R/HullWhite.R

In loftina/FinMod: Financial Models for Variance and Volatility

#' @title Hull-White Realized Variance and Volitility
#' @description The Hull White model is used to estimate the deviation from the long term average of the log return of the stock market
#' @usage HullWhite(ticker, start_date, end_date, prediction_period, days_to_drop, kappa, zeta, variance, testing)
#' @param ticker Stock ticker
#' @param start_date Starting date of data to use
#' @param end_date Ending date of data to use
#' @param prediction_period How many days to predict using the model
#' @param days_to_drop Number of days of realized variances to drop when fitting the model
#' @param kappa The drift coefficient of the model.
#' @param zeta The Diffusion coefficient of the model.
#' @param variance Boolean where true predicts variance and false predicts volitility
#' @param testing Whether to use prediction_period days prior to end_date as the prediction period. If false, the prediction period takes place after end_date
#' @return Realized variance or volitility according to the Hull-White model when the correct parameter values are used
HullWhite <- function(ticker, start_date, end_date, prediction_period, days_to_drop, kappa, zeta, variance, testing) {
  stock <- getSymbols(Symbols = ticker, from = start_date, to = end_date, auto.assign = F)
  close_price <- stock[,4]
  log_returns <- diff(log(close_price))[-1]
  time <- 1:length(log_returns)

  realized_variance <- cumsum(log_returns^2)/time
  if (days_to_drop > 0)
  {
    realized_variance <- realized_variance[seq(-1,-days_to_drop)]
  }
  realized_volitility <- sqrt(realized_variance)

  time <- time[1:(length(time)-days_to_drop)]

  if(testing) {
    train_realized_variance <- realized_variance[1:(length(realized_variance)-prediction_period)]
    train_realized_volitility <- realized_volitility[1:(length(realized_volitility)-prediction_period)]
    train_time <- time[1:(length(time)-prediction_period)]
    full_time <- time
  }
  else {
    train_realized_variance <- realized_variance
    train_realized_volitility <- realized_volitility
    train_time <- time
    full_time <- 1:(length(time)+prediction_period)
  }

  nlm.rvar <- nlsLM(formula = train_realized_variance~HullWhiteRealizedVariance(rep(train_realized_variance[1], length(train_time)), train_time, k), start = c(k=kappa))

  dates <- rep(time(stock)[nrow(stock)], prediction_period)
  dates <- dates + seq(1,prediction_period)
  dates

  predicted_realized_variance <- HullWhiteRealizedVariance(rep(train_realized_variance[1], length(full_time)), full_time, coef(nlm.rvar))

  CIvar <- sqrt(mean((train_realized_variance-predict(nlm.rvar))^2))

  UCvar <- predicted_realized_variance + 1.96*CIvar
  LCvar <- predicted_realized_variance - 1.96*CIvar

  nlm.rvol <- nlsLM(formula = train_realized_volitility~HullWhiteRealizedVolitility(rep(train_realized_variance[1],length(train_time)), train_time, k, z), start = c(k=kappa, z=zeta))

  predicted_realized_volitility <- HullWhiteRealizedVolitility(rep(train_realized_variance[1], length(full_time)), full_time, coef(nlm.rvol)[1], coef(nlm.rvol)[2])

  CIvol <- sqrt(mean((train_realized_volitility-predict(nlm.rvol))^2))

  UCvol <- predicted_realized_volitility + 1.96*CIvol
  LCvol <- predicted_realized_volitility - 1.96*CIvol

  if(variance) {
    plot(time, realized_variance,
         ylim = c(min(LCvar), max(realized_variance)),
         xlim = c(0,length(full_time)),
         main=ticker,
         ylab="Realized Variance",
         xlab="Date")
    lines(full_time, predicted_realized_variance, type='l', col='green')
    lines(full_time, UCvar, col='red')
    lines(full_time, LCvar, col='red')
    abline(v=length(train_time), lty=2, col='blue')
  }
  else {
    plot(time, realized_volitility,
         ylim = c(min(LCvol), max(realized_volitility)),
         xlim = c(0,length(full_time)),
         main=ticker,
         ylab="Realized Volitility",
         xlab="Date")
    lines(full_time, predicted_realized_volitility, type='l', col='green')
    lines(full_time, UCvol, col='red')
    lines(full_time, LCvol, col='red')
    abline(v=length(train_time), lty=2, col='blue')
  }

  if (variance) {
    return_data <- data.frame(dates=1:prediction_period,
                              realized_variance=predicted_realized_variance[(length(predicted_realized_variance)-prediction_period+1):length(predicted_realized_variance)],
                              UC95=UCvar[(length(UCvar)-prediction_period+1):length(UCvar)],
                              LC95=LCvar[(length(LCvar)-prediction_period+1):length(LCvar)])
  }
  else {
    return_data <- data.frame(dates=1:prediction_period,
                              realized_volitility=predicted_realized_volitility[(length(predicted_realized_volitility)-prediction_period+1):length(predicted_realized_volitility)],
                              UC95=UCvol[(length(UCvol)-prediction_period+1):length(UCvol)],
                              LC95=LCvol[(length(LCvol)-prediction_period+1):length(LCvol)])
  }

  return(return_data)
}

#' @title Hull-White Realized Variance
#' @description The Hull-White model's approximation for realized variance.
#' @usage HullWhiteRealizedVariance(realizedVar_0, time, kappa)
#' @param realizedVar_0 Initial realized variance of a stock repeated T times
#' @param time Number of realized variance observations
#' @param kappa The drift coefficient of the model.
#' @return Realized variance according to the Hull-White model when the correct kappa value is used
HullWhiteRealizedVariance <- function(realizedVar_0, time, kappa) {
  return (realizedVar_0*(exp(kappa*time)-1)/(kappa*time))
}


#' @title Hull-White Variance of Realized Variance
#' @description The variance of the realized variance of a stock according to
#' the Hull-White model.
#' @usage HullWhiteVariance(realizedVar_0, time, kappa, zeta)
#' @param realizedVar_0 Initial realized variance of a stock repeated T times
#' @param time Number of realized variance observations
#' @param kappa The drift coefficient of the model.
#' @param zeta The Diffusion coefficient of the model.
#' @return The variance of the realized variance according to the Hull-White model when correct kappa and zeta values are used
HullWhiteVariance <- function(realizedVar_0, time, kappa, zeta) {
  return ( ((2*realizedVar_0^2)/(time^2*(kappa+zeta^2)))*(((1-exp(kappa*time))/kappa)-((1-exp((2*kappa+zeta^2)*time))/(2*kappa+zeta^2)))-((realizedVar_0/(kappa*time))*(exp(kappa*time)-1))^2 )
}


#' @title Hull-White Realized Volitility
#' @description The Hull-White model's approximation for realized volitility.
#' @usage HullWhiteRealizedVolitility(realizedVar_0, time, kappa, zeta)
#' @param realizedVar_0 Initial realized variance of a stock repeated T times
#' @param time Number of realized variance observations
#' @param kappa The drift coefficient of the model.
#' @param zeta The Diffusion coefficient of the model.
#' @return Realized volitility according to the Hull-White model when the correct kappa and zeta values are used
HullWhiteRealizedVolitility <- function (realizedVar_0, time, kappa, zeta) {
  return ( sqrt(HullWhiteRealizedVariance(realizedVar_0, kappa, time))-HullWhiteVariance(realizedVar_0, time, kappa, zeta)/(8*((realizedVar_0/(kappa*time))*(exp(kappa*time)-1))^(3/2)) )
}