1. -- TANMOY GANGULI
Assistant Manager,
Financial Service Analytics,
Genpact (Kolkata)
A DETERMINISTIC DESCRIPTION
OF STOCK MARKET BUBBLES IN
ASSET MARKETS

2. TABLE OF CONTENTS
• Why study asset price bubbles?
• Major strands of thought in bubble
theories
• Need for deterministic bubble
theories
• A review of literature in
Deterministic asset price bubbles.
• A deterministic representation of
asset price bubble
• Major characteristics of
deterministic representation

3. WHY STUDY ASSET PRICE BUBBLES?
• Loosely defined, bubbles in asset prices
refer to a sudden increase in the price of an
asset over and above its fundamental value,
followed by a crash.
• Bursting stock market bubbles lead to huge
losses of wealth. Several instances from the
distant and the recent past have re-iterated
the severity of such bursts.
• The review of two most recent stock market
crashes show the type of financial damage
that the burst of bubbles can bring about -
Dotcom bubble burst (1999-2002) and the
Sub-prime bubble burst (2007-2009).
• Following the dotcom crash NASDAQ lost
78% of its value in a span of 2 years (2000-
2002). The markets lost $7 trillion in
market value.
• Sub-prime crisis (2007-2009) resulted in
S&P 500 to decline 57% from October 2007
to March 2009.

4. WHY STUDY ASSET PRICE BUBBLES?
• Bursts of asset bubbles abruptly increase
the riskiness associated with the economic
system, as they result in huge losses of
savings of households and corporations,
who invested in the asset .
• The frequency of such bubble bursts have
increased over time. The frequent number
of such bursts have increased the riskiness
of the financial systems of countries.
• Data suggests that from 2 crashes in 1600-
1700, there has been 10 crashes between
2000-2014. The average span of market
crashes have reduced from 14 years in the
seventeenth century to 1.4 years in the first
two decades of the 21st century.

5. WHY STUDY ASSET PRICE BUBBLES?
Time Period No . of Crashes Avg. Gap of time
between two
crashes
1600 - 1700 2 14 years
1700-1800 7 10 years
1800-1900 14 5 years
1900-2000 12 3 years
2000-2014 10 1.4 years

6. WHY STUDY ASSET PRICE BUBBLES?
• Given enhanced financial integration among
the nations in 21st century, the impacts of
bubble spill overs can be severe.
• Bubble bursts cause stock markets to crash.
The impacts of these crashes in one country
passes on to other integrated.
• The Indian experience during the most
recent financial crisis following the crash of
the U.S. stock markets, during 2007-2009,
following the burst of the U.S. Housing
Bubble has been significantly tough.
• Subbarao (2009) defined financial
integration as the ratio of total external
transactions to the GDP. This had doubled
from 46.8% in 1997-98 to 117.4% in 2007-
2008.
• Following the crisis, India was affected
through three main channels: (i) Trade
Channel (ii) Financial Channel (iii)
Confidence Channel

7. WHY STUDY ASSET PRICE BUBBLES?
• India was affected through three main
channels: Trade channel (expected shock
to GDP up to 1.5%), the financial channel
and confidence channel (Subbarao,
2009)
• The enhanced financial integration
affected India in three related ways: (i)
Reducing India’s access to overseas
finance, hence, lower or no external
borrowings (ii) Lower domestic
liquidity, given lower accretion of
reserves to reserve ratio (iii) falling
stock index, as a result of net sales by
FIIs. (Ram Mohan T.T; 2010)
• The chances of loss of wealth from stock
market crashes have become very high.
And, given the financial integration of
international markets, economies are
susceptible to periods of downturn in a
chain like fashion.

8. MAJOR STRANDS OF THOUGHT IN
BUBBLE THEORIES
• Bubble: Bubble exists for a stock if its
price deviates from its fundamental price
for a significant period. A bubble in a stock
price can be defined as :
𝑃𝑡 = 𝐹𝑡 + 𝐵𝑡 + 𝑒𝑡 (1)
Where, 𝑃𝑡 = Price of the asset at time t.
𝐵𝑡 = Bubble component at time t.
𝑒𝑡 = White noise error term.
• Fundamental value: The fundamental
value of an asset is defined as the present
discounted value of expected future
dividends, where the discount rate is the
rate of return of the risk-free asset. This
definition has been used primarily in the
empirical works Taylor (1977), Shiller
(1978), Blanchard and Watson (1982),
Diba and Grossman (1984), West
(1987), Phillips, Wu and Yu (2011) etc.

9. MAJOR STRANDS OF THOUGHT IN
BUBBLE THEORIES
• Rationale behind the definition of the
fundamental value: Dividend yield
provides a compact measure of how stocks
are valued vis-à-vis their fundamentals.
Low dividend yield indicates that stocks are
over-priced relative to their earning
capacity. So, if price of a stock increases at a
greater rate relative to the increase in the
dividends then the increase is not driven by
fundamentals. So for stable growth, the
stock prices must grow at a slower rate
than the growth of the dividend (Campbell
and Shiller, 1988).
• Literature describes the existence of the
following main strand of thoughts in
bubble theorization:
 Rationalist strand of Bubble Theory
 Irrational Bubble Theory
 Semi-Rational or Quasi-Rational Bubble

10. MAJOR STRANDS OF THOUGHT IN
BUBBLE THEORIES
• Rationalist strand of bubble
theorisation (Rational Bubbles):
Brock (1974), Taylor (1977) and Shiller
(1978) explains that the rational expectation
models of stock pricing are characterised by
an indeterminate aspect, which arises when
the current decision of agents depends both
on the market price of the stock as well as
their expectation of future prices. As current
stock price is determined by the expectations
of the future price and future price on the
current price, any simple demand-supply
model cannot calculate the equilibrium
market price of a stock. At best, a sequence
of prices can be obtained. Only one
sequence is the market fundamental price and
all the other have price bubbles.

11. MAJOR STRANDS OF THOUGHT IN
BUBBLE THEORIES
Blanchard (1970) pointed out that the
equilibrium amount of stock purchased by a
risk-neutral individual is the result of an
arbitrage between stocks and riskless asset, the
arbitrage equation being:
𝐸 𝑝 𝑡+1 𝐼 𝑡 −𝑝 𝑡
𝑝 𝑡
+
𝑑 𝑡
𝑝 𝑡
= r
The arbitrage stops when the expected return
from stock equals the return on the riskless
asset:
𝑝𝑡 = a E 𝑝𝑡+1 𝐼𝑡 + a 𝑑 𝑡
Where a =
1
1+𝑟
. Solving recursively up to T
periods:
𝑝𝑡 = a. 𝑖=0
𝑇
𝑎 𝑖E 𝑑 𝑡+𝑖 𝐼𝑡 +𝑎 𝑇+1 E[ 𝑝𝑡+𝑇+1 𝐼𝑡]

12. MAJOR STRANDS OF THOUGHT IN
BUBBLE THEORIES
• Asset prices are at its fundamental when 𝑎 𝑇+1
E[ 𝑝𝑡+𝑇+1 𝐼𝑡]  0 as time tends to infinity. For
the convergence of the asset prices to occur we
must have the condition that expectation of
dividends should not grow faster than the rate of
returns on the safe asset, else bubbles occur.
• Irrational bubble theory:
Minsky (1972), Kindleberger (1989) have
explained bubbles as a manifestation of the irrational
mob psychology. Camerer (1989) attributed the
occurrence of bubbles to the presence of
heterogeneous expectations of agents, which caused
some investors to believe that the bubble will last
longer than their contemporaries believe and hence
they run high chances of getting stuck with the risky
asset. Shiller (1984) attributed the occurrence of
bubbles to the epidemic and contagious nature of
‘fads’. Phillips, Wu, Yu (2011) identified the
existence of cognitive bias in the irrational dot-com
bubble.

13. MAJOR STRANDS OF THOUGHT IN
BUBBLE THEORIES
• Semi-rational bubbles: Closely related
to irrational bubbles, is another class of
bubbles, called semi-rational bubbles.
These models of stock market contain a
mix of well-informed agents and ill-
informed (noise) agents. These models
mainly comprise of a ‘fundamentalist’
who trade on fundamentals and ‘noise
traders’ who chase trends
Goodman(1968), Day and Huang
(1990). Some models formulated by De
Long, Schleifer, Summers and
Waldmann (1990), Gennotte and
Leland (1990) include a third agent
who influences the market price of a
stock through speculation. This agent is
also known by names as ‘rational
speculators’, ‘supply-informed investors’
etc.

14. NEED FOR A DETERMINISTIC BUBBLE
THEORY
1. The econometric models of bubble
identification use the existence of the
random error in empirical exercise. It
basically estimates the fundamental
component and the bubble component
under the assumption of Rational
Expectations. It does not consider the role
of the endogenous turbulence that can be
generated as a result of the interaction of
agents with heterogenous expectations.
2. Samuelson (1957) pointed out that
rational bubbles are a theoretical
possibility. Tirole (1982) established that
there can be no bubbles if there are a
finite numbers of risk-averse, infinitely
lived agents, with common prior
information and beliefs, trading a finite
number of assets with real returns in
discrete time periods. Bubbles are mostly
irrational. It is easy to model irrational
bubbles using a deterministic model.

15. REVIEW OF LITERATURE ON
DETERMINISTIC BUBBLES:
• The deterministic models of asset price
bubble tries to explain the turbulence in
the asset price generated endogenously
as a result of the interaction of traders
with heterogeneous agents.
• Day and Huang (1990), Gennotte and
Leland (1990) defined three types of
agents who interact in purchasing and
selling assets :
• (1) The Rational Speculators – These
traders use the available information in
making their trading decisions
• (2) The Noisy Traders – These traders
are the ones who believe that
information collection is expensive and
hence they follow the trends. If prices
are going up, they believe that this would
continue and they can make a profit out
of it by selling, once prices rise in the
future.

16. REVIEW OF LITERATURE ON
DETERMINISTIC BUBBLES
(3) The Market Makers: These are
institutions which adjust the price of the
asset, depending on the excess demand for
the asset.
• The interaction of these agents create a
turbulence in the dynamics of the asset
price. The turbulence is observed as
switching over from a bull market to a
bear market and vice-versa. Sudden
switches explain the points where a
bubble burst has occurred

17. A DETERMINISTIC REPRESENTATION
OF ASSET PRICE BUBBLE:
• OBJECTIVE: To provide a representation of
an irrational bubble in a deterministic
framework. By a deterministic frame work,
we imply that there are no exogeneous
white noise in this representation. Any
stochasticity within the model is generated
endogenously.
• THE FRAMEWORK: The framework of an
asset price bubble that is referred here is
the one provided by Rosser (1994). A
synopsis is provided below:
(a) Initial displacement from the fundamental
values of the stocks, usually upwards
where the displacements were brought
about by system behaviour or human
error.
(b) Speculation develops as buyers start
making money and many such buyers
follow suit. Prices rise and ‘euphoria’
emerges.

18. A DETERMINISTIC REPRESENTATION
OF ASSET PRICE BUBBLE:
(c) The presence of Cognitive bias in taking
investment decisions has been observed in all
of these bubbles. A cognitive bias refers to a
pattern of deviation in judgement which
occurs in particular situations which may
lead to perceptual distortions, inaccurate
judgement, or what is called irrationality.
They mainly arise from lack of appropriate
mental mechanisms (such as bounded
rationality) or from limited capacity of
information processing on part of the
economic agents.
(d) The euphoria is followed by a period
of distress where expectations undergo
major change. ‘Bad news’ follows and the
bubble crashes where the stocks return to
the fundamental value. This change is very
discreet in nature.

19. A DETERMINISTIC REPRESENTATION
OF ASSET PRICE BUBBLE:
ASSUMPTIONS: This representation is
based on the following assumptions:
1. The financial asset is assumed to have a
fundamental value of zero. So any
increase in the price of the asset would
be due to non- fundamental factors.
2. Investors form their expectations about
the future price of the asset based on
the decisions of their fellow investors
whom they assume to be relatively
well-informed.
3. There are two types of investors in this
model : (i) Rationalists (ii) Noisy
traders. Rationalists buy because they
know that they would make a profit by
selling their holdings in the immediate
future. The noisy traders observe the
behaviour of the rationalists and also
start purchasing the share. Noisy
traders behave in a naïve fashion in this
representation.

20. A DETERMINISTIC REPRESENTATION
OF ASSET PRICE BUBBLE:
MAJOR EQUATIONS USED IN THE
REPRESENTATION:
1. P(0) = 0 – A major challenge in identifying
bubbles in the existing literature is to
distinguish between price rises caused by
fundamentals and those that are caused by
non-fundamental factors. Shiller (2000)
pointed out that pointing out explosive
behaviour in the price data does not imply
the existence of bubble. To represent an
irrational bubble the response of the noisy
agents. So, fundamental price can be
assumed off.
2. 𝑬 𝑹 𝑷(𝟏) 𝑰 𝟎 = 𝑬 𝑵 𝑷(𝟏) 𝑰 𝟎 = Ԑ - R implies
‘Rational Speculators’ and N implies ‘Noisy
Traders’. The Rational Speculators know
the actual price of the asset, and they know
that they can make a profit by selling it off.
As they start buying based on their
expectations of the future price, the price
increases. Noisy traders thus buy it.

21. A DETERMINISTIC REPRESENTATION
OF ASSET PRICE BUBBLE:
3. P(t+1) = α E 𝑷(𝒕 + 𝟏) 𝑰 𝒕 - Investors buy
assets in the current period in the hope of
selling it in the future for a higher price. This
acts as a self fulfilling process- the expected
asset price movements in the present period
result in actual price movement in the next
period (t+1). Excess demand for the asset
would be generated if E 𝑃(𝑡 + 1) 𝐼𝑡 > P (t).
Hence α > 1.
4. E 𝑷 (𝒕 + 𝟏) 𝑰 𝒕 = 𝜹 𝒕 P (t) -- Investors are
driven most strongly by the price increases in the
most recent period than in the relatively distant past,
since the latter is remembered less distinctly
compared to the former. This is a simple expectation
formation rule for the investors on aggregate. This
says that every increase in the asset prices causes
the investors to scale up their expectations at an
increasing rate.

22. A DETERMINISTIC REPRESENTATION
OF ASSET PRICE BUBBLE:
5. Necessary substitutions show that :
P (t+1) = P (t)/ a , where a = (1/ 𝜹 𝒕). This
shows the positive feedback pattern, which
would continue as long as investors would
expect the prices to rise at an increasing
rate. After a terminal time (𝑡 𝑐 ) the asset
prices stop rising and the prices come back
to its fundamental value.
6. A reverse feedback pattern replaces the
positive pattern once a maximum
sustainable price is obtained. As ‘Insiders’
start bailing out and the ‘outsiders’ start
buying the asset. When the bail off occurs
faster than the buying activities of the
‘outsiders’ the asset price starts falling. After
some calculations we obtained the following
equation for the negative feedback pattern:
P (t+1) = [1 – P (t)]/ [1 – a] a < P (t) ≤ 1,
a є (0, 1)

23. MAJOR CHARACTERISTICS OF THE
REPRESENTATION:
• The pair of equations that were obtained
to represent an irrational bubble is
known as Tent- Map Recursions. These
maps are Chaotic in nature.
• A dynamic variable is Chaotic in nature if:
(1) If the path generates trajectories that
locally diverge away from each other
(2) If small changes in the initial
conditions build up exponentially fast
into large changes.
(3) If the error of the dataset either
remains small or grows exponentially.
This distinguishes the determinism from
any stochastic models where the error
term is random.
So in the presence of chaos small
difference in the value of initial
conditions or parameters build up
through the model.

24. MAJOR CHARACTERISTICS OF THE
REPRESENTATION:
• Some major characteristics of the equation
were checked previously which showed:
1. It was checked that this equation
generates the same correlation
coefficients as the first order
autoregressive process:
V (t+1) = (ka-1) V (t) + u (t+1), k is a
constant
2. Despite an initial displacement from its
fundamental value, the asset price
converges to the latter over a finite period
of time. The average displacement in the
price of the asset from its fundamental
value over a sufficiently long period of
time is zero.

25. THANK YOU

26. APPENDIX

27. APPENDIX