Stochastic Approaches to Asset Price Analysis

Paper: arXiv 2407.06745

Abstract

In this project, we propose to explore the Kalman filter’s performance for estimating asset prices. We begin by introducing a stochastic mean-reverting processes, the Ornstein-Uhlenbeck (OU) model. After this we discuss the Kalman filter in detail, and its application with this model. After a demonstration of the Kalman filter on a simulated OU process and a discussion of maximum likelihood estimation (MLE) for estimating model parameters, we apply the Kalman filter with the OU process and trailing parameter estimation to real stock market data. We finish by proposing a simple day-trading algorithm using the Kalman filter with the OU process and backtest its performance using Apple’s stock price. We then move to the Heston model, a combination of Geometric Brownian Motion and the OU process. Maximum likelihood estimation is commonly used for Heston model parameter estimation, which results in very complex forms. Here we propose an alternative but easier way of parameter estimation, called the method of moments (MOM). After the derivation of these estimators, we again apply this method to real stock data to assess its performance.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper involves advanced mathematics including stochastic differential equations, Ito calculus, and Kalman filter derivations, but also includes real-world data applications, backtesting, and algorithmic trading strategies.

Research Flowchart

  flowchart TD
  A["Research Goal: <br>Asset Price Analysis using Stochastic Models"] --> B["Data Inputs: <br>Simulated OU Process & Apple Stock Data"]
  B --> C["Model 1: Ornstein-Uhlenbeck (OU) <br>+ Kalman Filter"]
  C --> D{"Parameter Estimation <br>Method?"}
  D --> E["MLE (Simulation) <br>Trailing Est. (Real Data)"]
  E --> F["Outcome: <br>Day-Trading Algorithm & Backtest"]
  C --> F
  B --> G["Model 2: Heston Model <br>(GBM + OU Process)"]
  G --> H["Method of Moments (MOM) <br>for Parameter Estimation"]
  H --> I["Outcome: <br>Model Performance on Real Data"]

Cited References (11)

1. Forecasting trends with asset prices (2017) — link
2. Fast strong approximation Monte Carlo schemes for stochastic volatility models (2006) — link
3. The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution (2020)
4. Theory of Speculation (1900)
5. Estimation II (2001) — link
6. Online Kalman Filter Tutorial — link
7. A pairs trading strategy based on linear state space models and the Kalman filter (2016) — link
8. Filtering Complex Turbulent Systems (2012) — link

0 of 11 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.