General Compound Hawkes Processes in Limit Order Books

Abstract

In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regard to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for several specific variations. We apply several of these FCLTs to limit order books to study the link between price volatility and order flow, where the volatility in mid-price changes is expressed in terms of parameters describing the arrival rates and mid-price process.

Multivariate General Compound Point Processes in Limit Order Books

“…As for AMZN, although we derive a remarkable improvement from 2-state model (74.60% error) to 7-state model (28.29% error), we cannot make the error smaller than 5% or 10%. This is to say, MGCPP model may not be able to capture the full dynamics for AMZN data, but it still can be a strong candidate for modeling the mid-price, which is consistent with the conclusion of compound Hawkes model in Swishchuk and Huffman (2020). Table 7.…”

“…We applied the method in Swishchuk and Huffman (2020) to calculate the state values a(X i,k ) for each stock. First, we compute the changes of mid-price and separate the data into two sets by positive increments or negative increments.…”

Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models

“…Good introduction into Hawkes processes and their properties may be found in (Laub et al 2015). GCHP and regime-switching GCHP were first introduced in (Swishchuk 2017b) and studied in details using real data in Swishchuk and He 2019;Swishchuk 2020;Swishchuk and Huffman 2020). Risk model based on GCHP was first introduced in (Swishchuk 2017a) and described in details in (Swishchuk 2018).…”

“…Merton optimal investment and consumption stochastic problem is one of the most studied classical problem in finance (Merton 1969(Merton , 1971(Merton , 1990Bjork 2009;Karatzas and Shreve 1998). In this paper, we will show how to solve the Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance, i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound Hawkes process (GCHP) (Swishchuk 2020;Swishchuk and Huffman 2020;Swishchuk 2017b), and for a capital R(t) of an insurance company with the amount of claims described by risk model based on GCHP (Swishchuk 2018;.…”

“…where G(t) := ∑ N(t) i=1 a(X i ) is the GCHP, X i is a discrete-time Markov chain (MC) with finite or infinite states, r > 0 is the interest rate, and a(x) is a continuous and finite function on X. We note, that the justification of using HP in finance may be found in (Da Fonseca and Zaatour 2013), and using GCHP that based on HP N(t) and a(X i ) may be found in (Swishchuk and He 2019;Swishchuk and Huffman 2020). The model for a stock price S(t) that based on GCHP is a new and original in this paper.…”