability lower than 1/2
...
Let us work out a second example where common knowledge is needed to reach the
no-trade outcome
...
However,
this time there is a countable number of states ω0 , ω1 , ω2 ,
...
and
PB = {ω0 }, {ω1 }, {ω2 , ω3 }, {ω4 , ω5 }, {ω6 , ω7 },
...
Therefore,
ruling out one of the two candidates enables an agent to correctly infer the true state
...
26
ω0
θ=0
1/4
θ=1
(1/4) + (2/3)ε
ω1
1/8
(1/8) − ε
ω2
1/16
(1/16) + (1/2)ε
ω3
1/32
(1/32) − (1/4)ε
ω4
...
...
...
(1/64) + (1/8)ε
...
...
...
1
2n+2
ωn
2n+2
...
...
...
Ex
...
18 Verify that
+∞
n=0
1
2n+2
=
1
2
+∞
(−1)n
and
n=1
1
2
=−
2n−1
3
and check that this is a well-defined probability distribution
...
In state ω2 , Ann needs a second round of deductive thinking
to know that the state is not ω1 (therefore it must be ω2 ) and hence to decline trade
...
For short, Ann knows that Bob knows that the state is
not ω1
...
The crucial step is
that Bob must be able to deduce that Ann knows that the state is not ω2
...
Continuing like this, it turns out that it takes n rounds for Ann (if n is even) or Bob (if
n is odd) to deduce that the state is unfavorable and hence decline to trade
...
Since the number
of states is infinite, for no trade to occur in any state, they must run an infinite chain of
deductions
...
5
...
Here is an example with heterogenous priors
...
However, this
time trade fails because of information asimmetry
...
Ann gets a binary signal which she can reveal
27
to Bob
...
When
Ann gets signal u, she assigns probability pu to a rise in the interest rate; when she gets
1
signal d, she assigns probability pd to a decrease in the interest rate
...
Gains from trade are possible if pu = pu or pd = pd
...
Then Ann could agree to reveal the signal to Bob
p1
2
2
with the understanding that she would pay her 2 if the signal is correct and be paid 5
from him if it is incorrect
...
On the other hand, if Ann reveals him her signal, Bob can expect for either signal
an average gain of (3/4)(+2) + (1/4)(−5) = 1/4 if the signal is truthful and (3/4)(−5) +
(1/4)(+2) = −13/4 if it is not
...
Unfortunately, Ann can gain more by misreporting his signal
...
Even if Ann prefers being truthful and sell
her signal to no trade, once the deal is signed she gains more by lying
...
So trade breaks down due to informational
asimmetry: if Bob could monitor the truthfulness of Ann’s report, trade would resume
...
Milgrom and N
...
[2] S
...
[3] M
...
Osborne and A
...
[4] S
...
Bhattacharya and G
...
), Theory of valuation: Frontiers in modern financial theory, Totowa (NJ): Rowman
and Littlefield
...
Herding and informational cascades
6
...
The prior probability for A (respectively, B) being better is 51% (respectively,
49%)
...
Each agent knows
the prior probability and, before going to the theater, receives a private (independent) signal
about which movie is better
...
Suppose that 99 people receive a signal favoring B and only one a signal favoring A
...
Consider what could happen when aggregation of information is not possible because
people arrive in sequence and do not talk to each other
...
Clearly,
Primus will go to A
...
Since the two signals are of equal quality, they cancel out, and the rational choice
is to go by the prior probabilities and pick A
...
Therefore, Secunda chooses A regardless of her signal
...
Facing exactly the same choice as Secunda, he will
also choose A and so on
...
(Food for thought: have you ever wondered
why producers heavily advertise a movie only in its opening days?)
A group dynamics where most people coordinate on the same choice is called “herding”
...
This lecture
illustrates what may rationally start and maintain this kind of imitative behavior, as well
as which consequences herding may have on the valuation of financial assets
...
2
Some terminology
Herding behavior is associated with people blindly following others’ choices
...
For our purposes, however,
we are interested in situations where herding is the outcome of a rational choice
...
In the first case, we say that there is a payoff externality; for
example, think of the your last decision to upgrade to the ump-th version of Word (packed
with many new frills absolutely worthless to you) only because the old version cannot read
files prepared with the new version
...
29
In many cases, both types of externalities are present and they may oppose or reinforce
each other
...
Then the information externality pushing for A-herding would
have been opposed by a payoff externality driving later customers to choose the (seemingly
slightly worse but) less crowded movie show
...
The state space can be discrete or continuous, as well as the choice set
...
6
...
Suppose
that market participants hold perfectly accurate information when aggregated and that
the offer price must be uniform
...
However, if distribution channels are limited, the underwriter needs time to approach
interested investors
...
Suppose that the issuer and the n investors are risk-neutral
...
Neither the issuer nor the investors know
θ
...
The issue has sufficient shares to offer one to each investor; each investor can only
afford one share
...
Each investor receives an independent binary signal, which is H igh with probability θ
and Low with probability 1 − θ
...
We examine three different scenarios
...
In the first scenario investors can freely communicate
...
By (7), the highest price that the issuer can charge (if k out of n
investors receive an H-signal) is
k+1
p(k) =
...
, n
...
When choosing a price p(k), the issuer is implicitly betting on the event that at least k
investors will get an H-signal
...
Hence
the IPO will certainly go through and yield a sure profit of n/(n + 2)
...
n+2
This is maximized for k ∗ = n/2
...
Since this is greater than the profit n/(n + 2),
p∗ = 1/2 is the optimal price
...
6
...
Perfect communication from early to late investors
...
Now the profits to the seller are path-dependent
...
If the first investor gets an H-signal
and the second investor an L-signal, the issuer gets a profit of 2 · (2/5)
...
When the circulation of information is restricted, the profits to the seller may depend on
the order in which signals are revealed to the investors
...
In the third scenario, late investors knows only whether
early investors purchased or not; that is, only actions are observable
...
For instance, suppose that at some point investor i has an H-signal and
finds that it is not in his interest to invest
...
Therefore, she will not be able to learn anything from i’s action
...
By induction, so will all later investors
...
Suppose that p = 1/2 − ε and the first three investors observe (in
this order) an H-signal and two L-signals
...
Consequently, Secunda infers that the value of the project is 1/2 and
she purchases as well
...
Like her, he will purchase even with an L-signal
...
Consider now what happens if the first three investors observe (in this order) L, L and H
...
Regardless of his signal, the third
31
investor does not purchase
...
The example shows that the order in which people with different signals move can
determine the success or the failure of the IPO
...
For
instance, imagine the case where there are 100 investors: 99 have observed an L-signal and
only one has had an H-signal
...
However, if it so happens that the sequencing of moves has the investor
with the H-signal in first place, everybody will purchase and an overpriced offering will be
entirely sold
...
6
...
For p < 2/3, there is
positive probability that an offering of ultimate value θ > 0 (and in particular an overpriced
issue) succeeds perfectly (all investors buy) because of a positive cascade
...
Other phenomena that can be explained by this sort of models include: clumping on an
incorrect decision, low consensus in opinions (when polled) but high uniformity in actions,
fragility (a little bit of information can reverse a long-standing cascade), and strong dependence on initial conditions
...
4
Excursions
Herding can arise in different ways
...
Devenow and
Welch (1996) surveys models based on iii), iv) and v)
...
References
[1] A
...
Banerjee (1992), “A simple model of herd behavior,” Quarterly Journal of Economics 107, 797–817
...
Devenow and I
...
[3] I
...
[4] I
...
32
7
...
1
Introduction
Suppose that Primus is interested in taking actions that depend on an unknown parameter
y
...
If Primus is a
(subjective) expected utility maximizer, he should pick the action which maximizes the expected value of his utility computed with respect to his (subjective) probability distribution
for y
...
Then he should update his prior probability distribution to a posterior probability
distribution and use this latter one to choose an optimal action
...
This basic format for taking decisions under uncertainty when new information is revealed is the tenet of Bayesian rationality
...
This is made for analytical convenience
...
7
...
For short, we write Y ∼ N (m, sy )
...
Each signal x is independently and identically distributed according to a normal
distribution with mean y and standard deviation sx > 0; that is, a signal is an (iid) draw
from X ∼ N (y, sx )
...
If we denote by g(y) the prior distribution for Y and by f (x|y) the conditional
distribution of the signal, we have respectively
g(y) = √
1
1
exp − 2 (y − m)2
2sy
2πsy
f (x|y) = √
1
1
exp − 2 (x − y)2
2sx
2πsx
By Bayes’ rule, the posterior density function for Y given a signal X = x is given by
g(y|x) =
f (x|y) · g(y)
...
(10)
Three properties are worth being noted
...
If
we begin with a normal prior and the signal is normally distributed, the posterior remains
normal
...
Second, we can simplify (10) by defining the precision of a normally distributed signal
as the inverse of its variance
...
Then (10) can be written as
Y |x ∼ N
mτy + xτx
1
,
τy + τx
τy + τ x
...
In the following, we make
frequent use of this simple method for computing the expected value of a posterior belief
...
7
...
Suppose
that Primus receives a second (iid) signal X = x2 and derive his new posterior distribution for Y |{x1 , x2 }
...
, xn
...
7
...
What is the distribution of Y |x when the signal X has infinite precision?
Third, note that the Bayesian posterior beliefs converge to the truth as the number of
signals increase
...
, xn , the variance of the posterior goes to zero
while the Strong Law of Large Numbers implies that the posterior mean converges to m
...
3
Cara preferences in a normal world
If Primus is an expected utility maximizer with constant absolute risk aversion, his utility
function must be linear or exponential
...
Suppose that Primus has preferences which satisfy these assumptions and that his beliefs
are normally distributed so that W ∼ N (µ, σ)
...
3 in Lecture 4
to check that his expected utility can be written
Eu(W ) =
−e−kw · √
1
1
1
exp − 2 (w − µ)2 dw = − exp − kµ − k 2 σ 2
2σ
2
2πσ
34
...
Since − exp(−kw) is a strictly increasing function of w, Primus’ preferences in a cara–normal
world are more simply expressed by the functional
1
V (µ, σ) = µ − kσ 2
2
and we can characterize them using simply the two statistics µ and σ 2 and the coefficient
of absolute risk aversion k
...
7
...
Derive his (posterior) expected utility
...
4
Demand for a risky asset
Suppose that Primus is an expected utility maximizer with constant absolute risk aversion
k > 0
...
The bond has a current price normalized to 1 and will pay a riskless amount
(1 + r) at the end of the period
...
Primus’ current endowment is w
...
Assuming that short sales are allowed, he can invest in any portfolio of α
stocks and β bonds such that αp + β = w
...
Note that α and β are
unrestricted in sign and may not add to 1
...
Hence, the expected utility of Primus of a portfolio
with α stocks is
1
αm + (w − αp)(1 + r) − kα2 s2
...
ks2
(13)
Note that the demand for the risky stock is separately monotone in each of its arguments;
for instance, it is increasing in the mean m and decreasing in the variance s2
...
7
...
35
8
...
1
Introduction
The main question addressed by rational expectations models is what happens when people
with different information decide to trade
...
The fundamental
insight is that prices serve two purposes: they clear markets and they aggregate information
...
Let us begin with an example
...
Primus receives a binary signal about the true value of widgets: if the signal is
H igh, his demand for widgets is p = 5 − q; if the signal is Low, his demand is p = 3 − q
...
Secunda receives no signal and offers an unconditional supply of widgets p = 1+q
...
When Secunda is sufficiently naive, the following situation occurs
...
If he receives an L-signal,
his demand equates the supply from Secunda at a price of pL = 2 (and q = 1 widget is
exchanged)
...
This outcomes, however, presumes that Secunda does not understand that prices also
convey information
...
Thus, if Secunda sees that markets clear at a price of p = 3 she can infer that Primus has
received an H-signal and this suffices to let her change the supply function to p = 1+3q
...
Similarly,
if the market-clearing price would be pL = 2, Secunda would understand that Primus got
an L-signal and her supply would switch to p = 1, making this the market-clearing price
...
If she exploits the information embedded in different prices, the prices
will be pH = 4 and pL = 1
...
Market-clearing equilibria
with rational agents require that the information embedded in prices is fully exploited, and
this is what the notion of rational expectations equilibrium is about
...
2
An example
We consider a two-asset one-period economy in which all random variables are independent
and normally distributed, with strictly positive standard deviations
...
The bond has a current price normalized to 1
and will pay a riskless amount (1+r) at the end of the period
...
For convenience,
denote by τy = 1/s2 the precision of Y
...
They have identical cara preferences expressed
by the utility function (12), defined over terminal wealth, with k = 1
...
Secunda has a
similar endowment w = m + Z2 , where Z2 is independent but identically distributed to Z1
...
Primus is an informed trader, while Secunda is uninformed
...
By (11), Primus’ posterior distribution is
Y |x ∼ N
mτy + xτx
1
,
τy + τx
τy + τ x
...
The uninformed agent receives no signal
...
If she knew how this trading affects the market-clearing
prices, she could extract information about the informed trader’s signal from the market
price
...
In equilibrium, the pricing rule must be correct; that is, Secunda must
extract information that is consistent with Primus’ signal
...
That is, assume
p = am + bx − cZ
(14)
for some appropriate coefficients a, b, c to be determined as part of the equilibrium
...
b
b
√
Since Z ∼ N (0, 2sz ), this implies that η is an unbiased (but garbled) estimate of the signal
x actually received by Primus
...
Indeed, as we see from the right-hand side, η ∼ N (m, sx + (c/b) 2sz )
...
Let τη be the precision associated with sη = sx + (c/b) 2sz ; of course, τη < τx
...
Again by (13), Secunda’s demand for the risky asset is
α2 =
mτy +ητη
τy +τη
− p(1 + r)
1
τy +τη
= mτy + ητη − p(1 + r)(τy + τη )
...
This yields (after substituting for η)
p=
2mτy + x(τx + τη ) − Z[1 + (c/b)τη ]
...
(1 + r)(2τy + τx + τη )
This confirms that the pricing rule conjectured in (14) is linear and concludes the example
...
We say that the rational expectations equilibrium is fully
revealing if prices can be used to infer exactly what are the signals
...
The rational expectations equilibrium is partially revealing if only a partial inference
is possible, as in the example discussed, where prices reflect both private information and
exogenous noise
...
e
...
e
...
Ex
...
24 Assume that the aggregate supply is known to be z
...
38
8
...
Each trader observes a signal Xi (i =
1, 2,
...
The construction of a rational expectations equilibrium (REE) can be outlined in five steps
...
Specify each trader’s prior beliefs and propose a pricing rule (which for the moment is
only a conjecture) P c mapping the traders’ information to the prices of the assets
...
, Xn , ε) may incorporate some noise ε
...
The pricing rule must be determined in equilibrium; at this
stage, it is parameterized by undetermined coefficients because the true equilibrium
price is not known yet
...
Derive each trader’s posterior beliefs, given the parameterized price conjectures and
the important assumption that all traders draw inferences from prices
...
g
...
3
...
4
...
Since individual demands depend on traders’ beliefs, so do
prices
...
, Xn , ε) which provides the
actual relationship between traders’ signals and the prices
...
Impose rational expectations; that is, make sure that the conjectured pricing rule
P c coincides with the actual pricing rule P a
...
8
...
1
...
2
...
3
...
On the other sides, there are three important difficulties with the notion of a REE
...
If the number of possible signals is finite, then
for a generic set of economies there exists a REE
...
Similarly, for j the number of assets in the
economy, if the dimension S of the space of signals is lower than j − 1, then for a generic
set of economies there exists a REE
...
On the other hand,
39
if S = j − 1, there is an open set of economies for which no REE exists
...
Finally, for S > j − 1, all weird
things can happen
...
In a REE, traders must
know the pricing rule that specifies the equilibrium prices
...
One possible explanation is that it is learned over time,
but learning to form rational expectations is not an easy task
...
Third, there is the issue of price formation
...
This implicit auctioneer collects the “preliminary orders” and uses them to find the marketclearing prices
...
In
the next lecture, we will see that the microstructure literature improves on the rational
expectations model by making explicit the process of price determination
...
The common approach circumvents these difficulties by using specific examples
...
References
[1] S
...
Grossman and J
...
Stiglitz (1980), “On the impossibility of informationally efficient
markets,” American Economic Review 70, 393–408
...
K
...
4
...
O’Hara (1995), Market microstructure theory, Oxford: Basil Blackwell
...
Market microstructure: Kyle’s model
9
...
In auction models (also known as “order-driven”), the best available
price is defined by the submitted orders
...
The rational expectations model can be seen as an example of an (implicit) auction
model
...
In particular, Kyle assume batch-clearing; that is, all orders are fulfilled
simultaneously at the same price
...
Moreover, the market-maker can take trading positions and
has privileged access to information on the order flow
...
The
market-maker must set prices using only the information which is available to him, which
is determined by the trading protocol
...
Besides the market-maker, Kyle assumes that there is one informed agent (named
Primus) and a number of liquidity traders in the market
...
The informed trader chooses those transactions
which maximize the value of his private information
...
Thus price reflects
both the trading protocol and the strategic behavior of the informed trader
...
2
The model
We consider a one-asset one-period economy, with a zero riskless interest rate
...
Primus
and Secunda are risk-neutral expected utility maximizers
...
The only
available asset is a risky stock which will pay a risky amount Y ∼ N (m, sy ) at the end of
the period
...
Secunda is the uninformed market-maker, who knows
only the prior distribution of Y
...
The noise traders submit a random demand Qu ∼ N (0, su ); if this is negative, they
are on balance selling
...
The market-maker receives an aggregate demand Q = Qi + Qu ; she knows the sum but
not who demanded what
...
This is consistent
with free entry of competing market-makers, which impairs any monopoly power of the
single market-maker
...
Primus places an order Qi which maximizes his profit E[(Y − p)Qi |Y = y] = (y − p)Qi
...
At the same time, his
demand Qi affects the price quoted
...
Kyle proves that there exists a linear equilibrium for this model such that the marketmaker pricing rule is
P (Q) = m + αQ
(15)
and Primus’ trading rule is
Qi = β(Y − m),
where
α=
1 sy
2 su
and
β=
(16)
su
...
Before
that, a quick commentary may be useful
...
When α is high and orders have a significant price impact,
then β is low because Primus trades less aggressively (to avoid the impact of his own
trades)
...
When su is high, Primus’ order is less likely to be
a conspicuous component of the total order flow, and therefore he can afford to trade more
aggressively
...
First,
that the best reply of Secunda to an insider using the trading rule (16) is precisely (15)
...
We prove only the first fact, and leave the second as an exercise
...
Secunda’s prior for Y is that Y ∼ N (m, sy )
...
As
Qi = β(Y −m), the total order flow can be written Q = β(Y −m)+Qu , with Qu ∼ N (0, su )
...
This
can be used to make inferences about Y
...
Looking at E(Y |η) and substituting for τη and τy , we find
ms2 + ηs2 β 2
mτy + ητη
u
y
=
...
2
Secunda sets her price equal to her best estimate of Y ; that is, P = E(Y |η)
...
Hence, the price set by Secunda is:
P = E (Y |η) =
1
2
m+
Q
+m
β
=m+
1
Q
...
9
...
1
...
She
uses information gathered by the traders as transmitted by their demands
...
We can actually measure how much of the trader’s information is revealed in Kyle’s
model by looking at the variance of Secunda’s posterior distribution for Y
...
After she observes Q and updates her prior, the
variance becomes (check the steps!)
V (Y |η) =
s2 s2
s2
1
y u
y
= 2 2
=
...
Regardless of the exact value, the important
y
message is that the updated variance is somewhere in between the prior variance and a
zero variance
...
If it were to drop to zero, Primus would make no profit
because all his trades would clear at the perfectly revealing price Y
...
Since the market-maker cannot separate informative from uninformative trade, the
transmission of information is noisy and the informed trader can use this to his advantage
...
3
...
Efficient markets tend to gravitate
towards constant liquidity, defined as the price impact of orders
...
Then if Primus buys
10’s worth of stock each day in the next month, each purchase will push the price of the
stock progressively upward
...
Then,
when at the end of next month liquidity has returned to normality, Primus could suddenly
sell the 300’s worth of stock purchased during the month with almost no price impact and
make a riskless excessive profit similar to an arbitrage opportunity
...
Kyle (1985) considers a multiple-period extension of his model which generates constant
liquidity in equilibrium
...
9
...
This accelerates price discovery
and reduces insider profits
...
References
[1] S
...
S
...
B
...
[2] A
...
[3] S
...
[4] R
...
Lyons (2001), The microstructure approach to exchange rates, Cambridge (MA):
The MIT Press, Chap
...
[5] M
...
44
10
...
1
Introduction
The market in Kyle’s model is “order-driven”: the market-maker sees the order flow and
sets a price which clears the market in a single batch
...
This lecture considers a different trading protocol, possibly closer
to reality, which is “quote-driven” and involves sequential trades
...
All trades involve a dealer, who posts bid and ask prices
...
Thus orders are fulfilled sequentially at (possibly)
different prices
...
At each trade, the current trader
may be informed or uninformed so the model can accomodate more than one informed
trader
...
First,
there is an explicit bid-ask spread, as opposed to the single market-clearing price of Kyle’s
model
...
Third, since
trades occur sequentially, it is possible to analyze explicitly how the information content of
each trade affects the bid-ask spread
...
2
The model
We consider a one-asset one-period economy, with a zero riskless interest rate
...
Both the dealer and the informed traders are risk-neutral expected
utility maximizers, while the uninformed traders trade only for liquidity or hedging reasons
...
For simplicity, we assume that it can take only the high value Y = 1 or
the low value Y = 0
...
All the informed traders know the realization of Y (because they have received a perfectly
informative signal about it), while Primus does not
...
The pool of potential traders is
given and the dealer knows that q of it are informed traders and (1 − q) are uninformed
traders
...
The informed trader can buy, sell, or pass at her discretion
...
As in Kyle’s model, the dealer sets prices such that the expected profit on any trade is
zero
...
) This implies that
Primus must set prices equal to his conditional expectation of the asset’s value given the
type of transaction taking place
...
(18)
This rule takes explicitly into account the effect that the sale/purchase of one unit would
have on Primus’ expectations
...
Such “regret-free” price-setting behavior makes sure that prices incorporate the information revealed by a trade
...
This generates a sequence of
bid-ask prices {bt , at } that change over time, paralleling the evolution of Primus’ beliefs
...
10
...
Denote by Bt and St respectively the event that at the trading
opportunity t there is a buy or a sale
...
(20)
In order to find what a1 should be, we need to compute P (Y = 1 | B1 )
...
P (Y = 1) · P (B1 | Y = 1) + P (Y = 0) · P (B1 | Y = 0)
(21)
Since we know by assumption that P (Y = 1) = P (Y = 0) = 1/2, it suffices to determine
P (B1 | Y = 1) and P (B1 | Y = 0)
...
On the other hand, the informed trader know Y and therefore buy only if
Y is high and sell only if Y is low
...
Therefore, conditional on Y = 1, the probability of a buy is 1/2 if it comes from an
uninformed trader and 1 if it comes from an informed trader
...
By a similar reasoning, P (B1 | Y = 0) = (1/2)·(1/2)+(1/2)·0 =
1/4
...
(1/2) · (3/4) + (1/2) · (1/4)
4
46
Ex
...
25 By a similar reasoning, compute P (S1 | Y = 1) and P (S1 | Y = 0) and deduce
that P (Y = 1 | S1 ) = 1/4
...
4
4
Suppose that there actually occurs a buy at a price of 3/4
...
This acts as a new prior for the next trading opportunity
...
We need to compute P (Y = 1 | B1 , B2 ) and P (Y = 1 | B1 , S2 )
...
Ex
...
26 Check that P (Y = 1 | B1 , S2 ) = 1/2 and deduce that b2 = 1/2
...
Therefore, the fact that the first transaction is a buy or a sale
reveals information
...
The exercise can be repeated
...
For instance, if the first four events are B1 , S2 , B3 , and B4 , the
sequence will be
at
trade
t bt
1 1/4 3/4 B1
2 1/2 9/10 S2
3 1/4 3/4 B3
4 1/2 9/10 B4
5 3/4 27/28 etc
...
For instance, if
βt − σt = 0, then bt = 1/4 and at = 3/4
...
10
...
Prove that b1 = 3/8 and a1 = 5/8
...
4
Comments on the model
The assumptions about the trading protocol are crucial
...
An informed trader prefers
to trade as much (and as often) as possibile
...
This cannot occur in the model because the only trader who is allowed to trade is chosen
randomly and she can only buy or sell one unit of stock
...
The probabilistic selection process dictates that the population of traders facing the
dealer is always the same as the population of potential traders
...
Moreover, it
implies that plausible trading scenarios are ruled out
...
This cannot occur
here
...
That is,
the best predictor for the transaction price in t + 1 (given the information It available after
trade t) is the transaction price in t: E(pt+1 |It ) = pt
...
The current price is pt = at = 3/4
...
The probability that it will be a buy is 1/2 if the next trader is uninformed and 3/4
if he is informed
...
Hence, the next price will be 9/10 with probability 5/8
and 1/2 with probability 3/8
...
The martingale property dictates that prices respect semi-strong efficiency, in the sense
that they reflect all the information available to the dealer
...
10
...
1
...
The equilibrium spread in this model is such that when
the dealer trades with an informed trader he loses money (as in Kyle’s model, the informed
trader knows exactly the value of the asset)
...
The equilibrium spread balances
these losses and gains exactly so that expected profits are zero
...
Learning takes place over time, as it involves the sequential arrival of distinct orders
...
However, if a preponderance of sales takes place over
time, the dealer adjusts her beliefs and prices downward
...
3
...
The dealer must uncover the private information hidden behind trades before
market prices can be efficient
...
Glosten and P
...
[2] R
...
Lyons (2001), The microstructure approach to exchange rates, Cambridge (MA):
The MIT Press, Chap
...
[3] M
...
49
11
...
1
Introduction
In Glosten and Milgrom’s (1985) model, spreads arise to protect the dealer from the risk of
trading with an informed trader
...
If so, they might quit the market, exposing the dealer to a sure loss against the
remaining informed traders
...
11
...
There are three
types of agents: an informed trader (Primus), a single market maker (Secunda), and nine
uninformed traders
...
We call
these two types of uninformed traders respectively “big” and “little”
...
We assume that Y is randomly distributed over the interval [0, 4] with unconditional
mean m = 2
...
Finally, we assume that both the uninformed traders are price sensitive
...
On the other hand, the four big uninformed traders are willing to accept a higher
ask price if they decide to place a big order
...
3 each; U2 up is willing to buy it up to a price of p = 2
...
6 each; and U4 up to a price of p = 2
...
For simplicity, suppose that Primus knows that the true value of the asset is 4 and
therefore wants to buy the asset
...
She
must a set a price for little orders of 1 unit and a (possibly different) price for big orders
of 5
...
First, assume that Primus trades big
...
Since there is a one in five chance that the big trade
comes from Primus, the price p must solve the equation (1/5)(p − 4) + (4/5)(p − 2) = 0,
from which p = 2
...
However, if Secunda were to quote a price p = 2
...
This would make the number of uninformed big traders
50
drop to three, in which case the competitive equilibrium price should solve the equation
(1/4)(p − 4) + (3/4)(p − 2) = 0, from which p = 2
...
At such a price, U2 would drop out
rising the new competitive price (with only two big uninformed traders left) to p = 2
...
At such a price, U3 would drop out and the new competitive price (with only one big
uninformed trader left) should be p = 3
...
At this price, however, Primus’ profit would be zero and therefore he
would rather trade a small quantity and make a positive profit
...
Thus, assume now that Primus trades a small quantity
...
1
...
Therefore, they would quit the market leaving Secunda with no
uninformed customer to offset the sure loss caused by Primus’ order
...
11
...
However,
while trading halts occasionally occur, they are not usual
...
One possible
answer is that some characteristics of the trading mechanism make it less responsive to
these difficulties
...
The basic idea is to compare a perfectly competitive dealer market against the “specialist
system” of the New York Stock Exchange, where each investor has to trade through a single
monopolist
...
This implies that the expected profit
on each and every trade must be zero
...
In the monopolist case, the specialist does not face competition and can use a different
pricing rule
...
Secunda may willingly accept to be losing money over large trades provided that she can
make enough positive profits on small trades to offset these losses
...
)
Under asymmetric information, because of the greater protection afforded to the market
maker, a monopolistic regime can lead to a greater social welfare than a competitive regime
...
4
The basic steps in the model
We consider a one-asset one-period economy in which all random variables are independent
(unless otherwise mentioned) and normally distributed, with strictly positive standard deviations
...
51
Let τy denote the precision of Y
...
Each trader is a
risk-averse expected utility maximizer with the same constant coefficient of (absolute) risk
aversion k > 0
...
The market maker (named Secunda) is a risk-neutral expected utility
maximizer
...
Finally, each trader receives a noisy
signal X = Y +ε, where ε ∼ N (0, sx )
...
A trader may
seek a trade for speculative reasons (taking advantage of his information) or (regardless of
his signal) for liquidity or hedging reasons, such as rebalancing his portfolio
...
Then, given the pricing rule p(Q) chosen by the market maker,
Primus’s decision problem is to choose Q so as to maximize the expected utility of his final
wealth
W1 = [c − Q · p(Q)] + (W0 + Q)Y
...
2
2
Assuming a differentiable pricing rule p(Q) (which is yet to be determined), this leads to
the first-order condition
[Q · p (Q) + p(Q) − m] (τy + τx ) + kQ
kW0
=X−
− m
...
Since Secunda sees
Q, she can calculate the left-hand side of (22) and add m to obtain the signal
η=X−
kW0
,
τx
which is normally distributed with mean X and standard deviation sx + sw
...
Using this
information (revealed by Primus’ order flow), Secunda sets her pricing rule p(Q)
...
Moreover, and more importantly, a necessary
condition for a solution to exist is α > (1/2)
...
52
11
...
Higher orders are filled at higher (and, hence,
worse) prices
...
When a trader transacts for portfolio-rebalancing reasons, the greater the slope, the more
costly is for him to achieve his optimal holding
...
Glosten (1989) proves that ex ante utility
is strictly inferior under asymmetric information
...
This condition requires that the risk of informed trading is not so
large as to overwhelm the market maker’s ability to set market-clearing prices which let
her break-even
...
On the other hand, if we suppose that the market maker acts as a monopolist, she can
choose a pricing rule p(Q) to maximize her expected profits
...
With this pricing strategy, the specialist quotes prices such that she loses money on large trades but makes money
on small trades
...
This implies that prices for small trades are higher than they would be in a competitive
market
...
On the
other hand, these monopolistic prices makes trade possible where it would otherwise be
not
...
References
[1] M
...
Brunnermeier (2001), Asset pricing under asymmetric information: Bubbles,
crashes, technical analysis and herding, Oxford: Oxford University Press
...
Glosten (1989), “Insider trading, liquidity, and the role of the monopolist specialist,”
Journal of Business 62, 211–236
...
O’Hara (1995), Market microstructure theory, Oxford: Basil Blackwell
...
Noise trading: limits to arbitrage
12
...
Somehow, these people
fail to behave optimally
...
When there are many noise traders, the market is more liquid in the sense of having
frequent trades that allow people to observe prices
...
The more noise trading, the less information gets into prices,
and therefore the less efficient the market
...
This limits
their willingness to take large positions and slows down the process by which adjustment to
fundamental values takes place
...
In the words of Black (1986), “noise creates the opportunity to trade profitably, but at
the same time makes it difficult to trade profitably”
...
2
A model with noise trading
We begin with a known result
...
Assume that there are two assets, one
of which is a risky stock with a current price of p and a random payoff Y ∼ N (m, s) and
the other is a riskless bond with a current price of 1 and a riskless payoff of (1 + r)
...
Each agent chooses a portfolio of two assets in the first period (when young)
and consumes the payoff from his portfolio in the second period (when old)
...
In each period, there are old timers who sell the assets in their
portfolios to youngsters
...
The risky asset is a stock that costs pt in period t
and pays a fixed dividend r in the next period
...
Hence, in period t,
youngsters should regard the stock as normally distributed with payoff Y ∼ N (Et (p)+r, st )
...
However, there are two
types of agents in the economy with respect to beliefs
...
The remaining λ are noise traders, who in period t misperceive the
expected price of the stock Et (p) by an identical amount εt and wrongly believe that the
payoff to the stock is Y + εt ∼ N (Et (p) + r + εt , st )
...
We assume that the noise traders’ misperception is a normal random variable εt ∼
N (µ, σ) independent and identically distributed over time
...
Each agent maximizes his expected utility
...
ks2
kst
t
When noise traders overestimate expected returns, they demand more of the risky asset than
sophisticated investors; when they underestimate, they demand less
...
To calculate the equilibrium prices, note that the old sell to the young and so in each
n
i
period λαt + (1 − λ)αt = 1
...
We consider only steady-state equilibria by imposing the requirement that the unconditional distribution of pt+1 be identical to the distribution of pt
...
1+r
r
r
Since all terms but the second one are constant, the one-step-ahead variance of Y is
s2 =
t
λ2 σ 2
...
+
−
1+r
r
r(1 + r)2
(26)
Ex
...
28 Explicitly carry out the recursive substitution leading to Equation (25)
...
The last three terms in (26) represent the impact of noise trading on the
price of the stock
...
The second term captures the fluctuations in price due to variations in noise traders’
misperceptions
...
Moreover, the higher the fraction of noise
traders, the higher the volatility of the price of the stock
...
If noise traders are bullish on average, there is a
positive “pressure” which raises the price above its fundamental value
...
Both noise traders and
sophisticated investors in period t believe that the stock is mispriced, but the uncertainty
about pt+1 makes them unwilling to bet too much on this mispricing
...
In a sense, noise traders
“create their own space”, driving the price of the stock down and its return up
...
3
Relative returns
The model can also be used to show that the common belief that noise traders earn lower
returns than sophisticated investors and therefore are eventually doomed to disappear may
not be true
...
Hence, assuming equal
initial wealth, the difference between noise traders’ and sophisticated investors’ total returns
is the product of the difference in their holdings of the stock and the excess return paid by
a unit of the stock
n
i
∆R = (αt − αt ) [pt+1 + r − pt (1 + r)]
...
kλ2 σ 2
ks2
t
Note that this difference becomes very large as λ becomes small
...
56
Substituting from (24), the expected value of the excess return (conditional on the
information available in t) is
Et pt+1 + r − pt (1 + r) = ks2 − λεt =
t
kλ2 σ 2
− λεt
...
kλσ 2
Taking the unconditional expectation, we have
E ∆R = µ −
(1 + r)2 µ2 + (1 + r)2 σ 2
,
kλσ 2
(27)
which can be positive for intermediate (and positive) values of µ
...
According to De Long et alii (1990), Equation (27) may be interpreted in terms of
four effects
...
For µ > 0, this increases noise traders’ expected returns relative to sophisticated investors
because noise traders on average hold more of the risky asset and thus earn a larger share
of the rewards to risk bearing
...
As noise
traders become more bullish, they demand more of the risky asset and drive up its price
...
The second term in the numerator represents the “buy high–sell low” effect
...
The more variable their
beliefs are, the more damage their poor market timing does to their returns
...
A higher variability in noise
traders’ beliefs increases the price risk
...
Being risk averse, however, they reduce
the extent to which they bet against noise traders in response to this increased risk
...
Ex
...
29 Suppose σ 2 = 1
...
12
...
According
to Friedman (1953), “people who argue that speculation is generally destabilizing seldom
realize that this is largely equivalent to saying that speculators lose money, since speculation
57
can be destabilizing in general only if speculators on average sell when currency is low in
price and buy when it is high
...
De Long et alii (1991) shows that, when noise traders follow positive feedback strategies,
sophisticated investors may prefer to anticipate the bandwagon
...
Shleifer and Vishny (1997),
shows that the existence of noise traders is enough to limit the positions of sophisticated
investors in an agency context where professional arbitrageurs must attract outside funding
...
The model by De Long et alii (1990) discussed here, instead, shows that under reasonable
conditions noise traders can dominate rational traders
...
Thus, the second argument alone is not conclusive either
...
5
Excursions
De Long (1990) shows that noise traders may earn higher expected returns than rational
investors
...
Palomino (1996) shows that in an
imperfectly competitive market even a higher utility may occur
...
Black (1986), “Noise,” Journal of Finance 41, 529–543
...
Blume and D
...
[3] E
...
Easley (1993), “Economic natural selection”, Economics Letters 42,
281–289
...
B
...
Shleifer, L
...
Summers, and R
...
Waldmann (1990), “Noise trader
risk in financial markets”, Journal of Political Economy 98, 703–738
...
B
...
Shleifer, L
...
Summers, and R
...
Waldmann (1991), “Positive feedback
investment strategies and destabilizing rational speculation”, Journal of Finance 45,
379–395
...
Friedman (1953), Essays in positive economics, Chicago: University of Chicago Press
...
Palomino (1996), “Noise trading in small markets”, Journal of Finance 51, 1537–
1550
...
Shleifer and R
...
Vishny (1997), “The limits of arbitrage”, Journal of Finance 52,
35–55
...
Noise trading: simulations
13
...
For convenience, we can distinguish three main cases
...
For instance, if a trader cares about a longer horizon than the one studied in a
one-period model, his choices may appear nonoptimal in the framework of one-period, even
though they are so over the longer horizon
...
In some of the earlier lectures, we have implicitly relied on this interpretation
when considering a few examples in which noise trading affects informed agents’ plans
...
This
may occur for many reasons, but it is a reasonable approximation that in order to do so
they must be using “wrong” preferences or “wrong” beliefs
...
Whatever the reason impairing the rationality of their choices, it
remains true that non-noise agents must formulate their plans by taking into account the
actions of the noise traders
...
Third, agents’ choices may be rule-driven
...
Consider feedback traders, who base their trading on a rule
driven by the past price dynamics
...
Of
course, this might also be the outcome of a deliberate rational choice
...
This
lecture discusses two models which fall into this latter category
...
2
A simple dynamic model
Consider a one-asset economy, with a zero riskless interest rate
...
We assume that the value of the asset is
constrained to lie between a minimum m and a maximum M
...
For convenience, we assume that all traders of the same type behave identically
and that they are all risk-neutral
...
Besides the current price pt , Primus uses all the additional information It available to
reach his best estimate ut of the long-term value of the asset
...
Dropping subscripts for simplicity, when u > p a capital gain is expected
and hence Primus buys; similarly, he sells if u − p < 0
...
To model this second effect, we assume that
there is a bimodal positive function π over [m, M ] achieving its minimum in p = u; the
higher π, the higher the (expected) profitability and hence Primus’ willingness to trade
...
The parameter a > 0 measures the strength of fundamentalists’ demand
...
) This matches Black’s (1986) hypothesis that “the farther the price of a stock is from its (investment) value, the more aggressive the information
traders become
...
”
While Primus is a sophisticated investor who closely monitors the market, Secunda is
a more simple-minded operator who follows simpler rules
...
Using a simple extrapolative rule, Secunda responds only to the spread between the current price and the (long-term) investment
value
...
Given that noise traders chase prices, their excess demand can be modelled as
D2 (p) = k · (u − p) = b · (p − y)
for p in [m, M ] and 0 otherwise
...
Note how Secunda “chases” prices up and down,
following a pattern which may potentially sustain both bull and bear markets
...
He announces a price pt in each period, and then executes all orders
received at that price filling out excess demand or absorbing excess supply
...
Given that the aggregate demand
is D1 + D2 , we assume that he sets prices such that
pt+1 = pt + c[D1 + D2 ]
where c > 0 measures the rapidity of the price’s adjustments
...
Day and Huang
(1990) do so for u = y and find out that (for reasonable value of the parameters) the model
generates a time series matching a regime of irregular fluctuations around shifting levels
...
The intuition is the following
...
Secunda enters the
market, while Primus is not much willing to sell
...
This drives the prices up, initiating a bull
market until the price reaches a level at which Primus begins to sell consistent amounts
and creates an excess supply
...
This market admits two types of equilibria
...
In a temporary equilibrium, the aggregate demand is D1 + D2 = 0: Primus’
and Secunda’s demands exactly offset each other
...
For a pictorial
representation, draw the phase diagram of pt+1 versus pt on the interval [m, M ], with the
45-degree line and the price adjustment function which has a local minimum near m and
a local maximum near M (the function is initially convex and then concave)
...
For u = y, only Cases c) and d) may occur
...
If this is negative, the demand from Primus at p = y locally overwhelms the demand from
Secunda: we say that flocking is weak; otherwise, we say that it is strong
...
13
...
If flocking is weak, prices converge to y for c < c∗ =
−2/[D1 (y) + b] and locally unstable 2-period cycles around y arise for c > c∗
...
Case d) above occurs under strong flocking where, for instance, there are prices high
enough between the equilibrium price pM and M to make excess demand from Primus fall
so precipitously that the price is pulled below y
...
A similar fluctuation arises on the other side
...
13
...
Chaos: perpetual, erratic speculative fluctuations
...
Switching regimes: stock prices switch between bull and bear markets at random
intervals with irregular fluctuations around a low and a high level respectively
...
Ergodicity: the frequency of observed prices converges
...
Appearance of deceptive order: the price trajectory may pass close to cycles of varying
periodicities but, as these are unstable, it will eventually be driven away
...
For instance, if u > y, Primus has bullish expectations and we obtain the
bullish market described as Case a)
...
Of course, as u changes over time due to information arrival, the switch between bullish and bearish regimes may be driven by exogenous
shocks
...
However, in general, she will buy high and sell low, acting as the sheep to be sheared
by Primus
...
When Secunda buys in, the market usually goes up (except for a few
occasional sharp drops)
...
The possibility of cycles makes technical trading possible
...
Therefore, assuming that there are enough feedback traders,
technical trading may be temporarily effective, but sooner or later it is likely to go astray
...
3
An artificial stock market
A financial market involves a group of interacting agents, who adapt to the arrival of new
information and update their model of the world
...
LeBaron et alii (1999) is a good example
...
Consider an economy with a riskless bond and a risky stock, extending over several
periods
...
There are 25 shares of the stock and 25 agents trading, each of which is an expected
utility maximizer with the same (constant) coefficient of absolute risk aversion k
...
If stock prices were normally distributed, there would be a rational expectations
equilibrium where each agent holds one share and the price of the stock in one period is a
linear function of the dividend paid in that period: pt = αdt + β
...
In this simulated market, each
agent is endowed with the capacity of using his own several forecasting rules
...
These rules may apply at all times, or in certain
63
specific cases
...
After a chunk of trading periods, each agent reexamines the rules, eliminates
the worst ones and generate new ones
...
A (fictitious) auctioneer clears the market fixing a price which balances
the demand and the fixed supply of shares
...
Simulations are run under the assumption of slow or fast learning, which refers to the
frequency with which agents revise their forecasting rules
...
First, the stock price has a significantly greater kurtosis, especially under fast learning
...
Trading volume series are highly persistent and there are significant cross correlations
between volume and volatility, as is the case in actual stock markets
...
Black (1986), “Noise,” Journal of Finance 41, 529–543
...
H
...
Huang (1990), “Bulls, bears and market sheep,” Journal of Economic
Behavior and Organization 14, 299–329
...
LeBaron, W
...
Arthur, and R
...
64
14
...
1
Introduction
Most economic theory deals with an idealized homo œconomicus who embodies principles
of rational decision-making such as expected utility maximization, Bayesian learning and
rational expectations
...
The building blocks of the decision-making process of the homo œconomicus are beliefs
and preferences
...
By combining these together, he derives preferences about uncertain
options
...
These systematic errors of judgement are called biases
...
For instance, people tend to use heuristics and shortcuts to make their probabilistic
judgements and their preferences are often influenced by how a problem is framed
...
14
...
Most people cannot correctly calibrate their probabilistic beliefs
...
and then your prediction is compared against the actual outcome
...
If you are good at probabilistic judgements, you should expect to encounter
about 98% of outcomes inside the confidence interval and thus be “surprised” only 2% of
the times
...
Thus,
beware the investor who is 99% sure
...
After an event has occurred, people cannot properly reconstruct their state
of uncertainty before the event
...
It almost seems that the event was so inevitable that it
could have been easily predicted
...
First, it tends
to promote overconfidence, by fostering the illusion that the world is more predictable than
it is
...
65
Optimism
...
Suppose
you are asked:
Is your ability as a driver above or below the median?
and that the same question is posed to the members of a group
...
The combination of optimism and
overconfidence leads investors to overestimate their knowledge and underestimate risks,
leaving them vulnerable to statistical surprises
...
In a survey
admistered to 45 investors, De Bondt (1998) reports that 89% agree with the statement “I
would rather have in my stock portfolio just a few companies that I know well than many
companies that I know little about” whereas only 7% agree with the statement “Because
most investors do not like risk, risky stocks sell at lower market prices” and just 18% agree
with the statement that “The risk of a stock depends on whether its price typically moves
with or against the market
...
pose you are asked:
People tend to spot regularities even where there is none
...
Odean (1998) reports that, when individual investors sold a stock and
quickly bought another, the stock they sold on average outperformed the stock they bought
by 3
...
This costly overtrading may be
explained in terms of spurious regularities and overconfidence
...
De Bondt (1993) reports that the average gap between the percentage of investors
who is bullish and the percentage who is bearish increases by 1
...
14
...
People’s preferences are especially sensitive to
changes
...
Would you prefer an
additional gain of 5,000 for sure or a 50–50 chance for a gain of 10,000?
B: Imagine than you are richer by Euro 30,000 than you are today
...
Apparently, they tend to favor
the narrow framing based on gains and losses rather than the broader (and more relevant)
framing based on the final wealth
...
People’s sensitivity to losses is higher than their sensitivity to gains
...
If heads, you lose Euro 100
...
Probability weighting
...
Suppose you are asked:
Given a chance for a gain of Euro 20,000, would you pay more to raise the probability of
gain from 0 to 1%, from 41 to 42%, or from 99 to 100%?
While expected utility predicts that the answer should be the same, most people would
pay significantly less for raising the probability to 42%
...
Prospect theory
...
The major modifications are three
...
Second, the slope of the utility function
over changes in wealth is greater for losses than for gains
...
Note that
decision weights wi from the probability distribution and maximizes
the decision weights are not necessarily interpretable as probability weights
...
Shape and attractiveness
...
They have the same
expected value and the same number of possible outcomes
...
ty (%)
95
50
10
90
50
50
90
50
Payoff 2
105,000
15,000
11,000
91,000
18,000
20,000
118,000
25,000
Prob
...
Its preferability may be explained by a combination of the overweighting
67
of the small probability of a large gain and of the different risk attitudes over gains and
losses
...
4
Framing effects
Reference point
...
Suppose you are asked the question:
Primus owns 100 shares of an asset, originally bought at 100
...
The price of the share was 160 yesterday and it is
150 today
...
A consequence of this is the
disposition effect: an investor who needs cash and owns two stocks is much more likely to
sell the stock whose price has gone up; see Odean (1998a)
...
People often fail to take into account the joint effect of their
decisions
...
Examine both decisions and state the options you prefer
for each decision
...
Decision B: choose between a sure loss of 7,500 and a 75% chance of a loss of 10,000
...
Now, consider the following question:
Choose between a 25% chance to win 2,400 and a 75% chance to lose 7,600 versus a 25%
chance to win 2,500 and a 75% chance to lose 7,500
...
However, note that the pair chosen in the first question amounts to the inferior option, while
the pair rejected is equivalent to the dominating choice
...
A common effect is the use
of a “mental accounting” strategy by which people classify their savings (or expenses) in
different segments and deal with them in different ways; for instance, a windfall rise in
income rarely finds its way in the (mentally separated category of) retirement savings or
health insurance
...
F
...
De Bondt (1993), “Betting on trends: Intuitive forecasts of financial risk and
return”, International Journal of Forecasting 9, 355–371
...
F
...
De Bondt (1998), “A portrait of the individual investor”, European Economic
Review 42, 831–844
...
Kahneman and M
...
68
[4] D
...
Tversky (1979), “Prospect theory: An analysis of decision under
risk”, Econometrica 47, 263–291
...
Odean (1998), “Do investors trade too much?”, American Economic Review 89,
1279–1298
...
Odean (1998a), “Are investors reluctant to realize their losses?”, Journal of Finance
53, 1775–1798
...
Behavioral finance: asset pricing
15
...
Between 1926 and 1990, for instance, the annual real rate of return on U
...
stocks has been
about 7%, while the real return on U
...
treasury bills has been less than 1% percent
...
This has become
known as the equity premium puzzle
...
A coefficient of relative risk
aversion close to 30 corresponds to a situation like the following one: suppose that Primus is
offered a 50–50 gamble between USD 100,000 and USD 50,000; then his certainty equivalent
should be around USD 51,209
...
Reitz (1988) has argued that the equity premium may be a rational response to a
time-varying risk of economic catastrophe, as bonds protect capital investment better than
equity
...
Moreover, since the data from 1926 contain the
1929 crash, the catastrophe in question must be of greater magnitude
...
A different line of research has managed to explain part of the equity premium introducing nonexpected utility preferences
...
People become averse to reductions in their consumption and this may
be used to explain the equity risk premium
...
While this sort of models are probably on the
right track, emphasizing only consumption-based habit-forming neglects the weighty role of
pension funds, endowments, and very wealthy individuals with long horizons
...
15
...
They exploit two ideas from the psychological
evidence about of decision-making
...
This translates into a
slope of the utility function which is greater over wealth decrements than over increments
...
This bears relevance on how
outcomes are aggregated: because of loss aversion, aggregation rules may not be neutral
...
An investor named Primus
exhibits loss aversion, modelled by a utility function over wealth increments such as u(x) = x
if x ≥ 0 and u(x) = 2x if x < 0
...
Primus believes that their values one
year from now will be either 70 or 150 with equal probability and that the two distributions
are stochastically independent
...
The relevance of mental accounting for the equity premium puzzle can be seen by confronting Primus with the choice between a risky asset paying an expected 7% per year
with standard deviation of 20% and a safe asset yielding a sure 1%
...
The longer Primus’ investment horizon, the more attractive the risky
asset, provided that the investment is not evaluated frequently
...
15
...
Suppose that Primus’ preferences conform
to prospect theory
...
The second element necessary to build a multi-period model is a specification of the
length of time over which an investor aggregates returns; that is, his evaluation period
...
Consider a young investor saving for retirement 30
years off in the future who gets a newsletter from his mutual fund every quarter
...
Accordingly, he will behave as someone else whose investment
horizon is just one quarter
...
Benartzi and Thaler
(1995) approaches the puzzle by asking how long should be the evaluation period of an
investor with prospect theory preferences to explain the equity premium
...
The stock index is compared both with treasury
bills returns and with five-year bond returns, and these comparisons are done both in real
and nominal terms
...
And it is argued that nominal terms are
preferable because they are used in most annual reports (and because evaluation in real
terms would yield negative prospective utility over most periods)
...
It is found that the evaluation period that makes a portfolio of 100% stock indifferent
to a portfolio of 100% bonds in nominal terms is about 13 months
...
If bills are used in
place of bonds, this period is one month shorter
...
An obvious criticism to this findings is that most people prefer to invest in portfolios
containing both stocks and bonds
...
Portfolios
carrying between 30 and 55% of stocks all yield approximately the same prospective value
...
For instance, the most frequent allocation
in TIIA-CREF (a very large defined benefit retirement plan in U
...
) is 50-50
...
The actual equity
premium in the data used was 6
...
What happens if the evaluation period lengthens? With a two-year
evaluation period, the premium falls to 4
...
0%, and
with 20 years to 1
...
Therefore, assuming 20 years as the benchmark case, we can say
that the price of excessive vigilance is about 5
...
A common asset allocation for pension funds has about 60% in stocks and 40% in
bonds
...
A possible explanation links myopic loss
aversion with an agency problem
...
Their choice of a short horizon creates a
conflict of interest between the manager and the stockholders
...
The goals of maximizing the present value of spending
over an infinite horizon versus maintaining a steady operating budget compete against each
other
...
4
An equilibrium pricing model
Barberis et alii (2001) builds on Benartzi and Thaler (1995) to provide a behavioral explanation of the equity premium puzzle based on an equilibrium pricing model
...
Thaler and Johnson (1990) finds that a loss is less painful to people when it comes
after substantial earlier increases in wealth: the earlier gains “cushion” the subsequent loss
and make it more bearable
...
”
Starting from an underlying consumption growth process with low variance, the combination of prospect theory and the effect of prior outcomes can generate stock returns with
high mean, high volatility and significant predictability, while maintaining a riskless interest rate with low mean and volatility
...
Except for
the modifications in investors’ preferences, the (representative) investor (named Primus) is
fully rational and dynamically consistent
...
After a run-up
in prices, Primus is less risk averse because those gains will cushion any subsequent loss
...
This variation in risk aversion allows returns to be much more volatile than the underlying
dividends: an unusually good dividend raises prices, but this price increase also makes
Primus less risk averse, driving prices still higher
...
Moreover, since the high volatility of returns leads
to frequent losses for stocks, the loss averse investor requires a high equity premium to be
willing to hold stocks
...
Barberis and M
...
[2] N
...
Huang and T
...
[3] S
...
H
...
[4] J
...
Campbell and J
...
Cochrane (1999), “By force of habit: A consumption-based
explanation of aggregate stock market behavior,” Journal of Political Economy 107,
205–251
...
M
...
[6] R
...
C
...
[7] T
...
[8] R
...
Thaler and E
...
Johnson (1990), “Gambling with the house money and trying to
break even: The effects of prior outcomes on risky choice”, Management Science 36,
643–660
Title: Financial Market
Description: A financial market is a market in which people trade financial securities, commodities, and other fungible items of value at low transaction costs and at prices that reflect supply and demand.
Description: A financial market is a market in which people trade financial securities, commodities, and other fungible items of value at low transaction costs and at prices that reflect supply and demand.