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 1.0029e+01 1.0029e+01 1.9722e+02 1.9722e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 1.8394e-01 1.0290e+01 1.4600e-01 2.0587e+02 3.9810e+01 1.2720e-01 2.0000e+00
#> 2 1.0333e-01 1.0388e+01 4.2175e-02 2.1234e+02 3.3379e+01 8.3887e-03 4.0000e+00
#> 3 5.4322e-02 1.0369e+01 1.1838e-02 2.1705e+02 1.6591e+01 1.1764e-01 8.0000e+00
#> 4 2.8126e-02 1.0552e+01 3.2235e-03 2.1932e+02 1.3227e+01 2.4578e-01 1.6000e+01
#> 5 1.3857e-02 1.0318e+01 8.3669e-04 2.1392e+02 1.3113e+01 3.9635e-01 3.2000e+01
#> 6 7.1684e-03 1.0487e+01 2.0915e-04 2.1799e+02 1.1029e+01 2.5931e-01 6.4000e+01
#> 7 3.5655e-03 1.0418e+01 5.1754e-05 2.1059e+02 9.7300e+00 1.1593e-01 1.2800e+02
#> 8 1.7807e-03 1.0446e+01 1.3881e-05 2.2270e+02 1.3242e+01 4.0847e-02 2.5600e+02
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.963172 (exponent for MLMC weak convergence)
#> beta = 1.920125 (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 1.0456e+01 1.357e+07 3.041e+09 224.13 11906524 232615 91191 33305 12323 4397 1541 574 209
#> 0.0100 1.0448e+01 3.358e+06 3.594e+08 107.02 2957299 58040 24187 8085 2986 1072 388 130
#> 0.0200 1.0474e+01 8.373e+05 4.650e+07 55.54 739789 15333 5622 1982 806 291 98
#> 0.0500 1.0455e+01 1.324e+05 3.651e+06 27.57 117749 2356 823 375 135 47
#> 0.1000 1.0323e+01 3.216e+04 4.679e+05 14.55 28570 569 200 104 51
#>
#>
#> Exact value: 10.450584, MLMC value: 10.456468
#>
#> ---- 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 5.5951e+00 5.5951e+00 6.0102e+01 6.0102e+01 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 4.1848e-02 5.6599e+00 2.5574e-01 6.1727e+01 1.6183e+01 6.7128e-02 2.0000e+00
#> 2 4.1015e-02 5.6638e+00 3.8719e-02 6.1679e+01 1.3926e+01 1.0993e-01 4.0000e+00
#> 3 2.7105e-02 5.8340e+00 6.1034e-03 6.3818e+01 8.0283e+00 4.2362e-01 8.0000e+00
#> 4 1.5105e-02 5.7268e+00 1.2618e-03 6.3991e+01 8.9021e+00 3.5953e-01 1.6000e+01
#> 5 7.5520e-03 5.7032e+00 2.7444e-04 6.3187e+01 8.1746e+00 9.2026e-02 3.2000e+01
#> 6 3.9493e-03 5.7242e+00 6.3849e-05 6.2996e+01 7.4206e+00 5.0343e-02 6.4000e+01
#> 7 1.9923e-03 5.7760e+00 1.5791e-05 6.3790e+01 8.4057e+00 1.4766e-01 1.2800e+02
#> 8 9.9651e-04 5.7188e+00 3.7720e-06 6.3177e+01 7.4076e+00 1.7234e-01 2.5600e+02
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.820236 (exponent for MLMC weak convergence)
#> beta = 2.268800 (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 5.7605e+00 4.725e+06 8.626e+08 182.56 3873079 175297 48809 15168 4535 1444 485 166 53
#> 0.0100 5.7535e+00 1.182e+06 1.089e+08 92.10 968052 44224 12467 3893 1173 374 132 42
#> 0.0200 5.7370e+00 2.894e+05 1.344e+07 46.44 239448 10974 2833 894 284 95 30
#> 0.0500 5.7458e+00 4.528e+04 5.461e+05 12.06 37958 1862 436 140 46
#> 0.1000 5.6778e+00 1.039e+04 3.290e+04 3.17 8849 371 200
#>
#>
#> Exact value unknown, MLMC value: 5.760501
#>
#> ---- 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 1.7511e+01 1.7511e+01 1.8272e+02 1.8272e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 -1.1451e-01 1.7641e+01 1.4254e+00 1.9415e+02 1.0742e+01 4.0186e-01 2.0000e+00
#> 2 -1.3721e-01 1.7368e+01 4.9198e-01 2.0461e+02 8.3115e+00 2.2081e-01 4.0000e+00
#> 3 -9.4454e-02 1.7255e+01 1.4952e-01 2.1249e+02 8.0617e+00 3.0306e-02 8.0000e+00
#> 4 -5.4218e-02 1.7330e+01 4.0505e-02 2.1245e+02 7.6582e+00 2.0807e-01 1.6000e+01
#> 5 -2.8215e-02 1.7190e+01 1.0298e-02 2.1322e+02 7.4335e+00 1.8081e-01 3.2000e+01
#> 6 -1.4180e-02 1.7217e+01 2.6559e-03 2.1334e+02 9.3109e+00 6.6450e-02 6.4000e+01
#> 7 -7.2142e-03 1.7281e+01 6.7944e-04 2.1261e+02 7.8428e+00 1.1579e-01 1.2800e+02
#> 8 -3.5886e-03 1.7277e+01 1.6721e-04 2.1483e+02 7.2920e+00 1.4140e-03 2.5600e+02
#> 9 -1.7421e-03 1.7406e+01 4.1219e-05 2.1655e+02 7.2133e+00 2.1006e-01 5.1200e+02
#> 10 -8.5668e-04 1.7203e+01 1.0125e-05 2.1479e+02 7.5526e+00 3.2429e-01 1.0240e+03
#>
#> ******************************************************
#> *** Linear regression estimates of MLMC parameters ***
#> ******************************************************
#>
#> alpha = 0.854377 (exponent for MLMC weak convergence)
#> beta = 1.923630 (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 2.198e+07 1.173e+10 533.68 14705013 903557 378904 146464 53597 19602 6853 2548 928 335 119
#> 0.0100 1.7220e+01 5.439e+06 1.478e+09 271.80 3656467 225494 93830 36431 13514 4838 1751 625 229 84
#> 0.0200 1.7217e+01 1.333e+06 1.833e+08 137.50 903171 55947 23402 9207 3330 1206 429 135 56
#> 0.0500 1.7195e+01 2.276e+05 1.451e+07 63.77 158622 9132 3715 1505 513 218 79 28
#> 0.1000 1.7285e+01 4.837e+04 9.097e+05 18.81 34322 2144 884 354 126 43
#>
#>
#> Exact value: 17.216802, MLMC value: 17.220294
#>
#> ---- 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 -2.7040e+00 5.4204e+01 2.0283e-01 7.2524e+02 1.1215e+01 2.3129e-01 2.0000e+00
#> 2 -7.1971e-01 5.3474e+01 1.9122e+00 1.1903e+03 1.8316e+01 2.4219e-02 4.0000e+00
#> 3 -2.0226e-01 5.3362e+01 1.2522e+00 1.5093e+03 2.2352e+01 1.8000e-01 8.0000e+00
#> 4 -6.1953e-02 5.3186e+01 5.4529e-01 1.7271e+03 3.2836e+01 2.0907e-01 1.6000e+01
#> 5 -2.1556e-02 5.3342e+01 2.1123e-01 1.8701e+03 4.3469e+01 3.1090e-01 3.2000e+01
#> 6 -7.6433e-03 5.3245e+01 7.9410e-02 1.9777e+03 5.9358e+01 1.5181e-01 6.4000e+01
#> 7 -3.5706e-03 5.3145e+01 2.8310e-02 2.0528e+03 8.5312e+01 1.5978e-01 1.2800e+02
#> 8 -2.1199e-03 5.3293e+01 1.0033e-02 2.1033e+03 1.3585e+02 2.4542e-01 2.5600e+02
#>
#> WARNING: kurtosis on finest level = 135.851963
#> 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 = 1.502312 (exponent for MLMC weak convergence)
#> beta = 0.881623 (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 5.3252e+01 2.929e+06 1.688e+09 576.20 200 62778 136754 79398 36565 16168 8103
#> 0.0200 5.3233e+01 7.604e+05 4.219e+08 554.87 200 15902 34931 20474 9468 4287 2128
#> 0.0500 5.3287e+01 8.405e+04 3.192e+07 379.75 200 2165 4461 2581 1256 654
#> 0.1000 5.3398e+01 1.850e+04 3.685e+06 199.19 200 477 1192 770 401
#> 0.2000 5.3154e+01 3.136e+03 9.211e+05 293.73 200 200 204 105 55
#>
#>
#> Exact value: 53.232482, MLMC value: 53.251801
#>
#> ---- 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 9.4474e+00 9.4474e+00 1.8929e+02 1.8929e+02 0.0000e+00 0.0000e+00 1.0000e+00
#> 1 1.5951e-01 9.6066e+00 1.7046e-01 1.9977e+02 6.4497e+01 1.7977e-03 2.0000e+00
#> 2 1.4617e-01 9.7725e+00 1.5850e-01 2.0847e+02 9.9998e+01 1.0141e-01 4.0000e+00
#> 3 9.7979e-02 9.8558e+00 1.5455e-01 2.1263e+02 5.7430e+02 7.3983e-02 8.0000e+00
#> 4 4.6242e-02 9.9139e+00 4.6293e-02 2.1629e+02 1.1486e+03 5.9766e-02 1.6000e+01
#> 5 2.0943e-02 9.8764e+00 1.2250e-02 2.1494e+02 1.2541e+03 2.9562e-01 3.2000e+01
#> 6 9.8981e-03 9.9351e+00 3.0235e-03 2.1906e+02 1.1802e+03 2.4669e-01 6.4000e+01
#> 7 4.9817e-03 9.9004e+00 1.4311e-03 2.1603e+02 5.8747e+03 2.0031e-01 1.2800e+02
#> 8 2.3914e-03 1.0021e+01 5.2788e-04 2.1924e+02 2.1485e+04 5.9808e-01 2.5600e+02
#>
#> WARNING: kurtosis on finest level = 21484.500811
#> 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 = 0.926866 (exponent for MLMC weak convergence)
#> beta = 1.324364 (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 9.9474e+00 2.058e+07 5.987e+09 290.96 14279726 303808 210948 147438 67635 25971 9121 3809 1625 518
#> 0.0100 9.9457e+00 4.929e+06 7.484e+08 151.83 3502625 74757 52075 36492 14831 5822 2249 998 318
#> 0.0200 9.9472e+00 1.090e+06 9.217e+07 84.57 818077 15653 12652 7844 3382 1089 348 124
#> 0.0500 9.9445e+00 1.856e+05 7.477e+06 40.29 135789 3080 2359 1480 640 205 87
#> 0.1000 9.8066e+00 3.432e+04 4.614e+05 13.45 29190 628 330 171 74
#>
#>
#> Exact value: 9.949270, MLMC value: 9.947443
# }
# The level sampler can be called directly to retrieve the relevant level sums:
mcqmc06_l(l = 7, N = 10, option = 1)
#> $sums
#> [1] 5.965277e-02 1.405983e-03 4.231701e-05 1.389170e-06 1.190440e+02
#> [6] 5.613734e+03
#>
#> $cost
#> [1] 1280
#>