The document discusses probabilistic risk attitudes in financial markets as implied by index option prices. It provides context on credit risk, investor sentiment, and Minsky's financial instability hypothesis. Key points covered include the CBOE VIX index as a measure of implied volatility and investor fear, S&P 500 index options representing 36% of the global market for options, and the implied probability weighting function derived from option prices as a measure of global risk attitude.

1. G. Charles-Cadogan
School of Economics, University of Cape Town &
Institute for Innovation and Technology
Management, Ted Rogers School of
Management, Ryerson University

2. Market Instability, Investor Confidence, And
The Probability Weighting Functions
Implied By Index Option Prices
G. Charles-Cadogan
gocadog@gmail.com
School of Economics
Research Unit in Behavioural Economics and Neuroeconomics
Faculty of Commerce
University of Cape Town
Presentation
38th International Conference
Stochastic Processes and their Applications
Oxford-Man Institute of Quantitative Finance,
Oxford University
July 13-17, 2015

3. Credit and investor sentiment
The CBOE VIX (“VIX”) measures the markets perceived
future volatility (read: risk and uncertainty), most often
associated with a fear that the market will drop. More
specifically, the VIX measures the markets expectation of
future volatility implied by S&P 500 stock index (SPX)
option prices. While technically it does not measure the
probability that the market is going to drop in the near
future, at times it does represent a measure of fear that
it will.
“Commercial credit is the creation of modern times and
belongs in its highest perfection only to the most
enlightened and best governed nations. Credit is the vital
air of the system of modern commerce. It has done more
– a thousand times more - to enrich nations than all the
mines of the world.” Dr. R. Keith Sawyer, “Credit:
man’s confidence in man,” Huffington Post, June 29,
2013 (quoting Daniel Webster).

4. Motivation–Minsky’s Financial Instability Hypothesis
Financial instability ⇒ banking failures, intense asset-price
volatility or a collapse of market liquidity. The real sector is
affected due to its links to the financial sector
[South African Reserve Bank, 2014].
➼ “The first theorem of the financial instability hypothesis is
that the economy has financing regimes under which it is
stable, and financing regimes in which it is unstable.”
➼ “The second theorem of the financial instability hypothesis is
that over periods of prolonged prosperity, the economy
transits from financial relations that make for a stable system
to financial relations that make for an unstable system.”
[Minsky, 1994, p. 156].

5. Pertinent literature
➼ Great recession of 2008.
➼ History of manias, panics and crashes in financial
markets [Kindleberger and Aliber, 2011].
➼ Financial instability hypothesis
[Minsky, 1986, Minsky, 1994, Dymski, 2010, Keen, 2013,
Grasselli and Costa Lima, 2012].
➼ Contagion and integration of global financial markets
[King and Wadhwani, 1990, Bongaerts et al., 2014,
Schnabl, 2012, Ncube et al., 2012, Devereux and Yu, 2014].
➼ Stochastic Lyapunov exponent
[Nychka et al., 1992, Bougerol and Picard, 1992,
Whang and Linton, 1999, Park and Whang, 2012].
➼ Probability weighting function (pwf) implied by index
option prices [Polkovnichenko and Zhao, 2013].

6. Sources of probabilistic risk attitude in financial markets
Credit risk:CrCrCrCrCrCrCrCrCrCrCrCrCrCrCrCrCrCrededededededededededededededededititititititititititititititit rrrrrrrrrrrrrrrrrisisisisisisisisisisisisisisisisiskkkkkkkkkkkkkk:::::::::::::
sovereign,
CrCrCrCrCrCrCrCrCrCrCredededededededededededededededitititititititititititit rrrrrrrrrrisisisisisisisisisisisisisiskkkkkkkkk:::::
sosososososososososososososososovevevevevevevevevevevevevevevevevererererererererererererererereigigigigigigigigigigigigigigigignnnnnnnnnnsososososososososososososovevevevevevevevevevevevevererererererererererererereigigigigigigigigigigigigigignnnnnnnnnnnnnn,,,,,,
corporate,
sososososososososososososososososovevevevevevevevevevevevevevevererererererererererererereigigigigigigigigigigigigigigigigigigignnnnnnnn,,,,,,,,,,
cococococococococococococococococorprprprprprprprprprprprprprprpororororororororororororororatatatatatatatatatatatatatatatatatateeeeeeeeeeeeeeeecocococococococococococococorprprprprprprprprprprprprprpororororororororororororororororatatatatatatatatatatatatatatatateeeeeeeeeeee,,,,,,,
households
cocococococococococococococococorprprprprprprprprprprprprprprporororororororororororororororatatatatatatatatatatatatatatatatateeeeeeeeeeeee,,,,,,,,,,
hohohohohohohohohohohohohohohohoususususususususususususususususususeheheheheheheheheheheheheheheheheheheheholololololololololololololololololdsdsdsdsdsdsdsdsdsdsdsdsdsdsdsdsdsdshohohohohohohohohohohohohohohohohoususususususususususususehehehehehehehehehehehehehehehehololololololololololololdsdsdsdsdsdsdsdsdsdsdsdsdsdsds
S&P 500 indexS&S&S&S&S&S&S&S&S&S&S&S&S&S&S&S&S&S&S&PPPPPPPPPPPPPPP 505050505050505050505050505050505000000000000000 inininininininininininininininindededededededededededededededededededexxxxxxxxxxxxx
option =36% of
S&S&S&S&S&S&S&S&S&S&S&S&S&S&S&S&PPPPPPP 5050505050505050505050505050000000000000 inininininininininindedededededededededededededexxxxxxxxxxxx
opopopopopopopopopopopopoptitititititititititiononononononononononononononopopopopopopopopopopopopopopopoptititititititititititititititititionononononononononononononon ==================3636363636363636363636363636363636%%%%%%%%%%%%%%%%%3636363636363636363636363636363636%%%%%%%%%%%%%%%% ofofofofofofofofofofofofofofofofofofofofofofofofofofofofofofofof
global market
opopopopopopopopopopopopopopopoptitititititititititititititiononononononononononononononon 36363636363636363636363636363636%%%%%%%%%%%%%%%% ofofofofofofofofofofofofofofof
glglglglglglglglglglglglglglglglobobobobobobobobobobobobobobobalalalalalalalalalalalalalalglglglglglglglglglglglglglglglglglobobobobobobobobobobobobobobalalalalalalalalalalalalalalal mamamamamamamamamamamamamamamamamamarkrkrkrkrkrkrkrkrkrkrkrkrkrkrketetetetetetetetetetetetetetmamamamamamamamamamamamamamamamamarkrkrkrkrkrkrkrkrkrkrkrkrkrketetetetetetetetetetetetetetetet
capitalization
glglglglglglglglglglglglglglglglglglobobobobobobobobobobobobobobalalalalalalalalalalalalal mamamamamamamamamamamamamarkrkrkrkrkrkrkrkrkrkrkrkrkrketetetetetetetetetetetetetetetet
cacacacacacacacacacacacacacacacacacappppppppppppppppcacacacacacacacacacacacacacacappppppppppppppititititititititititititititititalalalalalalalalalalalalalalalalalalizizizizizizizizizizizizizizizizatatatatatatatatatatatatatatatatatioioioioioioioioioioioioioioioioioioioionnnnnnnnnnnnnnnnnitititititititititititititititalalalalalalalalalalalalalalizizizizizizizizizizizizizatatatatatatatatatatatatatatatioioioioioioioioioioioioioionnnnnnnnnnnn
CBOE VIX impliedCBCBCBCBCBCBCBCBCBCBCBCBCBCBCBCBCBCBOEOEOEOEOEOEOEOEOEOEOEOEOEOEOEOE VVVVVVVVVVVVVIXIXIXIXIXIXIXIXIXIXIXIXIXIX imimimimimimimimimimimimimimimimimplplplplplplplplplplplplplplplplieieieieieieieieieieieieieieieieieiedddddddddddddddd
volatility index:
CBCBCBCBCBCBCBCBCBCBCBCBOEOEOEOEOEOEOEOEOEOEOEOEOE VVVVVVVIXIXIXIXIXIXIXIXIXIX imimimimimimimimimimimimplplplplplplplplplplplplplplplplieieieieieieieieieieieieieieiedddddddddddddd
vovovovovovovovovovovovovovolalalalalalalalalalalalalalalalatititititititititilililililililitytytytytytytytytytytyty iiiiiiiindndndndndndndndndndndndndexexexexexexexexexexexexexexexexvovovovovovovovovovovovovovovolalalalalalalalalalalalalalalalalatitititititititititititititititililililililililililililililitytytytytytytytytytytytytyty iiiiiiiiiiiiiiindndndndndndndndndndndndndndndndndexexexexexexexexexexexex::::::::::::::::::::::::
Global risk attitude
vovovovovovovovovovovovovovovovolalalalalalalalalalalalalalalatitititititititititititititititililililililililililitytytytytytytytytytytytytytyty iiiiiiindndndndndndndndndndndndndndndexexexexexexexexexexexexexexexexexex:::::::::::
GlGlGlGlGlGlGlGlGlGlGlGlGlGlGlGlGlGlGlobobobobobobobobobobobobobobobobobobalalalalalalalalalalalalal rrrrrrrrisisisisisisisisisisisisisisiskkkkkkkkkkkkkGlGlGlGlGlGlGlGlGlGlGlGlGlGlGlGlGlobobobobobobobobobobobobobobalalalalalalalalalalalalalal rrrrrrrrrrrrrisisisisisisisisisisisisisisisisisiskkkkkkkkkk atatatatatatatatatatatatatatatattititititititititititututututututututututututudedededededededededededededededeatatatatatatatatatatatatatattitititititititititititititititutututututututututututututudededededededededededededededede
measure
obobobobobobobobobobobobalalalalalalalalalalalalal rrrrrrrrisisisisisisisisisisisisisisiskkkkkkkkk atatatatatatatatatatatattititititititititititititititutututututututututututututudedede
memememememememememememememememememememeasasasasasasasasasasasasasasasasasasururururururururururururururururureeeeeeeeeeeeeeeeemememememememememememememememememeasasasasasasasasasasasasasasasasasurururururururururururururureeeeeeeeeeeeeee
Implied probabilityImImImImImImImImImImImImImImImplplplplplplplplplplplplplplplieieieieieieieieieieieieieieieddddddddddddddd pppppppppppppprorororororororororororororororobababababababababababababababababibibibibibibibibibibibibibibibilililililililililililililitytytytytytytytytytytytytytyty
weighting function:
ImImImImImImImImImImImImImplplplplplplplplplplplplplplplplplieieieieieieieieieieieiedddddddddddd ppppppppppppppprororororororororororobabababababababababababababibibibibibibibibibibibibibibilililililililililitytytytytytytytytytytytytyty
weweweweweweweweweweweweweweweweigigigigigigigigigigigigigigigighththththththththththtinininininininininininggggggggggggggggg fufufufufufufufufufufufufufuncncncncncncncncncncncncncnctitititititititititititititionononononononononononononononon:::::::::::weweweweweweweweweweweweweweweigigigigigigigigigigigigigigighththththththththththththththtinininininininininininininininggggggggggggg fufufufufufufufufufufufufufufuncncncncncncncncncncncncncncncnctitititititititititititititititionononononononononononononononon:::::::::::
Probabilistic risk
weweweweweweweweweweweweweigigigigigigigigigigigigigigigigigighththththththththththththtininininininininininingggggggggggggggggg fufufufufufufufufufufufufufufuncncncncncncncncncncncncnctititititititititititititionononononononononononononon:::::::
PrPrPrPrPrPrPrPrPrPrPrPrPrPrProbobobobobobobobobobobobobobobobababababababababababababababababilililililililililililisisisisisisisisisisisisisisistitititititititititititicccccccccccccPrPrPrPrPrPrPrPrPrPrPrPrPrPrPrPrPrProbobobobobobobobobobobobobobobobababababababababababababababababililililililililililililisisisisisisisisisisisisistititititititititititititititiccccccccccccccc riririririririririririskskskskskskskskskskskskskskskririririririririririririririririsksksksksksksksksksksksksksksk
attitude measure
PrPrPrPrPrPrPrPrProbobobobobobobobobobobobobobobabababababababababababababababilililililililililisisisisisisisisisisisisisistitititititititititiccccccccccccc ririririririririsksksksksksksksksksksk
atatatatatatatatatatatatatatatatatatattitititititititititititititititititutututututututututututututututudededededededededededededededededeatatatatatatatatatatatatatatatatatatattitititititititititititititititutututututututututututututudedededededededededededededede memememememememememememememememememeasasasasasasasasasasasasasasasasasasasururururururururururururururururururureeeeeeeeeeeeeeeeeemememememememememememememememeasasasasasasasasasasasasasasasasurururururururururururureeeeeeeeeeeeee
Measures ofMeMeMeMeMeMeMeMeMeMeMeMeMeMeMeMeMeasasasasasasasasasasasasasasasasasasururururururururururururururururureseseseseseseseseseseseseseseses oooooooooooooooffffffffffffff
investor
MeMeMeMeMeMeMeMeMeMeMeMeMeMeasasasasasasasasasasasasasasasurururururururururururururururureseseseseseseseseseseseseses oooooooooooooffffffff
ininininininininininvevevevevevevevevevevevevevestststststststststststststststststororororororororororororororororininininininininininininininininvevevevevevevevevevevevevevestststststststststststststststorororororororororororororor
beliefs
ininininininininininininveveveveveveveveveveveveveststststststststststststststorororororororororororororor
bebebebebebebebebebebebebebebelililililililililililiefefefefefefefefefefefefefefefsssssssssssssssssbebebebebebebebebebebebebebebebelilililililililililililiefefefefefefefefefefefefefefefefefssssssssssssss
Credit event:CrCrCrCrCrCrCrCrCrCrCrCrCrCrCrCrCrCredededededededededededededededededititititititititititititititit eeeeeeeeeeeeeeeeveveveveveveveveveveveveveveventntntntntntntntntntntntntntntntnt:::::::::::::
bankruptcy,
CrCrCrCrCrCrCrCrCrCredededededededededededededededitititititititititititit eeeeeeeeeeeeveveveveveveveveveveveveventntntntntntntntntntntntntnt::::::::
bababababababababababababababanknknknknknknknknknknknknkrurururururururururururuptptptptptptptptptptptptptcycycycycycycycycycycycycycy,,,,,,bababababababababababababanknknknknknknknknknknknknkrururururururururururururururuptptptptptptptptptptptptptptcycycycycycycycycycycy
prepayment,
babababababababababababababababanknknknknknknknknknknknkrururururururururururururururuptptptptptptptptptptptptptptptcycycycycycycycycycycycycycycycy,,,,,,,,,,,,
prprprprprprprprprprprprprprprepepepepepepepepepepepepepepayayayayayayayayayayayayaymememememememememememememementntntntntntntntntntnt,,,,,,prprprprprprprprprprprprprprprprprepepepepepepepepepepepepepepepepayayayayayayayayayayayayayayayayaymemememememememememememememememementntntntntntntntntntntntnt
default
prprprprprprprepepepepepepepepepepepepepepayayayayayayayayayayayayayayaymemememememememememememememementntntntntntntntntntntntnt,,,,,,,,,
dededededededededededededededededefafafafafafafafafafafafafafafafafaulululululululululululululululttttttttttttttdededededededededededededededefafafafafafafafafafafafafafafafaululululululultttttttttttttttt
Credit
rating

7. Research questions posed
➼ Are investors’ probabilistic risk attitudes, i.e., beliefs, towards
financial decision making stable or unstable? If so, how and
when?
➼ Are there early warning signals of market crash discernible
from investors’ probabilistic risk attitudes towards the
underlying source of credit risk?
➼ Given that index option prices are correlated with credit
spreads, can we use the pwfs implied by index option prices in
a natural experiment to test Minsky’s financial instability
hypothesis?

8. Predictions of the theory
➼ Existence of invariant manifold, i.e., local n-dimensional space,
and hyperbolic fixed points for pwfs in financial markets.
➼ Existence of bifurcation for pwfs in financial markets.
➼ Concave-convex pwfs with fix pt around 0.36 are stable.
Convex-concave pwfs with fix pt around 0.36 are unstable.
➼ Criterion function for distribution of critical values of
probabilistic risk factors that predict eminent financial market
[in]stability in a seemingly stable market.
➼ Pwf implied by index option prices is a sufficient statistic for
Minsky’s Financial Instability Hypothesis.

9. Why the Standard and Poors 500 Stock Market Index?
➼ The S&P 500 stock market index contains the stocks of 500
large-cap corporations. It comprises over 70% of the total
market cap of all stocks traded in the U.S., and it constitutes
over 35% of the world’s stock market cap.
➼ CBOE VIX volatility index is a measure of the implied
volatility of the S&P 500 index. It is a measure of global risk
aversion [Hassan, 2014] and investor risk attitudes in financial
markets [Whaley, 2000]. S&P 500 data can be used to
conduct a natural experiment on investor psychology and
market instability.
➼ CBOE VIX highly correlated with credit spreads, i.e., gap
between interest on corporate bonds and risk free rate
[Merton, 1974, Che and Kapadia, 2012]. CBOE VIX volatility
transmitted to emerging markets like South Africa
[Ncube et al., 2012] and Peru [Schnabl, 2012].

10. Theories of probabilistic risk attitude
➼ Security, potential/ Aspiration (SP/A) theory based on Lorenz
curve analogy. Gini coeff. measures disparity in ranked lottery
payoffs. [Lopes, 1984, Lopes, 1987, Lopes, 1990, Lopes, 1995,
Lopes and Oden, 1999]
➼ Optimism and pessimism based on rank dependent utility (RDU)
theory [Quiggin, 1982, Quiggin, 1993] and curvature properties of
pwf first plotted by [Preston and Baretta, 1948].
➼ Confidence calibration: proportion of statements true =
probability assigned [Lichtenstein et al., 1982].
➼ Original prospect theory (OPT) and cumulative prospect theory
(CPT) based on frame dependent pwf
[Kahneman and Tversky, 1979, Tversky and Kahneman, 1992].
➼ Axiomatization of pwf to include fixed point probability
[Prelec, 1998].

11. Probabilistic risk attitudes in the lab [Wilcox, 2011]Figure 6: 80 individually estimated probability weighting functions.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Weightassociatedwithprobabilityq
Probability q of receiving high outcome in risky
Blue = 30 “Optimists”
(above identity line).
Red = 4 “Pessimists”
(below identity line).
Green = 14 “Prospect Theorists”
(initially above, then below
identity line).
Orange = 26 “Approximators”
(initially below, then above
identity line).
Yellow = 6 “Others” (crosses
identity line more than once).

12. Phase portrait of stable and unstable pwfs
Figure : Stable pwf
0 1
1
p
w(p)
Figure : Unstable pwf
0 1
1
p
w(p)
Phase diagrams for stable and unstable pwfs w(p). The behavioral
stochastic Lyapunov exponent process {λ(t, ω); t ≥ 0} in fixed
point (p⋆) probability neighbourhood Bδ(p⋆) predict the shape of
pwfs.

13. Hope and fear: AAII Weekly Sentiment Index
0%
10%
20%
30%
40%
50%
60%
70%
80%
Jun'87
Jun'88
Jun'89
Jun'90
Jun'91
Jun'92
Jun'93
Jun'94
Jun'95
Jun'96
Jun'97
Jun'98
Jun'99
Jun'00
Jun'01
Jun'02
Jun'03
Jun'04
Jun'05
Jun'06
Jun'07
Jun'08
Jun'09
Jun'10
Jun'11
Reported Bullish Reported Neutral Reported Bearish
Poly. (Reported Bullish) Poly. (Reported Neutral) Poly. (Reported Bearish)
Source: Amer. Assoc. Individual Investors weekly 6/1987-6/2011
Waves of market sentiment. Markets breakdown when bull and
bear sentiment coincide. For example, Japanese real estate crisis of
1990, and Great Recession of 2008 originating from US markets.
Japan real estate
crash 1990
Great Recession
2008

14. Stable investor beliefs implied by index option prices
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w(P)
FearFix0
FearFix1
FearFix2
FearFix3
P
Prelec 2-factor pwf
a=0.56 b=0.93fixed pt = 0.4
Source: Author’s plot of calibrated pwf for parameter estimates taken from
[Polkovnichenko and Zhao, 2013] for S&P 500 index option data 1996-2008

15. Unstable investor beliefs implied by index option prices
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
w(P)
HopeFix0
HopeFix1
P
0.37
0.36 fixed pt = 0.3679
Prelec 2-factor
pwf a=1.6
b=1.0
Source: Author’s plot of calibrated pwf for parameter estimates taken from
[Polkovnichenko and Zhao, 2013] for S&P 500 index option data 1996-2008

16. Local Lyapunov Exponent
Definition (Local Lyapunov exponent)
Let w(p) be a pwf such that the first derivative w′ exist. The
Lyapunov exponent of the orbit pn = w(pn−1), n ∈ N+ for p0 = p is
λ(p) := lim
n→∞
1
n
n
∑
j=1
ln |w′
(pj )|
provided the limit exist. In particular, since (p, w(p)) exist in a
bounded unit square n need not be infinite [Wolff, 1992, p. 356].
➼ Lyapunov exponent (λ) characterizes the rate of growth and
divergence over time of the effects of a small perturbation to a
system’s initial state
➼ λ < 0 implies exponential decay and a nonchaotic system
➼ λ > 0 implies exponential and potentially explosive growth and a
chaotic system

17. Stochastic Lyapunov Exponent Process
[Park and Whang, 2012, p. 64] Brownian functional representation
of Lyapunov exponent
λn(t) =
t
0
ln |mo
n (
√
nBn(s))|ds, t ∈ [0, 1]
➼ mo
n is the first derivative of a Nadaraya-Watson kernel
estimator for the nonparametric nonlinear function mn(·),
defined on C[0, 1]
➼ Bn(t) ∈ C[0, 1] is approximate Brownian motion
➼ [BenSa¨ıda, 2012, BenSa¨ıda, 2014] applied the tests above to
financial time series that include the S&P 500 index and failed
to find chaos in the data

18. Empirical Local Lyapunov Exponent Process
➼ We consider [Prelec, 1998, Prop. 1, pg. 503] 2-parameter
probability weighting function:
w(p) = exp(−β(− ln(p))α
), 0 < α < 1, β > 0
where alpha (curvature) and β (elevation) are risk attitude
factors.
➼ In a large sample of N agents the empirical LLE process we
derive is
d ¯λN (t; p, α, β) = ¯am,N (p; α, β)dt + σdW n,N (t),
¯λN (·) =
1
N
N
∑
j=1
λj
(·); ¯am,N (·) =
1
N
1
m
N
∑
j=1
m
∑
r=1
aj
(pr ; α, β)
W n,N (t) =
1
N
N
∑
j=1
W j
n(t)
¯λN (t; p, α, β) is the empirical LLE; ¯am,N(·) is a drift term, and
W n,N (t) is the background driving noise or Brownian motion.

19. Probability of market instability
➼ Lyapunov stability condition implies negative eigenvalues
[Hommes and Manzan, 2006], [Wiggins, 2003, p. 7]. Thus
sup
t
¯λN (t; p, α, β) < 0 ⇒ Pr sup
t
W n,N (t) < −
1
σ
¯am,N (p; α, β)t
= c0Φ −
¯am,N(p; α, β)
σ
√
Nt = ϕ(t, α, β, N, σ)
where c0 is a constant of proportionality, Φ(·) is the
cumulative normal distribution and ϕ(·) is a numerical
probability. W n,N (t) induces a Lyapunov-Perron effect
which stems from the notion of hyperbolic fixed points and
unstable manifolds [Wiggins, 2003, pp. 12, 50].
➼ Tail event probability of instability in a seemingly stable
system is given by 1 − ϕ(t, α, β, N, σ).
➼ Given α, β, σ the probability of instability increases in time t
and as the number of agents N gets larger.

20. Instability criteria for seemingly stable beliefs
Proposition (Tail Event Instability)
Given a large sample of heterogenous DMs with [Prelec, 1998]
2-parameter pwfs (α and β) in a dynamical system of confidence in
psychological space, the tail event probability 1 − ϕ that the
system becomes chaotic depends on either of the following
1. growth in sample size N;
2. risk attitude parameters α (curvature) and β (elevation) that
induce the range of confidence
0 < β(p) < max
α−1 + ln(− ln(p))
(− ln(p))α+1
, (− ln(p))−α
3. increased precision in σ for classifying measurement error by
DMs.

21. Market transition and distribution of critical risk factors
Figure : Pwfs for option prices
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w(P)
P
W_Asia_1997(p; a=0.56,b=0.93) W_US_2005(p; a=1.6,b=1) P
Figure : β(p)-instability
distribution
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b(p)
alpha=1.6, b=1 alpha=0.56, b=0.93
Pwf plots are for monthly S&P 500 index option prices between
1996-2008. The inverted S-shape curve depicts the state of investor
sentiment around the Asian financial crisis circa June 19, 1997. By
April 21, 2005 the state changed to optimism depicted by the skewed
S-shape curve during the real estate bubble in the US. So the option
market transitted from pessimism to optimism between 1997 and
2005. β(p) plots the distribution of β for market instability.

22. Mania, panics, and crashes predicted by criterion function
Figure : β(p) instability (0.6,
0.2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w(p)
w_Asia_crit(P;a=0.56,b=0.6) w_US_crit(P; a=1.6,b=0.2) P P
Figure : β(p) instability (0.7,
0.5)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w(P)
Pw_Asia_crit_(P;0.56,0.7) w_US_crit_(P;1.6,0.5) P
Concave pwf is pessimistic, convex is optimistic, concave-convex
cautiously hopeful, convex-concave incautiously hopeful
[Quiggin, 1993]. Markets crash when all investors are pessimistic
and uncertain about toxic assets [Akerlof, 1970].

23. Time series plot of probabilistic risk factors
Figure : Realization of α curvature and β elevation for index option prices
for [Prelec, 1998] pwf
Source: [Polkovnichenko and Zhao, 2013]

24. Market crash realized in Great Recession 2008
Figure : Risk attitudes at
market crash in 2008
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w(P)
exp(-b(-ln(P)^a)) P
a=1.2
b=0.9
P
Figure : Predicted β(p)
instability tipping point
b (P)= -3.8791 P2 + 3.7693 P - 0.0529
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
b(P)
P
Tipping Point
b=0.9041
When markets crashed in 2008 the pwf implied by index option
prices was concave as predicted by our stochastic LLE process.

25. Conclusion
➼ The probability weighting functions implied by index option
prices is a sufficient statistic for Minsky’s financial instability
hypothesis.
➼ Neuronal noise in fixed point neighbourhoods of pwfs implied
by index option prices induce an empirical Lyapunov process
that identify market instability.
➼ Potential application–monitor real time pwf implied by index
option prices for early warning signals of market instability.

26. Thank you

27. References I
Akerlof, G. A. (1970).
The market for lemons: quality uncertainty and the market mechanism.
Quarterly Journal of Economics, 84(3):488–500.
Bongaerts, D., Roll, R., R¨osch, D., Van Dijk, M. A., and Yuferova, D. (2014).
The Propagation of Shocks Across International Equity Markets: A
Microstructure Perspective.
Available at SSRN: http://ssrn.com/abstract=2475518.
Bougerol, P. and Picard, N. (1992).
Stationarity of GARCH processes and of some nonnegative time series.
Journal of Econometrics, 52(12):115 – 127.
Che, X. and Kapadia, N. (2012).
Understanding the Role of VIX in Explaining Movements in Credit Spreads.
Working Paper, Isenberg School of Management, U. Mass., Amherst.
Devereux, M. B. and Yu, C. (2014).
International financial integration and crisis contagion.
Working Paper 20526, National Bureau of Economic Research.
Dymski, G. A. (2010).
Why the subprime crisis is different: A Minskyian approach.
Cambridge Journal of Economics, 34(2):239–255.

28. References II
Grasselli, M. and Costa Lima, B. (2012).
An analysis of the Keen model for credit expansion, asset price bubbles and
financial fragility.
Mathematics and Financial Economics, 6(3):191–210.
Hassan, S. (2014).
Rand Volatility And Volatility Of Other Emerging Market Currencies.
Seminar presentation, School of Economics, University of Cape Town.
October.
Hommes, C. H. and Manzan, S. (2006).
Comments on “Testing for nonlinear structure and chaos in economic time
series”.
Journal of Macroeconomics, 28(1):169 – 174.
Special Issue: Nonlinear Macroeconomic Dynamics.
Kahneman, D. and Tversky, A. (1979).
Prospect theory: An analysis of decisions under risk.
Econometrica, 47(2):263–291.
Keen, S. (2013).
A monetary Minsky model of the Great Moderation and the Great Recession.
Journal of Economic Behavior & Organization, 86:221–235.

29. References III
Kindleberger, C. P. and Aliber, R. Z. (2011).
Manias, panics and crashes: A history of financial crises.
Palgrave Macmillan.
King, M. A. and Wadhwani, S. (1990).
Transmission of volatility between stock markets.
Review of Financial studies, 3(1):5–33.
Lichtenstein, S., Fischoff, B., and Phillips, L. D. (1982).
Calibration Of Probabilities: The State Of The Art To 1980.
In Kahneman, D., Slovic, P., and Tversky, A., editors, Judgment Under
Uncertainty: Heuristics and Biases, chapter 22, pages 306–334. Cambridge
University Press, New York, NY.
Lopes, L. L. (1984).
Risk and distributional inequality.
Journal of Experimental Psychology: Human Perception and Performance,
10(4):465 – 485.
Lopes, L. L. (1987).
Between Hope and Fear: The Psychology of Risk.
Advances in Experimental Social Psychology, 20(3):255–295.

30. References IV
Lopes, L. L. (1990).
Re-Modeling Risk Aversion: A Comparison Of Bernoullian and Rank Dependent
Value Approaches.
In von Furstenberg, G. M., editor, Acting Under Uncertainty: Multidisciplinary
Conceptions, Theory and Decision Library. Series A., Philosophy and
Methodology of The Social Science, chapter 11, pages 267–299. Kluwer
Academic Publishers, Dordrecht, Netherlands.
Lopes, L. L. (1995).
Algebra and Process in The Modeling of Risky Choice.
In Busemeyer, J., Hastie, R., and Medin, D. L., editors, Decision Making From A
Cognitive Perspective, volume Advances in Research and Theory of The
Psychology of Learning and Motivation, pages 177–220. Academic Press, Inc.,
San Diego, CA.
Lopes, L. L. and Oden, G. C. (1999).
The role of aspiration level in risky choice: A comparison of cumulative prospect
theory and sp/a theory.
Journal of Mathematical Psychology, 43(2):286 – 313.
Merton, R. C. (1974).
On the pricing of corporate debt: The risk structure of interest rates.
Journal of Finance, 29(2).
Minsky, H. (1986).
Stabilizing an Unstable Economy.
Yale Univ. Press, New Haven, CT.

31. References V
Minsky, H. (1994).
The financial instability hypothesis.
In Arestis, M. and Sawyer, M. C., editors, The Elgar Companion To Radical
Political Economy, pages 153–157. Elgar, Brookfield, VT.
Available at Jerome Levy Economics Institute, Bard College. Working Paper No.
74. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract id=161024.
Ncube, N., Ndou, E., and Gumata, N. (2012).
How are the U.S. financial shocks transmitted into South Africa? Structural
VAR evidence.
African Development Bank, WPS No. 157.
Nychka, D., Ellner, S., Gallant, A. R., and McCaffrey, D. (1992).
Finding Chaos in Noisy Systems.
Journal of the Royal Statistical Society. Series B (Methodological),
54(2):399–426.
Park, J. Y. and Whang, Y. (2012).
Random walk or chaos: A formal test on the Lyapunov exponent.
Journal of Econometrics, 169(1):61 – 74.
Recent Advances in Panel Data, Nonlinear and Nonparametric Models: A
Festschrift in Honor of Peter C.B. Phillips.
Polkovnichenko, V. and Zhao, F. (2013).
Probability weighting functions implied in options prices.
Journal of Financial Economics, 107(3):580 – 609.

32. References VI
Prelec, D. (1998).
The probability weighting function.
Econometrica, 60:497–528.
Preston, M. G. and Baretta, P. (1948).
An Experimental Study of the Auction Value of an Uncertain Outcome.
American Journal of Psychology, 61(2):183–193.
Quiggin, J. (1982).
A theory of anticipated utility.
Journal of Economic Behaviour and Organization, 3(4):323–343.
Quiggin, J. (1993).
Generalized Expected Utility Theory: The Rank Dependent Model.
Kluwer Academic Press, Boston, MA.
Schnabl, P. (2012).
The International Transmission of Bank Liquidity Shocks: Evidence from an
Emerging Market.
Journal of Finance, 67(3):897–932.
South African Reserve Bank (2014).
Financial Instability Review.
Technical report, South African Reserve Bank, Pretoria, South Africa.

33. References VII
Tversky, A. and Kahneman, D. (1992).
Advances in Prospect Theory: Cumulative Representation of Uncertainty.
Journal of Risk and Uncertainty, 5:297–323.
Whaley, R. E. (2000).
The Investor Fear Gauge: Explication of the CBOE VIX.
Journal of Portfolio Management, 26(3):12–17.
Whang, Y. and Linton, O. (1999).
The asymptotic distribution of nonparametric estimates of the Lyapunov
exponent for stochastic time series.
Journal of Econometrics, 91(1):1 – 42.
Wiggins, S. (2003).
Introduction to Applied Nonlinear Dynamical Systems and Chaos, volume 2 of
Texts in Applied Mathematics.
Springer, New York, NY, 2nd edition.
Wilcox, W. T. (2011).
A Comparison of Three Probabilistic Models of Binary Discrete Choice Under
Risk.
Work-in-Progress, Economic Science Institute, Chapman University.

34. References VIII
Wolff, R. C. L. (1992).
Local Lyapunov Exponents: Looking Closely at Chaos.
Journal of the Royal Statistical Society. Series B (Methodological),
54(2):353–371.