Financial options based on scalar geometric Brownian motion, similar to Mike Giles' MCQMC06 paper, Giles (2008), using a Milstein 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 \(N \times 2^l\) for level \(l\).
References
Giles, M. (2008) 'Improved Multilevel Monte Carlo Convergence using the Milstein Scheme', in A. Keller, S. Heinrich, and H. Niederreiter (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Berlin, Heidelberg: Springer, pp. 343–358. Available at: doi:10.1007/978-3-540-74496-2_20 .
Examples
# \donttest{
# These are similar to the MLMC tests for the MCQMC06 paper
# using a Milstein discretisation with 2^l timesteps on level l
#
# The figures are slightly different due to:
# -- change in MSE split
# -- change in cost calculation
# -- different random number generation
# -- switch to S_0=100
#
# Note the following takes quite a while to run, for a toy example see after
# this block.
N0 <- 200 # initial samples on coarse levels
Lmin <- 2 # minimum refinement level
Lmax <- 10 # maximum refinement level
test.res <- list()
for(option in 1:5) {
if(option == 1) {
cat("\n ---- Computing European call ---- \n")
N <- 20000 # samples for convergence tests
L <- 8 # 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 <- 20000 # samples for convergence tests
L <- 8 # 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 <- 20000 # samples for convergence tests
L <- 10 # levels for convergence tests
Eps <- c(0.005, 0.01, 0.02, 0.05, 0.1)
} else if(option == 4) {
cat("\n ---- Computing digital call ---- \n")
N <- 200000 # samples for convergence tests
L <- 8 # levels for convergence tests
Eps <- c(0.01, 0.02, 0.05, 0.1, 0.2)
} else if(option == 5) {
cat("\n ---- Computing barrier call ---- \n")
N <- 200000 # samples for convergence tests
L <- 8 # levels for convergence tests
Eps <- c(0.005, 0.01, 0.02, 0.05, 0.1)
}
test.res[[option]] <- mlmc.test(mcqmc06_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
B <- 0.85*K
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))
d3 <- (2*log(B/K) + (r+0.5*sig^2)*T) / (sig*sqrt(T))
d4 <- (2*log(B/K) + (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 <- K*( pnorm(d1) - exp(-r*T)*pnorm(d2) -
((K/B)^(1-1/k))*((B^2)/(K^2)*pnorm(d3) - exp(-r*T)*pnorm(d4)) )
}
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 = 20000 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.0030e+01 1.0030e+01 1.9725e+02 1.9725e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 1.8716e-01 1.0359e+01 1.6531e-01 2.0988e+02 9.8531e+01 2.3102e-01 2.0000e+00
#> 2 1.0142e-01 1.0191e+01 4.1839e-02 2.0683e+02 2.8282e+01 4.3754e-01 4.0000e+00
#> 3 5.5153e-02 1.0558e+01 1.2325e-02 2.1670e+02 2.1426e+01 5.0386e-01 8.0000e+00
#> 4 2.7904e-02 1.0468e+01 3.2321e-03 2.1466e+02 1.7647e+01 1.8893e-01 1.6000e+01
#> 5 1.3913e-02 1.0420e+01 8.2640e-04 2.1465e+02 1.2310e+01 9.9685e-02 3.2000e+01
#> 6 7.2131e-03 1.0479e+01 2.1421e-04 2.1745e+02 1.1577e+01 8.2241e-02 6.4000e+01
#> 7 3.5650e-03 1.0475e+01 5.2975e-05 2.1736e+02 1.0232e+01 1.0760e-02 1.2800e+02
#> 8 1.7857e-03 1.0545e+01 1.3436e-05 2.1571e+02 1.0093e+01 1.0816e-01 2.5600e+02
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.963583 (exponent for MLMC weak convergence)
#> beta = 1.937392 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.0458e+01 1.361e+07 2.945e+09 216.36 11917020 231958 98151 33333 12301 4375 1552 615 223
#> 0.0100 1.0439e+01 3.356e+06 3.710e+08 110.52 2953179 65020 21963 7800 2913 1068 386 136
#> 0.0200 1.0470e+01 8.371e+05 4.639e+07 55.41 740921 14276 5398 2185 776 293 106
#> 0.0500 1.0461e+01 1.312e+05 3.663e+06 27.93 117176 2060 829 365 132 48
#> 0.1000 1.0304e+01 3.137e+04 4.579e+05 14.60 28620 543 200 64 22
#>
#>
#> Exact value: 10.450584, MLMC value: 10.458105
#>
#> ---- Computing Asian call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 20000 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 5.6185e+00 5.6185e+00 6.0313e+01 6.0313e+01 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 4.1755e-02 5.6931e+00 2.6371e-01 6.2423e+01 2.3578e+01 9.5912e-02 2.0000e+00
#> 2 3.9962e-02 5.7058e+00 3.7704e-02 6.1951e+01 1.2360e+01 8.0573e-02 4.0000e+00
#> 3 2.7370e-02 5.7290e+00 6.0579e-03 6.2682e+01 8.1099e+00 1.2570e-02 8.0000e+00
#> 4 1.5138e-02 5.7909e+00 1.2289e-03 6.4331e+01 7.6054e+00 1.3822e-01 1.6000e+01
#> 5 7.5845e-03 5.7123e+00 2.6881e-04 6.3121e+01 7.4456e+00 2.5437e-01 3.2000e+01
#> 6 3.8863e-03 5.6738e+00 6.2498e-05 6.1495e+01 7.9930e+00 1.2635e-01 6.4000e+01
#> 7 1.9756e-03 5.7571e+00 1.5521e-05 6.2941e+01 7.8415e+00 2.4282e-01 1.2800e+02
#> 8 1.0065e-03 5.8278e+00 3.8637e-06 6.4531e+01 7.4338e+00 2.0594e-01 2.5600e+02
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.818556 (exponent for MLMC weak convergence)
#> beta = 2.269413 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 5.7609e+00 4.738e+06 8.811e+08 185.94 3878099 176768 48852 15162 4375 1512 497 178 67
#> 0.0100 5.7537e+00 1.407e+06 1.074e+08 76.37 1189144 44761 13058 3745 1110 392 142 50
#> 0.0200 5.7585e+00 2.909e+05 1.312e+07 45.09 239760 10909 2985 928 289 104 32
#> 0.0500 5.7167e+00 4.541e+04 1.077e+06 23.72 37708 1729 451 149 48 15
#> 0.1000 5.6693e+00 1.250e+04 1.372e+05 10.98 10250 429 200 38 18
#>
#>
#> Exact value unknown, MLMC value: 5.760949
#>
#> ---- Computing lookback call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 20000 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 1.7603e+01 1.7603e+01 1.8757e+02 1.8757e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 -1.1974e-01 1.7458e+01 1.4071e+00 1.9666e+02 9.2816e+00 4.0921e-02 2.0000e+00
#> 2 -1.3427e-01 1.7475e+01 5.0942e-01 2.0891e+02 9.1129e+00 2.4391e-01 4.0000e+00
#> 3 -8.8887e-02 1.7226e+01 1.4282e-01 2.0632e+02 9.4423e+00 2.5918e-01 8.0000e+00
#> 4 -5.4996e-02 1.7188e+01 4.0886e-02 2.1052e+02 8.3295e+00 2.7305e-02 1.6000e+01
#> 5 -2.6679e-02 1.7198e+01 1.0245e-02 2.1394e+02 8.6646e+00 5.9360e-02 3.2000e+01
#> 6 -1.4201e-02 1.7086e+01 2.6724e-03 2.0766e+02 8.6317e+00 1.5823e-01 6.4000e+01
#> 7 -7.2594e-03 1.7233e+01 6.7415e-04 2.1055e+02 7.3269e+00 2.5092e-01 1.2800e+02
#> 8 -3.5000e-03 1.7275e+01 1.6549e-04 2.1712e+02 7.9473e+00 7.3983e-02 2.5600e+02
#> 9 -1.7809e-03 1.7173e+01 4.1676e-05 2.1442e+02 7.3364e+00 1.6080e-01 5.1200e+02
#> 10 -8.6176e-04 1.7440e+01 1.0424e-05 2.1902e+02 8.6798e+00 4.2827e-01 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.852889 (exponent for MLMC weak convergence)
#> beta = 1.920578 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 1.7220e+01 2.197e+07 1.196e+10 544.53 14695671 908426 378089 146440 54138 19500 6933 2469 917 332 113
#> 0.0100 1.7227e+01 5.350e+06 7.411e+08 138.53 3626992 222962 92265 36156 13314 4941 1741 615 223
#> 0.0200 1.7201e+01 1.334e+06 1.853e+08 138.92 902716 55846 23242 9121 3370 1208 430 150 55
#> 0.0500 1.7265e+01 2.030e+05 7.088e+06 34.92 141854 8734 3548 1379 508 186 68
#> 0.1000 1.7118e+01 5.179e+04 1.772e+06 34.21 35378 2295 934 393 137 50 18
#>
#>
#> Exact value: 17.216802, MLMC value: 17.220350
#>
#> ---- Computing digital call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 2e+05 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 5.6951e+01 5.6951e+01 1.0000e-10 1.0000e-10 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 -2.7021e+00 5.4263e+01 2.0431e-01 7.2354e+02 1.1150e+01 7.9904e-02 2.0000e+00
#> 2 -7.1604e-01 5.3580e+01 1.9385e+00 1.1897e+03 1.9612e+01 7.8338e-02 4.0000e+00
#> 3 -2.0194e-01 5.3346e+01 1.2403e+00 1.5085e+03 2.2141e+01 6.4013e-02 8.0000e+00
#> 4 -5.9328e-02 5.3352e+01 5.4685e-01 1.7222e+03 2.7503e+01 1.2010e-01 1.6000e+01
#> 5 -2.2149e-02 5.3216e+01 2.1220e-01 1.8724e+03 4.3452e+01 1.9886e-01 3.2000e+01
#> 6 -7.9524e-03 5.3246e+01 7.8848e-02 1.9789e+03 5.6893e+01 6.2962e-02 6.4000e+01
#> 7 -3.6181e-03 5.3389e+01 2.7991e-02 2.0497e+03 8.0117e+01 2.4439e-01 1.2800e+02
#> 8 -1.2441e-03 5.3354e+01 1.0449e-02 2.1024e+03 1.1769e+02 5.5995e-02 2.5600e+02
#>
#> WARNING: kurtosis on finest level = 117.692596
#> 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 = 1.561395 (exponent for MLMC weak convergence)
#> beta = 0.879603 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0100 5.3242e+01 3.755e+06 3.498e+09 931.64 200 71109 155815 88148 41078 17950 8299 4072
#> 0.0200 5.3237e+01 7.650e+05 4.222e+08 551.88 200 16210 34921 20467 9378 4267 2224
#> 0.0500 5.3256e+01 1.001e+05 3.195e+07 319.11 200 2404 4796 2991 1591 830
#> 0.1000 5.3214e+01 1.210e+04 3.674e+06 303.70 200 381 842 475 248
#> 0.2000 5.3294e+01 4.020e+03 9.185e+05 228.49 200 200 283 140 73
#>
#>
#> Exact value: 53.232482, MLMC value: 53.241652
#>
#> ---- Computing barrier call ----
#>
#> **********************************************************
#> *** Convergence tests, kurtosis, telescoping sum check ***
#> *** using N = 2e+05 samples ***
#> **********************************************************
#>
#> l ave(Pf-Pc) ave(Pf) var(Pf-Pc) var(Pf) kurtosis check cost
#> ---------------------------------------------------------------------------------------
#> 0 9.4564e+00 9.4564e+00 1.8871e+02 1.8871e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 1.6056e-01 9.6193e+00 1.7069e-01 1.9914e+02 6.1603e+01 1.2323e-02 2.0000e+00
#> 2 1.4474e-01 9.6700e+00 1.5745e-01 2.0464e+02 1.0612e+02 4.8690e-01 4.0000e+00
#> 3 9.8315e-02 9.8418e+00 1.4427e-01 2.1162e+02 3.9947e+02 3.7507e-01 8.0000e+00
#> 4 4.6888e-02 9.9143e+00 5.1685e-02 2.1567e+02 1.5447e+03 1.2963e-01 1.6000e+01
#> 5 2.0359e-02 9.9093e+00 1.0326e-02 2.1578e+02 1.2857e+03 1.2834e-01 3.2000e+01
#> 6 1.0032e-02 9.9355e+00 4.2701e-03 2.1714e+02 3.9267e+03 8.1491e-02 6.4000e+01
#> 7 4.8781e-03 9.9329e+00 1.1130e-03 2.1622e+02 1.7346e+03 3.7701e-02 1.2800e+02
#> 8 2.3827e-03 9.9133e+00 4.1529e-04 2.1690e+02 6.9372e+03 1.1119e-01 2.5600e+02
#>
#> WARNING: kurtosis on finest level = 6937.174649
#> 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 = 0.929256 (exponent for MLMC weak convergence)
#> beta = 1.357869 (exponent for MLMC variance)
#> gamma = 1.000000 (exponent for MLMC cost)
#>
#> *****************************
#> *** MLMC complexity tests ***
#> *****************************
#>
#> eps value mlmc_cost std_cost savings N_l
#> -----------------------------------------------------------
#> 0.0050 9.9433e+00 2.155e+07 5.923e+09 274.79 14584932 310729 219105 148864 56194 21897 10346 7173 2550 873
#> 0.0100 9.9355e+00 4.662e+06 7.403e+08 158.79 3394165 72037 49865 32362 13511 5756 2045 645 203
#> 0.0200 9.9761e+00 1.264e+06 9.225e+07 72.96 883634 18369 12864 8275 3246 2341 951 302
#> 0.0500 9.8703e+00 1.638e+05 7.412e+06 45.25 127079 2734 2077 943 407 169 54
#> 0.1000 9.7337e+00 3.815e+04 9.207e+05 24.13 30393 711 400 260 98 34
#>
#>
#> Exact value: 9.949270, MLMC value: 9.943268
# }
# The level sampler can be called directly to retrieve the relevant level sums:
mcqmc06_l(l = 7, N = 10, option = 1)
#> $sums
#> [1] 2.648764e-02 2.889784e-04 3.607043e-06 5.105926e-08 1.032765e+02
#> [6] 2.338980e+03
#>
#> $cost
#> [1] 1280
#>