1 / 34

Extending the VolatilityConcept to Point Processes David R. Brillinger Statistics Department

Extending the VolatilityConcept to Point Processes David R. Brillinger Statistics Department University of California Berkeley, CA brill@stat.berkeley.edu www.stat.berkeley.edu/~brill 2 π 1 ISI2007Lisbon. Introduction

Download Presentation

Extending the VolatilityConcept to Point Processes David R. Brillinger Statistics Department

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Extending the VolatilityConcept to Point Processes David R. Brillinger Statistics Department University of California Berkeley, CA brill@stat.berkeley.edu www.stat.berkeley.edu/~brill 2 π 1 ISI2007Lisbon

  2. Introduction Is there a useful extension of the concept of volatility to point processes? A number of data analyses will be presented

  3. Why bother with an extension to point processes? a) Perhaps will learn more about time series case b) Pps are an interesting data type c) Pps are building blocks d) Volatility often considered risk measure for time series. 27 July Guardian. “Down nearly 60 points at one stage, the FTSE recovered and put on the same amount again. But by the close it had slipped back, down 36.0 points.” “inject billions into the banking system”

  4. Volatility When is something volatile? When values shifting/changing a lot Vague concept Can be formalized in various ways There are empirical formulas as well as models

  5. Merrill Lynch. “Volatility. A measure of the fluctuation in the market price of the underlying security. Mathematically, volatility is the annualized standard deviation of returns. A - Low; B - Medium; and C - High.”

  6. Financial time series. Pt price at time t “Return” data, Yt = (Pt - Pt-1)/Pt-1 Empirical formula Realized volatility mean{[Ys – Ys-1]p | s near t}, p = 1 or 2

  7. Model based formula. GARCH Yt = μt + σtεt, ε zero mean, unit variance, t discrete σt2 = α0 + ∑ αi[Yt-i– μt-i]2 + ∑ βjσt-j2 α’s, β’s > 0 Volatility σt2 For μs, σs smooth mean{[Ys – Ys-1]2 | s near t} ~ σt2(εt - εt-1)2

  8. Crossings based. E{[Y(t) – Y(t-h)] 2} = 2[c(0) – c(h)] ≈ -2c"(0)h2 Recognize as 2c(0) π2 [E(#{crossings of mean})]2 for stationary normal Consider s #{crossings of mean | near t} as volatility measure

  9. Example. Standard & Poors 500. Weighted average of prices of 500 large companies in US stock market Events Great Crash Nov 1929 Asian Flu (Black Monday) Oct 1997 Other “crashes”: 1932, 1937, 1940, 1962, 1974, 1987

  10. Zero crossings

  11. S&P 500: realized volatility, model based, crossing based Tsay (2002) n= 792

  12. Point process case. locations along line: τ1 < τ2 < τ3 < τ4 < … N(t) = #{τj t} Intervals/interarrivals Xj = τj+1 - τj Stochastic point process. Probabilities defined Characteristics: rate, autointensity, covariance density, conditional intensity, … E.g. Poisson, doubly stochastic Poisson

  13. 0-1 valued time series. Zt = 0 or 1 Realized volatility ave{ [Zs – Zs-1]2 | s near t } Connection to zero crossings. Zt= sgn(Yt), {Yt} ordinary t.s. Σ [Zs – Zs-1]2 = #{zero crossings}

  14. Connecting pp and 0-1 series. Algebra Tj = <τj/h> <.> nearest integer, embed in 0’s h small enough so no ties Y(t) = N(t+h) – N(t) = tt+hdN(u) Stationary case E{Y(t)} = pN h cov{Y(t+u),Y(t)} = t+ut+u+h tt+hcov{dN(r),dN(s)} ~ pNδ{u}h + qNN(u)h2 as cov{dN(r),dN(s)} = [pN δ(r-s) + qNN(r-s)]dr ds, rate, pN, covariance density, qNN(∙), Dirac delta, δ(∙)

  15. Parametric models. Bernoulli ARCH. Cox (1970) Prob{Zt = 1|Ht} = πt logit πt = Σ αi Zt-i Ht history before t Fitting, assessment, prediction, … via glm() Bernoulli GARCH. logit πt = Σ αi Zt-i + Σ βj logit πt-j Volatility πt or πt(1 - πt)?

  16. Convento do Carmo

  17. “California” earthquakes magnitude 4, 1969-2003 N=1805

  18. Results of 0-1 analysis

  19. P.p. analysis. Rate as estimate of volatility Considervar{dN(t)} var{N(t)-N(t-h)} ≈ pNh + qNN(0)h2 Estimate of rate at time t. k(t-u)dN(u) / k(t-u)du k(.) kernel Variance, stationary case, k(.) narrow pN k(t-u)2du / [k(t-u)du]2 + qNN(0)

  20. Example. Euro-USA exchange rate

  21. Interval analysis. Xj = τj – τj-1 Also stationary

  22. California earthquakes

  23. Risk analysis. Time series case. Assets Yt and probability p VaR is the p-th quantile Prob{Yt+1 – Yt VaR } = p left tail If approxmate distribution of Yt+1 – Yt by Normal(0,σt) volatility, σt, appears Sometimes predictive model is built and fit to estimate VaR

  24. Point process case. Pulses arriving close together can damage Number of oscillations to break object (Ang & Tang) Suppose all points have the same value (mark), e.g. spike train. Consider VaR of Prob{N(t+u) – N(t) > VaR} = 1-p Righthand tail

  25. Examples. S&P500: p=.05 method of moments quantile VaR = $.0815 CA earthquakes: u = 7 days, p=.95 mom quantile VaR = 28 events

  26. Case with seasonality – US Forest Fires 1970 - 2005 n=8481

  27. Dashed line ~ seasonal

  28. Example. Volatility, simulate Poisson, recover volatility

  29. Conclusion. Returningto the question,“Why bother with extension? a) Perhaps will learn more about time series case b) Pps are an interesting data type c) Pps are building blocks d) Volatility often considered risk measure for time series.” The volatility can be the basic phenomenon

  30. Another question. “Is there a useful extension of the concept of volatility to point processes?” The running rate Gets at local behavior (prediction)

  31. Some references. Bouchard, J-P. and Potters, M. (2000). Theory of Financial Risk and Derivative Pricing. Cambridge Dettling, M. and Buhlmann, P. “Volatility and risk estimation with linear and nonlinear methods based on high frequency data” Tsay, R. S. (2002). Analysis of Financial Time Series. Wiley

More Related