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Rational Market Turbulence Kent Osband RiskTick LLC 27 March 2012 Inquire UK Conference

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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

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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?

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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?

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Recognizing Turbulence

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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

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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

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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

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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

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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

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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?

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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

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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

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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

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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)

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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

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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?

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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?

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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 Markets can’t know parameters without observation

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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

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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%

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Pandora’s Equation where is expected drift given beliefs is standard Brownian motion given beliefs is expected net inflow from regime switching Change in Conviction = Conviction x Idiosyncrasy x Surprise + Expected Regime Shift

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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

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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

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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

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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

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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?

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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?

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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

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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

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Turbulence?

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