A Markov-Switching Multi-Fractal Inter-Trade Duration Model, with Application to U.S. Equities

A Markov-Switching Multi-Fractal Inter-Trade Duration Model, with Application to U.S. Equities Fei Chen (HUST) Francis X. Diebold (UPenn) Frank Schorfheide (UPenn) April 28, 2014 1 / 39

Big Data Are big data informative for trends? No Are big data informative for volatilities? Yes: realized volatility. For durations? Yes: both trivially and subtly. Trivially: trade-by-trade data needed for inter-trade durations Subtly: time deformation links volatilities to durations. So big data inform us about vols which inform us about durations. 2 / 39

Why More Focus on Durations Now? We need better understanding of information arrival, trade arrival, liquidity, volume, market participant interactions, links to volatility, etc. Financial market roots of the Great Recession Purely financial-market events like the Flash Crash The duration point process is the ultimate process of interest. It determines everything else, yet it remains incompletely understood. Long memory is clearly present in calendar-time volatility and is presumably inherited from conditional intensity of arrivals in transactions time, yet there is little long-memory duration literature. 3 / 39

Stochastic Volatility Model r t = σ e ht ε t h t = ρh t 1 + η t ε t iidn(0, 1) η t iidn(0, ση) 2 ε t η t Equivalently, r t Ω t 1 N(0, σ 2 e ht ) 4 / 39

From Where Does Stochastic Volatility Come? Time-deformation model of calendar time (e.g., daily ) returns: e h t r t = i=1 r i h t = ρh t 1 + η t (trade-by-trade returns r i iidn(0, σ 2 ), daily volume e ht ) = r t Ω t 1 N(0, σ 2 e ht ) Volume/duration dynamics produce volatility dynamics Volatility properties inherited from duration properties 5 / 39

What Are the Key Properties of Volatility? In general: Volatility dynamics fatten unconditional distributional tails e.g., r t Ω t 1 N(0, σ 2 e ht ) = r t fat-tailed Volatility dynamics are persistent Volatility dynamics are long memory Elegant modeling framework that captures all properties: Calvet and Fischer (2008), Multifractal Volatility:Theory, Forecasting, and Pricing, Elsevier 6 / 39

Roadmap Empirical regularities in durations The MSMD model Preliminary empirics 7 / 39

Twenty-Five U.S. Firms Selected Randomly from S&P 100 Consolidated trade data extracted from the TAQ database 20 days, 2/1/1993-2/26/1993, 10:00-16:00 09:30-10:00 excluded to eliminate opening effects Symbol Company Name Symbol Company Name AA ALCOA ABT Abbott Laboratories AXP American Express BA Boeing BAC Bank of America C Citigroup CSCO Cisco Systems DELL Dell DOW Dow Chemical F Ford Motor GE General Electric HD Home Depot IBM IBM INTC Intel JNJ Johnson & Johnson KO Coca-Cola MCD McDonald s MRK Merck MSFT Microsoft QCOM Qualcomm T AT&T TXN Texas Instruments WFC Wells Fargo WMT Wal-Mart XRX Xerox Table: Stock ticker symbols and company names. 8 / 39

Overdispersion Figure: Citigroup Duration Distribution. We show an exponential QQ plot for Citigroup inter-trade durations between 10:00am and 4:00pm during February 1993, adjusted for calendar effects. 9 / 39

Persistent Dynamics 40 Duration 20 0 2500 5000 7500 10000 12500 15000 17500 20000 22500 Trade Figure: Citigroup Duration Time Series. We show a time-series plot of inter-trade durations between 10:00am and 4:00pm during February 1993, measured in minutes and adjusted for calendar effects. 10 / 39

Long Memory Autocorrelation.3.2.1.0 25 50 75 100 125 150 175 200 Displacement (Trades) Figure: Citigroup Duration Autocorrelations. We show the sample autocorrelation function of Citigroup inter-trade durations between 10:00am and 4:00pm during February 1993, adjusted for calendar effects. 11 / 39

Roadmap Empirical regularities in inter-trade durations! The MSMD model Empirics 12 / 39

A Dynamic Duration Model d i ɛ i λ i, ɛ i iidexp(1) Mixture of Exponentials Representation 13 / 39

Point Process Foundations {T i (ω) > 0} i 1,2,... on (Ω, F, P) s.t. 0 < T 1 (ω) < T 2 (ω) < N(t, ω) = 1(T i (ω) t) i 1 ( ) 1 λ(t) = lim t 0 t P[N(t + t) N(t) = 1 F t ] ( [ n ]) ti P(t 1,..., t n λ( )) = λ(t i ) exp λ(t)dt t i 1 i=1 If λ(t) = λ i on [t i 1, t i ), then: n P(t 1,..., t n λ( )) = (λ i exp [ λ i d i ]) i=1 d i ɛ i, ɛ i iidexp(1) λ i How to parameterize the conditional intensity λ i? 14 / 39

Markov Switching Multifractal Durations (MSMD) λ i = λ k M k,i k=1 λ > 0, M k,i > 0, k Independent intensity components M k,i are Markov renewal processes: { draw from f (M) w.p. γk M k,i = M k,i 1 w.p. 1 γ k f (M) is identical k, with M > 0 and E(M) = 1 15 / 39

Modeling Choices Simple binomial renewal distribution f (M): { m0 w.p. 1/2 M = 2 m 0 w.p. 1/2, where m 0 (0, 2] Simple renewal probability γ k : γ k = 1 (1 γ k) bk k γ k (0, 1) and b (1, ) 16 / 39

Renewal Probabilities Renewal Probability, γ k 0.00 0.05 0.10 0.15 0.20 3 4 5 6 7 k Figure: MSMD Intensity Component Renewal Probabilities. We show the renewal probabilities (γ k = 1 (1 γ k) bk k ) associated with the latent intensity components (M k ), k = 3,..., 7. We calibrate the MSMD model with k = 7, and with remaining parameters that match our subsequently-reported estimates for Citigroup. 17 / 39

All Together Now M k,i = { λ i = λ d i = ɛ i λ i k M k,i k=1 M w.p. 1 (1 γ k )bk k M k,i 1 w.p. (1 γ k) bk k M = { m0 w.p. 1/2 2 m 0 w.p. 1/2 ɛ i iid exp(1), k N, λ > 0, γ k (0, 1), b (1, ), m 0 (0, 2] parameters θ k = (λ, γ k, b, m 0) k-dimensional state vector M i = (M 1,i, M 2,i,... M k,i ) 2 k states 18 / 39

MSMD Overdispersion Figure: QQ Plot, Simulated Durations. N = 22, 988; parameters calibrated to match Citigroup estimates. 19 / 39

MSMD Persistent Dynamics Figure: Time-series plots of simulated M 1,i,..., M 7,i, λ i, and d i. N = 22, 988; parameters calibrated to match Citigroup estimates. 20 / 39

MSMD Long Memory Autocorrelation 0.0 0.1 0.2 0.3 0 25 50 75 100 125 150 175 200 Displacement (Trades) Figure: Sample Autocorrelation Function, Simulated Durations. N = 22, 988; parameters calibrated to match Citigroup estimates. 21 / 39

MSMD Long Memory The MSMD autocorrelation function satisfies ln ρ(τ) ln τ δ 1 0 as k sup τ I k δ = log b E( M 2 ) log b {[E( M)] 2 } M = 2m 1 0 m 1 0 +(2 m 0 ) 1 w.p. 1 2 2(2 m 0 ) 1 m 1 0 +(2 m 0 ) 1 w.p. 1 2 22 / 39

Literature I: Mean Duration vs. Mean Intensity Mean Duration: d i = ϕ i ɛ i, ɛ i iid(1, σ 2 ) ACD: Engle and Russell (1998),... MEM: Engle (2002),... d i ɛ i λ i, Mean Intensity: ɛ i iidexp(1) MSMD Bauwens and Hautsch (2006) Bowsher (2006) 23 / 39

Literature II: Observation- vs. Parameter-Driven Models Observation-Driven: Ω t 1 observed (like GARCH) ACD MEM as typically implemented GAS Parameter-Driven: Ω t 1 latent (like SV) MSMD SCD: Bauwens and Veredas (2004) 24 / 39

Literature III: Short Memory vs. Long Memory Short Memory: Quick (exponential) duration autocorrelation decay ACD as typically implemented MEM as typically implemented Long-Memory: Slow (hyperbolic) duration autocorrelation decay MSMD FI-ACD: Jasiak (1999) FI-SCD: Deo, Hsieh and Hurvich (2010) 25 / 39

Literature IV: Reduced-Form vs. Structural Long Memory Reduced Form: (1 L) d y t = v t, v t short memory FI-ACD FI-SCD Structural: y t = v 1t +... + v Nt, v it short memory MSMD 26 / 39

Roadmap Empirical regularities in inter-trade durations! The MSMD model! Empirics 27 / 39

Twenty-Five U.S. Firms Selected Randomly from S&P 100 Consolidated trade data extracted from the TAQ database 20 days, 2/1/1993-2/26/1993, 10:00-16:00 09:30-10:00 excluded to eliminate opening effects Symbol Company Name Symbol Company Name AA ALCOA ABT Abbott Laboratories AXP American Express BA Boeing BAC Bank of America C Citigroup CSCO Cisco Systems DELL Dell DOW Dow Chemical F Ford Motor GE General Electric HD Home Depot IBM IBM INTC Intel JNJ Johnson & Johnson KO Coca-Cola MCD McDonald s MRK Merck MSFT Microsoft QCOM Qualcomm T AT&T TXN Texas Instruments WFC Wells Fargo WMT Wal-Mart XRX Xerox Table: Stock ticker symbols and company names. 28 / 39

Count 0 2 4 6 8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Coefficient of Variation Figure: Distribution of Duration Coefficients of Variation Across Firms. We show a histogram of coefficients of variation (the standard deviation relative to the mean), as a measure of overdispersion relative to the exponential. For reference we indicate Citigroup. 29 / 39

Autocorrelation 0.0 0.1 0.2 0.3 0.4 0 25 50 75 100 125 150 175 200 Displacement (Trades) Figure: Duration Autocorrelation Function Profile Bundle. For each firm, we show the sample autocorrelation function of inter-trade durations between 10:00am and 4:00pm during February 1993, adjusted for calendar effects. For reference we show Citigroup in bold. 30 / 39

MSMD Likelihood Evaluation (Using k = 2 for Illustration) Each M k, k = 1, 2 is a two-state Markov switching process: [ ] 1 γk /2 γ P(γ k ) = k /2 γ k /2 1 γ k /2 Hence λ i is a four-state Markov-switching process: λ i {λs 1 s 1, λs 1 s 2, λs 2 s 1, λs 2 s 2 } P λ = P(γ 1 ) P(γ 2 ) (by independence of the M k,i ) Likelihood function: n p(d 1:n θ k ) = p(d 1 θ k ) p(d i d 1:i 1, θ k ) i=2 Conditional on λ i, the duration d i is Exp(λ i ): p(d i λ i ) = λ i e λ i d i Weight by state probabilities obtained by the Hamilton filter. 31 / 39

Log Likelihood Differential 100 80 60 40 20 0 20 3 4 5 6 7 k Figure: Maximized Log Likelihood Profile Bundle. We show likelihood profiles for all firms as a function of k, in deviations from the value for k = 7, which is therefore identically equal to 0. For reference we show Citigroup in bold. 32 / 39

Count 0 2 4 6 8 10 12 Count 0 2 4 6 8 10 12 Count 0 2 4 6 8 10 Count 0 2 4 6 8 10 12 14 1.2 1.3 1.4 1.5 0 5 10 15 20 25 30 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 m 0 b λ γ k Figure: Distributions of MSMD Parameter Estimates Across Firms, k = 7. We show histograms of maximum likelihood parameter estimates across firms, obtained using k = 7. For reference we indicate Citigroup. 33 / 39

Renewal Probability, γ k 0.0 0.2 0.4 0.6 0.8 1.0 3 4 5 6 7 Figure: Estimated Intensity Component Renewal Probability Profile Bundle, k = 7. For reference we show Citigroup in bold. k 34 / 39

1.0 0.8 0.6 Citi CDF 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 P-Value Figure: Empirical CDF of White Statistic p-value, k = 7. For reference we indicate Citigroup. 35 / 39

Count 0 2 4 6 8 0 100 200 300 400 500 600 D = BIC MSMD BIC ACD Figure: Distribution of Differences in BIC Values Across Firms. We use BIC/2 = ln L k ln(n)/2, and we compute differences as MSMD(7) - ACD(1,1). We show a histogram. For reference we indicate Citigroup. 36 / 39

Count 0 5 10 15 Count 0 2 4 6 8 10 12 Count 0 2 4 6 8 1.0 0.5 0.0 1 Step RMSE Difference 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 5 Step RMSE Difference 4 3 2 1 0 20 Step RMSE Difference Figure: Distribution of Differences in Forecast RMSE Across Firms. We compute differences MSMD(7) - ACD(1,1). For reference we indicate Citigroup. 37 / 39

Roadmap Empirical regularities in inter-trade durations! The MSMD model! Empirics! 38 / 39

Future Directions Additional model assessment Current data (using ultra-accurate time stamps) Panel of trading months. Structural change? 39 / 39