Importance sampling and active subspace method in quasi-Monte Carlo

Abstract: The quasi-Monte Carlo method is widely used in computational finance, whose efficiency strongly depends on the smoothness and effective dimension of the integrand. In this work, we investigate the combination of importance sampling and the active subspace method under the quasi-Monte Carlo framework and propose a three-step approach, referred to as the IS-AS-preintegration method, which sequentially applies importance sampling, active subspace, and preintegration. The proposed method is applied to the option pricing and sensitivity analysis problems in finance, and its performance is evaluated through extensive numerical experiments. The results demonstrate that the proposed method is highly competitive compared with existing popular methods. In particular, for out-of-the-money and deep out-of-the-money options, the proposed approach overcomes the limitations of the preintegration via active subspace method and achieves superior variance reduction, while maintaining comparable performance for other moneyness cases.

• The paper introduces IS_AS_Preint, which applies importance sampling before active-subspace reduction and preintegration so gradient estimates remain informative in rare-event regions.
• The method delivers variance reduction factors up to 1.2×10^8 for deep out-of-the-money Asian options and remains effective where earlier active-subspace methods fail.
• The paper proves orthogonal invariance for ordinary and importance-sampled active subspaces, simplifying Brownian-motion implementations while highlighting open issues in uniqueness and gradient-estimation robustness.

The paper by Yu and Wang addresses a well-documented failure mode of gradient-based dimension reduction in quasi-Monte Carlo (QMC) methods: for out-of-the-money (OTM) and deep OTM option pricing problems, the gradient information matrix underlying the active subspace method is theoretically close to zero because sample points rarely fall in the rare-event region. The authors propose IS_AS_Preint, a three-step method that applies importance sampling (IS) first, then computes the active subspace (AS) of the importance-sampled integrand, and finally preintegrates with respect to the first active variable. The paper also establishes orthogonal invariance results showing that both ordinary and IS-induced active subspaces are independent of the Brownian motion generation matrix.

Motivation and background

QMC efficiency depends on the smoothness and effective dimension of the integrand rather than its nominal dimension. Existing remedies fall into two families: path generation methods such as Brownian bridge and PCA, which ignore integrand structure, and integrand-aware methods such as linear transformation, gradient PCA (GPCA), and the active subspace method. Two recent combinations of AS with preintegration exist: Preint_GPCA (Xiao and Wang), which preintegrates first and then applies GPCA to the $(d-1)$-dimensional smoothed function, and AS_Preint (Liu and Owen), which reverses this order. Zhang et al. additionally inserted optimal drift IS after preintegration (Preint_IS_GPCA).

The critical observation motivating the new method is that both Preint_GPCA and AS_Preint fail for OTM options: without IS, the estimated gradient information matrix $C = \mathbb{E}(\nabla g\,\nabla g^T)$ is nearly singular or near-zero, so eigenvector extraction is meaningless. The proposed ordering — IS before AS before preintegration — ensures that gradients carry information about the rare-event region while retaining the documented advantage of AS-before-preintegration over the reverse order.

The three-step method

Given a nonnegative integrand $g$ with $\mathbb{E}(g(\bm{Z}))$ under $\bm{Z}\sim N(\bm{0},I_d)$, the method proceeds as follows:

1. Importance sampling: solve $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$ for the optimal drift $\bm{\mu}^*$ (optimal drift IS), or additionally compute $\Gamma^*=(I_d-\nabla^2 F(\bm{\mu}^*))^{-1}$ for Laplace IS, transforming the problem into an expectation of $g^\mathrm{I}$.
2. Active subspace: estimate $C=\mathbb{E}(\nabla g^\mathrm{I}\,\nabla g^{I\,T})$ by finite differences over $C = \mathbb{E}(\nabla g\,\nabla g^T)$0 QMC samples, decompose $C = \mathbb{E}(\nabla g\,\nabla g^T)$1 with the sign convention that the first column of $C = \mathbb{E}(\nabla g\,\nabla g^T)$2 has nonnegative entries, and form $C = \mathbb{E}(\nabla g\,\nabla g^T)$3.
3. Preintegration: analytically integrate out the first active variable, exploiting monotonicity of the discontinuous indicator part in $C = \mathbb{E}(\nabla g\,\nabla g^T)$4.

The sign convention on $C = \mathbb{E}(\nabla g\,\nabla g^T)$5's first column is essential: combined with a nonnegative Brownian motion generation matrix $C = \mathbb{E}(\nabla g\,\nabla g^T)$6 (e.g., the standard Cholesky construction), it guarantees that $C = \mathbb{E}(\nabla g\,\nabla g^T)$7 has nonnegative first column, hence the discontinuity condition remains monotone in $C = \mathbb{E}(\nabla g\,\nabla g^T)$8 and admits a unique threshold $C = \mathbb{E}(\nabla g\,\nabla g^T)$9 via the implicit function theorem. This yields a closed-form preintegrated function expressed through Gaussian survival functions $g$0. For Asian call options under Black–Scholes, the optimal drift reduces to a scalar root-finding problem via recurrence relations, and explicit formulas for $g$1 are derived.

Orthogonal invariance results

The paper proves three structural results. First, a theorem establishes that if $g$2 for orthogonal $g$3, then $g$4, so $g$5 and $g$6 share identical active subspaces and identical expressions in the full-dimensional active subspace; statistical equivalence from prior work follows as a corollary. Second, analogous results hold for IS-active subspaces under both optimal drift and Laplace IS, using uniqueness of the solution to the optimal-drift equation — an assumption the paper states explicitly but does not verify case-by-case. Third, since any two generation matrices of Brownian motion differ by an orthogonal factor, corollaries show that only the standard construction needs to be considered when applying AS-preintegration or the three-step method to Brownian-driven problems. This substantially simplifies implementation and removes a degree of freedom that earlier work treated as a design choice.

Numerical results

Experiments use $g$7, $g$8, $g$9, $\mathbb{E}(g(\bm{Z}))$0, $\mathbb{E}(g(\bm{Z}))$1, strike prices $\mathbb{E}(g(\bm{Z}))$2 to $\mathbb{E}(g(\bm{Z}))$3, Sobol-type RQMC with $\mathbb{E}(g(\bm{Z}))$4 up to $\mathbb{E}(g(\bm{Z}))$5, repeated 50 times.

Convergence behavior: For deep ITM, ITM, and ATM cases ($\mathbb{E}(g(\bm{Z}))$6), IS_AS_Preint matches AS_Preint and outperforms Preint_IS_GPCA at small sample sizes. At $\mathbb{E}(g(\bm{Z}))$7, AS_Preint fails entirely; at $\mathbb{E}(g(\bm{Z}))$8, both AS_Preint and Preint_GPCA fail, while IS_AS_Preint achieves the best convergence among all six methods tested. The paper explains why AS_Preint fails more readily than Preint_GPCA: estimating a $\mathbb{E}(g(\bm{Z}))$9-dimensional active subspace is harder than GPCA on a $\bm{Z}\sim N(\bm{0},I_d)$0-dimensional preintegrated function, since reducing nominal dimension cannot increase effective dimension.

Variance reduction factors at $\bm{Z}\sim N(\bm{0},I_d)$1 illustrate the complementary contributions:

Strike $\bm{Z}\sim N(\bm{0},I_d)$2	AS_Preint	Preint_IS_GPCA	IS_AS_Preint
50	3.7e+07	1.1e+07	5.1e+07
100	7.2e+07	1.9e+07	5.2e+07
120	Failed	8.5e+06	4.6e+07
150	Failed	1.4e+07	1.2e+08

Among basic techniques, AS alone contributes most where applicable (VRFs around $\bm{Z}\sim N(\bm{0},I_d)$3), preintegration dominates deep ITM cases, and IS becomes relatively more important as moneyness decreases. Notably, IS_AS_Preint's VRF grows toward deep OTM strikes, reaching $\bm{Z}\sim N(\bm{0},I_d)$4 at $\bm{Z}\sim N(\bm{0},I_d)$5 — roughly an order of magnitude above Preint_IS_GPCA there — while remaining stable across moneyness levels.

Delta estimation: Using the same IS-active subspace as for pricing (justified by the argument that an IS density fitted only to the continuous component would be independent of $\bm{Z}\sim N(\bm{0},I_d)$6, and that gradients near discontinuities corrupt the estimated $\bm{Z}\sim N(\bm{0},I_d)$7), the method is applied to the pathwise Delta estimator with $\bm{Z}\sim N(\bm{0},I_d)$8, $\bm{Z}\sim N(\bm{0},I_d)$9, $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$0, $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$1. Compared with CPW_IS_GPCA, VRFs are comparable in the deep ITM case, about two orders of magnitude larger for ITM/ATM/OTM, and about four orders of magnitude larger in the deep OTM case ($\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$2 versus $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$3 at $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$4). These results indicate that reusing the pricing-oriented IS-active subspace for Greeks is not merely convenient but empirically superior.

Limitations and open questions

Several constraints bound the applicability of the results. The orthogonal invariance of IS-active subspaces relies on uniqueness of the optimal-drift equation's solution, which is assumed rather than proven for general payoffs. The closed-form preintegration step requires monotonicity of the discontinuous part in the first active variable, guaranteed here via nonnegativity of $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$5 and the first column of $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$6; whether this extends to payoffs with more complex discontinuity structures is not addressed. The gradient information matrix is estimated by finite differences with only $\nabla g(\bm{z})/g(\bm{z}) = \bm{z}$7 samples, and the sensitivity of the resulting subspace to this estimation error is not analyzed. Finally, the numerical evidence is confined to Asian options under Black–Scholes; performance on other payoff classes, models (e.g., stochastic volatility), or non-Brownian drivers remains untested. The authors also conjecture, without proof, that orthogonal invariance extends to uniform distributions on high-dimensional spheres.

Conclusion

The paper contributes a principled ordering of IS, active subspace computation, and preintegration within the QMC framework, together with orthogonally invariant formulations that make the method independent of the Brownian motion construction. The empirical findings are decisive for the regime where prior methods break down: for OTM and deep OTM Asian options, IS_AS_Preint delivers variance reduction factors exceeding competing methods by one to four orders of magnitude, while matching the best existing methods elsewhere. The main open questions concern the uniqueness assumption behind the invariance theory, robustness to noisy gradient estimation, and generalization beyond monotone-discontinuity payoffs.