mlmc (version 1.0.0)

opre_l: Financial options using an Euler-Maruyama discretisation

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.

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)