Multivariate General Compound Hawkes and Point Processes with Financial Applications
Date
2022-10-20
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Abstract
The Hawkes process (HP) significantly affected the financial modeling area in the past 15 years, especially high-frequency trading. This thesis focuses on various new Hawkes processes and considers their applications in the limit order book (LOB). Preexisting studies of the HP in the LOB showed that the arrivals of orders could be modeled by univariate or multivariate HP because of its long memory property and clustering effect. Therefore, we propose the multivariate general compound Hawkes process (MGCHP), a stochastic model for the mid-price in the LOB. For the MGCHP, we prove the Law of Large Numbers (LLN) and two Functional Central Limit Theorems (FCLT); the latter provides insights into the link between price volatilities and order flows in limit order books with several assets. The parameter estimation for the high-dimensional Hawkes process is always time-consuming. This motivates us to consider a generalization of the MGCHP. We replace the multivariate HP with a more general point process, and we call it the multivariate general compound point process (MGCPP). We also prove limit theorems for the MGCPP and compared numerical simulations for the MGCPP with the MGCHP. The MGCHP model provides us with a perfect framework for the stock price dynamics in the LOB. It’s natural to apply it to other financial applications. We extend the MGCHP to the exponential MGCHP (EMGCHP) and consider the corresponding asset-liability management problem. Risky assets are molded by the EMGCHP while the liability follows a Brownian motion with drift. We derive the Hamilton–Jacobi–Bellman equation and transformed it into a system of PDEs. With the FCLT, we can approximate the EMGCHP to a geometric Brownian motion in the LOB and apply Xie et al.’s results. Numerical simulations for the Hawkes-based model and comparisons with the Poisson-based model are also provided. In the last part of the thesis, we give an option pricing formula under the EMGCHP framework. We believe our study can provide a strong tool for not only researchers but also traders in the high-frequency market.
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Keywords
Hawkes processes, multivariate general compound Hawkes processes, limit order books, point processes, functional central limit theorems, law of large numbers, asset-liability management, option pricing
Citation
Guo, Q. (2022). Multivariate general compound Hawkes and point processes with financial applications (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.