High-frequency data modelling using Hawkes processes

Transcription

1 High-frequency data modelling using Hawkes processes Valérie Chavez-Demoulin 1 joint work J.A McGill 1 Faculty of Business and Economics, University of Lausanne, Switzerland Boulder, April 2016 Boulder, April / 25

2 High frequency financial data The availability of high frequency data on transactions, quotes and order flow has revolutionized data processing and has led to statistical modeling challenges At the same time, the frequency of submission of orders has increased and the time to execution of market orders has dropped from more than 25 milliseconds in 2000 to less than a millisecond in Average number Number of price of orders in 10s changes (June 26, 2008) Citigroup General Electric General Motors Table: Taken from R.Cont, 2011, Boulder, April / 25

3 Trade database structure SYMBOL DATE TIME PRICE SIZE G127 CORR COND NVDA :30: NVDA :30: NVDA :30: NVDA :30: Price :30:03 10:15:19 11:35:30 13:54:00 15:02:55 15:45:54 Presence of erroneous trades: moving average filtering Irregularly spaced in time: linear interpolation Boulder, April / 25

4 NVDA negative log-returns from 2006 to 2008 (15-minutes linearly interpolated) Negative log returns Time(15min intervals) NVDA, July to December 2008 negative returns exceedances : Negative log returns Time(15min intervals) Boulder, April / 25

5 Extreme Value Theory: Classical Peaks-Over-Threshold (iid case) x Time The number of exceedances over u has a Poisson distribution with mean λ Conditional on n exceedances, their sizes W j = Z j u are a random sample of size n from the generalized Pareto distribution (GPD) G ξ,σ (w) = { 1 (1 + ξw/σ) 1/ξ +, ξ 0, 1 exp( w/σ), ξ = 0. Boulder, April / 25

6 NVDA log-returns from 2006 to 2008 (15-minutes interpolated) time differences between exceedances over high threshold Difference in time Exponential does not seem exponentially distributed higher than expected frequencies for short distance between extremes Boulder, April / 25

7 High-frequency financial data features Returns not iid Absolute returns highly correlated Volatility appears to change randomly with time Returns are leptokurtic or heavy tailed Extremes appear in clusters Boulder, April / 25

8 Declustering methods exist: An example for daily returns Original data Returns in % Declustered data Returns in % Boulder, April / 25

9 Idea the classical Peaks-Over-Thresholds (POT) model of EVT assumes an homogeneous Poisson process the inter-event times would be independent exponential random variables the Figures above contradict the classical model Aim: Treats the negative log-returns above a high threshold u as a realization of a marked point process: Using a first model that combines a self-exciting process for the threshold exceedance times with Generalized Pareto model for the threshold excess sizes (mark sizes) Using a second model that considers a self-exciting POT model Boulder, April / 25

10 Overall loglikelihood (POT method) Since number and mark size for the threshold exceedances are assumed independent n l(λ, σ, ξ) = log P(N u = n) g ξ,σ (w j ) j=1. = n log λ λ n log σ (1 + 1/ξ) n log(1 + ξw j /σ) + so inference can be performed separately for the frequency of exceedances and their sizes. j=1 Boulder, April / 25

11 Marked Point Processes (1) consider an event process (T i, W i ), i Z observed over the period (0; t 0 ] gives data (t 1, w 1 ),..., (t n, w n ) let H t, entire history of process up to time t the joint density of the data seen over (0, t 0 ] can be written n f Ti,W i H ti 1 (t i, w i H ti 1 )P(T n+1 > t 0 H tn ) i=1 Boulder, April / 25

12 Marked Point Processes (2) Assuming independence of times T i and marks W i the conditional density function is f Ti,W i H ti 1 (t i, w i H ti 1 ) = f Ti H ti 1 (t i H ti 1 ) f Wi H ti 1 (w i H ti 1 ) leading to the log-likelihood l = { n log f Ti H i 1 (t i ) + log P(T n+1 > t 0 H tn ) i=1 { n + log f Wi H ti 1 (w i H ti 1 ) i=1 } B } A Boulder, April / 25

13 Hawkes process Combining separately a Hawkes process for the counting process of the times (part A) and a GPD model for the contributions from the marks (part B), the likelihood becomes { n ( t0 ) } l = log λ(t i H ti ) exp λ(u H u )du i=1 0 n n log σ (1 + 1/ξ) log(1 + ξw i /σ) i=1 with λ(t H t ) being the conditional intensity Boulder, April / 25

14 Self-exciting process We chose the following general form with a constant loading µ 0 λ(t H t ) = µ 0 + ω(t t j ; w j ), j:t j <t where µ 0 > 0 and the self-exciting function ω(s) has to be positive when s 0 and 0 elsewhere. For example, the conditional intensity model proposed by Ogata (1988) is λ(t H t ) = µ 0 + ψ where µ 0, ψ, δ, γ, ρ > 0. j:t j <t e δw j (t t j + γ) 1+ρ ETAS Boulder, April / 25

15 Applications to high-frequency data: 15-minutes linearly interpolated NVDA 2008 negative log-returns Negative log returns Time(15min intervals) Estimated intensity Boulder, April / 25

16 Estimated cumulative events number ˆΛ Ht (t) = t 0 ˆλ(u H u )du and two-sided 95% and 99% confidence limits based on the Kolmogorov-Smirnov statistic Cumulative number of extreme losses Transformed time Boulder, April / 25

17 Alternative models (1) Hawkes process with exponential decay λ(t H t ) = µ 0 + ψ exp{δw j γ(t t j )}. j:t j <t First order Markov chain for the marks W i W i 1 = w i GPD(ξ, σ i = exp(a + bw i 1 )) whose parameters a, b and ξ can be estimated by maximising n f Wi W i 1 (w i w i 1 )f w1 (w 1 ) i=2 Diagnostic for the marginal distribution based on the residuals R j = ˆξ 1 log (1 + ˆξw j /ˆσ j ), j = 1,..., n approximately independent unit exponential variables Boulder, April / 25

18 Alternative models (2): A self-exciting POT model A model with predictable marks λ (t, w) = τ + α ( 1ν(t) w u 1 + ξ σ + α 2 ν(t) σ + α 2 ν(t) ) 1/ξ 1 Where ν(t) = j:t j <t ω(t t j ) uses a self-exciting function ω. This uses the two-dimensional re-parametrisation of the POT: Homogeneous Poisson process for the exceedances of the level u with intensity τ = ln H(u; µ, β, ξ) = {1 + ξ(u µ)/β} 1/ξ +, where H(y; µ, β, ξ) is the GEV distribution and σ = β + ξ(u µ) Inhomogeneous Poisson process for their sizes with intensity λ(t, w) = λ(w) = 1 σ ( 1 + ξ w u ) 1/ξ 1 σ Boulder, April / 25

19 Estimated intensity of exceeding the threshold from self-exciting POT model for NVDA with Ogata function Marks amplitude in percent Time (15 min intervals) Intensity Time (15 min intervals) mean of the GPD Time (15 min intervals) Boulder, April / 25

20 Conditional risk measures The conditional approach models the intraday clustering of extremes and is used to calculate instantaneous conditional p =95% or 99% VaR over the next period z t+1 p = F 1 Z t+1 H (p), t F Zt+1 H (z) = P(Z t+1 > z H t ) which is t P(Z t+1 u > z Z t+1 > u, H t ) P(Z t+1 > u H t ). P(Z t+1 > u H t ) represents the conditional probability of an event in (t, t + 1) that is: ( t+1 ) 1 P{N(t, t + 1) = 0 H t } = 1 exp λ(u H u )du P(Z T +1 u > z Z t+1 > u, H t ) is estimated using the GPD model with parameters σ and ξ. t Boulder, April / 25

21 Conditional VaR and Expected Shortfall The VaR z t+1 p is then the solution of the equation P(Z T +1 > z t+1 p H t ) = 1 p, Using the self-exciting POT model the condition VaR is z t+1 p = u + σ + α 2ν(t+) ξ { ( ) 1 p ξ 1} τ + α 1 ν(t+) From the estimated conditional VaR it is straightforward to get an estimation of the conditional Expected Shortfall (ES) ES t+1 p = zt+1 p 1 ξ + σ ξu 1 ξ. with parameters replaced by their maximum likelihood estimates Boulder, April / 25

22 15-minutes 95%VaR (red) and 95%ES (blue) estimates for NVDA negative log-returns using self-exciting POT models and 95%VaR using classical POT (green) Negative log returns Times (15 min intervals) Boulder, April / 25

23 15-minutes 95%VaR and 95%ES estimates for NVDA negative log-returns using a competitor model (Chavez-Demoulin et al. (2014)) Negative log returns Time Boulder, April / 25

24 Backtesting results 0.95 Quantile Expected 25 POT 59 (0) ETAS ρ = 0 21 (0.24) ETAS ρ 0 22 (0.31) ETAS ρ = 0 24 (0.47) ETAS ρ 0 23 (0.39) exp 25 (0.55) NPOT 26 (0.39) 0.99 Quantile Expected 5 POT 14 (0) ETAS ρ = 0 6 (0.39) exp 6 (0.39) ETAS ρ = 0 6 (0.39) exp 7 (0.24) NPOT 9 (0.11) Boulder, April / 25

25 References A.G. Hawkes, 1971, Point spectra of some mutually exciting point processes, Biometrika. Y. Ogata, 1988, Statistical models for earthquake Occurrences and residuals analysis for point processes, JASA. E. Errais, K. Giesecke and L.R. Goldberg, 2010, Affine point processes and portfolio credit risk, SIAM J. Financial Math. Y. AIt-Sahalia, J. Cacho-Diaz, and R. J. A. Laeven, 2010, Modeling financial contagion using mutually exciting jump processes, The Review of Financial Studies. V. Chavez-Demoulin, A.C. Davison, A.J. McNeil, 2005, A point process approach to Value-at-Risk estimation, Quantitative Finance. V. Chavez-Demoulin, P. Embrechts, S.Sardy, 2014 Extreme-quantile tracking for financial time series Journal of Econometrics. Boulder, April / 25