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

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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

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Turbulence?