Last data update: 2014.03.03
R: Black-Scholes-Merton or geometric Brownian motion process...
Black-Scholes-Merton or geometric Brownian motion process conditional law
Description
Density, distribution function, quantile function, and
random generation for the conditional law X(t) | X(0) = x0
of the Black-Scholes-Merton process
also known as the geometric Brownian motion process.
Usage
dcBS(x, Dt, x0, theta, log = FALSE)
pcBS(x, Dt, x0, theta, lower.tail = TRUE, log.p = FALSE)
qcBS(p, Dt, x0, theta, lower.tail = TRUE, log.p = FALSE)
rcBS(n=1, Dt, x0, theta)
Arguments
x
vector of quantiles.
p
vector of probabilities.
Dt
lag or time.
x0
the value of the process at time t
; see details.
theta
parameter of the Black-Scholes-Merton process; see details.
n
number of random numbers to generate from the conditional distribution.
log, log.p
logical; if TRUE, probabilities p are given as log(p) .
lower.tail
logical; if TRUE (default), probabilities are P[X <= x]
;
otherwise, P[X > x]
.
Details
This function returns quantities related to the conditional law
of the process solution of
dX_t = theta[1]*Xt*dt + theta[2]*Xt*dWt.
Constraints: theta[3]>0 .
Value
Author(s)
Stefano Maria Iacus
References
Black, F., Scholes, M.S. (1973) The pricing of options
and corporate liabilities, Journal of Political Economy , 81, 637-654.
Merton, R. C. (1973) Theory of rational option pricing,
Bell Journal of Economics and Management Science , 4(1), 141-183.
Examples
rcBS(n=1, Dt=0.1, x0=1, theta=c(2,1))
Results