Estimates ES of American Put Option using binomial tree to price the option valuation tree and Monte Carlo simulation with a binomial option valuation tree nested within the MCS. Historical method to compute the VaR.

1 2 | ```
AmericanPutESSim(amountInvested, stockPrice, strike, r, mu, sigma, maturity,
numberTrials, numberSteps, cl, hp)
``` |

`amountInvested` |
Total amount paid for the Put Option and is positive (negative) if the option position is long (short) |

`stockPrice` |
Stock price of underlying stock |

`strike` |
Strike price of the option |

`r` |
Risk-free rate |

`mu` |
Expected rate of return on the underlying asset and is in annualised term |

`sigma` |
Volatility of the underlying stock and is in annualised term |

`maturity` |
The term to maturity of the option in days |

`numberTrials` |
The number of interations in the Monte Carlo simulation exercise |

`numberSteps` |
The number of steps over the holding period at each of which early exercise is checked and is at least 2 |

`cl` |
Confidence level for which VaR is computed and is scalar |

`hp` |
Holding period of the option in days and is scalar |

Monte Carlo Simulation VaR estimate and the bounds of the 95 confidence interval for the VaR, based on an order-statistics analysis of the P/L distribution

Dinesh Acharya

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

1 2 | ```
# Market Risk of American Put with given parameters.
AmericanPutESSim(0.20, 27.2, 25, .16, .2, .05, 60, 30, 20, .95, 30)
``` |

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