R/calibrate.BlackCox.R

In CreditRisk: Evaluation of Credit Risk with Structural and Reduced Form Models

#' @title Black and Cox model calibration to market CDS data
#'
#' @description
#' Compares CDS rates quoted on the market with theoric CDS rates calculeted by the function
#' \code{cds} and looks for the parameters to be used into \code{BlackCox}
#' for returning the default intensities corresponding to real market CDS rates performing the
#' minimization of the objective function.
#'
#' @param V0 firm value at time \code{t = 0}.
#' @param cdsrate CDS rates from the market.
#' @param r risk-free rate.
#' @param t a vector of debt maturity structure.
#' @param ... additional parameters used in \code{cds} function.
#'
#' @return
#' \code{calibrate.BlackCox} returns an object of class \code{data.frame} with calculated parameters of
#' the \code{BlackCox} model and the error occurred in the minimization procedure.
#'
#' @details
#' Inside \code{calibrate.BlackCox}, the function \code{objfn} takes the input a
#' vector of parameters and returns the mean error occurred estimating CDS rates with
#' \code{cds} function. The inputs used in \code{cds} are the default intensities calculated by
#' the \code{BlackCox} function with the calibrated parameters. In particular the error is
#' calculated as:
#' \deqn{\frac{1}{n}\sum_{i=1}^n (c^{ds}-c^{ds}_{mkt})^2.} This quantity is a function of
#' the default intensities and it is the objective function to be minimized in order to take
#' optimal solutions for intensities.
#'
#' @references
#' Damiano Brigo, Massimo Morini, Andrea Pallavicini (2013)
#' Counterparty Credit Risk, Collateral and Funding.
#' With Pricing Cases for All Asset Classes
#'
#' @examples
#' calibrate.BlackCox(V0 = 1, cdsrate = cdsdata$Par.spread, r = 0.005, t = cdsdata$Maturity)
#'
#'@export
#'
calibrate.BlackCox <- function(V0, cdsrate, r, t, ...){

  if(!is.numeric(V0)) stop("V0 must be a number")
  if(length(V0) != 1) stop("V0 must be a number and not a vector")
  if(!is.numeric(r)) stop("r must be a number")
  if(length(r) != 1) stop("r must be constant for all maturities")
  if(!is.numeric(cdsrate)) stop("cdsrate must be a numeric vector")
  if(!is.numeric(t)) stop("Time t must be a numeric vector")
  if(length(cdsrate) != length(t)) stop("cdsrate and t must have the same length")

	objfn <- function(par, cdsrate, r, t){

	      L <- par[1]
	      K <- par[1] - par[2]
        sigma <- abs(par[3])
    	  gamma <- par[4]

    		intensity <- BlackCox(L = L, K = K, V0 = V0, sigma = sigma, r = r, gamma = gamma, t = t)$Default.Intensity
    	  rate <- cds(t = t, int = intensity, r = r, ...)$Rate

    		err <- mean((rate - cdsrate)^2)
    		return(err)
	}

	par <- c(V0/2, (0.15 * V0/2), 0.2, 0.08)
	out <- stats::optim(par = par, fn = objfn, cdsrate = cdsrate, t = t, r = r, method = 'L-BFGS-B', lower = 0 * t)

	out <- c(V0, par[1], (par[1] - par[2]), abs(out$par[3]), out$par[4], out$value)
	names(out) <- c('V0', 'L', 'K', 'sigma', 'gamma', 'Error')

	return(data.frame(t(out)))
}