Rational Market Turbulence
Kent Osband
RiskTick
LLC
27 March 2012
Inquire UK Conference
Rational Market Turbulence
Financial markets analogous to fluids
Both adjust to their containers, but rarely adjust smoothly
Common driver explains both smoothness and turbulence
Rational learning breeds market turbulence
Volatility of each cumulant of beliefs depends on cumulant one
order higher, so computable solutions are rare
Disagreements fade given stability but flare up under sharp
regime change
Profound implications
No
deus
ex
machina
needed to explain heterogeneity of beliefs
Financial system must withstand turbulence
Outline
I.
How has physics explained turbulence in fluids?
II.
How has economics explained turbulence in markets?
III.
Why does rational learning breed turbulence?
IV.
What can we learn from turbulence?
Outline
I.
How has physics explained turbulence in fluids?
II.
How has economics explained
turbulence in markets?
III.
Why does rational learning breed turbulence?
IV.
What can we learn from turbulence?
Recognizing Turbulence
Brief History of Turbulence
Fluids are materials that conform to their containers
Liquids, gases, and plasmas are fluids; some solids are semi

fluid
Gradients of response depending on viscosity (internal friction)
Fluids can adjust shape smoothly but rarely do
“Laminar” = smooth flows
“Turbulent” = messy flows
Sharp contrast suggests different drivers
Ancients attributed turbulence to deities
Poseidon’s wild moods drove the seas
Various gods of the winds
Turbulence still associated with divine wrath
Brief Analysis of Turbulence
Turbulence considered mysterious well into 20
th
century
Feynman: Turbulence “the most important unsolved problem
of classical physics”
Lamb (1932): “[W]hen I die and go to heaven, there are two
matters on which I hope for enlightenment. One is quantum
electrodynamics, and the other is the turbulent motion of
fluids. And about the former I am rather optimistic.”
Modern view traces all flows to
Navier

Stokes equation
(Newton’s 2
nd
law applied to fluids)
Videos of supercomputer simulations key to persuasion
Analytic connection involves a moment/cumulant hierarchy
Moment/Cumulant Hierarchy
Adjustment of each moment of the particle distribution
depends on moment one order higher
McComb
,
Physics of Fluid Turbulence
: “[C]losing the moment
hierarchy … is the underlying problem of turbulence theory”
Common to
Navier

Stokes, Fokker

Planck equation for
diffusion, and BBGKY equations for large numbers of particles
Often expressed more neatly as cumulant hierarchy
Cumulants are Taylor coefficients of log characteristic function,
which add up for sums of independent random variables
Mean, variance, skewness, kurtosis = (standardized) cumulants
No end to non

zero cumulants unless distribution is Gaussian
Hierarchy explains both laminar flow and turbulence
Key determinant is Reynolds ratio of velocity to viscosity
Implications of Turbulence
Limited predictability
Neighboring particles can behave
very differently
Dynamics can magnify
importance of small outliers
Forecasts decay rapidly with
space and time
Track with high

powered
computing to adjust short term
Need to build in extra
robustness
Turbulence Isn’t All Bad
Accelerates mixing
Much faster than diffusion
Crucial to efficient combustion in
gasoline

powered engine
Amplifying or reducing drag
changes impact
Dimpling a golf ball increases
turbulence yet more than
doubles flight
Major practical challenge for
engineers
Outline
I.
How has physics explained turbulence in fluids?
II.
How has economics explained turbulence in
markets?
III.
Why does rational learning breed turbulence?
IV.
What can we learn from turbulence?
Two Faces of Market Adjustment
Financial markets adjust to capital

weighted forecasts
Prices as net present values discounted for time and risk
Local martingales (fair games) as equilibria
Financial markets rarely adjust smoothly
Seem driven by “animal spirits” or “irrational exuberance”
Price behavior looks “turbulent” (Mandelbrot,
Taleb
)
How can we make sense of this?
Focus on long

term adjustment (orthodox finance)
Focus on human quirks (behavioral finance)
“As long as it makes dollars, who cares if it makes sense?”
Focus on uncertainty and disagreement
Honored
Views on Turbulence
Orthodox theory looks ahead to calm water and emphasizes
that turbulence fades
Behavioral finance looks behind to white water and
emphasizes that turbulence re

emerges
Nobel
prizes awarded in each
field!
Unsolved: How do rational
and irrational
coexist long

term?
Rational
Water
Irrationally
Exuberant
Water
Uncertain Explanations
Knight and Keynes highlighted uncertainty
Uncertainty is “
unmeasurable
” (Knight) risk with “no scientific
basis on which to form any calculable probability” (Keynes)
Knight: Accounts for “divergence between actual and
theoretical computation” of anticipated profit [risk premium]
Keynes: Fluctuating animal spirits drive economic cycles
Shortcomings
Denial of quantification, although more qualified than it appears
No clear linkage between uncertainty and observed risk
“Rational expectations” revolution sidelined this approach
Subsumed uncertainty under risk
Unexpected Doubts
Many puzzles that rational expectations can’t explain
Risk premium too high, markets too volatile, etc.
GARCH behavior not linked to financial valuation
Breeds behaviorist reaction
Kurz
and rational beliefs
Rational expectations presumes underlying process is known
Rational beliefs weakens that to consistency with evidence
Resolves host of puzzles but hasn’t gained broad traction
Growing literature on financial learning
Explores reactions to Markov switching processes with known
parameters though unknown regime (David,
Veronesi
)
Importance of small doubts (
Barro
, Martin)
Agreement on Disagreement
Empirical importance of uncertainty and disagreement
Rich literature relating asset returns to VIX and variance risk
premium on equities to disagreement over fundamentals
Mueller,
Vedolin
and Yen (2011) extend to bonds
Theorists’ growing emphasis on heterogeneity of beliefs
Hansen (2007, 2010), Sargent (2008) and
Stiglitz
(2010) have
each bashed models based on single representative agent
Great puzzle: Why doesn’t Bayes’ Law homogenize beliefs?
Various theories on how heterogeneity can regenerate
Everlasting fountain of wrong

headedness
Different info sources or multiple equilibria
Rational equilibrium not achievable
Outline
I.
How has physics explained turbulence in fluids?
II.
How has economics explained turbulence?
III.
Why does rational learning breed turbulence?
IV.
What can we learn from turbulence?
Ebb and Flow of Uncertainty
In basic Bayesian analysis, disagreement fades over time
However, this presumes a stable risk regime
In finance, God sometimes changes dice without telling us
Disagreements soar following abrupt regime shift
How many tails in row before relaxing assumption of fair coin?
How to reassess probability of tails after?
0.5
0.6
0.7
0.8
0.9
1.0
0
5
10
15
20
25
30
Probability of Heads
Number of Heads in a Row
Fundamentals of Financial Uncertainty
Brownian motion is main foundation for finance modeling
Displacement = drift + noise
Drift and variance of noise assumed linear in time
Dilemmas of measurement
Observations from different assets or times may not be
relevant to current motion
Observations over short period can identify
vol
but not drift
dx dt dz
Markets can’t know parameters
without observation
Quantifying Uncertainty
Core motion is Brownian or Poisson but …
Multiple possible drifts, and drifts can change without warning
Inferences from observation are rational and efficient
Model as
Multiple regimes with various drifts or
default
rates
Markov switching for drift at rates
Uncertainty as probabilistic beliefs over regimes
Bayesian updating of beliefs using latest evidence
dx
Reinterpretation of fair asset price
No single
fair
price, but a probabilistic cloud of fair prices, each
conditional on a believed set of future risks
Asset prices weight the cloud by current convictions
i
ij
i
p
Simplest Example
Posit two Brownian regimes with negligible switching
rates, equal volatility
and
opposite drifts
For beliefs
p
and observation density
f
, Bayes’ Rule implies
New evidence never changes differences in perceived log odds
but differences in
p
can diverge before they converge
If you start with
p
+
=10

6
, I start with p
+
=10

9
, and drift is
positive, then someday your
p
+
>95% while my p
+
<5%
2
log log ( ) ( ) 2
d p p f dx f dx dx
Pandora’s Equation
where
is expected drift given beliefs
is standard Brownian motion given beliefs
is expected net inflow from regime switching
i
i i i
dp p dW dt
i i
dp
dx dt
dW
Change in Conviction =
Conviction x Idiosyncrasy x Surprise
+ Expected Regime Shift
i ji j
dp
Pandora’s Equation Treasures
Core equation of learning, analogous to
Navier

Stokes
Discovered by
Wonham
(1964) and
Liptser
and
Shirayev
(1974)
Applies with reinterpretation to jump (default) processes too
Most popular machine

learning rules are special cases
Exponentially Weighted
Average: Beliefs always Gaussian with
constant variance
Kalman
Filter:
Gaussian
with changing variance
Normalized Least Squares:
Gaussian about regression beta
Sigmoid:
Beliefs beta

distributed between two extremes
i
i i i
dp p dW dt
Pandora’s Equation Troubles
Need to update continuum of probabilities every instant
Hard to identify regime switching
parameters
Even in simple two

regime model, discrete
approximations can cause significant errors
Best hope is to transform to a countable and hopefully
finite set of moments or cumulants
i
i i i
dp p dW dt
Laws of Learning
Change in mean belief is roughly proportional to variance
Same news affects markets more when we’re uncertain
Wisdom of the hive hinges on robust differences
Dangers of groupthink
Analogy to Fisher’s Fundamental Theorem of evolution
Mean fitness adjusts proportionally to variance
Static fitness can conflict with adaptability
Variance changes
with skewness
Explains GARCH behavior
var
var( )
news
d beliefs d regime
news
The Uncertainty of Uncertainty
Good news: Cumulant expansion
yields simple recursive formula above
Slight modifications for Poisson jumps
Bad news: Recursion moves in wrong
direction!
Errors in estimating a higher cumulant
percolate down below
Outliers can have nontrivial impact on
central values
1
volatility
n
n
cumulant
cumulant
Smooth or Turbulent Adjustment
Cumulant hierarchy predicts both types of behavior
When regime is stable, higher cumulants eventually fade
Given sufficient evidence of abrupt change, disagreements will
flare up with highly volatile volatility
Might here be counterpart to Reynolds number?
Cumulant hierarchy explains heterogeneity of beliefs
Miniscule differences in observation or assessment of
relevance can flare into huge disagreements
In practice no one can be perfectly rational or fall short in
exactly the same way
To what extent does a market of varied believers resemble a
single analyst with varied beliefs?
Outline
I.
How has physics explained turbulence in fluids?
II.
How has economics explained turbulence in markets?
III.
Why does rational learning breed turbulence?
IV.
What can we learn from turbulence?
Lessons from Financial Turbulence
We’ll always seem wildly moody
Don’t need to justify heterogeneity; it comes for free
Orthodox/behaviorist rift founded on false dichotomy
Financial markets will always be hard to predict
Forecast quality decays rapidly with horizon, like the weather,
although better math and computing can help
Justifies additional risk premium
Financial institutions need to withstand turbulence
Can’t regulate turbulence away
Systemic risks have highly non

Gaussian tails
Turbulence Can Breed Confidence
Memory as fading weights
over past experience
Fast decay speeds adaptation
Slow decay stabilizes
Turbulence is key to quick
recovery after crisis
Encourages
short

term focus
Short

term
focus is only way
to renew confidence
quickly
“This time must seem
different” to restart lending
Observation Weight
Time Elapsed Since Observation
Faster decay
Slower decay
Turbulence?
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