opre_l: Financial options using an Euler-Maruyama discretisation

In louisaslett/mlmc: Multi-Level Monte Carlo

Description

Financial options based on scalar geometric Brownian motion and Heston models, similar to Mike Giles' original 2008 Operations Research paper, using an Euler-Maruyama discretisation

Usage

opre_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 = Heston model.

Details

This function is based on GPL-2 'Matlab' code by Mike Giles.

Author(s)

Louis Aslett <[e-mail]>

Mike Giles <[e-mail]>

Tigran Nagapetyan <[e-mail]>

References

M.B. Giles. Multilevel Monte Carlo path simulation. Operations Research, 56(3):607-617, 2008.

Examples

## Not run: 
# These are similar to the MLMC tests for the original
# 2008 Operations Research paper, using an Euler-Maruyama
# discretisation with 4^l timesteps on level l.
#
# The differences are:
# -- the plots do not have the extrapolation results
# -- two plots are log_2 rather than log_4
# -- the new MLMC driver is a little different
# -- switch to X_0=100 instead of X_0=1

M    <- 4 # refinement cost factor
N0   <- 1000 # initial samples on coarse levels
Lmin <- 2 # minimum refinement level
Lmax <- 6 # maximum refinement level

test.res <- list()
for(option in 1:5) {
  if(option==1) {
    cat("\n ---- Computing European call ---- \n")
    N      <- 2000000 # samples for convergence tests
    L      <- 5 # 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      <- 2000000 # samples for convergence tests
    L      <- 5 # 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      <- 2000000 # samples for convergence tests
    L      <- 5 # levels for convergence tests
    Eps    <- c(0.01, 0.02, 0.05, 0.1, 0.2)
  } else if(option==4) {
    cat("\n ---- Computing digital call ---- \n")
    N      <- 3000000 # samples for convergence tests
    L      <- 5 # levels for convergence tests
    Eps    <- c(0.02, 0.05, 0.1, 0.2, 0.5)
  } else if(option==5) {
    cat("\n ---- Computing Heston model ---- \n")
    N      <- 2000000 # samples for convergence tests
    L      <- 5 # levels for convergence tests
    Eps    <- c(0.005, 0.01, 0.02, 0.05, 0.1)
  }

  test.res[[option]] <- mlmc.test(opre_l, M, 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

  d1  <- (r+0.5*sig^2)*T / (sig*sqrt(T))
  d2  <- (r-0.5*sig^2)*T / (sig*sqrt(T))

  if(option==1) {
    val <- K*( pnorm(d1) - exp(-r*T)*pnorm(d2) )
    cat(sprintf("\n Exact value: %f, MLMC value: %f \n", val, test.res[[option]]$P[1]))
  } else if(option==3) {
    k   <- 0.5*sig^2/r
    val <- K*( pnorm(d1) - pnorm(-d1)*k - exp(-r*T)*(pnorm(d2) - pnorm(d2)*k) )
    cat(sprintf("\n Exact value: %f, MLMC value: %f \n", val, test.res[[option]]$P[1]))
  } else if(option==4) {
    val <- K*exp(-r*T)*pnorm(d2)
    cat(sprintf("\n Exact value: %f, MLMC value: %f \n", val, test.res[[option]]$P[1]))
  }

  # plot results
  plot(test.res[[option]])
}

## End(Not run)

# The level sampler can be called directly to retrieve the relevant level sums:
opre_l(l=7, N=10, option=1)

louisaslett/mlmc documentation built on May 21, 2019, 7:49 a.m.