rcBS: Black-Scholes-Merton or geometric Brownian motion process conditional law

Description

Density, distribution function, quantile function, and random generation for the conditional law $X(t) | X(0) = x_0$ 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)

Value

x: a numeric vector

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].

Author

Stefano Maria Iacus

Details

This function returns quantities related to the conditional law of the process solution of $${\rm d}X_t = \theta_1 X_t {\rm d}t + \theta_2 X_t {\rm d}W_t.$$

Constraints: $\theta_3>0$.

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

Run this code

rcBS(n=1, Dt=0.1, x0=1, theta=c(2,1))

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