LSMRealOptions (version 0.2.1)

GOU_simulate: Simulate the geometric Ornstein-Uhlenbeck (GOU) stochastic process through Monte Carlo 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

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: $$ dS(t)/ = reversion_rate(equilibrium - S(t)) dt + \sigma dW(t)$$

$$\frac{dS(t)}{S(t)} = equilibrium - (reversion_rate * S(t) + risk_premium) dt + \sigma dW(t)$$

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

# NOT RUN {
## 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)
# }