## How to perform a Monte Carlo Simulation?
## First Step:
# Write a function to generate the option's innovations.
# Use scrambled normal Sobol numbers:
sobolInnovations = function(mcSteps, pathLength, init, ...) {
# Create Normal Sobol Innovations:
innovations = rnorm.sobol(mcSteps, pathLength, init, ...)
# Return Value:
innovations }
## Second Step:
# Write a function to generate the option's price paths.
# Use a Wiener path:
wienerPath = function(eps) {
# Note, the option parameters must be globally defined!
# Generate the Paths:
path = (b-sigma*sigma/2)*delta.t + sigma*sqrt(delta.t)*eps
# Return Value:
path }
## Third Step:
# Write a function for the option's payoff
# Example 1: use the payoff for a plain Vanilla Call or Put:
plainVanillaPayoff = function(path) {
# Note, the option parameters must be globally defined!
# Compute the Call/Put Payoff Value:
ST = S*exp(sum(path))
if (TypeFlag == "c") payoff = exp(-r*Time)*max(ST-X, 0)
if (TypeFlag == "p") payoff = exp(-r*Time)*max(0, X-ST)
# Return Value:
payoff }
# Example 2: use the payoff for an arithmetic Asian Call or Put:
arithmeticAsianPayoff = function(path) {
# Note, the option parameters must be globally defined!
# Compute the Call/Put Payoff Value:
SM = mean(S*exp(cumsum(path)))
if (TypeFlag == "c") payoff = exp(-r*Time)*max(SM-X, 0)
if (TypeFlag == "p") payoff = exp(-r*Time)*max(0, X-SM)
# Return Value:
payoff }
## Final Step:
# Set Global Parameters for the plain Vanilla / arithmetic Asian Options:
TypeFlag <- "c"; S <- 100; X <- 100
Time <- 1/12; sigma <- 0.4; r <- 0.10; b <- 0.1
# Do the Asian Simulation with scrambled random numbers:
mc = MonteCarloOption(delta.t = 1/360, pathLength = 30, mcSteps = 5000,
mcLoops = 100, init = TRUE, innovations.gen = sobolInnovations,
path.gen = wienerPath, payoff.calc = arithmeticAsianPayoff,
antithetic = TRUE, standardization = FALSE, trace = TRUE,
scrambling = 2, seed = 4711)
# Plot the MC Iteration Path:
par(mfrow = c(1, 1))
mcPrice = cumsum(mc)/(1:length(mc))
plot(mcPrice, type = "l", main = "Arithmetic Asian Option",
xlab = "Monte Carlo Loops", ylab = "Option Price")
# Compare with Turnbull-Wakeman Approximation:
# TW = TurnbullWakemanAsianApproxOption(TypeFlag = "c", S = 100, SA = 100,
# X = 100, Time = 1/12, time = 1/12, tau = 0 , r = 0.1, b = 0.1,
# sigma = 0.4)
# print(TW)
# abline(h = TW, col = 2)
