fExpressCertificates (version 1.3)

GeometricBrownianMotion: Simulate paths from a Arithmetic or Geometric Brownian Motion

Description

Simulate one or more paths for an Arithmetic Brownian Motion \(B(t)\) or for a Geometric Brownian Motion \(S(t)\) for \(0 \le t \le T\) using grid points (i.e. Euler scheme).

Usage

BM(S0, mu=0, sigma=1, T, N)
GBM(S0, mu, sigma, T, N)
GeometricBrownianMotionMatrix(S0, mu, sigma, T, mc.loops, N)

Arguments

S0

start value of the Arithmetic/Geometric Brownian Motion, i.e. S(0)=S0 or B(0) = S0

mu

the drift parameter of the Brownian Motion

sigma

the annualized volatility of the underlying security, a numeric value; e.g. 0.3 means 30% volatility pa.

T

time

mc.loops

number of Monte Carlo price paths

N

number of grid points in price path

Value

a vector of length N+1 with simulated asset prices at \((i * T/N), i=0,\ldots,N\).

References

Iacus, Stefan M. (2008). Simulation and Inference for Stochastic Differential Equations: With R Examples Springer

Examples

# NOT RUN {
# Simulate three trajectories of the Geometric Brownian Motion S(t)
T        <- 1
mc.steps <- 100
dt       <- T/mc.steps
t        <- seq(0, T, by=dt)
S_t      <- GBM(S0=100, mu=0.05, sigma=0.3, T=T, N=mc.steps)
plot(t, S_t, type="l", main="Sample paths of the Geometric Brownian Motion")
for (i in 1:2) {
 S_t      <- GBM(S0=100, mu=0.05, sigma=0.3, T=T, N=mc.steps)
 lines(t, S_t, type="l")
}
# }