rcBS function

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

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.

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

Constraints: $theta[3]>0$.

Returns

• x: a numeric vector

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.

Author(s)

Stefano Maria Iacus

Examples

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