R/RcppExports.R

In mlmc: Multi-Level Monte Carlo

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#' Financial options using a Milstein discretisation
#'
#' Financial options based on scalar geometric Brownian motion, similar to Mike Giles' MCQMC06 paper, Giles (2008), using a Milstein discretisation.
#'
#' This function is based on GPL-2 C++ code by Mike Giles.
#'
#' @param l
#'        the level to be simulated.
#' @param N
#'        the number of samples to be computed.
#' @param option
#'        the option type, between 1 and 5.
#'        The options are:
#'        \describe{
#'          \item{1 = European call;}{}
#'          \item{2 = Asian call;}{}
#'          \item{3 = lookback call;}{}
#'          \item{4 = digital call;}{}
#'          \item{5 = barrier call.}{}
#'        }
#'
#' @return A named list containing: \describe{
#'           \item{\code{sums}}{is a vector of length six \eqn{\left(\sum Y_i, \sum Y_i^2, \sum Y_i^3, \sum Y_i^4, \sum X_i, \sum X_i^2\right)} where \eqn{Y_i} are iid simulations with expectation \eqn{E[P_0]} when \eqn{l=0} and expectation \eqn{E[P_l-P_{l-1}]} when \eqn{l>0}, and \eqn{X_i} are iid simulations with expectation \eqn{E[P_l]}.
#'                              Note that only the first two components of this are used by the main \code{\link[=mlmc]{mlmc()}} driver, the full vector is used by \code{\link[=mlmc.test]{mlmc.test()}} for convergence tests etc;}
#'           \item{\code{cost}}{is a scalar with the total cost of the paths simulated, computed as \eqn{N \times 2^l} for level \eqn{l}.}
#'         }
#'
#' @author Louis Aslett <louis.aslett@durham.ac.uk>
#' @author Mike Giles <Mike.Giles@maths.ox.ac.uk>
#'
#' @references
#' Giles, M. (2008) 'Improved Multilevel Monte Carlo Convergence using the Milstein Scheme', in A. Keller, S. Heinrich, and H. Niederreiter (eds) \emph{Monte Carlo and Quasi-Monte Carlo Methods 2006}. Berlin, Heidelberg: Springer, pp. 343–358. Available at: \doi{10.1007/978-3-540-74496-2_20}.
#'
#' @examples
#' \donttest{
#' # These are similar to the MLMC tests for the MCQMC06 paper
#' # using a Milstein discretisation with 2^l timesteps on level l
#' #
#' # The figures are slightly different due to:
#' # -- change in MSE split
#' # -- change in cost calculation
#' # -- different random number generation
#' # -- switch to S_0=100
#' #
#' # Note the following takes quite a while to run, for a toy example see after
#' # this block.
#'
#' N0   <- 200 # initial samples on coarse levels
#' Lmin <- 2 # minimum refinement level
#' Lmax <- 10 # maximum refinement level
#'
#' test.res <- list()
#' for(option in 1:5) {
#'   if(option == 1) {
#'     cat("\n ---- Computing European call ---- \n")
#'     N      <- 20000 # samples for convergence tests
#'     L      <- 8 # levels for convergence tests
#'     Eps    <- c(0.005, 0.01, 0.02, 0.05, 0.1)
#'   } else if(option == 2) {
#'     cat("\n ---- Computing Asian call ---- \n")
#'     N      <- 20000 # samples for convergence tests
#'     L      <- 8 # levels for convergence tests
#'     Eps    <- c(0.005, 0.01, 0.02, 0.05, 0.1)
#'   } else if(option == 3) {
#'     cat("\n ---- Computing lookback call ---- \n")
#'     N      <- 20000 # samples for convergence tests
#'     L      <- 10 # levels for convergence tests
#'     Eps    <- c(0.005, 0.01, 0.02, 0.05, 0.1)
#'   } else if(option == 4) {
#'     cat("\n ---- Computing digital call ---- \n")
#'     N      <- 200000 # samples for convergence tests
#'     L      <- 8 # levels for convergence tests
#'     Eps    <- c(0.01, 0.02, 0.05, 0.1, 0.2)
#'   } else if(option == 5) {
#'     cat("\n ---- Computing barrier call ---- \n")
#'     N      <- 200000 # samples for convergence tests
#'     L      <- 8 # levels for convergence tests
#'     Eps    <- c(0.005, 0.01, 0.02, 0.05, 0.1)
#'   }
#'
#'   test.res[[option]] <- mlmc.test(mcqmc06_l, N, L, N0, Eps, Lmin, Lmax, option = option)
#'
#'   # print exact analytic value, based on S0=K
#'   T   <- 1
#'   r   <- 0.05
#'   sig <- 0.2
#'   K   <- 100
#'   B   <- 0.85*K
#'
#'   k   <- 0.5*sig^2/r;
#'   d1  <- (r+0.5*sig^2)*T / (sig*sqrt(T))
#'   d2  <- (r-0.5*sig^2)*T / (sig*sqrt(T))
#'   d3  <- (2*log(B/K) + (r+0.5*sig^2)*T) / (sig*sqrt(T))
#'   d4  <- (2*log(B/K) + (r-0.5*sig^2)*T) / (sig*sqrt(T))
#'
#'   if(option == 1) {
#'     val <- K*( pnorm(d1) - exp(-r*T)*pnorm(d2) )
#'   } else if(option == 2) {
#'     val <- NA
#'   } else if(option == 3) {
#'     val <- K*( pnorm(d1) - pnorm(-d1)*k - exp(-r*T)*(pnorm(d2) - pnorm(d2)*k) )
#'   } else if(option == 4) {
#'     val <- K*exp(-r*T)*pnorm(d2)
#'   } else if(option == 5) {
#'     val <- K*(                             pnorm(d1) - exp(-r*T)*pnorm(d2) -
#'               ((K/B)^(1-1/k))*((B^2)/(K^2)*pnorm(d3) - exp(-r*T)*pnorm(d4)) )
#'   }
#'
#'   if(is.na(val)) {
#'     cat(sprintf("\n Exact value unknown, MLMC value: %f \n", test.res[[option]]$P[1]))
#'   } else {
#'     cat(sprintf("\n Exact value: %f, MLMC value: %f \n", val, test.res[[option]]$P[1]))
#'   }
#'
#'   # plot results
#'   plot(test.res[[option]])
#' }
#' }
#'
#' # The level sampler can be called directly to retrieve the relevant level sums:
#' mcqmc06_l(l = 7, N = 10, option = 1)
#'
#' @export
mcqmc06_l <- function(l, N, option) {
    .Call('_mlmc_mcqmc06_l', PACKAGE = 'mlmc', l, N, option)
}