R/Xi_call_price_convex.R

In mociepa/ShortfallRiskHedging: Shortfall Risk Hedging for european option

#' @title Delta hedging for modified european call option
#'
#' @description
#' The Xi_call_price_convex function takes parameters from Black-Scholes model and returns a number of stock needed to fully hedge modified european call option.
#'
#' @usage Xi_call_price_convex(asset, strike, rate, vol, drift, p, time, End_Time, L, L2 = NA)
#'
#' @param asset a numeric vector of asset prices.
#' @param strike numeric value, strike price for call or put option.
#' @param rate numeric value, risk free rate in the model, r >= 0.
#' @param vol numeric value, volatility of the model, vol > 0.
#' @param drift numeric value, drift of the model.
#' @param time a numeric vector of actual time, time > 0.
#' @param p numeric value, power of the loss function, p > 1.
#' @param End_Time end time of the option, End_time >= time.
#' @param L numeric value, determines option payoff, L > 0.
#' @param L2 numeric value, determines option payoff, if L2 = NA, but is needed, function finds it with Newton's algorithm.
#' @return A numeric vector, number of asset to hedge modification of european call option using convex loss function.
#'
#'
#' @examples
#' Xi_call_price_convex(100, 100, 0, 0.5, 0.05, 2,  0, 1, 105)
#' Xi_call_price_convex(c(100, 120), 100, 0, 0.3, 0.05, 2, 0, 1, 105)
#' Xi_call_price_convex(c(100, 120), 100, 0, 0.3, 0.05, 2, c(0, 0.5), 1, 105)
#'
#'
#'
#' @export

Xi_call_price_convex <- function(asset, strike, rate, vol, drift, p, time, End_Time, L, L2 = NA){
  if (p <= 1){
    stop("Wrong p argument. p > 1")
  }
  m = drift - rate
  k = m / ( vol^2*(p-1) )
  l = 0.5 * vol^2 - rate + 0.5*m / (p-1)
  tau = End_Time - time

  if (length(tau) == 1){
    tau <- rep(tau, length(asset))
  }

  result <- c()
  for (i in 1:length(asset)) {
    if(asset[i] == 0){
      asset[i] <- 1e-100
    }


  }

  if (-k <= 1 & m != 0){
    result1 <- Xi_call_price(asset, L, rate, vol, time, End_Time) - k*(strike - L)*L^k*asset^(-k - 1)*exp(-rate*tau)*exp(k*tau*l)*pnorm( d2(asset, L, rate, vol, time, End_Time) - m*sqrt(tau)/(vol*(p-1)) )

    result2 <- Xi_call_price(asset, L, rate, vol, time, End_Time) + 1/(vol*sqrt(tau))*dnorm( d1(asset, L, rate, vol, time, End_Time) )*(1 - strike/L) -
      k*(strike - L)*L^k*asset^(-k - 1)*exp(-rate*tau)*exp(k*tau*l)*pnorm( d2(asset, L, rate, vol, time, End_Time) - m*sqrt(tau)/(vol*(p-1)) ) +
      (strike - L)*(L/asset)^k*exp(-rate*tau)*exp(k*tau*l)*dnorm( d2(asset, L, rate, vol, time, End_Time) - m*sqrt(tau)/(vol*(p-1)) )*1/(asset*vol*sqrt(tau))
    result <- ifelse(tau == 0, result1, result2)
  }

  else if (-k > 1 & m != 0){

    if ( is.na(L2) ){
      warning("In this case L2 is needed, while parameter L2 = NA. Calculated L2 argument by the Newton method (call_Newton_convex function).")
      L_bis <- call_Newton_convex(L, strike, drift, rate, vol, p)[1]
      L1 <- min(L, L_bis)
      L2 <- max(L, L_bis)
    }
    else{
      L1 <- min(L, L2)
      L2 <- max(L, L2)
    }

    result1 <- pnorm( -d1(asset, L2, rate, vol, time, End_Time) ) - pnorm( -d1(asset, L1, rate, vol, time, End_Time) ) -
      k*(strike - L1)*L1^k*asset^(-k - 1)*exp(-rate*tau)*exp(k*tau*l)*(pnorm( - d2(asset, L2, rate, vol, time, End_Time) + m*sqrt(tau)/(vol*(p-1)) ) - pnorm( -d2(asset, L1, rate, vol, time, End_Time) + m*sqrt(tau)/(vol*(p-1)) ) )

    result2 <- pnorm( -d1(asset, L2, rate, vol, time, End_Time) ) - pnorm( -d1(asset, L1, rate, vol, time, End_Time) ) - 1/(vol*sqrt(tau))*(dnorm( -d1(asset, L2, rate, vol, time, End_Time) ) - dnorm( -d1(asset, L1, rate, vol, time, End_Time) )) +
      strike/(vol*sqrt(tau))*(1/L2*dnorm( -d1(asset, L2, rate, vol, time, End_Time)) - 1/L1*dnorm( -d1(asset, L1, rate, vol, time, End_Time) )) -
      k*(strike - L1)*L1^k*asset^(-k - 1)*exp(-rate*tau)*exp(k*tau*l)*(pnorm( - d2(asset, L2, rate, vol, time, End_Time) + m*sqrt(tau)/(vol*(p-1)) ) - pnorm( -d2(asset, L1, rate, vol, time, End_Time) + m*sqrt(tau)/(vol*(p-1)) ) ) -
      (strike - L1)*(L/asset)^k*exp(-rate*tau)*exp(k*tau*l)*1/(asset*vol*sqrt(tau))*(dnorm( - d2(asset, L2, rate, vol, time, End_Time) + m*sqrt(tau)/(vol*(p-1)) ) - dnorm( -d2(asset, L1, rate, vol, time, End_Time) + m*sqrt(tau)/(vol*(p-1)) ) )
    result <- ifelse(tau == 0, result1, result2)
  }

  else{
    result <- Xi_call_price(asset, L, rate, vol, time, End_Time)
  }

  return(result)
}