# 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

1 2 3 4 |

### Arguments

`x` |
vector of quantiles. |

`p` |
vector of probabilities. |

`Dt` |
lag or time. |

`x0` |
the value of the process at time |

`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 |

`lower.tail` |
logical; if TRUE (default), probabilities are |

### 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 |