Brownian motion, Brownian bridge, and geometric Brownian motion simulators

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Description

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

Usage

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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].

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), t0 <= t <= T | B(t0)=x, B(T)=y)

The function BM returns a trajectory of the translated Brownian motion (B(t), t >= t0 | B(t0)=x); i.e., x+B(t-t0) 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))

Value

X

an invisible ts object

Author(s)

Stefano Maria Iacus

Examples

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