mcqmc06_l: Financial options using a Milstein discretisation

In mlmc: Multi-Level Monte Carlo

Financial options using a Milstein discretisation

Description

Financial options based on scalar geometric Brownian motion, similar to Mike Giles' MCQMC06 paper, Giles (2008), using a Milstein discretisation.

Usage

mcqmc06_l(l, N, option)

Arguments

l	the level to be simulated.
N	the number of samples to be computed.
option	the option type, between 1 and 5. The options are: 1 = European call; 2 = Asian call; 3 = lookback call; 4 = digital call; 5 = barrier call.

Details

This function is based on GPL-2 C++ code by Mike Giles.

Value

A named list containing:

sums

is a vector of length six \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 Y_i are iid simulations with expectation E[P_0] when l=0 and expectation E[P_l-P_{l-1}] when l>0, and X_i are iid simulations with expectation E[P_l]. Note that only the first two components of this are used by the main mlmc() driver, the full vector is used by mlmc.test() for convergence tests etc;

cost

is a scalar with the total cost of the paths simulated, computed as N \times 2^l for level l.

Author(s)

Louis Aslett <[e-mail]>

Mike Giles <[e-mail]>

References

Giles, M. (2008) 'Improved Multilevel Monte Carlo Convergence using the Milstein Scheme', in A. Keller, S. Heinrich, and H. Niederreiter (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Berlin, Heidelberg: Springer, pp. 343–358. Available at: \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-3-540-74496-2_20")}.

Examples

# 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)

mlmc documentation built on April 4, 2025, 4:14 a.m.