mc: Option Pricing via Monte-Carlo Simulation

In enricoschumann/NMOF: Numerical Methods and Optimization in Finance

Option Pricing via Monte-Carlo Simulation

Description

Functions to calculate the theoretical prices of options through simulation.

Usage

gbm(npaths, timesteps, r, v, tau, S0,
    exp.result = TRUE, antithetic = FALSE)
gbb(npaths, timesteps, S0, ST, v, tau,
    log = FALSE, exp.result = TRUE)

Arguments

npaths	the number of paths
timesteps	timesteps per path
r	the mean per unit of time
v	the variance per unit of time
tau	time
S0	initial value
ST	final value of Brownian bridge
log	logical: construct bridge from log series?
exp.result	logical: compute exp of the final path, or return log values?
antithetic	logical: if TRUE, random numbers for only npaths/2 are drawn, and the random numbers are mirrored

Details

gbm generates sample paths of geometric Brownian motion.

gbb generates sample paths of a Brownian bridge by first creating paths of Brownian motion W from time 0 to time T, with W_0 equal to zero. Then, at each t, it subtracts t/T * W_T and adds S0*(1-t/T)+ST*(t/T).

Value

A matrix of sample paths; each column contains the price path of an asset. Even with only a single time-step, the matrix will have two rows (the first row is S0).

Author(s)

Enrico Schumann

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")}

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). http://enricoschumann.net/NMOF.htm#NMOFmanual

See Also

vanillaOptionEuropean

Examples

## price a European option
## ... parameters
npaths <- 5000   ## increase number to get more precise results
timesteps <- 1
S0   <- 100
ST   <- 100
tau  <- 1
r <- 0.01
v   <- 0.25^2

## ... create paths
paths <- gbm(npaths, timesteps, r, v, tau, S0 = S0)

## ... a helper function
mc <- function(paths, payoff, ...)
    payoff(paths, ...)

## ... a payoff function (European call)
payoff <- function(paths, X, r, tau)
    exp(-r * tau) * mean(pmax(paths[NROW(paths), ] - X, 0))

## ... compute and check
mc(paths, payoff, X = 100, r = r, tau = tau)
vanillaOptionEuropean(S0, X = 100, tau = tau, r = r, v = v)$value


## compute delta via forward difference
## (see Gilli/Maringer/Schumann, ch. 9)
h <- 1e-6                 ## a small number
rnorm(1)                  ## make sure RNG is initialised
rnd.seed <- .Random.seed  ## store current seed
paths1 <- gbm(npaths, timesteps, r, v, tau, S0 = S0)
.Random.seed <- rnd.seed
paths2 <- gbm(npaths, timesteps, r, v, tau, S0 = S0 + h)

delta.mc <- (mc(paths2, payoff, X = 100, r = r, tau = tau)-
             mc(paths1, payoff, X = 100, r = r, tau = tau))/h
delta <- vanillaOptionEuropean(S0, X = 100, tau = tau,
                               r = r, v = v)$delta
delta.mc - delta




## a fanplot
steps <- 100
paths <- results <- gbm(1000, steps, r = 0, v = 0.2^2,
                        tau = 1, S0 = 100)

levels <- seq(0.01, 0.49, length.out = 20)
greys  <- seq(0.9,  0.50, length.out = length(levels))

## start with an empty plot ...
plot(0:steps, rep(100, steps+1), ylim = range(paths),
     xlab = "", ylab = "", lty = 0, type = "l")

## ... and add polygons
for (level in levels) {

    l <- apply(paths, 1, quantile, level)
    u <- apply(paths, 1, quantile, 1 - level)
    col <- grey(greys[level == levels])
    polygon(c(0:steps, steps:0), c(l, rev(u)),
            col = col, border = NA)

    ## add border lines
    ## lines(0:steps, l, col = grey(0.4))
    ## lines(0:steps, u, col = grey(0.4))
}

enricoschumann/NMOF documentation built on April 13, 2024, 12:16 p.m.