rational market turbulence kent osband risktick llc 27 march 2012 inquire uk conference
TRANSCRIPT
Rational Market Turbulence
Kent Osband RiskTick LLC
27 March 2012Inquire 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
20th 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 2nd 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 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829300.5
0.6
0.7
0.8
0.9
1.0
Number of Heads in a Row
Pro
babilit
y o
f H
eads
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 ip
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%
2log log ( ) ( ) 2d 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
ii i idp p dW dt
i idp dx dt
dW
Change in Conviction =Conviction x Idiosyncrasy x
Surprise+ Expected Regime Shift
i ji jdp
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
ii i idp 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
ii i idp 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
varvar( )
newsd 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
1volatility nn
cumulantcumulant
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
Faster decay
Time Elapsed Since Ob-servation
Ob
se
rvati
on
Weig
ht
Turbulence?