# GOU_simulate: Simulate the geometric Ornstein-Uhlenbeck (GOU) stochastic... In LSMRealOptions: Value American and Real Options Through LSM Simulation

## Description

The geometric Ornstein-Uhlenbeck process is a member of the general affine class of stochastic process. The Ornstein-Uhlenbeck process is a Gaussian process, a Markov process, is temporally homogeneous and exhibits mean-reverting behaviour. The `IGBM_simulate` function utilizes antithetic variates as a simple variance reduction technique.

## Usage

 `1` ```GOU_simulate(n, t, reversion_rate, sigma, equilibrium, risk_premium, S0, dt) ```

## Arguments

 `n` The total number of price paths to simulate `t` The forecasting period, in years `reversion_rate` The reversion rate term of the GOU process `sigma` The volatility term of the GOU process `equilibrium` The equilibrium term of the GOU process `risk_premium` The risk premium of the GOU process `S0` The initial value of the underlying asset `dt` The discrete time step of observations, in years A stochastic process S(t) is an IGBM that follows the following continuous-time stochastic differential equation: dS(t) = reversion_rate(equilibrium - S(t)) dt + sigma dW(t) dS(t)/S(t) = dS(t)/S(t) = equilibrium + (- reversion_rate * S(t) - risk_premium) dt + sigma dW(t) Where 'reversion_rate' is the rate of reversion term, 'equilibrium' is the equilibrium value the process reverts towards, 'risk_premium' is the risk premium of the process, 'sigma' the volatility term and W(t) is defined as a Weiner process.

## Value

A matrix of simulated price paths of the GOU process. Each column corresponds to a simulated price path, and each row corresponds to a simulated observed price of the simulated price paths at each discrete time period.

## Examples

 ```1 2 3 4 5 6 7 8 9``` ```## 100 simulations of 1 year of monthly price paths: Simulated <- GOU_simulate(n = 100, t = 1, reversion_rate = 1, sigma = 0.2, equilibrium = 100, risk_premium = 0.05, S0 = 100, dt = 1/12) ```

LSMRealOptions documentation built on June 26, 2021, 5:06 p.m.