We develop a rational expectations model of financial bubbles and study ways in which a generic risk-return interplay is incorporated into prices. We retain the interpretation of the leading Johansen-Ledoit-Sornette model, namely, that the price must rise prior to a crash in order to compensate a representative investor for the level of risk. This is accompanied, in our stochastic model, by an illusion of certainty as described by a decreasing volatility function. As the volatility function goes to zero, crashes can be seen to represent a phase transition from stochastic to deterministic behaviour in prices.financial crashes; super-exponential growth; illusion of certainty; housing-bubble

Fry, J. M.

English

Research Papers in Economics

Bubbles and crashes in ¯nance: A phase
transition from random to deterministic
behaviour in prices
John M. Fry ¤
September 2010
Abstract
We develop a rational expectations model of ¯nancial bubbles and study ways
in which a generic risk-return interplay is incorporated into prices. We retain the
interpretation of the leading Johansen-Ledoit-Sornette model, namely, that the price
must rise prior to a crash in order to compensate a representative investor for
the level of risk. This is accompanied, in our stochastic model, by an illusion of
certainty as described by a decreasing volatility function. As the volatility function
goes to zero, crashes can be seen to represent a phase transition from stochastic to
deterministic behaviour in prices.
Keywords: ¯nancial crashes, super-exponential growth, illusion of certainty, housing-
bubble.
1 Introduction
Rational expectations models were introduced with the work of Blanchard and Watson to
account for the possibility that prices may deviate from fundamental levels [1]. We take
as our main starting point the somewhat controversial subject of log-periodic precursors
to ¯nancial crashes [2]-[11], with a fundamental aim of our approach being relatively easy
¤UEL Royal Docks Business School, Docklands Campus, University of East London, London E16 2RD
E-mail: frymaths@googlemail.com
1calibration of our model to empirical data. Additional background on log-periodicity and
complex exponents can be found in [12]. A ¯rst-order approach in [3] and subsequent
extensions in [13] state that prior to a crash the price must exhibit a super-exponential
growth in order to compensate a representative investor for the level of risk. However, this
approach concentrates solely on the drift function and ignores the underlying volatility
°uctuations which typically dominate ¯nancial time series [14]. Similar in spirit to [3], we
derive a second-order condition which incorporates volatility °uctuations and enables us
to combine insights from a rational expectations model with a stochastic model [15]-[16].
Our model gives two important characterisations of bubbles in economics. Firstly, a
rapid super-exponential growth in prices. Secondly, an illusion of certainty as described by
a decreasing volatility function prior to the crash. As the volatility function goes to zero
bubbles and crashes can be seen to represent a phase transition from stochastic to purely
deterministic behaviour in prices. This clari¯es the oft cited link in the literature between
phase transitions in critical phenomena and ¯nancial crashes. Further, this recreates
the phenomenology of the Sornette-Johansen paradigm: namely that prices resemble a
deterministic function prior to a crash. We explore a number of di®erent applications of
our model and the potential relevance to recent events is striking.
The layout of this paper is as follows. In Section 2 we introduce the basic model
and derive the crash-size distribution, the post-crash dynamics, simple estimates of
fundamental-value and speculative-bubble components. Section 3 describes an empirical
application to the UK housing bubble of the early to late 2000s [17]. Section 4 is a brief
conclusion.
2 The model
In this section we give an alternative formulation of the model solution in [3]. This leads
naturally to a stochastic generalisation of the original model, which is then solved in
full to give empirical predictions for the distribution of crash-sizes, post-crash dynamics,
fundamental values and the level of over-pricing.
We o®er an alternate derivation of the basic model in [3] as follows. Let P(t) denote
the price of an asset at time t. Our starting point is the equation
P(t) = P1(t)(1 ¡ ·)
j(t); (1)
2where P1(t) satis¯es
dP1(t) = ¹(t)P1(t)dt + ¾(t)P1(t)dWt; (2)
where Wt is a Wiener process and j(t) is a jump process satisfying
j(t) =
(
0 before the crash
1 after the crash.
(3)
When a crash occurs ·% is automatically wiped o® the value of the asset. Prior to a
crash P(t) = P1(t) and Xt = log(P(t)) satis¯es
dXt = ~ ¹(t)dt + ¾(t)dWt + ln[(1 ¡ ·)]dj(t); (4)
where ~ ¹ = ¹(t) ¡ ¾2(t)=2. If a crash has not occurred by time t, we have that
E[j(t + ±) ¡ j(t)] = h(t)dt + o(dt); (5)
Var[j(t + ±) ¡ j(t)] = h(t)dt + o(dt); (6)
where h(t) is the hazard rate. We compare (4) with the prototypical Black-Scholes model
for a stock price:
dXt = ¹dt + ¾dWt; (7)
and use (7) as our model for \fundamental" or purely stochastic behaviour in prices.
The ¯rst-order condition see e.g. [1], [3], suggests that ~ ¹(t) in (4) grows in order
to compensate a representative investor for the risk associated with a crash. The
instantaneous drift associated with (4) is
~ ¹(t) + (ln(1 ¡ ·))h(t): (8)
For (7) the instantaneous drift is ¹. Setting (8) equal to ¹, it follows that in order for
bubbles and non-bubbles to co-exist
~ ¹(t) = ¹ ¡ (ln(1 ¡ ·))h(t): (9)
If we ignore volatility °uctuations by setting ¾(t) = ¾, then our pre-crash model for an
asset price becomes
dXt = (¹ ¡ ln(1 ¡ ·)h(t))dt + ¾dWt: (10)
3However, this is actually a rather poor empirical model [18], failing to adequately account
for the volatility °uctuations in (4). Under a Markowitz interpretation, means represent
returns and variances/standard deviations represent risk. Suppose that in (4) ¾(t) adapts
in an analogous way to ¹(t) so as to compensate a representative investor for bearing
additional levels of risk. The instantaneous variance associated with (4) is
¾
2(t) + (ln(1 ¡ ·))
2h(t): (11)
For (7) the instantaneous variance is ¾2. Setting (11) equal to ¾2, the second-order
condition for co-existence of bubbles and non-bubbles becomes
¾
2(t) = ¾
2 ¡ (ln(1 ¡ ·))
2h(t): (12)
(12) illustrates an illusion of certainty { a decrease in the volatility function { which
arises as part of a bubble process. Intuitively, in order for a bubble to occur not only
must returns increase but the volatility must also decrease. If this does not happen (7)
with an instantaneous variance of ¾2 would represent a more attractive and less risky
investment than a market described by (10) and bubbles could not occur. We use (7) as
a model of a `fundamental' or purely stochastic regime, as in Black-Scholes theory. From
(12), our model for prices under a bubble regime becomes
dXt = [¹ ¡ ln(1 ¡ ·)h(t)]dt +
p
¾2 ¡ (ln(1 ¡ ·))2h(t)dWt: (13)
The simplest h(t) considered in [3] is
h(t) = B(tc ¡ t)
¡®; (14)
where it is assumed that ® 2 (0;1) and tc is a critical time when the hazard function
becomes singular, by analogy with phase transitions in statistical mechanical systems
[19]. Here, we choose on purely statistical grounds
h(t) =
¯t¯¡1
®¯ + t¯; (15)
which is the form corresponding to a log-logistic distribution and is intended to capture
the essence of the previous approach as the hazard rate has both a relatively simple form
and, for ¯ > 1, has a non-trivial mode at t = ®(¯ ¡1)
1
¯, with modal point (¯ ¡1)
1¡ 1
¯=®.
For these reasons, the log-logistic distribution is commonly used in statistics [20]. The
4log-logistic distribution has probability density
f(x) =
¯®¯x¯¡1
(®¯ + x¯)2; (16)
on the positive half-line. The cumulative distribution function is
F(x) = 1 ¡
®¯
®¯ + x¯ (17)
The model (13) with h(t) given by (15) has the solution
Xt = X0 + ¹t + v ln
µ
1 +
t¯
®¯
¶
+
Z t
0
r
¾2 ¡ v2 ¯t¯¡1
®¯ + t¯dWu: (18)
where v = ¡ln(1 ¡ ·) with v > 0. From (18) the conditional densities can be written as
XtjXs»N(¹tjs;¾
2
tjs); (19)
where
¹tjs = Xs + ¹(t ¡ s) + v ln
µ
®¯ + t¯
®¯ + s¯
¶
; (20)
¾
2
tjs = ¾
2(t ¡ s) ¡ v
2 ln
µ
®¯ + t¯
®¯ + s¯
¶
: (21)
Under the fundamental equation (7) these expressions are simply ¹tjs = Xs+¹(t¡s) and
¾2
tjs = ¾2(t ¡ s). Thus, we see that under the bubble model the incremental distributions
demonstrate a richer behaviour over time.
The fundamental or purely stochastic non-bubble model (7) corresponds to the case
that · = 0, or equivalently that v = 0. We can test for bubbles by testing the null
hypothesis v = 0 (no bubble) against the alternative hypothesis v > 0 (bubble). This
can be simply done using a (one-sided) t-test since maximum likelihood estimates, and
estimated standard errors, can be easily calculated numerically from (19). A range of
further implications of our bubble model can be derived as we describe below.
Crash-size distribution. Suppose that prices are observed up to and including time t and
that a crash has not occurred by time t. The crash-size distribution resists an analytical
description but a Monte Carlo algorithm to simulate the crash-size C is straightforward
and reads as follows:
1. Generate u from U » Log-logistic(®;¯) with the constraint u¸t.
2. C»·eZ,
5where
Z»N
µ
Xt + ¹u + v ln
µ
®¯ + u¯
®¯ + t¯
¶
;¾
2u ¡ v
2 ln
µ
®¯ + u¯
®¯ + t¯
¶¶
(22)
We note that simulating u from the log-logistic distribution is straight-forward and from
(17) possible via inversion using
F
¡1(x) = ®
µ
x
1 ¡ x
¶ 1
¯
or F
¡1(x) =
µ
®¯ + t¯
1 ¡ x
¡ ®
¯
¶ 1
¯
with constraint u¸t.
Post-crash increase in volatility. Before a crash equation (18) applies. After a crash,
the price reverts to the fundamental price dynamics (7). Suppose the crash occurs at time
C. At t = C we have that
Var(Xt+hjXt) = ¾
2h; (23)
but for t < C
Var(Xt+hjXt) = ¾
2h ¡ v
2 ln
µ
®¯ + (t + h)¯
®¯ + (t)¯
¶
(24)
Thus, our model predicts an increase in volatility following a crash given by
·
2 ln
µ
®¯ + (t + h)¯
®¯ + (t)¯
¶
: (25)
Fundamental values. The above model suggests a simple approach to estimate
fundamental value. Under the fundamental dynamics (7)
PF(t) := E(P(t)) = P(0)e
¹t; (26)
and we use (26) to estimate fundamental value in our empirical application in Section
7. This approach recreates the widespread phenomenology of approximate exponential
growth in economic time series (see e.g. Chapter 7 in [21]).
Estimated bubble component.De¯ne
H(t) =
Z t
0
h(u)du: (27)
Under the fundamental model E(P(t)) is given by (26). Under the bubble model, since
6Xt = log(Pt) satis¯es
Xt » N
¡
X0 + ~ ¹t + vH(t);¾
2t ¡ v
2H(t)
¢
; (28)
it follows that
PB(t) := E(P(t)) = P(0)e
¹t+
³
v¡ v2
2
´
H(t): (29)
This motivates the following estimate for the proportion of observed prices which can be
attributed to a speculative bubble:
1 ¡
1
T
Z T
0
PF(t)
PB(t)
dt = 1 ¡
1
T
Z T
0
µ
1 +
t¯
®¯
¶¡(v¡v2=2)
dt: (30)
3 Empirical application
As an empirical application we consider the UK housing bubble from 2002-2007 by
modelling a monthly time series of average UK house prices. The null hypothesis of
no bubble is a test of the hypothesis v = 0. This can be tested using a one-sided t-test
{ dividing the estimate ^ v by its estimated standard error and comparing to a normal
distribution. For this data set we obtain a t-statistic of 3.66 and a p value of 0.0001 to
give strong evidence of a bubble in this data.
From our ¯t of the bubble model (18) we use PF(t) = P(0)e¹t in (26) as a simple
estimate of fundamental value. A plot of UK house prices together with estimated
fundamental values and associated 95% con¯dence intervals is shown in Figure 1. Prices
appear to be well in excess of fundamental values, with prices lying above the upper
con¯dence limits of the estimated fundamental values throughout the sample (Figure 1
to the left of the vertical line). We then estimate fundamental value for the years 2008-
2009 using data from 2002-2007 only and compare with the actual historically observed
prices. That is, we use our model to provide estimates of fundamental value out of
sample. The results are shown to the right of the vertical line in Figure 1 and show prices
reverting towards fundamental values { moving inside the con¯dence intervals constructed
for fundamental value. From the second half of 2008 observed prices are statistically
indistinguishable from estimated fundamental value. Finally, the estimated speculative
bubble component is 0.202 suggesting that the bubble accounts for around 20% of the
observed prices. This compares reasonably with similar estimates of 12-25% in [22] and
28-53% in [23].
72002 2004 2006 2008 2010
1
0
0
0
0
0
1
2
0
0
0
0
1
4
0
0
0
0
1
6
0
0
0
0
1
8
0
0
0
0
Time
A
v
e
r
a
g
e
 
U
K
 
h
o
u
s
e
 
p
r
i
c
e
Figure 1: Plot of average UK house-prices and estimated fundamental value (dashed line)
and associated 95% con¯dence intervals (dots). Estimation takes place over the period
2002-2007 (to the left of the dashed vertical line). Out-of-sample estimates of fundamental
value are then compared to historically observed prices (to the right of the dashed vertical
line).
4 Conclusions
This paper has provided a stochastic version of the model in [3]. Crash precursors are a
super-exponential growth accompanied by an \illusion of certainty", characterised by a
decrease in the volatility function prior to the crash. A range of potential applications to
economics were discussed including statistical tests for bubbles, crash-size distributions,
predictions of a post-crash increase in volatility and simple estimates of fundamental-value
and speculative-bubble components. As a brief empirical application we consider the UK
housing bubble in the early to mid 2000s. Prices appear to be in excess of fundamental
levels, with the speculative bubble component accounting for around 20% of observed
prices. In addition, prices are seen to revert towards estimated fundamental values out of
sample over the period 2008-2009.
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