Financial options based on scalar geometric Brownian motion and Heston models, similar to Mike Giles' original 2008 Operations Research paper, Giles (2008), using an Euler-Maruyama discretisation
Value
A named list containing:
sumsis a vector of length six \(\left(\sum Y_i, \sum Y_i^2, \sum Y_i^3, \sum Y_i^4, \sum X_i, \sum X_i^2\right)\) where \(Y_i\) are iid simulations with expectation \(E[P_0]\) when \(l=0\) and expectation \(E[P_l-P_{l-1}]\) when \(l>0\), and \(X_i\) are iid simulations with expectation \(E[P_l]\). Note that only the first two components of this are used by the main
mlmc()driver, the full vector is used bymlmc.test()for convergence tests etc;costis a scalar with the total cost of the paths simulated, computed as \(N \times 4^l\) for level \(l\).
References
Giles, M.B. (2008) 'Multilevel Monte Carlo Path Simulation', Operations Research, 56(3), pp. 607–617. Available at: doi:10.1287/opre.1070.0496 .
Author
Louis Aslett <louis.aslett@durham.ac.uk>
Mike Giles <Mike.Giles@maths.ox.ac.uk>
Tigran Nagapetyan <nagapetyan@stats.ox.ac.uk>
Examples
# \donttest{
# 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
#
# Note the following takes quite a while to run, for a toy example see after
# this block.
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 <- 1000000 # 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 <- 1000000 # 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 <- 1000000 # 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 <- 4000000 # 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, 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
k <- 0.5*sig^2/r;
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) )
} else if(option == 2) {
val <- NA
} else if(option == 3) {
val <- K*( pnorm(d1) - pnorm(-d1)*k - exp(-r*T)*(pnorm(d2) - pnorm(d2)*k) )
} else if(option == 4) {
val <- K*exp(-r*T)*pnorm(d2)
} else if(option == 5) {
val <- NA
}
if(is.na(val)) {
cat(sprintf("\n Exact value unknown, MLMC value: %f \n", test.res[[option]]$P[1]))
} else {
cat(sprintf("\n Exact value: %f, MLMC value: %f \n", val, test.res[[option]]$P[1]))
}
# plot results
plot(test.res[[option]])
}
#>
#> ---- Computing European call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 1e+06 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0190e+01 1.0190e+01 1.6101e+02 1.6101e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 2.1145e-01 1.0421e+01 4.4649e+00 2.0152e+02 2.0357e+01 2.2485e-01 4.0000e+00
#> 2 3.0094e-02 1.0443e+01 1.0645e+00 2.1265e+02 1.2213e+01 9.1521e-02 1.6000e+01
#> 3 5.6597e-03 1.0442e+01 2.7227e-01 2.1523e+02 7.6011e+00 7.8199e-02 6.4000e+01
#> 4 1.5063e-03 1.0456e+01 6.8945e-02 2.1684e+02 6.3145e+00 1.4833e-01 2.5600e+02
#> 5 5.9454e-04 1.0444e+01 1.7243e-02 2.1583e+02 5.9149e+00 1.4228e-01 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 2.126907 (exponent for MLMC weak convergence)
#> beta = 1.998164 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.0447e+01 4.622e+07 2.961e+09 64.05 19926135 1657315 404995 102366 25924
#> 0.0100 1.0452e+01 8.464e+06 1.837e+08 21.70 4264911 354826 86738 21742
#> 0.0200 1.0479e+01 2.144e+06 4.592e+07 21.42 1073618 89427 21938 5647
#> 0.0500 1.0450e+01 2.462e+05 1.815e+06 7.37 152423 12210 2811
#> 0.1000 1.0471e+01 6.718e+04 4.536e+05 6.75 38401 3195 1000
#>
#>
#> Exact value: 10.450584, MLMC value: 10.447168
#>
#> ---- Computing Asian call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 1e+06 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 5.1039e+00 5.1039e+00 4.0293e+01 4.0293e+01 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 6.0032e-01 5.7066e+00 1.5915e+01 5.7602e+01 4.8836e+00 4.4714e-02 4.0000e+00
#> 2 4.9725e-02 5.7615e+00 1.4242e+00 6.2242e+01 5.9414e+00 1.0466e-01 1.6000e+01
#> 3 5.6281e-03 5.7766e+00 1.5273e-01 6.3428e+01 6.2011e+00 1.9430e-01 6.4000e+01
#> 4 7.9738e-04 5.7513e+00 2.4887e-02 6.3392e+01 5.9353e+00 5.4184e-01 2.5600e+02
#> 5 2.2619e-04 5.7534e+00 5.3805e-03 6.3495e+01 5.6904e+00 3.8895e-02 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 2.871053 (exponent for MLMC weak convergence)
#> beta = 2.889940 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 5.7643e+00 2.728e+07 2.165e+08 7.94 7656043 2406391 360110 66135
#> 0.0100 5.7582e+00 6.791e+06 5.413e+07 7.97 1908674 599483 89531 16443
#> 0.0200 5.7394e+00 1.718e+06 1.353e+07 7.88 481200 151668 22704 4170
#> 0.0500 5.7950e+00 1.952e+05 5.311e+05 2.72 64857 20311 3066
#> 0.1000 5.7332e+00 5.266e+04 1.328e+05 2.52 16090 5142 1000
#>
#>
#> Exact value unknown, MLMC value: 5.764282
#>
#> ---- Computing lookback call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 1e+06 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 2.0658e+01 2.0658e+01 1.7529e+02 1.7529e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 -2.5116e+00 1.8108e+01 1.3943e+01 1.9878e+02 4.4978e+00 4.0534e-01 4.0000e+00
#> 2 -6.9227e-01 1.7453e+01 4.8871e+00 2.0963e+02 4.0202e+00 3.9640e-01 1.6000e+01
#> 3 -1.8064e-01 1.7268e+01 1.4316e+00 2.1166e+02 3.7982e+00 4.7191e-02 6.4000e+01
#> 4 -4.4695e-02 1.7256e+01 3.8803e-01 2.1360e+02 3.7314e+00 3.6932e-01 2.5600e+02
#> 5 -1.1395e-02 1.7228e+01 1.0062e-01 2.1308e+02 3.7048e+00 1.8273e-01 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 1.952122 (exponent for MLMC weak convergence)
#> beta = 1.788375 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0100 1.7218e+01 4.764e+07 2.909e+09 61.06 10553225 1490221 439115 118801 31047 8353
#> 0.0200 1.7230e+01 1.199e+07 7.273e+08 60.67 2649178 375181 110072 29909 7829 2107
#> 0.0500 1.7222e+01 1.299e+06 2.916e+07 22.45 348339 49126 14458 3950 1054
#> 0.1000 1.7344e+01 3.310e+05 7.291e+06 22.03 88103 12417 3620 1018 274
#> 0.2000 1.7377e+01 5.614e+04 4.515e+05 8.04 17523 2469 1000 199
#>
#>
#> Exact value: 17.216802, MLMC value: 17.217921
#>
#> ---- Computing digital call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 4e+06 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 5.6990e+01 5.6990e+01 2.1732e+03 2.1732e+03 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 -2.7915e+00 5.4128e+01 2.5774e+02 2.2190e+03 3.2106e+01 4.2415e-01 4.0000e+00
#> 2 -6.9927e-01 5.3474e+01 1.6554e+02 2.2271e+03 5.4194e+01 2.7650e-01 1.6000e+01
#> 3 -1.6958e-01 5.3303e+01 8.6449e+01 2.2291e+03 1.0456e+02 7.1852e-03 6.4000e+01
#> 4 -4.6158e-02 5.3230e+01 4.3383e+01 2.2300e+03 2.0854e+02 1.7451e-01 2.5600e+02
#> 5 -1.0844e-02 5.3216e+01 2.1435e+01 2.2301e+03 4.2211e+02 2.4322e-02 1.0240e+03
#>
#> WARNING: kurtosis on finest level = 422.114439
#> indicates MLMC correction dominated by a few rare paths;
#> for information on the connection to variance of sample variances,
#> see http://mathworld.wolfram.com/SampleVarianceDistribution.html
#>
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 1.993719 (exponent for MLMC weak convergence)
#> beta = 0.910770 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0200 5.3256e+01 7.140e+08 7.612e+09 10.66 71927300 12390737 4961618 1801584 632022 230472
#> 0.0500 5.3235e+01 5.430e+07 3.045e+08 5.61 7934620 1367595 549185 204028 74401
#> 0.1000 5.3231e+01 1.320e+07 7.612e+07 5.77 1955414 337337 135584 49074 17895
#> 0.2000 5.3392e+01 1.362e+06 4.755e+06 3.49 314162 53918 20684 7827
#> 0.5000 5.2787e+01 2.450e+05 7.609e+05 3.11 53156 9434 3822 1452
#>
#>
#> Exact value: 53.232482, MLMC value: 53.256370
#>
#> ---- Computing Heston model ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 2e+06 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0207e+01 1.0207e+01 1.6089e+02 1.6089e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 2.1326e-01 1.0420e+01 3.5529e+00 1.9071e+02 1.4748e+01 2.8790e-03 4.0000e+00
#> 2 3.4245e-02 1.0438e+01 3.6981e+00 1.9162e+02 7.3462e+00 2.6720e-01 1.6000e+01
#> 3 7.2058e-03 1.0447e+01 1.7474e+00 1.9185e+02 6.1217e+00 3.7705e-02 6.4000e+01
#> 4 1.0090e-03 1.0458e+01 5.1187e-01 1.9171e+02 5.5768e+00 1.5339e-01 2.5600e+02
#> 5 9.6171e-05 1.0474e+01 1.3363e-01 1.9202e+02 5.4543e+00 2.7100e-01 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 2.731437 (exponent for MLMC weak convergence)
#> beta = 1.231829 (exponent for MLMC variance)
#> gamma = 2.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.0454e+01 6.432e+07 6.548e+08 10.18 23506656 1743584 890673 305993
#> 0.0100 1.0450e+01 1.610e+07 1.637e+08 10.17 5881570 436419 222705 76735
#> 0.0200 1.0440e+01 1.968e+06 1.022e+07 5.19 1047180 75445 38698
#> 0.0500 1.0471e+01 3.138e+05 1.635e+06 5.21 163782 12814 6170
#> 0.1000 1.0431e+01 8.048e+04 4.088e+05 5.08 42297 3191 1589
#>
#>
#> Exact value unknown, MLMC value: 10.454337
# }
# The level sampler can be called directly to retrieve the relevant level sums:
opre_l(l = 7, N = 10, option = 1)
#> $sums
#> [1] -1.894700e-03 8.694623e-03 -1.054083e-04 2.305858e-05 3.726061e+01
#> [6] 3.477253e+02
#>
#> $cost
#> [1] 163840
#>