# Price one or more american-exercise options

### Description

Use a control-variate scheme to simultaneously estimate the present values of a collection of one or more American-exercise options under a default model with survival probabilities not linked to equity prices.

### Usage

1 2 3 4 5 6 7 | ```
american(callput, S0, K, time, const_short_rate = 0,
const_default_intensity = 0, discount_factor_fcn = function(T, t, ...) {
exp(-const_short_rate * (T - t)) }, survival_probability_fcn = function(T,
t, ...) { exp(-const_default_intensity * (T - t)) },
default_intensity_fcn = function(t, S, ...) { const_default_intensity +
0 * S }, ..., num_time_steps = 100, structure_constant = 2,
std_devs_width = 5)
``` |

### Arguments

`callput` |
1 for calls, -1 for puts (may be a vector of the same) |

`S0` |
initial underlying price |

`K` |
strike (may be a vector) |

`time` |
Time from |

`const_short_rate` |
A constant to use for the instantaneous interest rate in case |

`const_default_intensity` |
A constant to use for the instantaneous default intensity in case |

`discount_factor_fcn` |
A function for computing present values to
time |

`survival_probability_fcn` |
(Implied argument) A function for probability of survival, with
arguments |

`default_intensity_fcn` |
A function for computing default intensity
occurring at a given time, dependent on time and stock price, with
arguments |

`...` |
Further arguments passed on to |

`num_time_steps` |
Number of steps to use in the grid solver. Can usually be set quite low due to the control variate scheme. |

`structure_constant` |
The maximum ratio between time intervals |

`std_devs_width` |
The number of standard deviations, in |

### Details

The scheme uses find_present_value() to price the options and their European-exercise equivalents. It then compares the latter to black-scholes formula output and uses the results as an error correction on the prices of the American-exercise options.

### Value

A vector of estimated option present values

### See Also

Other American Exercise Equity Options: `american_implied_volatility`

,
`control_variate_pairs`

Other Equity Independent Default Intensity: `american_implied_volatility`

,
`black_scholes_on_term_structures`

,
`blackscholes`

,
`equivalent_bs_vola_to_jump`

,
`equivalent_jump_vola_to_bs`

,
`implied_volatilities_with_rates_struct`

,
`implied_volatilities`

,
`implied_volatility_with_term_struct`

,
`implied_volatility`

### Examples

1 2 3 4 5 6 |

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