# rcBS: Black-Scholes-Merton or geometric Brownian motion process... In sde: Simulation and Inference for Stochastic Differential Equations

## 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

 ```1 2 3 4``` ```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

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

### Example output

```Loading required package: MASS

Attaching package: 'fda'

The following object is masked from 'package:graphics':

matplot

Attaching package: 'zoo'

The following objects are masked from 'package:base':

as.Date, as.Date.numeric

sde 2.0.15
Companion package to the book
'Simulation and Inference for Stochastic Differential Equations With R Examples'
Iacus, Springer NY, (2008)
To check the errata corrige of the book, type vignette("sde.errata")
[1] 1.066929
```

sde documentation built on May 31, 2017, 3:58 a.m.