mc function

Option Pricing via Monte-Carlo Simulation

Functions to calculate the theoretical prices of options through simulation.

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).

Returns

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).

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")

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

Author(s)

Enrico Schumann

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))
}