Analysis of Quasi-Monte Carlo Efficiency for Asian Option Pricing

Introduction: The Asian Option Pricing Challenge

Asian options are especially difficult to price because their payoffs involve the average price of an underlying asset over a period of time. In comparison to simpler European options, arithmetic (or average) Asian options cannot be easily handled to result in a closed-form solution due to their path dependence. Subsequently, approximating arithmetic Asian options requires numerical methods, most frequently, Monte Carlo (MC) simulation.

In standard MC simulations, the model would simulate a large number of asset price paths, where each price path is generated by a different set of pseudorandom numbers, average the resulting discounted payoffs, and hope for convergence on the value of the option using the law of large numbers. However, the rate of convergence is \(O(N^{-1/2})\) as derived from the Central Limit Theorem, which is a rather slow rate of convergence for estimation. For practical purposes, this means you would need to have a rather costly, large simulation just to obtain a decent estimate.

Quasi-Monte Carlo (QMC) methods provide an alternative to MC methods by substituting deterministic low-discrepancy sequences (e.g., Sobol’, Halton) or point sets (e.g., lattice rules) in place of pseudorandom numbers. This results in more uniform coverage of the simulation space through the use of deterministic points and faster convergence rates (approximately \(O(N^{-1})\)), but only if the integrands are sufficiently smooth. In this work, we investigate the actual gains in efficiency provided by QMC over MC methods in the context of pricing an arithmetic Asian call option, providing median performance from several different simulation runs to ensure the findings are robust.

Mathematical Framework

We consider an arithmetic Asian call option with strike \(K\) and expiry \(T\). The underlying asset price \(S(t)\) is observed at \(d\) discrete times \(t_j = jT/d\). The discounted payoff is:

where \(r\) is the risk-free rate. The goal is to estimate the expected value of this payoff.

Asset prices are modeled using Geometric Brownian Motion (GBM) under the risk-neutral measure:

where \(\sigma\) is volatility and \(W(t)\) is a Wiener process.

Simulation Setup

We used QMCPy to compare standard MC (IID) against QMC (Sobol’, Halton, Lattice) for pricing an Asian call option with parameters:

• \(S_0 = \$120\)
• \(K = \$130\)
• \(T = 1\) year
• \(r = 0.02\)
• \(\sigma = 0.50\)
• \(d = 12\) monthly observations

The integration dimension is \(d = 12\).

We used adaptive algorithms (CubMCG, CubQMCSobolG, CubQMCCLT, CubQMCLatticeG) to achieve target absolute error tolerances (\(\epsilon\)) from \$0.50 to as low as \$0.0005. For each method and tolerance, we ran the simulation 25 times to collect data on sample needs and processing time. The numbers we report for sample counts and timings show the median value across these runs. We include the 25th and 75th percentiles in graphs to show the variability in results.

QMCPy Sobol’ Example

The following example shows how to use Sobol’ sequence in QMCPy for an Asian option pricing simulation.

import qmcpy as qp

# Point Generators (re-instantiated for each run)
SobolPoints = qp.Sobol(dimension=d)
IIDPoints = qp.IIDStdUniform(dimension=d)  # etc.

# Define Asian Option Measure (Sobol’)
ArithMeanCallSobol = qp.AsianOption(
    integrand=qp.Sobol(dimension=12),
    volatility=0.50,
    start_price=120,
    strike_price=130,
    interest_rate=0.02,
    t_final=1,
    call_put='call',
    mean_type='arithmetic'
)
# Define similarly for IID, Halton, Lattice...

# Run Integration (example for one run, one tolerance)
price, data = qp.CubQMCSobolG(ArithMeanCallSobol, abs_tol=0.01, rel_tol=0).integrate()
print(f"Sobol’ Price: {price:.4f}, Samples: {data.n_total}")
# Example output: Sobol’ Price: 10.3173, Samples: 4096
# Full script iterates this over tolerances and multiple runs.

Results: QMC Efficiency Demonstrated

As Table 1 demonstrates, different methods required different numbers of samples to achieve the desired accuracy. Because they are deterministic, QMC techniques like Sobol’ and Lattice, when combined with particular integrators, typically have predetermined sample counts for a given tolerance. These medians show the central tendency across a range of results for IID (MC) and Halton (with CubQMCCLT). Sobol’ and Lattice perform better than Halton in part because of their stopping criteria: CubQMCSobolG and CubQMCLatticeG use the decay of Fourier/Walsh coefficients to calculate sample sizes, whereas CubQMCCLT for Halton uses multiple randomizations, which may result in larger sample requirements.

Table 1: Median samples required for target absolute error (\(\epsilon\)) over 25 runs.

The final high-precision estimates (at \(\epsilon = 0.0005\)) from the last simulation run were consistent across QMC methods: Sobol’ (\$10.3173), Lattice (\$10.3173), Halton (\$10.3172). Standard MC could not produce a reliable estimate at \(\epsilon = 0.05\) in the last simulation run, indicating practical limitations at higher precision levels.

Figure 1 visually underscores the efficiency gap, plotting median requirements with 25th-75th percentile ranges to indicate variability.

• Figure 1: Median Samples vs. Tolerance
  
• Figure 2: Median Time vs. Tolerance
  

Several significant conclusions can be drawn from the results. Since both Sobol’ and Lattice rules required significantly fewer samples (median) to achieve error tolerance \(\epsilon=0.0695\), utilizing roughly 618 times fewer paths (median \(N_{\text{Sobol’}} = 2,048\) vs. median \(N_{\text{IID}} \approx 1,265,302\)), the comparisons demonstrate a significant QMC performance benefit. At higher levels of precision, the efficiency difference increases significantly. Both approaches are equally effective for this problem, as evidenced by the identical median sample counts produced by the Sobol’ and Lattice methods. They used the smallest median sample size tested, 1,024, and obtained moderate accuracy levels of \(\epsilon = 0.1864\). Compared to Sobol’ and Lattice, Halton performed better than MC but needed more samples (median) at all tested tolerances. This was partially because Halton relied on multiple randomizations in the stopping criterion. Because standard MC’s slow convergence rate \(O(N^{-1/2})\) prevented it from achieving tolerances below 0.0424 without an excessive number of samples, its practical limit became evident. In line with theoretical predictions regarding faster convergence rates for smooth integrand functions when using low-discrepancy points, the empirical results based on median performance provide strong support for QMC. The variability plots (quantile ranges) show that QMC methods are more stable than MC, especially when it comes to sample counts.

Performance Edge and Practical Implications

The better sample efficiency of QMC, evidenced by median performance across 25 runs, shows many practical advantages. The main advantage is lower computation time for pricing and risk analysis. When speed increases, high precision becomes computationally feasible and thus can be used to make the model more reliable when standard MC would have been too slow. QMC uses less CPU time and can provide cost savings in computation, especially when dealing with cloud computing resources. The expense of generating QMC points is relatively minor compared to the expense of running the simulation, especially while using efficient libraries like QMCPy to generate QMC points. QMC methods are world of difference for this problem with a dimension \(d = 12\) and none of the downsides were significant.

Conclusion

When pricing arithmetic Asian options using simulation, quasi-Monte Carlo has clear and distinct advantages over Monte Carlo. This remains true even when assessing medians across the 25 runs. Quasi-Monte Carlo (Sobol’ sequences and Lattice rules) also produced some outstandingly accurate results with many more orders of magnitude greater accuracy over standard MC using medians, with orders of magnitude fewer samples and computations. Halton sequences provided some improvement over standard MC; however, the limitations were due in part to their stopping criterion being based on more than 1 randomization. The standard MC approach does not make sense to deliver precision as indicated here, and quasi-Monte Carlo using something such as QMCPy (which is extremely easy to use) is recommended for practitioners looking to value path-dependent derivatives more quickly and accurately and more efficiently. Repeated runs using medians and quantiles provided more confidence in the conclusions.

References

• Choi, S.-C. T., et al., “QMCPy: A Quasi-Monte Carlo Python Library,” https://qmcpy.org.

Karm Dave

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Undergraduate student in Computer Science working on computational mathematics, finance, and machine learning.

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