Wirtschaftswissenschaftliche Fakultt
der Eberhard-Karls-Universitt Tbingen
An Overreaction Implementation of the
Coherent Market Hypothesis
and Option Pricing
Rainer Schbel
Jochen Veith
Tbinger Diskussionsbeitrag Nr. 306
April 2006
Wirtschaftswissenschaftliches Seminar
Mohlstrae 36, D-72074 Tbingen
An Overreaction Implementation of the
Coherent Market Hypothesis
and Option Pricing
Rainer Schbel and Jochen Veith"
April 2006
Abstract
Inspired by the theory of social imitation (Weidlich 1970) and its adap-
tation to financial markets by the Coherent Market Hypothesis (Vaga 1990),
we present a behavioral model of stock prices that supports the overreaction
hypothesis. Using our dynamic stock price model, we develop a two factor
general equilibrium model for pricing derivative securities. The two factors
of our model are the stock price and a market polarization variable which
determines the level of overreaction. We consider three kinds of market sce-
narios: Risk-neutral investors, representative Bernoulli investors and myopic
Bernoulli investors. In case of the latter two, risk premia provide that herding
as well as contrarian investor behaviour may be rationally explained and jus-
tified in equilibrium. Applying Monte Carlo methods, we examine the pricing
of European call options. We show that option prices depend significantly on
the level of overreaction, regardless of prevailing risk preferences: Downward
overreaction leads to high option prices and upward overreaction results in
low option prices.
JEL Classification: G12, G13
Keywords: behavioral finance, coherent market hypothesis, market polarization,
option pricing, overreaction, chaotic market, repelling market
"
Department of Corporate Finance, Faculty of Economics and Business Administration, Eber-
hard Karls University Tbingen, Mohlstrasse 36, 72074 Tbingen, Germany. Phone: +49-7071-
2977088 or -2978206. E-Mail: rainer.schoebel@uni-tuebingen.de or jochen.veith@uni-tuebingen.de
1
1 Introduction
Since more than thirty years, the widely accepted hypothesis about the behavior of
asset prices in perfect markets is the Efficient Market Hypothesis (EMH) or, in its
continuous-time form, the geometric Brownian motion hypothesis. This implies
that asset prices are log-normally distributed. However, in the last twenty years,
new studies of security prices have reversed some of the earlier evidence favoring
the EMH. Behavioral finance has emerged as an alternative view of the financial
markets suggesting, that investing in speculative assets is a social activity. It thus
seems plausible that investors behavior and hence prices of speculative assets would
be influenced by social movements. A historically important study in this context
is Shiller s (1981) work on stock market volatility. Shiller found, that even though
the price of the aggregate stock market is importantly linked to its dividends, stock
prices seem to show far too much volatility to be in accordance with a simple dividend
discount model. He offered the following explanation: This behavior of stock prices
may be consistent with some psycological models. Psychologists have shown in
experiments that individuals may continually overreact to superficially plausible
evidence even when there is no statistical basis for their reaction . Other studies
by DeBondt and Thaler (1985, 1987), Campbell and Shiller (1988) or Poterba and
Summers (1988) provided some evidence that stock returns are mean reverting.
De Bondt and Thaler also propose an overreaction hypothesis, whereby financial
markets weigh recent information more heavily than prior information. Markets
eventually realize that they formed biased expectations and prices revert back to
their fundamental value.
In this paper, we follow the intensions of Shiller and DeBondt et al. and suggest
a two-factor stock price model that supports the overreaction hypothesis whereby
overreaction is due to herd behavior. The first factor of our model may be regarded
as the present value of future dividends as a proxy of the fundamental value of the
stock market. The second one is of psycological nature describing the behavior of
investors and how they esteem the fundamental value in their current psycological
environment. To model the behavior of investors, we use a theory of social imitation
originally developed by Weidlich (1970). The theory was adopted to the financial
markets by Vaga (1990) called the coherent market hypothesis (CMH). The CMH
is a nonlinear statistical model using terms such as order parameters to describe
the behavior of investors at a macroscopic level. Introducing two key parameters,
the level of crowd behavior and the prevailing economic fundamentals, the model
allows for a variety of associated market states ranging from efficient over coherent to
chaotic. However, Vaga assumed a direct proportional relationship between market
polarization and stock returns. The distributions of the polarization are therefore
simply viewed as steady state return distributions, which is in contradiction with
empirical evidence.
In this paper, we take a step further and formulate a continuous-time version of
the CMH. This opens the door to apply dynamical financial theory, especially op-
2
tion pricing. Starting from the Fokker Planck equation describing investor sentiment
in terms of polarization levels, we are able to extract and analyze the underlying
stochastic process. We also find that the process exhibits a special kind of mean re-
version, which can be used to model overreaction in stock prices when superimposed
on a geometric Brownian motion. The resulting return distributions offer the same
risk-return characteristics associated to the various states of investor sentiment as
proposed by Vaga, but overcome the deficiency of a steady state perspective. Based
on our stock price process, we develop a general equilibrium model for valuing deriv-
ative securities using the CIR (1985a) framework and apply it to the valuation of
European call options.
The remaining part of the paper is organized as follows: In the second Section,
we present the main ideas and properties of Weidlichs s theory of social imitation
and Vaga s adoption to financial markets, the coherent market hypothesis. In sec-
tion 3, we introduce a continuous-time version of the CMH and formulate a stock
price process in accordance with the overreaction hypothesis. Furthermore, we take
this stock price process and develop three different equilibrium models for valuing
derivative securities. Section 4 illustrates the characteristics of these models and
reports some results of a comparative static analysis. In section 5 we summarize
and provide some concluding remarks.
2 The Coherent Market Hypothesis
2.1 A Theory of Social Imitation
The starting point of our work is a theory of social imitation, originally provided
by W. Weidlich (1970). The general idea of Weidlich was, that there is a structural
similarity between a physical ensemble of interacting individual systems in thermal
equilibrium and the interactions in a society of human individuals. In particular,
Weidlich extends the well known Ising model (1925) of ferromagnetism to the phe-
nomenon of polarization of opinion in social groups.
Two important characteristics determine the behavior of an Ising system: The
interactions (e.g. attracting forces) between fluctuating individual subsystems and
a collective parameter determining the degree of the fluctuations. As a consequence,
the ensemble as a whole behaves macroscopically in a complete different way above
and below a certain critical phase transition parameter.
To transfer this basic approach to human behavior, Weidlich applies the following
framework: There is a group of n individuals who may influence each other with
respect to their decisions and each individual can decide between two attitudes: Up
(+) or down (-). The probability of finding, at time t, n+ individuals with attitude
+, and n- with attitude - is f[n+,n-;t] and the transition probabilities per unit
time for an individual of changing from attitude + to - or vice versa are p+-(n+,n-)
and p-+(n+,n-), respectively. Then, the rate of change of the probability density
3
function f[n+,n-;t] can be expressed by a master equation of the form1
"f(n+,n-; t)
= (n- +1)p-+(n+ - 1,n- +1)f[n+ - 1,n- +1; t]
"t
+(n+ +1)p+-(n+ +1,n- - 1)f[n+ +1,n- - 1;t] (1)
-n+ p+-(n+,n-) f[n+,n-;t] - n- p-+(n+,n-)f[n+,n-; t].
which simply sums up all probability currents to and from the considered state
(n+,n-) within the short time interval "t. To describe the market s excess opinion
(positive or negative), we introduce a new state variable which we call the market
polarization q defined as
"
1 1
q =(n+ - n-)/2n; q " - , (2)
2 2
and normalised to 2n.
After some algebraic manipulation, equation (1) can be expressed in terms of
q and a Taylor expansion up to the second order finally yields the Fokker Planck
equation2
" 1 "2
ft(q,t) =- [K(q)f(q,t)] + [Q(q)f(q,t)] . (3)
"q 2 "q2
Note, that the drift K(q) and the diffusion coefficient Q(q) still depend on the yet
unspecified transition probabilities p+- and p-+. The stationary solution of equation
(3) can be found by standard methods with the result
" #
Z
q
c K(q)
fst(q) = exp 2 dq (4)
1
Q(q) Q(q)
-
2
where c is a normalization constant.
The general time-dependent solution of (3) describes the evolution of the prob-
ability density from its initial condition (e.g. Dirac s delta function at t =0) to the
stationary distribution. Since there exists no closed expression, a series expansion
of the form
"
X
f(q; t) =fst(q) + ci exp [-i(t - t0)] Ći(x) (5)
i=1
may be applied. Here, the i form a denumerable sequence of eigenvalues and
Ći(x) are an orthonormal system of eigenfunctions to the Fokker Planck operator
"
-1 "2 Q(q) + K(q).
2 "q2 "q
In order to further evaluate fst(q) we need explicit assumptions about the transi-
tion probabilities p+-(q) and p-+(q). At this point, Weidlich employes the intuitively
1
For an illustration see figure 8 in appendix A.1.
2
For a detailed derivation see appendix A.1.
4
appealing analogy between the behavior of individuals in a social group and the ele-
mentary magnets within a ferromagnet. As in the Ising model, he assumes that the
individuals are subject to two forces: An internal field created by the individuals
themselves and proportional to the excess opinion of the group and an external field
independent of the behavior of the individuals. The transition probabilities then
depend exponentially on these force fields and adapted to our case, they are3
p+-(q) = ą exp [-(kq + h)] (6)
p-+(q) = ą exp [+(kq + h)].
The adaption parameter k describes the willingness of the individuals to align to
the group and preference parameter h measures the influence of the external force.
Applying these transition probabilities, explicit expressions for the drift K(q) and
diffusion coefficient Q(q) can be derived as
K(q) = ą [sinh(kq + h) - 2q cosh(kq + h)] (7)
ą
Q(q) = [cosh(kq + h) - 2q sinh(kq + h)].
n
Substituting (7) into (4), the solution for the stationary case can be evaluated easily
using numerical methods. The resulting distributions are shown in figure 1. There
are two regimes of behavior, ordered and disordered. For low adaption of the indi-
viduals to the group k ż kcrit, we find almost balanced states (q around zero) with
a centered distribution. If the adaption parameter k approaches its critical value,
a phase transition to more ordered states occures and the distribution undergoes a
bifurcation. For an increasing preference parameter h the distributions are distorted
additionally whereby the response to a change in h is largest near the transition.
[Insert figure 1]
2.2 Five Market States
The coherent market hypothesis was suggested by Vaga (1990) in contrast to the
efficient market hypothesis. Vaga carried over Weidlich s concepts to the financial
markets. He recognized, that the theory was able to drop the premise of rational
investors and therefore relaxes the assumption of approximately normally distributed
stock returns. Vaga re-interpreted the control parameters k and h and proposed
to use k as a measure of the level of crowd behavior or market sentiment and h
as a measure of the prevailing economic fundamentals. Each set of parameters
(k,h) corresponds to a different market state. He assumed a direct proportional
relationship between market polarization and stock returns. The distributions of the
polarization derived by Weidlich are therefore simply viewed as return distributions.
According to Vaga, there are four types of market states which can be described
by the model under reasonable assumptions about the control parameters:
3
For a detailed explanation see appendix A.2.
5
1. Efficient market (0 d" k ż kcrit,h =0): This market state can be obtained in
choosing neutral fundamentals and market sentiment well below the critical
threshold. In the limiting case k =0, the return distribution is similar to the
normal distribution (fig. 2.2 a).
2. Coherent market (k H" kcrit,h ż 0 or h Ą 0): If we presume crowd behavior in
conjunction with strong bullish or bearish fundamentals, the model allows to
create coherent market states. As we have seen in the last section, the return
distribution will be skewed to the left or to the right. The expected return is
different from zero, accompanied with an unusual small standard deviation.
In this state, the traditional risk-return tradeoff is inverted and investors can
earn above-average returns with below-average risk (fig. 2.4 a).
3. Chaotic market (k H" kcrit,h . 0 or h & 0): This market state can be modeled
in supposing crowd behavior with only slightly bearish or bullish fundamentals
which leads to a bimodal return distribution. This is the worst of all worlds
with low return for above-average risk (fig. 2.5 a and fig. 2.6 a).
4. Unstable transitions: These are all intermediate market states that cannot be
assigned to one of the former states (fig. 2.3 a).
We would like to extend this list of market states by the following case:
5. Repelling market (k <0,h =0): In this market state, we would observe just
the opposite of crowd behavior, namely a repelling behavior among investors.
Any investor would try to avoid to have the same opinion than the majority.
As a result, the return distribution becomes concentrated around q =0 and
eventually a Dirac s delta function as k approaches minus infinity.
(fig. 2.1 a).
All things considered, the CMH offers a framework to evaluate the risk-reward
potential for a given market. However, this is only achieved in a qualitative sense.
Our main point of criticism is the rigorous re-interpretation of Weidlichs polarization
distributions as return distributions. It is obvious, that Vaga s distributions f(q,t)
are unable to fulfill the most basic demands on financial return distributions since
they end up in a steady state.
6
3 The Continuous-Time CMH
3.1 Dynamics of Market Polarization
The Fokker Planck equation (3) describes a diffusion process for the polarization
q(t) with diffusion coefficients K(q) and Q(q). There are two basically different
approaches to the class of diffusion processes. On the one hand, one can define
them in terms of the conditions on the transition probabilities f(q,t), which is what
we did in the previous sections. On the other hand, one can study the variable q(t)
itself and its variation with respect to time. This leads to a stochastic differential
equation for q. In fact, under the usual regularity conditions, it can be shown
that there exists a Wiener process w(t) such that the polarization q(t) follows the
stochastic differential equation
p
dq = K(q)dt + Q(q)dw(t). (8)
To gain further insight into the behavior of the stochastic process dq, we have
to analyze the drift K(q), the diffusion coefficient Q(q) and the associated sample
paths in more detail. In figure 2 we plot the graphs of the functions K(q) and Q(q)
for different market states together with the associated distributions f(q,t). The
drift exhibits kind of mean reversion towards the stable nodes which in turn leads to
the maxima of the distributions f(q,t). The diffusion coefficient has its maximum at
q =0 and diminishes for growing |q|. This assures that the distribution is bounded
Ł ń
1
on the interval -1, . Figure 3 shows some sample paths, which are stationary.
2 2
As is apparent from figure 3 b), the sample path for the efficient market state
(k = 0,h = 0) has nothing in common with a random walk (e.g. no growing
variance over time), except that it has the same type of distribution at a given point
in time.
[Insert figure 2]
[Insert figure 3]
3.2 Stock Price Dynamics
In this section, we formulate a stock price process in accordance with the overreaction
hypothesis using the properties of the coherent market hypothesis. Two empirical
discoveries are the guidelines of our stock price model: The observed variation in
expected discounted dividends is too low to justifiy the volatility in stock price
movements and stock prices tend to mean revert. To adress these characteristics, we
assume that stock price changes are driven by two factors: The first one describes
the fundamental value D of the company or market index and can be regarded
as the value of all future dividends discounted to the present. The second factor
is the market polarization q, which we use to describe how investors esteem the
7
fundamental value in their current psycological environment. Since the fundamental
value is not a stationary target, we model its dynamics by a geometric Brownian
motion
dD = DDdt + DDdw2 (9)
a commonly used assumption in asset pricing. The return of the asset is then
modeled by superimposing the process dq in the following way4:
d(ln S) =d(ln D) +dq (10)
Here, S denotes the market price of the asset and describes the impact of the
subjective perception of the investors. From It s lemma, we derive the joint Markov
process for polarization q and stock price S as follows:
p
dq = K(q)dt + Q(q)dw1
p
1
dS =[D + K(q) + 2Q(q)]Sdt + Q(q)Sdw1 + DSdw2. (11)
2
As can be seen in figure 4 (solid lines), the return distributions assigned to the process
dS retain the characteristics of the density distributions of the process dq. Further-
more, dS shows the desired growing variance over time and therefore overcomes the
deficiencies of Vaga s return distributions without loosing its main attractions. We
can see this from the contour plots in figure 5 which illustrate the evolution of the
return distributions over a time period of two years.
[Insert figure 4]
[Insert figure 5]
3.3 Pricing Derivative Securities
In the following, we develop several equilibrium models for valuing derivative securi-
ties based on our stock price process. We consider an economy with a fixed number
of identical investors which receive only capital income and can choose between
immediate consumption or investment. The investors trade in perfectly compet-
itive, continuous markets with no transaction costs. Trading takes place only at
4
In terms of absolute values, equation (10) can be written as
S = D exp(q) (10a)
so we can see that a high level of positive coherence leads to high stock valuations and vice versa.
A loose interpretation may be that the stock price is most overvalued when everybody is too
optimistic about it (q at high levels). Eventually, no more buyers can be found and it is more
probable for the stock price to decrease than to increase. One major advantage of this assumption
about the stock price is that D drops out of the sde for S.
8
equilibrium prices and the representative investor is assumed to have time-additive
preferences of the form
"
Z
"
Et U(C(s),s)ds . (12)
t
Et is an expectation operator conditional on current wealth W at state (q, S) of the
economy and C(s) is the consumption flow at time s. There are three investment
assets: A stock which underlies the speculative forces described above, a risk free
asset and a derivative security written on the stock price S.
The representative investor s decision problem is equivalent to maximizing (12)
subject to the budget constraint
dS dF
dW =[(1 - P - F ] rWdt + PW + FW - Cdt (13)
S F
by selecting an optimal level of consumption and fractions P and F of wealth
invested in stock and the derivative instrument respectively.
Assuming, that the value of the derivative security will not depend on wealth
W, it can be shown, that the price of the derivative security F must satisfy the
fundamental valuation equation
Ł ń
1
0
tr(Fxx 0) +Fx - + Ft - rF =0 (14)
2
" "
q K(q)
Ł ń
with x = , drift vector (x,t) = and correla-
1
S
p
" D + K(q) + 22Q(q) S
pQ(q) 0 . denotes the market price of risk and
tion matrix (x,t) =
Q(q)S DS
r is the riskless rate of return.
Since the market polarization q is not a traded asset, we cannot use arbitrage
arguments to fully eliminate investors risk preferences from the derivative pricing
problem. To close the model, we make additional assumptions regarding risk pref-
erences by considering the following three types of investors:
Case 1 General equilibrium and risk neutrality.
Setting =0, equation (14) does no longer depend on risk preferences explicitly.
Thus we can calculate the price of the derivative security by assuming that risk
neutrality prevails. The riskless rate of return then simply equals the drift of the
stock price process
1
r(q) =D + K(q) + 2Q(q) (15)
2
and the fundamental partial differential equation (PDE) for valuing derivative se-
curities reduces to
1 1
0 = Q(q)Fqq + (2Q(q) +2 )S2 FSS (16)
D
2 2
+K(q)Fq + r(q)SFS + Ft - r(q)F.
9
Case 2 General equilibrium and a representative Bernoulli investor with log utility.
This case corresponds to the general equilibrium case of CIR (1985b) with r
derived endogenously. We assume, that investors are endowed with a logarithmic
utility function over consumption
U(C(t),t) =exp(-t)lnC(t) (17)
where is the parameter of time preference. Since in macroeconomic equilibrium
all investments must be held with a positive fraction in the representative investors
portfolio, the market clearing condition is P =1. That is, the aggregatedwealthis
always fully invested in stock. The risk free asset as well as the derivative security
are both in zero net supply and serve for risk allocation purposes only. Solving the
corresponding Bellman equation for this problem leads to the risk free interest rate
1
r(q) =D + K(q) - 2Q(q) - 2 (18)
D
2
and the vector of factor risk prices
" p
Q(q)
(q) = . (19)
D
The required risk premium S(q) regarding stock price risk becomes the lower, the
larger |q| and the higher the prevailing crowd behavior k (see figure 6a). Thus, opti-
mal guidance forces investors to reduce their risk aversion in times of high coherence
and strong crowd behavior. The fundamental PDE is now given by
1 1
0 = Q(q)Fqq + (2Q(q) +2 )S2 FSS (20)
D
2 2
+[K(q) - Q(q)] Fq + r(q)SFS + Ft - r(q)F.
Note that in both cases so far, r is endogenous and represents the instantaneously
riskless real interest rate in a pure barter economy, which can become negative in a
recession or a depression.
Case 3 Partial equilibrium and a myopic Bernoulli investor with log utility.
In this case, there is a small individual investor who seeks to optimize the com-
position of his entire portfolio. His transactions exert no influence on the formation
of prices and he acts as a price taker. As above, this investor has logarithmic utility
but the risk free interest rate is exogenously given and elastically supplied:
r = const (21)
10
Accordingly, the market clearing condition is now given by P + r =1. On the
macroeconomic level, only the derivative security is in zero net supply (F = 0).
Solving the Bellman equation under this clearing condition, we get
Ą ó
" p
1
D + K(q) + 2Q(q) - r
Q(q)
2
(q) = (22)
D
2Q(q) +2
D
for the vector of factor risk prices and the fundamental PDE becomes
1 1
0 = Q(q)Fqq + (2Q(q) +2 )S2 FSS
D
2 2
" ś
Q(q) 1
+ K(q) - D + K(q) + 2Q(q) - r Fq (23)
2Q(q) +2 2
D
+rSFS + Ft - rF.
We can see from (22) that the vector of factor risk prices (as well as optimal demand
" ) is independent of the instantaneous covariance between realized returns and
P
state variables. Therefore, investors decisions do not take into account the effect of
economic state variables on future realized returns and their behavior is described
as myopic5.
The risk premium S(q) is shown in figure 6b. We can see that below the
critical threshold kcrit, investors require high risk premia at negative polarization
levels q, indicating high risk aversion. By contrast, investors would even pay to take
risk (negative risk premia), if q is at positive levels. Above the critical threshold
kcrit, investors get more courageous in pessimistic polarization states by lowering
their required risk premia and they become cautious in optimistic ones by raising
the risk premia.
[Insert figure 6]
4 Model Analysis
In this section, we solve the various fundamental valuation equations for the case of
European call options using Monte Carlo methods.
4.1 Computational considerations
According to Feynman and Kac, the solution of (14) is equivalent to the discounted
expectation of the boundary condition under the equivalent martingale measure:
b
F(S, q, t) =e-r(T-t)Et [g(S(T))] (24)
5
See Feldman (1992). Albeit, we should keep in mind that investors still use this covariance to
calculate the risk-adjusted drift through the risk premium .
11
Since we want to price European call options, the boundary condition is given by g =
max(S(T)-X, 0). Using equation (24), we are able to derive a solution of (14) doing
Monte Carlo simulations. For this purpose, we employ the Euler approximation of
the stochastic processes dq and dS:
h i
p p
qt = qt-1 + K(qt-1) - q(qt-1) Q(qt-1) "t + Q(qt-1) "w1,t (25)
p
St = St-1 + r(qt-1)St-1"t + Q(qt-1)St-1"w1,t + DSt-1"w2,t
To improve the efficiency of the simulation procedure, we use the antithetic variates
technique. For every pair of random variables ("w1,t,"w2,t) the four stock prices
St1("w1,t,"w2,t), St2("w1,t,-"w2,t), St3(-"w1,t, "w2,t) and St4(-"w1,t, -"w2,t)
are calculated to get the antithetic variates estimate
1
F (St) = {F(St1) +F(St2) +F(St3) +F(St4)}.
4
For our nonlinear problem, it prooved to be efficient to increase the number of sample
paths at the expense of time step "t.6 Therefore we use a time step of "t =1/252
corresponding to one trading day and to ran 100 000 sample paths for each option
series .
4.2 Sample option prices
For each of the three considered types of investors, sample option prices based on
the stock price distributions S(q,t) are presented in tables 1-3. The maturity of the
options is six month, but general results appear to be true for all maturities. The
tables are divided into panels, labeled A-E, covering a broad range of parameter
values including most of the scenarios introduced in chapter 2. Panel A investigates
the influence of the degree of crowd behavior k on option prices. Panels B and C
contrast option prices in different economic climates h. Panels D and E examine the
effect of varying initial overreaction. For each option, three numbers are calculated:
A new price corresponding to our simulated true distribution, a Black-Scholes
price equivalent and the Black-Scholes implied volatility associated with the new
price. The latter serves as a proxy for the skewness of the implied return distribu-
tion. To calculate the Black-Scholes price, we use for all three models the volatility
of the simulated terminal distribution. In case of the risk neutral investor and
representative Bernoulli investor, we use the risk free rate that is consistent with
put-call parity. In case of the myopic Bernoulli investor, we fix r at ln(1.1) = 0.0953.
Independent of risk preferences, we set S =100, D =0.1, D =0.1, =0, ą =50.
6
For example, the standard error of F for an at the money option (myopic Bernoulli investor,
1
efficient market state) using 100 000 sample paths at time step "t = is 0.291%. The standard
252
1
error of F using only 10 000 sample paths at the finer time step "t2 = "t is 0.710%. Since
10
the number of calculated increments is the same in both settings, they take the same amount of
computing time.
12
Case 1 General equilibrium and risk neutrality.
Here, our aim is to find out, how options would be priced in the original sense
of the coherent market hypothesis. By assuming that risk neutrality prevails, no
risk-adjustment is needed and we can simply use the original return distributions
to price options. On panel A of table 1, we present option prices based on repelling
market, efficient market and chaotic market environments. Our first observation
is, that rising crowd behavior leads to higher implied volatilities but also higher
interest rates. As a result, option prices are the higher, the higher crowd behavior
k. Second, option prices in repelling and efficient markets don t differ much from
their Black-Scholes equivalents, while in chaotic markets, deviations are significant.
Third, implied volatilities exhibit a strong volatility frown for positive k and a
slight volatility smile for negative k, indicating the presence of short and fat tails
respectively. In figure 7a, we have plotted the differences between the new option
price and the Black-Scholes equivalent at different levels of crowd behavior. The
more positive k, the more overprices Black-Scholes far-in- and far-out-of-the-money
options while at-the-money options are underpriced. For negative k, effects of mean
reversion and stochastic volatility are mostly offsetting each other.
[Insert Figure 7]
Panels B and C encompass the values that appear to characterize options in
coherent market states, allowing h to range from-0.1 to 0.1 while crowd behavior
is either below (k =1.8) or above (k =2.1) the critical threshold. As we can see,
coherent bear markets are associated with low interest rates and low option prices
while coherent bull markets come along with high interest rates and high option
prices7. In weak coherent markets, deviations from the Black-Scholes equivalent
are rather small. In strong coherent bear markets, out of-the-money options are
underpriced by Black-Scholes, whereas in strong coherent bull markets, out of-the-
money options are overpriced.
The most interesting results are obtained by varying initial overreaction by the
choice of polarization q0. As indicated in panels D and E, we find, that a downside
overreaction (q < 0) exerts an upward influence on option prices while an upside
overreaction (q >0) has the opposite effect. Figure 7b shows the difference between
the new option price and the Black-Scholes price in a downward versus an upward
overreaction. Upward overreaction leads to positive differences in-the-money and
negative differences out-of-the-money, indicating the presence of a left skewed im-
plied return distribution. In a downward overreaction, the reverse is true.
7
This is not an unusual quality. Our model suggests, that positive fundamentals should come
along with high interest rates and vice versa. Thus, our model is in accordance with the commonly
observable behaviour of a central bank, which regulates interest rates by switching to expansive
fiscal policy in poor economic climates and to restrictive fiscal policy in boom times.
13
Case 2 General equilibrium and a representative Bernoulli investor with log
utility.
The difference between the representative Bernoulli investor and the risk neutral
investor lies in the risk adjustment of expected returns. Since the representative
Bernoulli s risk premium is always positive, risk adjusted expected returns are lower
(see dotted lines in figure 4). This in turn leads to a reduction of all option prices
(compare option prices of table 2 with those of table 1). Surprisingly, risk adjustment
has no influence on the qualitative pricing features already discussed above (figures
7c and 7d). Implied volatility patterns as well as general pricing behavior regarding
variations in k, h and q remain the same as for the risk neutral investor.
Case 3 Partial equilibrium and a myopic Bernoulli investor with log utility
The myopic Bernoulli investor accounts for overreaction by expanding the risk-
adjusted return distribution beyond the borders of the original return distribution,
both, to the left and to the right (see dashed lines in figure 4). Risk preferences
provide, that drift uncertainty is fully transfered into volatility uncertainty. The
drift of the adjusted stock price process becomes a constant and we have to deal
with a purely stochastic volatility model.
As can be seen from panel A of table 3, volatility and hence option prices are
decreasing with rising crowd behavior k. Thus, options are priced lower in chaotic
markets than in efficient markets. Second, observations regarding implied volatility
remain the same as discussed above, except that the volatility smile for negative k
is more pronounced while the volatility frown for positive k is reduced. If we have
a look at figure 7e, we can see how this translates into differences between the new
option price and equivalent Black-Scholes price. Since there is no mean reversion,
deviations between the model price and Black-Scholes price extend over a much
larger range of strike prices.
Looking at Panels B and C, we find that the influence of economic bias h on
option prices is rather small, independent of prevailing crowd behavior. Thus, co-
herent markets don t matter much in the risk adjusted world of the myopic Bernoulli
investor.
Regarding overreaction, we have similar pricing patterns as for the other two
types of investors: Downward (upward) overreaction is associated with right (left)
skewed return distributions and high (low) option prices (see panels D and E). Here,
the intuition behind can be explained isolated from effects due to mean reversion:
Consider the case in which q0 is below zero. Higher q (still negative) is associated
p
with higher volatilities Q(q) but also with higher stock prices. As q rises, S
rises and the probability of large positive changes in S increases. This means,
that very high stock prices become more probable. On the other hand, lower q
is associated with lower stock price volatilities and stock prices. If q and hence S
falls, it becomes less likely that large changes take place and terminal stock price
14
will be low. The net effect is, that the terminal return distribution is positively
skewed. When q0 is positive, the reverse is true. Thus, the underlying mechanism
is very similar to traditional stochastic volatility models where positive or negative
correlation between volatility and stock price leads to right or left skewed return
distributions respectively8.
5 Conclusion
Motivated by the theory of social imitation originally developed by Weidlich, we
present a behavioral model of stock prices that supports the overreaction hypothesis.
According to Weidlichs theory, group polarization can be described in a probabilistic
manner. Vaga was the first who recognized the theory s usefulness to describe return
distributions of financial markets. He re-interpreted Weidlich s control parameters
as level of crowd behavior and prevailing economic fundamentals respectively and
formulated the so called coherent market hypothesis. This hypothesis offers a rich
variety of market states where the relationship between risk and return is solely
explained by investor sentiment.
In this paper, we take a step further and formulate the continuous-time ver-
sion of the CMH. This opens the door to established formalism of financial theory,
especially option pricing. Starting from the Fokker Planck equation describing in-
vestor sentiment in terms of polarization levels, we are able to extract and analyze
the underlying stochastic process. We find that the associated distributions of this
diffusion end in a steady state and therefore are unable to represent return distribu-
tions. We also find that the process exhibits a special kind of mean reversion, which
can be used to model overreaction in stock prices when superimposed on a geomet-
ric Brownian motion. The resulting return distributions offer the same risk-return
characteristics associated to the various states of investor sentiment as proposed by
Vaga, but overcome the deficiency of a steady state perspective.
Using our overreaction stock price model, we develop a two factor general equi-
librium model for pricing derivative securities. The two factors of our model are
the stock price and market polarization which determines the level of overreaction.
We examine three competing hypothesis about investor preferences under which
either the market price of risk is zero or derived endogenously and option prices
are consistent with the absence of arbitrage: Risk-neutral investor, representative
Bernoulli investor and myopic Bernoulli investor. In case of the latter two types
of investors, we are able to draw conclusions about their pattern of behavior from
their required risk premia. The representative Bernoulli investor s risk premium
diminishes as market polarization and hence overreaction increases (in either di-
rection). This implicates that he acts contrarian in pessimistic but contagious in
8
In traditional stochastic volatility models, skewness is introduced by assuming an instantaneous
correlation between the driving Wiener processes: dwSdw = dt. For an analytical approach,
where volatility follows an Ornstein-Uhlenbeck process, see Schbel and Zhu (1999).
15
optimistic market environments. Optimal behavior in case of the myopic Bernoulli
investor is quite different. Below the critical level of crowd behavior, this investor
exhibits strong herding behavior: When market polarization is negative, he wants
to be compensated for risk, whereas when it is positive, he even pays for taking risk.
By contrast, above the critical threshold he mostly takes contrarian positions by re-
ducing the risk premium at positive polarization levels and increasing it at negative
levels. Overall, we could show that herding as well as contrarian investor behavior
may be the optimal course of action in response to an apparently irrational stock
price process driven by investor sentiment.
Applying Monte Carlo simulations, we examine European call options. Although
our three model specifications behave quite multi-faceted regarding the various pric-
ing parameters, they exhibit one common feature: Option prices depend significantly
on the level of overreaction. Upward overreaction leads to low option prices and
downward overreaction leads to high option prices. The main reason behind this
lies in the model s intrinsic assumption about stochastic volatility: Upward overre-
action is accompanied by a negative correlation between stock price and volatility,
whereas in a downward overreaction, this correlation is positive. As a result, the
skewness of the terminal return distribution can be negative or positive, providing
the observed pricing biases.
6 Appendices
6.1 Derivation of the Fokker-Planck equation and its sta-
tionary solution
Let p+-(n+,n-) and p-+(n+,n-) be the transition probabilities per unit time for
an individual of changing from attitude + to - and vice versa. The temporal change
of the probability f(n+,n-;t) to find the group in the state (n+,n-) is the difference
between gains and losses of probability per unit time. As we see from figure 8, this
is kind of master equation and can be written as
"f(n+,n-; t) gains - losses
=
"t "t
= (n- +1)p-+(n+ - 1,n- +1)f[n+ - 1,n- +1; t]
+(n+ +1)p+-(n+ +1,n- - 1)f[n+ +1,n- - 1;t] (26)
-n+ p+-(n+,n-) f[n+,n-;t] - n- p-+(n+,n-)f[n+,n-; t].
[Insert Figure 8]
If we introduce
"
1 1
q =(n+ - n-)/2; q " - ,
2 2
16
as a measure of the polarization of the group, we can rewrite the transition proba-
bilities p in terms of q:
1
w+-(q) :=n+p+-(n+,n-) =n( + q)p+-(q)
2
1
w-+(q) :=n-p-+(n+,n-) =n( - q)p-+(q)
2
1
With these definitions and "q = , equation (26) simplifies to
n
"f(q; t)
= w+-(q + "q)f(q + "q,t) +w-+(q - "q)f(q - "q,t)
"t
-w+-(q)f(q,t) - w-+(q)f(q,t) .
A series expansion of the first two terms on the right hand side up to the second
order in "q leads to
d " 1 "2
f(q, t) = w+-(q)f(q,t) + w+-(q)f(q,t)"q + w+-(q)f(q,t)"q2
dt "q 2 "q2
" 1 "2
+w-+(q)f(q, t) - w-+(q)f(q,t)"q + w-+(q)f(q,t)"q2
"q 2 "q2
-w+-(q)f(q,t) - w-+(q)f(q,t)
" 1 "2
= [w+-(q) - w-+(q)]f(q,t)"q + [w+-(q) +w-+(q)]f(q,t)"q2
"q 2 "q2
If we define the drift K(q) and the diffusion coefficient Q(q) as follows
K(q) = [w-+(q) - w+-(q)]"q
Q(q) = [w+-(q) +w-+(q)]"q2
we get in the limit "q 0 a Fokker Planck equation for the probability density:
d " 1 "2
f(q,t) =- [K(q)f(q,t)] + [Q(q)f(q,t)] (27)
dt "q 2 "q2
To solve this partial differential equation, it is convenient to introduce the probability
current
1 "
j(q; t) :=K(q)f(q,t) + [Q(q)f(q,t)] .
2 "q
Using j(q; t), equation(27) becomes a continuity equation for the probability density
function f(q,t)
d "
f(q,t) + j(q;t) =0.
dt "q
Since there is no probability current across the boundaries at q = ą1, the boundary
2
conditions are
1
j(q = ą ;t) =0. (28)
2
17
The condition for the steady state solution is
d
fst(q,t) =0.
dt
"
But this means that jst(q;t) =0 and therefore jst(q;t) =const. . Together with
"q
the boundary conditions (28) it follows that
1 "
jst(q;t) =0 =K(q)fst(q,t) + [Q(q)fst(q,t)].
2 "q
This is an ordinary differential equation and the solution can be obtained easily by
integration:
" #
Z
q
c K(q)
fst(q) = exp 2 dq
1
Q(q) Q(q)
-
2
with the normalization constant c.
6.2 Derivation of the transition probabilities p+- and p-+
Ising introduced an idealized model of the ferromagnet that consists of a lattice
of elementary magnets (more precise magnetic moments, so-called spins i), which
can adopt only two orientations: Parallel (i =1) to an external field H or anti-
parallel (i = -1). The total energie E of a magnet configuration is assumed to
depend on the external field and on an internal field created by the magnet-magnet
interaction in the following way:
!
X X X
E = Ei = -Hi - Iijij
i i j=i
6
Iij are the spin-spin interaction parameters and is the magnitude of a magnetic
moment. As in every thermodynamic system, the total energie E is the central item
that determines various other thermodynamic variables. For example, the proba-
bility of finding the system in the spin-configuration is given by the Boltzmann
distribution
w(E) =cexp(-E/kBT)
with the Boltzmann constant kB and temperature T. Even though the system is in
thermal equilibrium it is not in a static state. Spins may flip from one orientation
to the other, changing the total energy. The probability per unit time of such a
flipping process is governed by the equation
" "-(W + Hi + P Iijij)
A -W + Ei A
j=i
6
Pr(i -i) = exp = exp
kBT kBT
(29)
18
where A, and W are appropriately chosen constants.
The characteristics of the Ising model are well documented9: There exists a phase
transition temperature Tc defined by the formula
sinh(2I/kBTc) =1.
Therefore, three phases can be distinguished: For high temperatures (T > Tc), the
spins fluctuate almost independently of their neighbours entailing a mean polariza-
tion of zero. For temperatures around the phase transition temperature (T H" Tc)
there are big clusters of aligned spins, while the mean polarization is still zero. In
the case of very low temperatures (T < Tc), one of the clusters grows at the cost
of the oppositely orientated spins until all spins point in the same direction. The
mean polarisation is then close to one.
The basic idea of Weidlich was to formulate the transition probabilities p+- and
A W
p-+ in complete analogy to Ising model. Setting ą = exp(- ), ąH =
kBT
Pthe
Hi and ąI(n+ - n-) = Iijij we get from (29)
j=i
6
ś
-I(n+ - n-) - H
p+-(q) = ą exp
kBT
!
I
- q - H
2n
= ą exp = ą exp [-(kq + h)]
kBT
ś
+I(n+ - n-) +H
p-+(q) = ą exp
kBT
!
I
+2nq + H
= ą exp = ą exp [+(kq + h)] .
kBT
I
For further simplification we introduced the adaption parameter k = and the
2nkBT
H
preference parameter h = .
kBT
6.3 Derivation of the Fundamental Valuation Equation
According to (11), the returns of the risky asset are governed by the stochastic
differential equation
dS(x,t)
= ą(x,t)dt + (x,t)dw(t) (30)
S
with drift
1
ą = D + K(q) + 2Q(q)
2
9
For an analytic investigation see Cohen (1968) and for a computer simulation see Ogita et al.
(1969).
19
and diffusion vector
Ł p ń
= Q(q) D .
"
q
Let x = be the state vector which follows the two-dimensional Markov process
S
dx = (x,t)dt + (x,t)dw(t) (32)
where
"
K(q)
Ł ń
(x,t) =
1
D + K(q) + 2Q(q) S
2
is the drift vector and
p
"
pQ(q) 0
(x,t) =
Q(q)S DS
is the correlation matrix of the subordinated processes dS and dq. Let further
dF(x,t)
= (x,t)dt + (x,t)dw(t) (33)
F
be the stochastic process of the returns of the derivative security which may depend
on all state variables, but not on wealth W.
We want to maximize
"
Z
"
J(W,x,t) =Et U(C)ds (34)
t
with the utility function
U(C) =exp(-s)lnC(s) (35)
subject to the budget constraint
dS dF
dW =[(1 - P - F ] rWdt + PW + FW - Cdt. (36)
S F
The last equation can be rewritten using equations (30) and (33), so we get
Ł ń
dW =[r + P (ą - r) +F ( - r)] Wdt + P + F Wdw - Cdt. (37)
The necessary optimality condition for (34) is the Bellman equation
"
Et[dJ]
max U(C) + =0. (38)
C,P ,F
dt
We can write Et[dJ] as follows:
1
0
Et[dJ] = Jtdt + JWEt[dW] +JxEt[dx] + tr(Jxx Et[dxdx0])
2
1
+JxWEt[dxdW] + JWWEt[dW2] +o(dt)
2
20
with the expectations
E[dx] = dt
Et[dxdx0] = 0dt
Ą ó
Et[dxdW] = P0 + F0 Wdt
Et[dW] = [r + P (ą - r) +F ( - r)]Wdt - Cdt
Ą ó
Et[dW2] = 2 0 +2PF0 + 2 0 W2dt.
P F
Ignoring terms of order o(dt), we can now substitute into the Bellman equation to
get
0 = max [U(C) +Jt + JW ([r + P (ą - r) +F ( - r)] W - C)
C,P ,F
Ą ó
+Jx + JxW P0 + F0 W
Ą ó
1 1
0
+ JWW 2 0 +2PF0 + 2 0 W2 + tr(Jxx 0)].
P F
2 2
The first order conditions are
UC - JW = 0 (39)
0
JW (ą - r) +0JxWW + JWW (P0 + F0)W = 0 (40)
0
JW ( - r) +0JxWW + JWW (P0 + F0)W = 0. (41)
As shown in Merton (1971), the investor s value function J is partially separable
and has the form
exp(-t)
J(W, x,t) = ln(W) +H(x,t). (42)
With this value function, Equation (39) becomes the usual consumption optimality
condition derived by Merton
C = W. (43)
Case 1 Market clearing condition P =1:
In this case, r = F =0 and we can solve equation (40) for r:
r = ą - 0 (44)
Substituting this result into (41) yields
= r + 2 (45)
with market price of risk
2 = 0. (46)
21
Case 2 Market clearing condition P + r =1:
In this case, only the derivative security is in zero net supply (F =0) and we
obtain the optimal portfolio decision " from (40):
P
1
" = (ą - r). (47)
P
0
If we substitute " into (41), the drift of the derivative security can be written as
P
= r + 3 (48)
where
0
3 = (ą - r) (49)
0
is the vektor of factor risk prices.
To derive the fundamental valuation equation we have to apply Ito s lemma to
F(x,t) with the result
ś
1
0
dF = Ft + Fx + tr(Fxx 0) dt + Fxdw (50)
2
= (x,t)Fdt + (x,t)Fdw(t).
Ą ó
1 1 1
0
Setting the identities (x,t) = Ft + Fx + tr(Fxx 0) and (x,t) = (Fx)
F 2 F
into equations (45) and (48) respectively, we get
Ą ó
1
0
tr(Fxx 0) +Fx - i + Ft - rF =0. (51)
2
Resubstituting the expressions for x, , , ą, and i finally leads to our valuation
equations for derivative securities.
References
[1] Campbell, John Y. and Robert L. Shiller (1988): Stock Prices, Earn-
ings, and Expected Dividends , Journal of Finance 43, 661-676
[2] Cohen, E. G. D. (ed.) (1968): Fundamental Problems in Statistical Me-
chanics, Vol. 2 , Amsterdam: North-Holland.
[3] Cox John C., Jonathan E. Ingersoll Jr. and Stephen A. Ross
(1985a): An Intertemporal General Equilibrium Model of Asset Prices ,
Econometrica 53 (2), 363-384
[4] Cox John C., Jonathan E. Ingersoll Jr. and Stephen A. Ross
(1985b): A Theory of the Term Structure of Interest Rates , Econometrica
53 (2), 385-407
22
[5] De Bondt, Werner F. M. and Richard H. Thaler (1985): Does the
Stock Market Overreact ? , Journal of Finance 40, July, 793-808
[6] De Bondt, Werner F. M. and Richard H. Thaler (1987): Further
Evidence on Investor Overreaction and Stock Market Seasonality , Journal of
Finance 42, July, 557-581
[7] Feldman, D. (1992): Logarithmic Preferences, Myopic Decisions and Incom-
plete Information , Journal of Financial and Quantitative Analysis 27, Decem-
ber, 619-29
[8] Merton, Robert C. (1971): Optimum Consumption and Portfolio Rules in
a Continuous-Time Model , Journal of Economic Theory 3, 373-413
[9] Ogita, N., et al. (1969). Suppl. Jap. phys. Soc. 26, 145
[10] Poterba James M. and Lawrence H. Summers (1988): Mean Reversion
in Stock Prices , Journal of Financial Economics 22, 27-59
[11] Shiller, Robert J. (1981): Do Stock Prices Move Too Much to be Justified
by Subsequent Changes in Dividends? , American Economic Review 71, 421-
436
[12] Schbel, Rainer and Jianwei Zhu. (1999): Stochastic Volatility with
an Ornstein-Uhlenbeck Process: An Extension , European Finance Review 2,
23-46
[13] Vaga, Tonis (1990): "The Coherent Market Hypothesis", Financial Analysts
Journal, November-December, 36-49.
[14] Weidlich, Wolfgang (1971): The Statistical Description of Polarization
Phenomena in Society , British Journal of Math. Statist. Psychology 24, 251-
266
23
k = 0.0
0.07
k = 1.8
k = 2.0
0.06
k = 2.1
k = 2.4
0.05
0.04
0.03
0.02
0.01
0.00
-0.4 -0.2 0.0 0.2 0.4
Polarization q
Figure 1a: Phase transition in the case of no preferences (h =0.0).
k = 0.0
0.07
k = 1.8
k = 2.0
0.06
k = 2.1
k = 2.4
0.05
0.04
0.03
0.02
0.01
0.00
-0.4 -0.2 0.0 0.2 0.4
Polarization q
Figure 1b: Phase transition in the case of positive preferences (h =0.01).
24
Probability f(q;k,h=0.0)
Probability f(q;k,h=0.01)
0.05 0.015
(a) (a)
0.00 0.000
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
q q
5.0 5.0
(b) (b)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-5.0 -5.0
q q
0.25 0.25
(c) (c)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.05 -0.05
q q
Fig. 2.1: Repelling market Fig. 2.2: Efficient market
(k = -1.8, h =0.0) (k =1.8, h =0.0)
0.015 0.015
(a) (a)
0.000 0.000
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
q q
5.0 5.0
(b) (b)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-5.0 -5.0
q q
0.25 0.25
(c) (c)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.05 -0.05
q q
Fig. 2.3: Unstable transition Fig. 2.4: Coherent bull market
(k =2.0, h =0.0) (k =2.1, h =0.01)
25
Probability f(q)
Probability f(q)
Drift K(q)
Drift K(q)
Diffusion Q(q)
Diffusion Q(q)
Probability f(q)
Probability f(q)
Drift K(q)
Drift K(q)
Diffusion Q(q)
Diffusion Q(q)
0.015 0.015
(a) (a)
0.000 0.000
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
q q
5.0 5.0
(b) (b)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-5.0 -5.0
q q
0.25 0.25
(c) (c)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.05 -0.05
q q
Fig. 2.5: Chaotic market (neutral) Fig. 2.6: Chaotic market (bearish)
(k =2.1, h =0.0) (k =2.1, h = -0.003)
26
Probability f(q)
Probability f(q)
Drift K(q)
Drift K(q)
Diffusion Q(q)
Diffusion Q(q)
a) Repelling market
b) Efficient market
c) Unstable transition
d) Coherent bull market
Figure 3: Trajectories of the stochastic process dq.
27
e) Chaotic market (neutral)
f) Chaotic market (bearish)
Figure 3: Trajectories of the stochastic process dq (continued).
28
0.475 0.475
k = -1.8, h = 0.0 k = 1.8, h = 0.0
risk neutral investor risk neutral investor
representative Bernoulli inv. representative Bernoulli investor
myopic Bernoulli investor myopic Bernoulli investor
0 0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Return Return
0.475 0.475
k = 2.0, h = 0.0 k = 2.1, h = 0.015
risk neutral investor risk neutral investor
representative Bernoulli investor representative Bernoulli investor
myopic Bernoulli investor myopic Bernoulli investor
0 0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Return Return
0.475 0.475
k = 2.1, h = -0.003 k = 2.1, h = 0.0
risk neutral investor risk neutral investor
representative Bernoulli investor representative Bernoulli investor
myopic Bernoulli investor myopic Bernoulli investor
0 0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Return Return
Figure 4: Implied return distributions for risk-neutral investor (original), repre-
sentative Bernoulli investor and myopic Bernoulli investor in different market states.
29
Probability
Probability
Probability
Probability
Probability
Probability
1.0 1.0
Black-
k = -1.8
Scholes
h = 0.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.4 -0.4
-0.6 -0.6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Years Years
1.0 1.0
k = 1.8 k = 2.1
h = 0.0 h = 0.015
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.4 -0.4
-0.6 -0.6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Years Years
1.0 1.0
k = 2.1 k = 2.1
h = -0.003 h = 0.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.4 -0.4
-0.6 -0.6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Years Years
Figure 5: Contourplot of the evolution of the return distribution.
30
Return
Return
Return
Return
Return
Return
25
k = -1.8
k = 0.0
k = 1.8
k = 2.0
20 k = 2.1
k = 2.4
15
10
5
0
-0.50 -0.25 0.00 0.25 0.50
Polarization q
Fig. 6a: Risk premium subject to market polarization q and crowd behavior k
for a representative Bernoulli investor.
1000
k = -1.8
k = 0.0
800
k = 1.8
k = 2.0
600 k = 2.1
k = 2.4
400
200
0
-200
-400
-600
-800
-1000
-0.50 -0.25 0.00 0.25 0.50
Polarization q
Fig. 6b: Risk premium subject to market polarization q and crowd behavior k
for a myopic Bernoulli investor.
31
S
Risk Premium (
)
S
Risk Premium (
)
(a) (b)
Risk Neutral Risk Neutral
Investor Investor
q0 = -0.2
0.8 k = -1.8 0.8
q0 = 0.2
k = 0.0
0.6 k = 1.8 0.6
k = 2.0
0.4 k = 2.1 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
S/X S/X
(c)
(d)
Representative Representative
Bernoulli Investor Bernoulli Investor
q0 = -0.2
k = -1.8
0.8 0.8
q0 = 0.2
k = 0.0
k = 1.8
0.6 0.6
k = 2.0
k = 2.1
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
S/X S/X
(e) (f)
Myopic Myopic
Bernoulli Investor Bernoulli Investor
k = -1.8
q0 = -0.2
0.8 0.8
k = 0.0
q0 = 0.2
k = 1.8
0.6 0.6
k = 2.0
k = 2.1
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
S/X S/X
Figure 7: Differences between the model option prices and corresponding Black-
Scholes prices.
32
BS
BS
C - C
C - C
BS
BS
C - C
C - C
BS
BS
C - C
C - C
Gains in State Losses in
state (n+, n-) state (n+, n-)
f(n++1, n--1)
p+-(n++1, n--1) p-+(n+, n-)
f(n+, n-)
p-+(n+-1, n-+1) p+-(n+, n-)
f(n+-1, n-+1)
Figure 8: Probability currents to and from state (n+,n-).
33
X New B/S I. vol. New B/S I. vol. New B/S I. vol.
A. (k, h = 0.0, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
k = -1.8, r = 9.81% k = 1.8, r = 12.03% k = 2.1, r = 15.73%
80 23.84 23.84 0.1055 24.69 24.69 0.1504 26.28 26.42 0.2441
90 14.36 14.36 0.1050 15.53 15.53 0.1538 18.35 18.29 0.2720
100 5.89 5.89 0.1049 7.77 7.76 0.1545 12.07 11.60 0.2850
110 1.23 1.23 0.1049 2.86 2.85 0.1542 7.18 6.74 0.2802
120 0.11 0.11 0.1050 0.75 0.76 0.1531 3.57 3.60 0.2627
130 0.01 0.00 0.1055 0.13 0.15 0.1509 1.37 1.78 0.2412
repelling market efficient market chaotic market
B. (k = 1.8, h, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
h = -0.1, r = 3.93% h = 0.0, r = 12.03% h = 0.01, r = 19.98%
80 21.59 21.60 0.1484 24.69 24.69 0.1504 27.62 27.62 0.1523
90 12.35 12.37 0.1515 15.53 15.53 0.1538 18.67 18.67 0.1538
100 5.32 5.29 0.1533 7.77 7.76 0.1545 10.50 10.48 0.1533
110 1.64 1.59 0.1544 2.86 2.85 0.1542 4.50 4.51 0.1517
120 0.35 0.33 0.1541 0.75 0.76 0.1531 1.38 1.43 0.1498
130 0.05 0.05 0.1523 0.13 0.15 0.1509 0.29 0.33 0.1472
weak coherent bear mkt. efficient market weak coherent bull mkt.
C. (k = 2.1, h, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
h = -0.1, r = -9.48% h = 0.0, r = 15.73% h = 0.1, r = 38.94%
80 16.79 17.22 0.2041 26.28 26.42 0.2441 34.21 34.19 0.2539
90 9.61 9.58 0.2372 18.35 18.29 0.2720 26.30 26.12 0.2661
100 5.39 4.57 0.2646 12.07 11.60 0.2850 18.96 18.56 0.2606
110 2.84 1.88 0.2767 7.18 6.74 0.2802 12.32 12.08 0.2410
120 1.27 0.68 0.2742 3.57 3.60 0.2627 6.74 7.15 0.2156
130 0.44 0.22 0.2622 1.37 1.78 0.2412 2.86 3.85 0.1932
strong coh. bear mkt. chaotic market strong coh. bull mkt.
D. (k = 1.8, h = 0.0, q0, d = 0.1, d = 0.1, = 0.0, ą = 50)
q0 = -0.1, r = 32.00% q0 = 0.0, r = 12.03% q0 = 0.1, r = -7.93%
80 31.84 31.84 0.1517 24.69 24.69 0.1504 16.95 16.95 0.1536
90 23.33 23.34 0.1523 15.53 15.53 0.1538 8.13 8.12 0.1544
100 15.10 15.10 0.1539 7.77 7.76 0.1545 2.70 2.69 0.1541
110 8.06 8.05 0.1545 2.86 2.85 0.1542 0.59 0.60 0.1526
120 3.36 3.34 0.1543 0.75 0.76 0.1531 0.08 0.09 0.1504
130 1.06 1.06 0.1534 0.13 0.15 0.1509 0.01 0.01 0.1484
overreaction down fair valuation overreaction up
(efficient market) (efficient market) (efficient market)
E. (k = 2.1, h = 0.0, q0, d = 0.1, d = 0.1, = 0.0, ą = 50)
q0 = -0.2, r = 39.57% q0 = 0.0, r = 15.73% q0 = 0.2, r = -9.93%
80 34.37 34.40 0.1719 26.28 26.42 0.2441 17.59 17.23 0.2654
90 26.21 26.39 0.1943 18.35 18.29 0.2720 9.88 9.74 0.2467
100 18.58 18.92 0.2178 12.07 11.60 0.2850 4.19 4.79 0.2196
110 12.44 12.52 0.2423 7.18 6.74 0.2802 1.17 2.07 0.1963
120 8.09 7.61 0.2628 3.57 3.60 0.2627 0.20 0.80 0.1792
130 5.06 4.25 0.2742 1.37 1.78 0.2412 0.02 0.28 0.1680
overreaction down fair valuation overreaction up
(chaotic market) (chaotic market) (chaotic market)
Table 1: Call option prices in case of a risk neutral investor.
34
X New B/S I. vol. New B/S I. vol. New B/S I. vol.
A. (k, h = 0.0, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
k = -1.8, r = 8.63% k = 1.8, r = 8.31% k = 2.1, r = 5.98%
80 23.38 23.38 0.1063 23.28 23.28 0.1406 22.74 23.00 0.2148
90 13.86 13.86 0.1055 14.07 14.08 0.1519 15.02 14.99 0.2478
100 5.48 5.48 0.1049 6.60 6.59 0.1539 9.41 8.83 0.2750
110 1.08 1.08 0.1049 2.24 2.22 0.1545 5.34 4.71 0.2858
120 0.09 0.09 0.1049 0.54 0.54 0.1542 2.52 2.30 0.2796
130 0.00 0.00 0.1055 0.09 0.09 0.1528 0.91 1.04 0.2629
repelling market efficient market chaotic market
B. (k = 1.8, h, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
h = -0.1, r = 0.3% h = 0.0, r = 8.31% h = 0.01, r = 16.37%
80 20.17 20.18 0.1445 23.28 23.28 0.1406 26.30 26.30 0.1520
90 10.96 10.98 0.1484 14.07 14.08 0.1519 17.25 17.24 0.1533
100 4.37 4.33 0.1514 6.60 6.59 0.1539 9.22 9.21 0.1543
110 1.22 1.17 0.1533 2.24 2.22 0.1545 3.69 3.70 0.1534
120 0.24 0.22 0.1542 0.54 0.54 0.1542 1.05 1.08 0.1520
130 0.03 0.03 0.1538 0.09 0.09 0.1528 0.20 0.23 0.1500
weak coherent bear mkt. efficient market weak coherent bull mkt.
C. (k = 2.1, h, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
h = -0.1, r = -17.47% h = 0.0, r = 5.98% h = 0.1, r = 31.06%
80 13.68 14.10 0.1741 22.74 23.00 0.2148 31.60 31.60 0.2402
90 7.04 6.87 0.2023 15.02 14.99 0.2478 23.63 23.43 0.2646
100 3.65 2.73 0.2350 9.41 8.83 0.2750 16.52 16.09 0.2749
110 1.81 0.89 0.2594 5.34 4.71 0.2858 10.35 10.14 0.2676
120 0.76 0.25 0.2687 2.52 2.30 0.2796 5.42 5.85 0.2469
130 0.24 0.06 0.2654 0.91 1.04 0.2629 2.18 3.11 0.2222
strong coh. bear mkt. chaotic market strong coh. bull mkt.
D. (k = 1.8, h = 0.0, q0, d = 0.1, d = 0.1, = 0.0, ą = 50)
q0 = -0.1, r = 28.26% q0 = 0.0, r = 8.31% q0 = 0.1, r = -11.65%
80 30.55 30.55 0.1441 23.28 23.28 0.1406 15.48 15.49 0.1514
90 21.90 21.90 0.1484 14.07 14.08 0.1519 6.94 6.93 0.1538
100 13.63 13.64 0.1521 6.60 6.59 0.1539 2.11 2.09 0.1545
110 6.87 6.86 0.1539 2.24 2.22 0.1545 0.41 0.42 0.1541
120 2.66 2.64 0.1545 0.54 0.54 0.1542 0.05 0.06 0.1523
130 0.77 0.77 0.1544 0.09 0.09 0.1528 0.00 0.01 0.1504
overreaction down fair valuation overreaction up
(efficient market) (efficient market) (efficient market)
E. (k = 2.1, h = 0.0, q0, d = 0.1, d = 0.1, = 0.0, ą = 50)
q0 = -0.2, r = 31.25% q0 = 0.0, r = 5.98% q0 = 0.2, r = -18.15%
80 31.58 31.62 0.1520 22.74 23.00 0.2148 15.06 14.69 0.2714
90 23.11 23.33 0.1719 15.02 14.99 0.2478 8.06 7.90 0.2702
100 15.39 15.77 0.1914 9.41 8.83 0.2750 3.22 3.73 0.2491
110 9.60 9.62 0.2145 5.34 4.71 0.2858 0.84 1.56 0.2231
120 5.87 5.26 0.2387 2.52 2.30 0.2796 0.13 0.59 0.2012
130 3.49 2.59 0.2579 0.91 1.04 0.2629 0.01 0.21 0.1846
overreaction down fair valuation overreaction up
(chaotic market) (chaotic market) (chaotic market)
Table 2: Call option prices in case of a representative Bernoulli investor.
35
X New B/S I. vol. New B/S I. vol. New B/S I. vol.
A. (k, h = 0.0, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
k = -1.8, r = 9.53% k = 1.8, r = 9.53% k = 2.1, r = 9.53%
80 26.42 26.42 0.4299 26.10 26.08 0.4105 26.05 26.03 0.4070
90 19.65 19.70 0.4277 19.34 19.22 0.4140 19.29 19.15 0.4120
100 14.19 14.27 0.4268 13.88 13.71 0.4156 13.85 13.62 0.4143
110 10.00 10.09 0.4268 9.68 9.50 0.4156 9.64 9.40 0.4143
120 6.92 6.99 0.4275 6.57 6.42 0.4145 6.52 6.33 0.4127
130 4.72 4.76 0.4286 4.33 4.25 0.4124 4.27 4.18 0.4097
repelling market efficient market chaotic market
B. (k = 1.8, h, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
h = -0.1, r = 9.53% h = 0.0, r = 9.53% h = 0.01, r = 9.53%
80 26.09 26.07 0.4098 26.10 26.08 0.4105 26.11 26.08 0.4113
90 19.33 19.22 0.4136 19.34 19.22 0.4140 19.34 19.23 0.4144
100 13.88 13.70 0.4155 13.88 13.71 0.4156 13.89 13.72 0.4157
110 9.69 9.49 0.4157 9.68 9.50 0.4156 9.68 9.50 0.4154
120 6.58 6.41 0.4148 6.57 6.42 0.4145 6.55 6.43 0.4141
130 4.35 4.25 0.4130 4.33 4.25 0.4124 4.32 4.26 0.4118
weak coherent bear mkt. efficient market weak coherent bull mkt.
C. (k = 2.1, h, q0 = 0.0, d = 0.1, d = 0.1, = 0.0, ą = 50)
h = -0.1, r = -9.53% h = 0.0, r = 9.53% h = 0.1, r = 9.53%
80 26.03 26.02 0.4061 26.05 26.03 0.4070 26.06 26.03 0.4079
90 19.28 19.14 0.4115 19.29 19.15 0.4120 19.30 19.16 0.4125
100 13.84 13.61 0.4141 13.85 13.62 0.4143 13.85 13.63 0.4144
110 9.65 9.39 0.4144 9.64 9.40 0.4143 9.64 9.41 0.4141
120 6.53 6.32 0.4131 6.52 6.33 0.4127 6.50 6.34 0.4122
130 4.29 4.17 0.4105 4.27 4.18 0.4097 4.25 4.18 0.4090
strong coh. bear mkt. chaotic market strong coh. bull mkt.
D. (k = 1.8, h = 0.0, q0, d = 0.1, d = 0.1, = 0.0, ą = 50)
q0 = -0.1, r = 9.53% q0 = 0.0, r = 9.53% q0 = 0.1, r = 9.53%
80 25.97 26.01 0.4022 26.10 26.08 0.4105 26.16 26.08 0.4144
90 19.20 19.13 0.4080 19.34 19.22 0.4140 19.36 19.23 0.4151
100 13.79 13.59 0.4119 13.88 13.71 0.4156 13.84 13.71 0.4141
110 9.64 9.38 0.4142 9.68 9.50 0.4156 9.57 9.50 0.4118
120 6.59 6.31 0.4152 6.57 6.42 0.4145 6.41 6.42 0.4087
130 4.40 4.15 0.4151 4.33 4.25 0.4124 4.16 4.26 0.4049
overreaction down fair valuation overreaction up
(efficient market) (efficient market) (efficient market)
E. (k = 2.1, h = 0.0, q0, d = 0.1, d = 0.1, = 0.0, ą = 50)
q0 = -0.2, r = 9.53% q0 = 0.0, r = 9.53% q0 = 0.2, r = 9.53%
80 25.63 25.78 0.3798 26.05 26.03 0.4070 26.11 25.94 0.4112
90 18.79 18.79 0.3895 19.29 19.15 0.4120 19.20 19.03 0.4080
100 13.40 13.19 0.3975 13.85 13.62 0.4143 13.55 13.47 0.4032
110 9.34 8.95 0.4034 9.64 9.40 0.4143 9.16 9.25 0.3972
120 6.38 5.91 0.4076 6.52 6.33 0.4127 5.93 6.19 0.3907
130 4.28 3.80 0.4103 4.27 4.18 0.4097 3.68 4.05 0.3841
overreaction down fair valuation overreaction up
(chaotic market) (chaotic market) (chaotic market)
Table 3: Call option prices in case of a myopic Bernoulli investor.
36
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