Modeling Ultra-High-Frequency Multivariate Financial Data by Monte Carlo Simulation Methods

Outline Modeling Ultra-High-Frequency Multivariate Financial Data by Monte Carlo Simulation Methods Ph.D. Student: Supervisor: Marco Minozzo Dipartimento di Scienze Economiche Università degli Studi di Verona Ph.D. Student Presentation Verona, May 2

Outline Outline Ultra-High-Frequency Data (UHF) 2 Marked Doubly Stochastic Poisson Processes (DSPP) Financial Models For Equally spaced data Marked Point Process (MPP) and UHF Data Literature review for MPP Marked DSPP A DSPP model with a common intensity 3 Modeling framework for logreturns Logreturn Models Simulation studies Brief Conclusion 4 Filtering by RJMCMC Markov Chain Monte Carlo Reversible jump MCMC

.UHF 2.Model 3.Return 4.Filtering Motivation Interest in understanding financial markets, which are the source of UHF data. Most of the econometric literatures is dealing with regularly spaced data (5-min; 5-min; daily). Evidence of strong co-movements in individual trading rate. Correlation measures between returns has direct application in portfolio management. Methodology Our modeling framework for Multi-UHF data is based on Doubly Stochastic Poisson Processes, developing some ideas from Bauwens(26). Our model will allow to capture co-movements and common component through the implementation of RJMCMC algorithm.

Outline.UHF 2.Model 3.Return 4.Filtering Ultra-High-Frequency Data (UHF) 2 Marked Doubly Stochastic Poisson Processes (DSPP) Financial Models For Equally spaced data Marked Point Process (MPP) and UHF Data Literature review for MPP Marked DSPP A DSPP model with a common intensity 3 Modeling framework for logreturns Logreturn Models Simulation studies Brief Conclusion 4 Filtering by RJMCMC Markov Chain Monte Carlo Reversible jump MCMC

.UHF 2.Model 3.Return 4.Filtering UHF Data Definition (UHF) UHF data also known as tick-by-tick data: each tick is one logical unit of information, like a quote or a transaction price, time of event. Data Source: Borsa di Milano. 7 Italian banks: Banco Popolare (POP), Mediobanca (MED),Banca popolare di Milano (MIL), MPS Banca (MPS), Banca Intesa SanPaolo (ISP), UBI Banca (UBI), Unicredit Banca (UCD). Period: 5 business days from 27//28 to 4//28; market open from 9:5 to 7:25 (3s).

.UHF 2.Model 3.Return 4.Filtering The total number and the average transactions for each asset during 27//28 and 4//28 (5 business days). Asset Total number Frequency per of transactions business day Banco Popolare 22993 533 Medio Banca 4434 962 Banca popolare di Milano 44 96 MPS Banca 48 938 Intesa San Paolo Banca 5626 3442 UBI Banca 522 2 Unicredit Banca 64444 4297

.UHF 2.Model 3.Return 4.Filtering Characteristics of UHF Data Irregular time spacing Price Discreteness Negative first order autocorrelation of logreturns Empirical synchronized correlation (Epps effect) 9 8.8 First 5 minute transactions Banco Popolare Medio Banca 8.6 price(euro) 8.4 8.2 8 7.8 7.6 7.4 2 4 6 8 2 4 6 8 2 22 24 26 28 3 time(second)

.UHF 2.Model 3.Return 4.Filtering Negative First-Order Autocorrelation.5 Autocorrelation for Banco Popolare.5 Autocorrelation for Medio Banca.5 2 3 4 5 time lag Autocorrelation for MPS Banca.5.5 2 3 4 5 time lag.5 Autocorrelation for Banca Popolare di Milano.5 2 3 4 5 time lag Autocorrelation for Banca di Intesa SanPaolo.2.5 2 3 4 5 time lag.5 Autocorrelation for UBI Banca.2 2 3 4 5 time lag Autocorrelation for Unicredit Banca.2.5 2 3 4 5 time lag.2 2 3 4 5 time lag

.UHF 2.Model 3.Return 4.Filtering Synchronization Methods Linear Interpolation and Previous-tick Interpolation Linear interpolation: to interpolate time between t j and t j + with the following form: Y (t + i t) = Y j + t + i t t j (Y j t j + t + Y j ) j where the j refer to the most recent time index of time t + i t. Previous-tick interpolation: take the most recent value of the raw data, in formula, Y (t + i t) = Y j where the j refer to the most recent time index of time t + i t.

.UHF 2.Model 3.Return 4.Filtering A graphic example of interpolation Graphic example of interpolation 9 8.5 8 7.5 7 price 6.5 6 5.5 5 4.5 4 5 5 2 25 time

.UHF 2.Model 3.Return 4.Filtering Previous-tick synchronization-logreturn aggregation r logp logp r2 logp2 r3 logp3 r4 logp4 r5 logp5 r6 logp6 logreturn aggregation r t = logp 3 logp =logp 3 logp 2 +logp 2 logp +logp logp = r 3 + r 2 + r

.UHF 2.Model 3.Return 4.Filtering Epps effect: Intesa Sanpaolo Banca with others CORRELATION ISP POP CORRELATION ISP MED.5.5.5.5 2 2.5 3 x 4 CORRELATION ISP MPS.5.5 2 2.5 3 x 4 CORRELATION ISP MIL.5.5.5.5 2 2.5 3 x 4 CORRELATION ISP UBI.5.5 2 2.5 3 x 4 CORRELATION ISP UCD.5.5.5.5 2 2.5 3.5.5 2 2.5 3 x 4 x 4

.UHF 2.Model 3.Return 4.Filtering Linear dependence: Scatter plot of ISP-UCD.. ISP UCD(m) ISP UCD(4m).5.5.5.5..4.2.2.4.6.8...5.5...5 ISP UCD(m).5. ISP UCD(2m).5.5..5..5.5..5..5.5..5..5.5 ISP UCD(3m)..5.5. ISP UCD(6m)..5..5.5..5..5.5.

.UHF 2.Model 3.Return 4.Filtering Epps effect: Medio Banca with others CORRELATION MED POP CORRELATION MED MPS.5.5.5.5 2 2.5 3 x 4 CORRELATION MED MIL.5.5 2 2.5 3 x 4 CORRELATION MED ISP.5.5.5.5 2 2.5 3 x 4 CORRELATION MED UBI.5.5 2 2.5 3 x 4 CORRELATION MED UCD.5.5.5.5 2 2.5 3 x 4.5.5 2 2.5 3 x 4

.UHF 2.Model 3.Return 4.Filtering Epps effect: Banco Popolare with others CORRELATION POP MED CORRELATION POP MPS.5.5.5.5 2 2.5 3 x 4 CORRELATION POP MIL.5.5 2 2.5 3 x 4 CORRELATION POP ISP.5.5.5.5 2 2.5 3 x 4 CORRELATION POP UBI.5.5 2 2.5 3 x 4 CORRELATION POP UCD.5.5.5.5 2 2.5 3 x 4.5.5 2 2.5 3 x 4

.UHF 2.Model 3.Return 4.Filtering Epps effect: MPS Banca with others CORRELATION MPS POP CORRELATION MPS MED.5.5.5.5 2 2.5 3 x 4 CORRELATION MPS MIL.5.5 2 2.5 3 x 4 CORRELATION MPS ISP.5.5.5.5 2 2.5 3 x 4 CORRELATION MPS UBI.5.5 2 2.5 3 x 4 CORRELATION MPS UCD.5.5.5.5 2 2.5 3.5.5 2 2.5 3 x 4 x 4

.UHF 2.Model 3.Return 4.Filtering Epps effect: Banca Popolare di Milano with others CORRELATION MIL POP CORRELATION MIL MED.5.5.5.5 2 2.5 3 x 4 CORRELATION MIL MPS.5.5 2 2.5 3 x 4 CORRELATION MIL ISP.5.5.5.5 2 2.5 3 x 4 CORRELATION MIL UBI.5.5 2 2.5 3 x 4 CORRELATION MIL UCD.5.5.5.5 2 2.5 3.5.5 2 2.5 3 x 4 x 4

.UHF 2.Model 3.Return 4.Filtering Epps effect: UBI Banca with others CORRELATION UBI POP CORRELATION UBI MED.5.5.5.5 2 2.5 3 x 4 CORRELATION UBI MPS.5.5 2 2.5 3 x 4 CORRELATION UBI MIL.5.5.5.5 2 2.5 3 x 4 CORRELATION UBI ISP.5.5 2 2.5 3 x 4 CORRELATION UBI UCD.5.5.5.5 2 2.5 3.5.5 2 2.5 3 x 4 x 4

.UHF 2.Model 3.Return 4.Filtering Epps effect: Unicredit Banca with others CORRELATION UCD POP CORRELATION UCD MED.5.5.5.5 2 2.5 3 x 4 CORRELATION UCD MPS.5.5 2 2.5 3 x 4 CORRELATION UCD MIL.5.5.5.5 2 2.5 3 x 4 CORRELATION UCD ISP.5.5 2 2.5 3 Time Scale(Seconds) x 4 CORRELATION UCD UBI.5.5.5.5 2 2.5 3.5.5 2 2.5 3 x 4 x 4

.UHF 2.Model 3.Return 4.Filtering Epps effect Epps effect: the correlation of returns decrease dramatically as the time intervals enter the intra-hour level. It is extensively observed in the high frequency data. Epps effect.9.8.7 correlation coefficient.6.5.4.3.2. 2 3 4 5 6 7 8 9 time scale

Outline.UHF 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Ultra-High-Frequency Data (UHF) 2 Marked Doubly Stochastic Poisson Processes (DSPP) Financial Models For Equally spaced data Marked Point Process (MPP) and UHF Data Literature review for MPP Marked DSPP A DSPP model with a common intensity 3 Modeling framework for logreturns Logreturn Models Simulation studies Brief Conclusion 4 Filtering by RJMCMC Markov Chain Monte Carlo Reversible jump MCMC

Literature review.uhf 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Standard Econometric Models Engle(982): ARCH Model Bollerslev(986): GARCH

.UHF 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Marked Point Processes (MPP) and UHF Data T = T T 2 Z 2 Z 3 T 3 T 4 Marked Point Process A marked point process (MPP) is a sequence of pairs Z Z 4 φ = (T i, Z i ) i N. Time Ultra-High-Frequency (UHF) Data With ultra-high-frequency (UHF) data, all trades (or quotes) of a financial asset are recorded (time of event, (bid-ask) price, volume, etc.). In this case, T i and Z i, i =, 2,..., could be the time and the size of the ith logprice change.

.UHF 2.Model 3.Return 4.Filtering Literature review for MPP 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Autoregressive Conditional Duration (ACD) Models Engle and Russell (998): Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data ; Zhang, Russell and Tsay (2): A Nonlinear Autoregressive Conditional Duration Model with Applications to Financial Transaction Data ; Bauwens and Giot (2): Asymmetric ACD Models: Introducing Price Information in ACD Models.

.UHF 2.Model 3.Return 4.Filtering Literature Review for MPP 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Marked Doubly Stochastic Poisson Processes (DSPP) Rydberg and Shephard (2): A modelling framework for the prices and times of trades made on the New York stock exchange. ; Frey (2): Risk-minimization with incomplete information in a model for high-frequency data ; Frey and Runggaldier (2): A nonlinear filtering approach to volatility estimaiton with a view towards high frequency data ; Centanni and Minozzo (26): Estimaiton and filtering by reversible jump MCMC for a doubly stochastic Poisson model for ultra-high-frequency data ;

.UHF 2.Model 3.Return 4.Filtering Literature Review for MPP 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Multivarite Point Process Bowsher(22): Modelling Security Markets in Continuous Time: Intensity based, Multivariate Point Process Models ; Hall and Hautsch(24): A Continuous-Time Measurement of the Buy-Sell Pressure in a Limite Order Book Market ; Hall and Hautsch(26): Order Aggressiveness and Order Book Dynamics ; Hautsch(25): The Latent Factor VAR Model: Testing for a Common Component in the Intraday Trading Process ; Bauwens and Hautsch(26): Stochastic Conditional Intensity Process.

.UHF 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Marked Doubly Stochastic Poisson Processes (DSPP) Definition (Last and Brandt, 995) Given a probability space (Ω, F, P) and a filtration {F t } t R +, an MPP φ adapted to the filtration is a DSPP if there exists a F -measurable random measure ν on R + R such that ( P µ ( (s, t] A ) ) (υ ( (s, t] A )) k = k F s = k! ( ) e υ (s,t] A, almost surely, for every A B(R), where µ denotes the counting measure associated to the MPP φ, that is, µ ( ω, (, t] A ) N t = i= {Zi A}.

.UHF 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Marked Doubly Stochastic Poisson Processes (DSPP) Intensity of a Marked DSPP DSPPs are point processes in which the number of jumps N t N s in any time interval (s, t] is Poisson distributed, given another positive stochastic process λ, called intensity P ( N t N s = k λ ) = ( t s λ udu k! ) k ( t ) exp λ u du. s Moreover, given λ, the number of jumps in disjoint time intervals are independent. Marked DSPP With Intensity Driven by MPP We assume the intensity λ depends itself on another MPP, that is, λ is of the form λ t = h(t, ψ t), where ψt is the restriction of another MPP ψ = (τ j, X j ) j N to [, t].

.UHF 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Marked DSPP With Shot Noise Intensity (Basic Model) Stochastic Differential Equation Stochastic noise intensity is assumed as the solution of the following Stochastic Differential Equation: dλ t = kλ t dt + dj t where k >,,J = N t j= X j, and N t = #j : τ j t. Solution of SDE N t λ t = λ exp( kt) + X j exp( k(t τ j )), t j= where (τ j ) j=,2,... is a Poisson process with constant parameter ν, and X j are exponentially distributed with parameter γ.

.UHF 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Simulated Shot Noise Intensity with parameter ν =., k =., γ =.2 35 Intensity 3 25 Lambda 2 5 5 2 3 4 5 6 7 8 9 Time

.UHF 2.Model 3.Return 4.Filtering Common Factor Intensity Model 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Model Description λ s t = λ s t + α s λ t s =, 2,...,m. where λ s t denotes s-specific individual intensity and λ t denotes a common component. The coefficient s is a scaling parameter driving the common intensity impact on the s-individual intensity. Bivariate case λ t λ 2 t = λ t + α λ t = λ 2 t + α 2 λ t

.UHF 2.Model 3.Return 4.Filtering 2..ARCH 2.2.MPP 2.3.MPP 2.4.DSPP 2.5.Common Fact Simulated common intensity with parameter ν =.,k =.,γ =.2,ν =.5,k =.5,γ =,ν 2 =,k 2 =.4,γ 2 =. 3 Lambda Lambda 2 2 4 6 8 Time Lambda Lambda2 Lambda 5 Lambda 5 2 4 6 8 Time.5 Transaction time 2 4 6 8 Time.5 Transaction time2.5.5 2 4 6 8 2 4 6 8 Time Time

Outline.UHF 2.Model 3.Return 4.Filtering 3..Return 3.2.ρ 3.3.Summary Ultra-High-Frequency Data (UHF) 2 Marked Doubly Stochastic Poisson Processes (DSPP) Financial Models For Equally spaced data Marked Point Process (MPP) and UHF Data Literature review for MPP Marked DSPP A DSPP model with a common intensity 3 Modeling framework for logreturns Logreturn Models Simulation studies Brief Conclusion 4 Filtering by RJMCMC Markov Chain Monte Carlo Reversible jump MCMC

.UHF 2.Model 3.Return 4.Filtering Modelling framework for logreturns 3..Return 3.2.ρ 3.3.Summary Independent Wiener Process R Ti = µ + ξ Ti, ξ Ti N (( ) ( )) σ, (T i T i ) σ 22 Concurrently correlated Wiener Process R Ti = µ + ξ Ti, (( ) ( )) σ σ 2 ξ t N, (T i T i ) σ 2 σ 22 where σ 2 = σ 2.

.UHF 2.Model 3.Return 4.Filtering Modelling framework for logreturns 3..Return 3.2.ρ 3.3.Summary Bi-AR() without cross-correlation (( ) ( )) σ σ 2 R Ti = µ + βr Ti + ξ Ti, ξ Ti N, σ 2 σ 22 where β = ( β ) β 22 Bi-AR() R Ti = µ + βr Ti + ξ Ti, (( ) ( )) σ σ 2 ξ T i N, σ 2 σ 22 where β ij, i, j =, 2.

Pooled time.uhf 2.Model 3.Return 4.Filtering 3..Return 3.2.ρ 3.3.Summary t t2 t3 t4 t5 t t2 t3 T T2 T3 T4 T5 T6 T7 T8 R R2 R3 R4 R5 R6 R7 R8 R R2 R3 R4 R5 R6 R7 R8

µ =.UHF 2.Model 3.Return 4.Filtering 3..Return 3.2.ρ 3.3.Summary i)independent Wiener Process with σ = σ 22 =., σ 2 = σ 2 =.3 Independent.25.2.5..5.5..5.2 5 5 time scale

µ =.UHF 2.Model 3.Return 4.Filtering 3..Return 3.2.ρ 3.3.Summary ii) Concurrently correlated Wiener Process with σ =., σ 22 =.2, σ 2 = σ 2 =. Temporally Correlated.8.6.4.2.8.7.6.5.4.3.2 5 5 Temporally Correlated time scale Temporally Correlated.8.6.4.2. 5 5 time scale 5 5 time scale

µ =.UHF 2.Model 3.Return 4.Filtering 3..Return 3.2.ρ 3.3.Summary iii) Bi-AR() without cross correlation: σ =., σ 22 =.2, σ 2 = σ 2 =., β =., β 22 =.2, β 2 = β 2 = VAR() without cross correlation.8.6.4.2.8 5 5 time scale VAR() without cross correlation VAR() without cross correlation.8.6.6.4.4.2.2 5 5 time scale 5 5 time scale

µ =.UHF 2.Model 3.Return 4.Filtering 3..Return 3.2.ρ 3.3.Summary iv) Bi-AR(): σ =., σ 22 =.2, σ 2 = σ 2 =., β =.2, β 22 =., β 2 =.3, β 2 =.6 VAR().9.8.7.6.5.4.3.2 5 5 time scale

Brief Conclusion.UHF 2.Model 3.Return 4.Filtering 3..Return 3.2.ρ 3.3.Summary Common factor model of intensity has no significant impact on Epps effect, in general on empirical synchronized correlation function; To obtain the empirical correlation, we need not only concurrent correlation but also cross-serial correlation; Epps effect caused by asynchronization. logp r2 r logp logp2 r3 logp3 r4 logp4 r5 logp5 r6 logp6 logreturn aggregation r t = logp 3 logp =logp 3 logp 2 +logp 2 logp +logp logp = r 3 + r 2 + r

.UHF 2.Model 3.Return 4.Filtering Cross-Autocorrelation 3..Return 3.2.ρ 3.3.Summary logp logp4 r logp r2 logp2 r3 r4 logp3 r5 logp5 r6 logp6 News logp logp logp2 logp3 News

Outline.UHF 2.Model 3.Return 4.Filtering 4..MCMC 4.2.RJMCMC Ultra-High-Frequency Data (UHF) 2 Marked Doubly Stochastic Poisson Processes (DSPP) Financial Models For Equally spaced data Marked Point Process (MPP) and UHF Data Literature review for MPP Marked DSPP A DSPP model with a common intensity 3 Modeling framework for logreturns Logreturn Models Simulation studies Brief Conclusion 4 Filtering by RJMCMC Markov Chain Monte Carlo Reversible jump MCMC

.UHF 2.Model 3.Return 4.Filtering Markov Chain Monte Carlo(MCMC) 4..MCMC 4.2.RJMCMC Introduction MCMC methods are a class of algorithms for sampling from probability distribution based on constructing a Markov chain that has the target distribution (π) as its stationary distribution. In order for π to be the stationary distribution of the chain, the transition probabilities must satisfy the detailed balance condition π i P ij = π j P ji MCMC Two important algorithm to construct Markov Chain: Metropolis-Hasting algorithm; Gibbs sampling (which is the special case of M-H algorithm).

.UHF 2.Model 3.Return 4.Filtering Metropolis-Hasting Algorithm 4..MCMC 4.2.RJMCMC Suppose we have a Markov Chain in state x. We want to simulate a draw from the transition kernel p(x, y) with stationary distribution π, but we do not know the form of p(x, y). We do know how to compute a function proportional to π, f (x) = kπ(x). Assume that we can draw y from an arbitrary distribution q(x, y). Consider using this q as a transition kernel. If we construct α(x, y) such that π(x)q(x, y)α(x, y) = π(y)q(y, x)α(y, x) then we will have a reversible transition kernel with stationary distribution π. We can take: α(x, y) = min{, π(y)q(y, x) π(x)q(x, y) } Although we do not know π(x), we do know f (x) = kπ(x), so we can calculate α(x, y).

.UHF 2.Model 3.Return 4.Filtering Metropolis-Hasting Algorithm 4..MCMC 4.2.RJMCMC In summary, the Metropolis-Hasting algorithm proceeds as follows. Given x j, to move to x j+ we:. Generate a draw y from q(x j,.) 2. Calculate α(x j, y) 3. Generate u U[, ] 4. if α(x j, y) > u, then x j+ = y. Otherwise x j+ = x j. After a large number of iterations, the marginal distribution of x j will converge to π. Example Suppose we have three intensity processes, λ = ν x, λ 2 = ν 2 x 2, λ = ν x, where x i Gamma(α i, β i ), and ν i, α i and β i are parameters, i =,, 2.

.UHF 2.Model 3.Return 4.Filtering 4..MCMC 4.2.RJMCMC 4 Lambda=v*x 4 Lambda2=v2*x2 Lambda 3 2 Lambda2 3 2 5 5 2 25 3 Time 5 4.5 5 5 2 25 3 Time Lambda=v*x Lambda 4 3.5 3 4 2 2.5 5 5 2 25 3 Lambda=Lambda+a*Lambda Time Lambda2=Lambda2+a2*Lambda 4 2 Return Return2 2 2 4 5 5 2 25 3 Time 4 5 5 2 25 3 Time

.UHF 2.Model 3.Return 4.Filtering 4..MCMC 4.2.RJMCMC Run of Metropolis-Hasting algorithm for iterations with 5 burn-in period. Parameter ν =,ν = 2, ν 2 =, α = 3, β =, α = 2, β = 2, α 2 = 5, β 2 =.5,a =, a 2 =.5. 8 Histogram of X 6 4 2 2 3 4 5 6 7 8 9 x = 3.8524 6 Histogram of X 4 2 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 x = 6.323 8 Histogram of X2 6 4 2 x 2 = 2.579.5.5 2 2.5 3 3.5 4 4.5

.UHF 2.Model 3.Return 4.Filtering Reversible jump MCMC 4..MCMC 4.2.RJMCMC Introduction RJMCMC algorithm is a generalization of the Metropolis-Hasting algorithm that supplies a sample from a target distribution π on the spaces of the form C = n= C n, where C n = {n} R kn, k n N, in which subspaces of different dimention k n are combined. RJMCMC We define r transition moves to choose the next state of the chain, where some of the moves involve vectors of different dimensions. Thus, RJMCMC algorithm update proceeds with the following steps: i) choose one of the moves; ii) propose a new state; iii) accept the new state with a certain probability A such that the Markov chain converges.

o t x x 5 3 x 4 =x* o τ 4 =τ* x4 x 4 t t t λ x 4 o t t.uhf 2.Model 3.Return 4.Filtering Move types of RJMCMC 4..MCMC 4.2.RJMCMC λ t = λ exp( kt) + N t j= X j exp( k(t τ j )) birth x x 2 height position current τ τ 2 τ 3 τ 5 ¹ x x 2 x 3 x 4 x 4 τ τ 2 τ 3 τ 4 x x 2 x 3 x 4 ¹ λ x x 2 x 3 τ τ 2 τ 2 τ 3 τ 4 τ τ 2 τ 3 τ 4 ¹ x x 2 x 3 death x 2 x 3 x τ τ 2 τ 3 τ 4 τ τ 2 τ 3 τ 4 starting value

.UHF 2.Model 3.Return 4.Filtering 4..MCMC 4.2.RJMCMC Component 3 3 3 (s).3 (h).3.3.5 (p).5 (b) (d) (s).3 (h).3.3 (p).5.5 (b) (d) (s).3.3 (h).3 2 (p).5.5 (b) (d)

.UHF 2.Model 3.Return 4.Filtering 4..MCMC 4.2.RJMCMC Bivariate DSPP model filtering by RJMCMC Filtering of the Intensity by RJMCMC,T=,run for iteration with 5 burn in a= a2= 2 True(simulated)trajectories of the joint intensity Filtering expectation of the joint intensity 5 Lambda () 5 2 3 4 5 6 7 8 9 Time 25 2 True(simulated)trajectories of the joint intensity Filtering expectation of the joint intensity Lambda (2) 5 5 2 3 4 5 6 7 8 9 Time

.UHF 2.Model 3.Return 4.Filtering 4..MCMC 4.2.RJMCMC Bivariate DSPP model filtering by RJMCMC Filtering of the Intensity by RJMCMC,T=,run for iteration with 5 burn in a= a2=. True(simulated)trajectories of the common intensity Filtering expectation of the common intensity 8 Lambda 6 4 2 Lambda 4 2 8 6 4 2 2 3 4 5 6 7 8 9 Time True(simulated)trajectories of the first intensity Filtering expectation of the first intensity Lambda 25 2 5 5 True(simulated)trajectories of the second intensity Filtering expectation of the second intensity 2 4 6 8 Time 2 4 6 8 Time

5.Conclusions Conclusion and Further research Bibliography A new model for multivariate DSPP based on a common component for the intensities. The intensities of DSPPs are not observable; RJMCMC methods are applied for filtering. VAR() provides correlations between two assets close to empirical ones (EPPs effect). Further research: Parameter estimation by Monte Carlo Expection-Maximization (MCEM) methods based on the RJMCMC algorithm.

Bibliography 5.Conclusions Bibliography L. Bauwens and N. Hautsch (26). Stochastic conditional intensity Process. Journal of Financial Econometrics, 4, 45-493. S. Centanni and M. Minozzo (26). Estimation and filtering by reversible jump MCMC for a doubly stochastic Poisson model for ultra-high-frequency financial data. Statistical Modelling, 6, 97 8. S. Centanni and M. Minozzo (26). A Monte Carlo approach to filtering for a class of marked doubly stochastic Poisson processes. Journal of the American Statistical Association,, 582 597. R. Engle and J. Russell (998). Autoregressive conditional duration: A new model of irregularlt spaced transaction data. Econometrica, 66, 27-62. R. Frey (2). Risk-minimization with incomplete information in a model for high-frequency data. Mathematical Finance,, 25-25. R. Frey and W. Runggaldier (2). A nonlinear filtering approach to volatility estimation with a view towards high frequency data. International Journal of Theoretical and Applied Finance, 4, 27-3.

Bibliography 5.Conclusions Bibliography A. Hall and N. Hautsch (24). A continuous-time measurement of the buy-sell pressure in a limit order book market. Discussion Paper, 4-7, Department of Economics, University of Copenhagen, Copenhagen. A. Hall and N. Hautsch (26). Order aggressiveness and order book dynamics. Empirical Economics, 3, 973-5. N. Hautsch (25). The latent factor VAR model: testing for a common component in the intraday trading process. Discussion Paper, Finance Research Unit, Department of Economics, University of Copenhagen, Copenhagen. T. Rydberg and N. Shephard (2). A modelling framework for the prices and times of trades made on the New York stock exchange. Cambridge University Press, 27-246. M. Zhang, J. Russell and R. Tsay (2). A nonlinear autoregressive conditional duration model with applications to financial transaction data. Journal of Econometrics, 4, 79-27.