1. Statistical Analysis for Partially-Observed Markov Processes with Marked Point Process Observation
PI: Professor Yong Zeng
Department of Mathematics and Statistics
University of Missouri at Kansas City
LEAD DEVELOPER: Junqi Yin
ORNL
PRESENTER:  Mitch Horton
Application Acceleration Center of Excellence (AACE) 
Established by Joint Institute for Computational Sciences (JICS)
April 18, 2017
XSEDE Symposium

2. The particle Markov Chain Monte Carlo (PMCMC) [Jian-hui (James) HUANG]method is applied to
estimate ultra-high frequency data models in which the underlying intrinsic value process follows:
geometric Brownian motion (GBM), and
2) Heston Stochastic Volatility (HSV), under 1/8 and 1/100 tick size rules. 
Numerical studies through simulation and real data show that PMCMC method is able to yield reasonable estimates
for model parameters.

3. We have two models for streaming financial ultra-high frequency data.
For each model, there are unknown parameters.
For each model, we have two methods for estimating unknown parameters.
We evaluate at 1/8 and 1/100 tick mark size.
For each solution, there are three versions:
1) Fortran
2) Fortran, accelerated using single GPU
3) Fortran, accelerated using multiple GPUs

4. We have two models for streaming financial ultra-high frequency data.
  1) Geometric Brownian Motion (GBM),
  2) Heston Stochastic Volatility (HSV).
  The models consider the explicit structure of market microstructure noise.

5. We have two models for streaming financial ultra-high frequency data.
  1) Geometric Brownian Motion (GBM),

6. We have two models for streaming financial ultra-high frequency data.
  2) Heston Stochastic Volatility (HSV).

7. We have two models for streaming financial ultra-high frequency data.
For each model, we have two methods for estimating unknown parameters.
  1) Bayesian filtering estimation method. The method is a recursive
  algorithm relying on the Markov chain approximation method to compute
  the approximate posterior and then the Bayes estimator
  2) Particle Markov Chain Monte Carlo (PMCMC)

8. The model is based on the intuition that trading price should arise from an intrinsic price process in 
combination with market noise from trading activities at trading times. The model consists of 
three parts: trading time series, Microstructure noise in observed price and the unobservable intrinsic process.

9. Trading Times
Trading time series {t_i : i ≥ 1} are modeled as a doubly-stochastic
Poisson process including a Poisson process with constant intensity.

10. Market Microstructure Noise
1) Discrete noise, which is generated from trading mechanism, exists because intraday prices move discretely, that is,
tick by tick, and the smallest tick size for trading is set by security exchanges.
2) Clustering noise means prices gather more on some ticks, instead of distributing evenly on all ticks.
3) All other noise is modeled as non-clustering noise.

11. Market Microstructure Noise
At trading time t_i, there is a unobservable intrinsic value {S(t_i)} of an asset. S(t) or S_t is commonly modeled by a
certain stochastic process.
The price at trading time t_i,Y(t_i), is constructed from the intrinsic value S(t_i) by incorporating the three kinds of noise.
Step 1. Incorporate price discreteness by rounding off S_t to its closest tick: Round[S(t_i), 1/M], where M is tick size.
Step 2. Incorporate non-clustering noise by adding V_i: Y'(t_i) = Round[S(t_i)+ V_i, 1/M], where V is an i.i.d. random
variable and independent of the intrinsic value S(t).
A good candidate is the doubly geometric distribution with parameter rho and M = 100.
Step 3. Incorporate clustering noise by biasing Y'(t_i). The biasing function b(.) moves Y'(t_i) to some close ticks according to
certain probability defined by parameters alpha, beta.

12. Results
In the first quarter, starting July 1, 2015, we  worked on collecting results by applying the code developed in the last ECSS
allocation to the real market data.
The raw data require some pre-processing, including trimming the 15 minutes at the beginning and before the end of each
trading day; removing outlier points; rounding trading intervals, etc. Also, since the runtime surpasses the wall time limit,
we implemented a checkpoint mechanism for the MPI-CUDA code.

13. Results
In the second quarter, we continued to collect results for stock market data, and results for GBM model looked promising.
We have exchanges and conference call with PI to discuss the details of the new algorithm—particle filtering via
simulations.
As the first step, for the new algorithm—particle filtering via simulations, we implemented the FORTRAN code on CPU for
GBM model with particle filtering, and the pseudo-code for possible GPU implementation for Heston model.

14. Results
In the third quarter, we worked with PI to validate the new code for GBM model. After some changes in the details of the
implementation of the algorithm, PI confirmed that the new method seemed to work correctly. Based the pseudo code of
the new algorithm and the CUDA code developed in the last allocation,  we wrote the first version of the multi-GPU code
for Heston model with particle filtering method, which runs properly but requires further testing.

15. Results
In the fourth quarter, we have completed the code development for the particle-filtering algorithm on multi-GPU. The new
algorithm outperforms the old one by several times. We have tested the simulated data for both algorithms, and sent the
results to PI. It seems the code is working as expected.

16. Technical Details
Unlike the PDE solver we implemented in the previous allocation, the particle filtering is a Monte Carlo method that
utilizes random number generator to simulate multiple paths for each parameter set and weights them based on likelihood
for the parameter. We use CURAND library to generate hundreds of thousands of random number simultaneously,
and each is used to simulate a new path and calculate the corresponding probability.