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

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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

x

a numeric vector

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

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

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