# 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

 `1` ```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 <[email protected]>

Mike Giles <[email protected]>

Tigran Nagapetyan <[email protected]>

## References

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76``` ```## 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, 2017, 2:39 p.m.