Estimating the Parameter of Geometric Brownian Motion in Hang Seng Index Using Bayesian Inference

Estimating the Parameter of Geometric Brownian Motion in Hang Seng Index Using Bayesian Inference

Content:

1. Introduction 3

2. Model description 3

Geometric Brownian motion 3

Bayesian Updating 5

Normal distribution: Natural conjugate priors for [pic] 5

Normal distribution: Natural conjugate priors for [pic] 6

Gibbs Sampling 7

Gibbs sampling algorithms 8

3. Performance evaluation 8

Simulating path and testing the performance 9

Implementation using Hang Seng Index 10

4. Conclusion 10

5. Appendix 10

File 10

Reference 10

6. Task Assignment 11

Abstract

This report proposes that we can use Bayesian updating and Gibbs sampling to estimate the parameter of geometric Brownian motion, which is normal distributed with two unknown parameter.

Introduction

As what Black and Scholes considered, the stock price should follow the geometric Brownian motion. Geometric Brownian motion is a good and reasonable estimation of asset price dynamics except unusual events.

However, if the asset price really follows the geometric Brownian motion, the parameter of the stochastic differential equation which are drift and volatility are hard to determine because of its randomness. The aim of this report is to approximate the parameters in real data using Bayesian updating.

Note that we need to estimate two parameters, so joint conjugate prior is needed. However, it is hard to construct such prior. So, I will use Gibbs sampling in this report. The idea of Gibbs sampling is to generate or update the parameter, conditional on other parameter fixed or updated. The pro of conditioning on other parameter is low-dimensional (only one dimensional). Also, the conditional conjugate prior has the same distributional structure as the conditional posterior.

Gibbs sampling is one of the most commonly used Markov Chain Monte Carlo methods. The Markov Chain Monte Carlo method is considered since it...