import matplotlib.pyplot as plt
import numpy as np
import qmcpy as qp
seed = 7
Comparison of multilevel (Quasi-)Monte Carlo for an Asian option problem¶
Compute the exact value of the Asian option with single level QMC, for an increasing number of time steps:
for level in range(5):
aco = qp.AsianOption(qp.Sobol(2*2**level, seed=seed), volatility=.2, start_price=100, strike_price=100, interest_rate=.05)
approx_solution, data = qp.CubQMCSobolG(aco, abs_tol=1e-4).integrate()
print("Asian Option true value (%d time steps): %.5f (to within 1e-4)"%(2*2**level, approx_solution))
Asian Option true value (2 time steps): 5.63591 (to within 1e-4)
Asian Option true value (4 time steps): 5.73171 (to within 1e-4)
Asian Option true value (8 time steps): 5.75526 (to within 1e-4)
Asian Option true value (16 time steps): 5.76113 (to within 1e-4)
Asian Option true value (32 time steps): 5.76260 (to within 1e-4)
This function compares 4 different algorithms: Multilevel Monte Carlo
(CubMCML), Multilevel Quasi-Monte Carlo (CubQMCML), continuation
Multilevel Monte Carlo (CubMCMLCont) and Multilevel Quasi-Monte
Carlo (CubQMCMLCont):
def eval_option(option_mc, option_qmc, abs_tol):
stopping_criteria = {
"MLMC" : qp.CubMCML(option_mc, abs_tol=abs_tol, levels_max=15),
"continuation MLMC" : qp.CubMCMLCont(option_mc, abs_tol=abs_tol, levels_max=15),
"MLQMC" : qp.CubQMCML(option_qmc, abs_tol=abs_tol, levels_max=15),
"continuation MLQMC" : qp.CubQMCMLCont(option_qmc, abs_tol=abs_tol, levels_max=15)
}
levels = []
times = []
for name, stopper in stopping_criteria.items():
sol, data = stopper.integrate()
levels.append(data.levels)
times.append(data.time_integrate)
print("\t%-20s solution %-10.4f number of levels %-6d time %.3f"%(name, sol, levels[-1], times[-1]))
return levels, times
Define the Multilevel Asian options:
option_mc = qp.MLCallOptions(qp.IIDStdUniform(seed=seed), option="asian")
option_qmc = qp.MLCallOptions(qp.Lattice(seed=seed), option="asian")
Run and compare each of the 4 algorithms for the Asian option problem:
eval_option(option_mc, option_qmc, abs_tol=5e-3);
MLMC solution 5.7630 number of levels 10 time 13.781
continuation MLMC solution 5.7583 number of levels 7 time 12.456
MLQMC solution 5.7613 number of levels 8 time 6.408
continuation MLQMC solution 5.7592 number of levels 7 time 2.073
Repeat this comparison for a sequence of decreasing tolerances, with 5 different random seeds each. This will allow us to visualize the asymptotic cost complexity of each method.
repetitions = 5
tolerances = 5*np.logspace(-1, -3, num=5)
levels = {}
times = {}
for t in range(len(tolerances)):
for r in range(repetitions):
print("tolerance = %10.4e, repetition = %d/%d"%(tolerances[t], r + 1, repetitions))
levels[t, r], times[t, r] = eval_option(option_mc, option_qmc, tolerances[t])
tolerance = 5.0000e-01, repetition = 1/5
MLMC solution 5.7654 number of levels 3 time 0.007
continuation MLMC solution 5.8128 number of levels 3 time 0.011
MLQMC solution 5.7229 number of levels 3 time 0.170
continuation MLQMC solution 5.7045 number of levels 3 time 0.000
tolerance = 5.0000e-01, repetition = 2/5
MLMC solution 5.9289 number of levels 4 time 0.005
continuation MLMC solution 5.7235 number of levels 3 time 0.009
MLQMC solution 5.7144 number of levels 3 time 0.208
continuation MLQMC solution 5.7164 number of levels 3 time 0.000
tolerance = 5.0000e-01, repetition = 3/5
MLMC solution 5.8284 number of levels 3 time 0.005
continuation MLMC solution 5.6069 number of levels 3 time 0.007
MLQMC solution 5.7251 number of levels 3 time 0.160
continuation MLQMC solution 5.6889 number of levels 3 time 0.000
tolerance = 5.0000e-01, repetition = 4/5
MLMC solution 5.7565 number of levels 4 time 0.005
continuation MLMC solution 5.7117 number of levels 3 time 0.007
MLQMC solution 5.6831 number of levels 3 time 0.162
continuation MLQMC solution 5.7106 number of levels 3 time 0.000
tolerance = 5.0000e-01, repetition = 5/5
MLMC solution 5.6210 number of levels 3 time 0.004
continuation MLMC solution 5.5126 number of levels 3 time 0.007
MLQMC solution 5.7067 number of levels 3 time 0.168
continuation MLQMC solution 5.7053 number of levels 3 time 0.000
tolerance = 1.5811e-01, repetition = 1/5
MLMC solution 5.8072 number of levels 6 time 0.021
continuation MLMC solution 5.7373 number of levels 4 time 0.017
MLQMC solution 5.7424 number of levels 4 time 0.287
continuation MLQMC solution 5.7085 number of levels 3 time 0.022
tolerance = 1.5811e-01, repetition = 2/5
MLMC solution 5.7893 number of levels 7 time 0.034
continuation MLMC solution 5.7419 number of levels 4 time 0.037
MLQMC solution 5.7329 number of levels 4 time 0.313
continuation MLQMC solution 5.6862 number of levels 3 time 0.030
tolerance = 1.5811e-01, repetition = 3/5
MLMC solution 5.7959 number of levels 5 time 0.025
continuation MLMC solution 5.6990 number of levels 4 time 0.030
MLQMC solution 5.7409 number of levels 4 time 0.423
continuation MLQMC solution 5.6983 number of levels 3 time 0.017
tolerance = 1.5811e-01, repetition = 4/5
MLMC solution 5.8389 number of levels 5 time 0.015
continuation MLMC solution 5.8064 number of levels 5 time 0.024
MLQMC solution 5.7378 number of levels 4 time 0.311
continuation MLQMC solution 5.7063 number of levels 3 time 0.025
tolerance = 1.5811e-01, repetition = 5/5
MLMC solution 5.7334 number of levels 4 time 0.018
continuation MLMC solution 5.8528 number of levels 4 time 0.020
MLQMC solution 5.7278 number of levels 4 time 0.312
continuation MLQMC solution 5.7241 number of levels 4 time 0.106
tolerance = 5.0000e-02, repetition = 1/5
MLMC solution 5.7680 number of levels 7 time 0.147
continuation MLMC solution 5.7065 number of levels 4 time 0.123
MLQMC solution 5.7489 number of levels 5 time 0.505
continuation MLQMC solution 5.7307 number of levels 4 time 0.032
tolerance = 5.0000e-02, repetition = 2/5
MLMC solution 5.7418 number of levels 7 time 0.159
continuation MLMC solution 5.7268 number of levels 4 time 0.078
MLQMC solution 5.7456 number of levels 5 time 0.493
continuation MLQMC solution 5.7492 number of levels 5 time 0.156
tolerance = 5.0000e-02, repetition = 3/5
MLMC solution 5.7405 number of levels 6 time 0.151
continuation MLMC solution 5.7331 number of levels 4 time 0.113
MLQMC solution 5.7449 number of levels 5 time 0.476
continuation MLQMC solution 5.7472 number of levels 5 time 0.109
tolerance = 5.0000e-02, repetition = 4/5
MLMC solution 5.7673 number of levels 7 time 0.134
continuation MLMC solution 5.7357 number of levels 5 time 0.147
MLQMC solution 5.7462 number of levels 5 time 0.435
continuation MLQMC solution 5.7324 number of levels 4 time 0.061
tolerance = 5.0000e-02, repetition = 5/5
MLMC solution 5.7583 number of levels 7 time 0.114
continuation MLMC solution 5.7191 number of levels 4 time 0.076
MLQMC solution 5.7464 number of levels 5 time 0.423
continuation MLQMC solution 5.7308 number of levels 4 time 0.025
tolerance = 1.5811e-02, repetition = 1/5
MLMC solution 5.7618 number of levels 8 time 1.363
continuation MLMC solution 5.7516 number of levels 6 time 1.279
MLQMC solution 5.7553 number of levels 6 time 1.293
continuation MLQMC solution 5.7479 number of levels 5 time 0.163
tolerance = 1.5811e-02, repetition = 2/5
MLMC solution 5.7630 number of levels 8 time 1.104
continuation MLMC solution 5.7576 number of levels 6 time 1.034
MLQMC solution 5.7563 number of levels 6 time 1.037
continuation MLQMC solution 5.7553 number of levels 6 time 0.318
tolerance = 1.5811e-02, repetition = 3/5
MLMC solution 5.7563 number of levels 9 time 1.476
continuation MLMC solution 5.7578 number of levels 6 time 1.081
MLQMC solution 5.7589 number of levels 7 time 1.409
continuation MLQMC solution 5.7566 number of levels 6 time 0.367
tolerance = 1.5811e-02, repetition = 4/5
MLMC solution 5.7648 number of levels 8 time 1.203
continuation MLMC solution 5.7546 number of levels 6 time 0.659
MLQMC solution 5.7551 number of levels 6 time 0.986
continuation MLQMC solution 5.7482 number of levels 5 time 0.170
tolerance = 1.5811e-02, repetition = 5/5
MLMC solution 5.7678 number of levels 8 time 1.490
continuation MLMC solution 5.7554 number of levels 6 time 0.581
MLQMC solution 5.7561 number of levels 6 time 0.854
continuation MLQMC solution 5.7474 number of levels 5 time 0.167
tolerance = 5.0000e-03, repetition = 1/5
MLMC solution 5.7635 number of levels 10 time 15.841
continuation MLMC solution 5.7588 number of levels 7 time 8.160
MLQMC solution 5.7609 number of levels 8 time 6.344
continuation MLQMC solution 5.7590 number of levels 7 time 2.422
tolerance = 5.0000e-03, repetition = 2/5
MLMC solution 5.7624 number of levels 10 time 14.806
continuation MLMC solution 5.7577 number of levels 7 time 12.720
MLQMC solution 5.7610 number of levels 8 time 5.639
continuation MLQMC solution 5.7601 number of levels 7 time 1.524
tolerance = 5.0000e-03, repetition = 3/5
MLMC solution 5.7651 number of levels 10 time 13.543
continuation MLMC solution 5.7605 number of levels 7 time 11.780
MLQMC solution 5.7610 number of levels 8 time 7.015
continuation MLQMC solution 5.7593 number of levels 7 time 3.229
tolerance = 5.0000e-03, repetition = 4/5
MLMC solution 5.7630 number of levels 10 time 12.791
continuation MLMC solution 5.7595 number of levels 7 time 11.755
MLQMC solution 5.7615 number of levels 8 time 5.914
continuation MLQMC solution 5.7594 number of levels 7 time 2.924
tolerance = 5.0000e-03, repetition = 5/5
MLMC solution 5.7611 number of levels 10 time 13.041
continuation MLMC solution 5.7587 number of levels 7 time 12.751
MLQMC solution 5.7613 number of levels 8 time 6.122
continuation MLQMC solution 5.7593 number of levels 7 time 3.543
Compute and plot the asymptotic cost complexity.
avg_time = {}
for method in range(4):
avg_time[method] = [np.mean([times[t, r][method] for r in range(repetitions)]) for t in range(len(tolerances))]
plt.figure(figsize=(10,7))
plt.plot(tolerances, avg_time[0], label="MLMC")
plt.plot(tolerances, avg_time[1], label="continuation MLMC")
plt.plot(tolerances, avg_time[2], label="MLQMC")
plt.plot(tolerances, avg_time[3], label="continuation MLQMC")
plt.xscale("log")
plt.yscale("log")
plt.xlabel("requested absolute tolerance")
plt.ylabel("average run time in seconds")
plt.legend();
max_levels = {}
for method in range(4):
levels_rep = np.array([levels[len(tolerances)-1, r][method] for r in range(repetitions)])
max_levels[method] = [np.count_nonzero(levels_rep == level)/repetitions for level in range(15)]
plt.figure(figsize=(14,10))
plt.subplot(2,2,1); plt.bar(range(15), max_levels[0], label="MLMC old", color="C0"); plt.xlabel("max level"); plt.ylabel("fraction of runs"); plt.ylim(0, 1); plt.legend()
plt.subplot(2,2,2); plt.bar(range(15), max_levels[1], label="MLMC new", color="C1"); plt.xlabel("max level"); plt.ylabel("fraction of runs"); plt.ylim(0, 1); plt.legend()
plt.subplot(2,2,3); plt.bar(range(15), max_levels[2], label="MLQMC old", color="C2"); plt.xlabel("max level"); plt.ylabel("fraction of runs"); plt.ylim(0, 1); plt.legend()
plt.subplot(2,2,4); plt.bar(range(15), max_levels[3], label="MLQMC new", color="C3"); plt.xlabel("max level"); plt.ylabel("fraction of runs"); plt.ylim(0, 1); plt.legend();
