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#' Estimates ES of American vanilla put using binomial option valuation tree and Monte Carlo
#' Simulation
#'
#' 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.
#'
#' @param amountInvested Total amount paid for the Put Option and is positive
#' (negative) if the option position is long (short)
#' @param stockPrice Stock price of underlying stock
#' @param strike Strike price of the option
#' @param r Risk-free rate
#' @param mu Expected rate of return on the underlying asset and is in
#' annualised term
#' @param sigma Volatility of the underlying stock and is in annualised
#' term
#' @param maturity The term to maturity of the option in days
#' @param numberTrials The number of interations in the Monte Carlo simulation
#' exercise
#' @param numberSteps The number of steps over the holding period at each
#' of which early exercise is checked and is at least 2
#' @param cl Confidence level for which VaR is computed and is scalar
#' @param hp Holding period of the option in days and is scalar
#' @return 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
#' @references Dowd, Kevin. Measuring Market Risk, Wiley, 2007.
#'
#' Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles,
#' Mathematics, Algorithms, Cambridge University Press, 2002.
#'
#' @author Dinesh Acharya
#' @examples
#'
#' # Market Risk of American Put with given parameters.
#' AmericanPutESSim(0.20, 27.2, 25, .16, .2, .05, 60, 30, 20, .95, 30)
#'
#' @export
AmericanPutESSim <- function(amountInvested, stockPrice, strike, r, mu, sigma,
maturity, numberTrials, numberSteps, cl, hp){
# Precompute Constants
annualMaturity <- maturity / 360 # Annualised maturity
annualHp <- hp / 360 # Annualised holding period
N <- numberSteps # Number of steps
dt <- annualHp / N # Size of time-increment equal to holding period
nudt <- (mu - .5 * sigma^2) * dt
sigmadt <- sigma * sqrt(dt)
lnS <- log(stockPrice)
M <- numberTrials
initialOptionPrice <- AmericanPutPriceBinomial(stockPrice, strike, r, sigma, maturity, N)
numberOfOptions <- abs(amountInvested) / initialOptionPrice
# Stock price simulation process
lnSt <- matrix(0, M, N)
newStockPrice <- matrix(0, M, N)
for (i in 1:M){
lnSt[i, 1] <- lnS + rnorm(1, nudt, sigmadt)
newStockPrice[i, 1] <- exp(lnSt[i, 1])
for (j in 2:N){
lnSt[i, j] <- lnSt[i, j - 1] + rnorm(1, nudt, sigmadt)
newStockPrice[i, j] <- exp(lnSt[i,j]) # New stock price
}
}
# Option calculation over time
newOptionValue <- matrix(0, M, N-1)
for (i in 1:M) {
for (j in 1:(N-1)) {
newOptionValue[i, j] <- AmericanPutPriceBinomial(newStockPrice[i, j],
strike, r, sigma, maturity - j * hp / N, N)
}
}
# Profit/Loss
profitOrLoss <- (newOptionValue - initialOptionPrice)*numberOfOptions
# Now adjust for short position
if (amountInvested < 0) {# If option position is short
profitOrLoss <- -profitOrLoss
}
# VaR estimation
ES <- HSESDFPerc(profitOrLoss, .5, cl) # VaR
confidenceInterval <- c(HSESDFPerc(profitOrLoss, .025, cl), HSESDFPerc(profitOrLoss, .975, cl))
return(list('ES' = ES, 'confidenceInterval' = confidenceInterval))
}
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