Rational Market Turbulence

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

2. 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 machinaneeded to explain heterogeneity of beliefs • Financial system must withstand turbulence

3. Outline • How has physics explained turbulence in fluids? • How has economics explained turbulence in markets? • Why does rational learning breed turbulence? • What can we learn from turbulence?

4. Outline • How has physics explained turbulence in fluids? • How has economics explained turbulence in markets? • Why does rational learning breed turbulence? • What can we learn from turbulence?

5. Recognizing Turbulence

6. 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

7. 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

8. 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

9. 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

10. 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

11. Outline • How has physics explained turbulence in fluids? • How has economics explained turbulence in markets? • Why does rational learning breed turbulence? • What can we learn from turbulence?

12. 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

13. 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? Irrationally Exuberant Water Rational Water

14. 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

15. 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)

16. 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

17. Outline • How has physics explained turbulence in fluids? • How has economics explained turbulence? • Why does rational learning breed turbulence? • What can we learn from turbulence?

18. 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?

19. 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

20. 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

21. Simplest Example • Posit two Brownian regimes with negligible switching rates, equal volatility andopposite 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%

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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?

28. Outline • How has physics explained turbulence in fluids? • How has economics explained turbulence in markets? • Why does rational learning breed turbulence? • What can we learn from turbulence?

29. 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

30. 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

31. Turbulence?