sde (version 2.0.9)

BM: Brownian motion, Brownian bridge, and geometric Brownian motion simulators

Description

Brownian motion, Brownian bridge, and geometric Brownian motion simulators.

Usage

BBridge(x=0, y=0, t0=0, T=1, N=100)
BM(x=0, t0=0, T=1, N=100)
GBM(x=1, r=0, sigma=1, T=1, N=100)

Arguments

x

initial value of the process at time t0.

y

terminal value of the process at time T.

t0

initial time.

r

the interest rate of the GBM.

sigma

the volatility of the GBM.

T

final time.

N

number of intervals in which to split [t0,T].

Value

• X
  an invisible ts object

Details

These functions return an invisible ts object containing a trajectory of the process calculated on a grid of N+1 equidistant points between t0 and T; i.e., t[i] = t0 + (T-t0)*i/N, i in 0:N. t0=0 for the geometric Brownian motion.

The function BBridge returns a trajectory of the Brownian bridge starting at x at time t0 and ending at y at time T; i.e., $${B(t), t_0 \leq t \leq T | B(t_0)=x, B(T)=y}.$$ The function BM returns a trajectory of the translated Brownian motion $B(t), t \geq 0 | B(t_0)=x$; i.e., $x+B(t-t_0)$ for t >= t0. The standard Brownian motion is obtained choosing x=0 and t0=0 (the default values).

The function GBM returns a trajectory of the geometric Brownian motion starting at x at time t0=0; i.e., the process $$S(t) = x \exp{(r-\sigma^2/2)t + \sigma B(t)}.$$

Examples

plot(BM())
plot(BBridge())
plot(GBM())