# Vectorized Black-Scholes pricing of european-exercise options

### Description

Price options according to the famous Black-Scholes formula, with the optional addition of a jump-to-default intensity and discrete dividends.

### Usage

1 2 | ```
blackscholes(callput, S0, K, r, time, vola, default_intensity = 0,
divrate = 0, borrow_cost = 0, dividends = NULL)
``` |

### Arguments

`callput` |
1 for calls, -1 for puts |

`S0` |
initial underlying price |

`K` |
strike |

`r` |
risk-free interest rate |

`time` |
Time from |

`vola` |
Default-free volatility of the underlying |

`default_intensity` |
hazard rate of underlying default |

`divrate` |
A continuous rate for dividends and other cashflows such as foreign interest rates |

`borrow_cost` |
A continuous rate for stock borrow costs |

`dividends` |
A |

### Details

Note that if the `default_intensity`

is set larger than zero then
put-call parity still holds. Greeks are reduced according to cumulated default
probability.

All inputs must either be scalars or have the same nonscalar shape.

### Value

A list with elements

`Price`

The present value(s)

`Delta`

Sensitivity to underlying price

`Vega`

Sensitivity to volatility

### See Also

Other Equity Independent Default Intensity: `american_implied_volatility`

,
`american`

,
`black_scholes_on_term_structures`

,
`equivalent_bs_vola_to_jump`

,
`equivalent_jump_vola_to_bs`

,
`implied_volatilities_with_rates_struct`

,
`implied_volatilities`

,
`implied_volatility_with_term_struct`

,
`implied_volatility`

Other European Options: `black_scholes_on_term_structures`

,
`implied_volatilities_with_rates_struct`

,
`implied_volatilities`

,
`implied_volatility_with_term_struct`

,
`implied_volatility`

### Examples

1 2 | ```
blackscholes(callput=-1, S0=100, K=90, r=0.03, time=1, # -1 is a PUT
vola=0.5, default_intensity=0.07)
``` |