Rational Market Turbulence

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22 Φεβ 2014 (πριν από 3 χρόνια και 3 μήνες)

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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?