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# Jump Diffusion Model
- URL: https://fsc.stevens.edu/jump-diffusion-model/
- Published: 2019-10-29T17:45:37.000Z
- Updated: 2026-05-16T22:41:39.000Z
- Author: Zheng Xing
- Tags: FE 800 Projects, 2018 Fall, Research, #Import 2024-05-30 10:26, #Import 2025-11-08 22:14

**Author:**

Tianxiang He

**Supervisor:**

Dr. Khaldoun Khashanah

### Abstract

To investigate the complex interaction among jumps on underlying assets, I employ Merton jump diffusion model to capture the jumps. The jumps are modeled by the continuous process and jump process. However, the BNS jump test technique shows that the jump definitely happen for every day, which is contrary to the conclusion from Merton jump diffusion model based on intraday data of SPY. On the other hand, in order to identify jumps the jumps based on intraday prices, I introduce BCH method. The result shows the “U” shape pattern and the property of clustering for jumps. Therefore, the Self-exciting Hawkes process is applied to capture these features. For extension, Bivariate Hawkes process is utilized to make an analysis on jumps from SPY and sector ETFs. Both self-exciting Hawkes process and Bivariate Hawkes exhibit the good fitting.

**Keywords**: Jump identification, Merton jump diffusion model, Hawkes process, BNS jump test, BCH method.

**Merton Jump Diffusion model (MJD)**

Merton assumes the dynamic of underlying assets St may be described by equation (4.1):