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)

Why More Focus on Durations Now?

I We need better understanding of information arrival, trade arrival, liquidity, volume, market participant interactions, links to volatility, etc.

I Financial market roots of the Great Recession

I Purely financial-market events like the Flash Crash

I The duration point process is the ultimate process of interest. It determines everything else, yet it remains incompletely understood.

I 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.

For What is “Big Data” Informative?

For trends? No

For volatilities? Yes: realized volatility.

Andersen, Bollerslev, Christoffersen and Diebold (2012)

For durations? Yes: both trivially and subtly. Hautsch (2012)

– Trivially: trade-by-trade data needed for inter-trade durations – Subtly: time deformation links volatilities to durations. So big data informs us about vols which inform us about durations.

Stochastic Volatility Model

rt = σ √ eht · ε_t ht= ρht−1+ ηt εt ∼ iidN(0, 1) ηt ∼ iidN(0, σ2η) εt ⊥ ηt Equivalently, rt|Ωt−1∼ N(0, σ2eht) 4 / 40

From Where Does Stochastic Volatility Come?

Time-deformation model of calendar time (e.g., “daily”) returns:

rt = eht X i =1 ri ht= ρht−1+ ηt

(trade-by-trade returns ri ∼ iidN(0, σ2), daily volume eht)

=⇒ rt|Ωt−1∼ N(0, σ2eht)

– Volume/duration dynamics produce volatility dynamics – Volatility properties inherited from duration properties

What Are the Key Properties of Volatility?

In general:

I Volatility dynamics fatten unconditional distributional tails e.g., rt|Ωt−1∼ N(0, σ2eht) =⇒ rt ∼ “fat-tailed”

I Volatility dynamics are persistent I Volatility dynamics are long memory

Elegant modeling framework that captures all properties:

Calvet and Fischer (2008),

Multifractal Volatility:Theory, Forecasting, and Pricing, Elsevier

Roadmap

I Empirical regularities in durations

I The MSMD model

Twenty-Five U.S. Firms Selected Randomly from S&P 100

I Consolidated trade data extracted from the TAQ database

I 20 days, 2/1/1993 - 2/26/1993, 10:00 - 16:00

I 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.

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.

Persistent Dynamics

0 20 40 2500 5000 7500 10000 12500 15000 17500 20000 22500 Trade Duration

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.

Long Memory

.0 .1 .2 .3 25 50 75 100 125 150 175 200 Displacement (Trades) Autocorrelation

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.

Roadmap

I Empirical regularities in inter-trade durations

!

I The MSMD model

I Empirics

A Dynamic Duration Model

di ∼

i

λi

, i ∼ iidExp(1)

Point Process Foundations

{T_i(ω) > 0}i ∈1,2,... on (Ω, F , P) s.t. 0 < T1(ω) < T2(ω) < · · · N(t, ω) =X i ≥1 1(Ti(ω) ≤ t) λ(t) = lim ∆t↓0  1 ∆tP[N(t + ∆t) − N(t) = 1 | Ft−]  P(t1, . . . , tn|λ(·)) = n Y i =1 λ(ti) exp " − Z ti ti −1 λ(t)dt #! If λ(t) = λi on [ti −1, ti), then: P(t1, . . . , tn|λ(·)) = n Y i =1 (λiexp [−λidi]) di ∼ i λi , i ∼ iidExp(1)

How to parameterize the conditional intensity λi?

Markov Switching Multifractal Durations (MSMD)

λi = λ ¯ k Y k=1 Mk,i λ > 0, Mk,i > 0, ∀k

Independent intensity components Mk,i

are Markov renewal processes:

Mk,i =



draw from f (M) w.p. γk

Mk,i −1 w.p. 1 − γk

Modeling Choices

Simple binomial renewal distribution f (M):

M = 

m0 w.p. 1/2

2 − m0 w.p. 1/2,

where m0 ∈ (0, 2]

Simple renewal probability γk:

γk = 1 − (1 − γ¯_k)b

k−¯k

γ¯_k ∈ (0, 1) and b ∈ (1, ∞)

Renewal Probabilities

k Rene w al Probability , γk 3 4 5 6 7 0.00 0.05 0.10 0.15 0.20

Figure: MSMD Intensity Component Renewal Probabilities. We

show the renewal probabilities (γk= 1 − (1 − γ¯k) bk−¯k

) associated with the latent intensity components (Mk), k = 3, ..., 7. We calibrate the MSMD model

with ¯k = 7, and with remaining parameters that match our subsequently-reported estimates for Citigroup.

All Together Now

di = i λi λi = λ ¯ k Y k=1 Mk,i Mk,i = ( M w.p. 1 − (1 − γ_k¯)b k−¯k Mk,i −1 w.p. (1 − γ_k¯)b k−¯k M =  m0 w.p. 1/2 2 − m0 w.p. 1/2 i ∼ iid exp(1), ¯k ∈ N, λ > 0, γ_k¯ ∈ (0, 1), b ∈ (1, ∞), m0 ∈ (0, 2] parameters θ_k¯ = (λ, γ¯_k, b, m0)0 ¯

k-dimensional state vector Mi = (M1,i, M2,i, . . . M¯_k,i)

2¯k states

MSMD Overdispersion

Figure: QQ Plot, Simulated Durations. N = 22, 988; parameters

MSMD Persistent Dynamics

Figure: Time-series plots of simulated M1,i, ..., M7,i, λi, and di.

N = 22, 988; parameters calibrated to match Citigroup estimates.

MSMD Long Memory

Displacement (Trades) A utocorrelation 0 25 50 75 100 125 150 175 200 0.0 0.1 0.2 0.3

Figure: Sample Autocorrelation Function, Simulated Durations.

MSMD Long Memory

Displacement (Trades) A utocorrelation 0 25 50 75 100 125 150 175 200 0.0 0.1 0.2 0.3

Figure: Sample Autocorrelation Function, Simulated Durations,

Filtered. N = 22, 988; parameters calibrated to match Citigroup estimates,

durations filtered by (1 − L).45.

MSMD Long Memory

The MSMD autocorrelation function satisfies

sup τ ∈I¯k     ln ρ(τ ) ln τ−δ − 1     → 0 as ¯k → ∞ δ = log_bE ( eM2_{) − log} b{[E ( eM)]2} e M =        2m−1₀ m−1₀ +(2−m0)−1 w.p. 1 2 2(2−m0)−1 m−1₀ +(2−m0)−1 w.p. 1 2

Literature I:

Mean Duration vs. Mean Intensity

Mean Duration:

di = ϕii, i ∼ iid (1, σ2)

– ACD: Engle and Russell (1998), ... – MEM: Engle (2002), ... Mean Intensity: di ∼ i λi , i ∼ iidExp(1) – MSMD

– Bauwens and Hautsch (2006) – Bowsher (2006)

Literature II:

Observation- vs. Parameter-Driven Models

Observation-Driven:

Ωt−1 observed (like GARCH)

– ACD

– MEM as typically implemented – GAS

Parameter-Driven:

Ωt−1 latent (like SV)

– MSMD

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)

Literature IV:

Reduced-Form vs. Structural Long Memory

Reduced Form: (1 − L)dyt= vt, vt ∼ short memory – FI-ACD – FI-SCD Structural: yt= v1t+ ... + vNt, vit ∼ short memory – MSMD

Roadmap

I Empirical regularities in inter-trade durations

!

I The MSMD model

!

I Empirics

Twenty-Five U.S. Firms Selected Randomly from S&P 100

I Consolidated trade data extracted from the TAQ database

I 20 days, 2/1/1993 - 2/26/1993, 10:00 - 16:00

I 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

Coefficient of Variation Count 1.0 1.2 1.4 1.6 1.8 2.0 2.2 0 2 4 6 8

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.

Displacement (Trades) A utocorrelation 0 25 50 75 100 125 150 175 200 0.0 0.1 0.2 0.3 0.4

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.

MSMD Likelihood Evaluation (Using ¯

k = 2 for Illustration)

Each Mk, k = 1, 2 is a two-state Markov switching process:

P(γ_k) = 

1 − γk/2 γk/2

γk/2 1 − γk/2



Hence λi is a four-state Markov-switching process:

λi ∈ {λs1s1, λs1s2, λs2s1, λs2s2}

Pλ = P(γ1) ⊗ P(γ2) (by independence of the Mk,i)

Likelihood function: p(d1:n|θ¯_k) = p(d1|θ_k¯) n Y i =2 p(di|d1:i −1, θ¯_k)

Conditional on λi, the duration di is Exp(λi):

p(di|λi) = λie−λidi

Weight by state probabilities obtained by the Hamilton filter.

k Log−Lik elihood Diff erential 3 4 5 6 7 −100 −80 −60 −40 −20 0 20

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.

m0 Count 1.2 1.3 1.4 1.5 0 2 4 6 8 10 12 b Count 0 5 10 15 20 25 30 0 2 4 6 8 10 12 λ Count 1 2 3 4 0 2 4 6 8 10 γk Count 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 14

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.

k Rene w al Probability , γk 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0

Figure: Estimated Intensity Component Renewal Probability Profile

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 CDF P-Value Citi

Figure: Empirical CDF of White Statistic p-Value, ¯k = 7. For

reference we indicate Citigroup.

D = BICMSMD − BICACD Count 0 100 200 300 400 500 600 0 2 4 6 8

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.

1−Step RMSE Difference Count −1.0 −0.5 0.0 0 5 10 15

5−Step RMSE Difference

Count −4.5 −4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 0 2 4 6 8 10 12

20−Step RMSE Difference

Count −4 −3 −2 −1 0 0 2 4 6 8

Figure: Distribution of Differences in Forecast RMSE Across Firms.

We compute differences MSMD(7) - ACD(1,1). For reference we indicate Citigroup.

Roadmap

I Empirical regularities in inter-trade durations

!

I The MSMD model

!

Future Directions

I Additional model assessment

I Current data (using ultra-accurate time stamps)

I Panel of trading months. Structural change?