Ornstein-Uhlenbeck Process

Overview

Ornstein-Uhlenbeck process is a stochastic process which uses the Random Number Generator (RNG) to generate locations for mesher. It uses a reference time point \(\Delta w\) and a time step \(\Delta t\) to calculate the drift and diffusion. The Ornstein-Uhlenbeck process is a simple stochastic processes, whose feature of interest is its mean-reverting drift term \(a(r-x)\) and its constant diffusion term \(\sigma\).

The Ornstein-Uhlenbeck process can be described by

\[dx=a(r-x_{t})dt+\sigma dW_{t}\]

Implementation

The implementation of OrnsteinUhlenbeckProcess contains a few methods. The implementation can be introduced as follows:

  1. init: The initialization process to set up arguments as below:

    a)speed, the spreads on interest rates;

    b)vola, the overall level of volatility;

    c)x0, the initial value of level;

    d)level, the width of fluctuation on interest rates.

  2. expectation: The expectation method returns the expectation of the process at time \(E(x_{t_{0}+\Delta t}|x_{t_{0}}=x_{0})\).

  3. stdDeviation: The stdDeviation method returns the standard deviation \(S(x_{t_{0}+\Delta t}|x_{t_{0}}=x_{0})\) of the process with a time period \(\Delta t\) according to the given discretization.

  4. variance: The variance method returns the variance \(V(x_{t_{0}+\Delta t}|x_{t_{0}}=x_{0})\) of the process with a time period \(\Delta t\) according to the given discretization.

  5. evolve: The evolve method returns the asset value after a time interval \(\Delta t\) according to the given discretization. It returns,

\[E(x_{0},t_{0},\Delta t)+S(x_{0},t_{0},\Delta t)*\Delta w\]

where \(E\) is the expectation and \(S\) the standard deviation.