Spectral Decomposition Technique

• Spectral decomposition technique is a mathematical method that expresses functions, operators, or matrices as sums over eigenfunctions and eigenvalues.
• It transforms complex PDEs and stochastic processes into decoupled ordinary differential equations, facilitating efficient analysis and option pricing.
• The approach unifies analytical and numerical strategies while effectively handling boundary conditions in both standard and exotic models.

Spectral decomposition technique refers broadly to mathematical and computational methods that express an object—such as a function, operator, matrix, network, or signal—as a sum, integral, or expansion in terms of elementary “spectral” components, most often associated with eigenfunctions, eigenvalues, or frequency modes. In applied mathematics, physics, financial engineering, and data science, spectral decomposition is used to transform complex systems governed by partial differential equations, stochastic processes, or high-dimensional data into forms amenable to analysis, efficient approximation, or precise computation. The approach is central to applications ranging from option pricing in stochastic volatility models to material decomposition in imaging and network analysis.

1. Foundational Principles of Spectral Decomposition

At its core, spectral decomposition exploits the fact that many linear operators, matrices, or differential equations admit a basis of eigenvectors or eigenfunctions—i.e., special functions $f$ that satisfy $Lf = \lambda f$ for an operator $L$ and eigenvalue $\lambda$. For finite-dimensional symmetric (Hermitian) matrices, the spectral theorem ensures diagonalizability: $A = V \Lambda V^{-1}$ where $\Lambda$ is a diagonal matrix of eigenvalues and $V$ contains the eigenvectors.

In the context of partial differential equations (PDEs) and stochastic processes, one seeks to diagonalize a spatial or infinitesimal generator $\mathcal{L}$, expanding the solution as a sum or integral over eigenfunctions: $u(t, x) = \sum_q A_q e^{\lambda_q t} \Psi_q(x)$ or, for continuous spectra, as

$$u(t, x) = \int A_\nu e^{\lambda_\nu t} \Psi_\nu(x)\, d\nu.$$

This “diagonalization” transforms the original evolution problem into a set of decoupled ordinary differential equations for the coefficients $A_q$, leveraging the boundary and initial conditions via orthogonality and completeness.

For problems with multiple scales or parameters, such as fast mean-reverting stochastic volatility, the operator $\mathcal{L}$ may itself depend on a small parameter $\epsilon$. Singular perturbation theory is then combined with spectral expansions by expanding both eigenfunctions $\Psi_q^{\epsilon}$ and eigenvalues $\lambda_q^\epsilon$ in powers of $\sqrt{\epsilon}$, facilitating asymptotic analysis and dimensional reduction.

2. Spectral Decomposition in Fast Mean-Reverting Stochastic Volatility

In “Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models” (Fouque et al., 2010), the technique is developed for option pricing under models where the volatility factor $Y_t^\epsilon$ evolves on a fast time scale, with small parameter $\epsilon \ll 1$. The un-discounted option price $u^\epsilon(t, x, y)$ solves a two-dimensional degenerate PDE: $$-\partial_t u^\epsilon + \mathcal{L}_{X,Y}^\epsilon u^\epsilon = 0,$$ where $\mathcal{L}_{X,Y}^\epsilon$ is the infinitesimal generator involving both the log-price $X_t$ and volatility $Y_t^\epsilon$.

Rather than traditional approaches, the method constructs an eigenfunction expansion: $$u^\epsilon(t, x, y) = \sum_q A_q e^{\lambda_q^\epsilon t} \Psi_q^\epsilon(x, y)$$ with $$(\mathcal{L}_{X,Y}^\epsilon \Psi_q^\epsilon) = \lambda_q^\epsilon \Psi_q^\epsilon$$, accommodating various boundary and initial conditions according to the option's payoff structure and the spatial domain.

A singular perturbation expansion is performed: $$\begin{aligned} \mathcal{L}_{X, Y}^{\epsilon} &= \frac{1}{\epsilon}\mathcal{L}^{(-2)} + \frac{1}{\sqrt{\epsilon}}\mathcal{L}^{(-1)} + \mathcal{L}^{(0)}, \ \Psi_q^\epsilon(x, y) &= \Psi_q^{(0)}(x) + \sqrt{\epsilon}\Psi_q^{(1)}(x) + \epsilon\Psi_q^{(2)}(x, y) + \cdots, \ \lambda_q^\epsilon &= \lambda_q^{(0)} + \sqrt{\epsilon}\lambda_q^{(1)} + \cdots, \end{aligned}$$ yielding at leading order ($1/\epsilon$) an averaging over the fast variable and, at higher orders, corrections that encode the effect of stochastic volatility and its correlation with price.

The coefficients $A_q^{(0)}$ and $A_q^{(1)}$ are determined by expanding the initial payoff in the appropriate eigenfunction basis, for example, via weighted inner products that naturally arise from the Sturm-Liouville formulation.

3. Implementation for Barrier and European Option Pricing

The framework is specialized for key option types:

• European Options: On the real line domain, the eigenfunctions reduce to Fourier modes:

$$\Psi_\nu^{(0)}(x) = e^{-c x} \sqrt{\frac{\sigma^2}{4\pi}} e^{i\nu x},$$

with $c$ set by drift and average volatility $\sigma^{*2} = \langle f^2 \rangle$. The option price recovers the Black–Scholes formula at leading order, matching previous singular perturbation results.

• Barrier Options: For domains $(l, r)$, as in double-barrier knock-out options, sine Fourier series are used:

$$\Psi_n^{(0)}(x) = e^{-c x} \sqrt{\frac{2}{r-l}} \sin\left(\alpha_n(x-l)\right),\quad \alpha_n = n\pi/(r-l).$$

This expansion automatically imposes the Dirichlet (knock-out) boundary conditions. First-order corrections account for the fast stochastic volatility and are obtained through an analogous asymptotic argument.

The spectral decomposition neatly builds boundary behavior into the representation, in contrast with classical perturbative methods that may require auxiliary steps to enforce boundary conditions.

4. Analytical and Numerical Advantages

The spectral method, when combined with singular perturbation theory:

• Reduces a complex two-dimensional PDE to a sequence of one-dimensional problems;
• Guarantees analytical tractability for leading and next-to-leading order pricing;
• Enables direct construction of approximations for both standard (European) and exotic (barrier) options within a unified framework;
• Allows the method to recover, at each asymptotic order, established results such as the Black–Scholes price ($u^{BS}(t,x) = u^{(0)}(t,x)$, Theorem 4.1) and the Fouque-Papanicolaou-Sircar correction ($u^{FPS}(t,x) = u^{(1)}(t,x)$, Theorem 4.2).

A salient outcome is that for a broad class of payoffs and domains with nontrivial boundaries, the spectral decomposition achieves the same accuracy as singular perturbative methods, but with improved clarity and extensibility—especially for high-dimensional or exotic contracts.

5. Mathematical Structure and Key Formulas

Central mathematical structures and formulas include:

• Spectral Expansion:

$$u^\epsilon(t,x,y) = \sum_q A_q e^{\lambda_q^\epsilon t} \Psi_q^\epsilon(x, y),$$

with $g_q^\epsilon(t) = \exp(\lambda_q^\epsilon t)$.

• Asymptotic Expansion:

\begin{align*} u^\epsilon(t,x,y) &= u^{(0)}(t,x) + \sqrt{\epsilon} u^{(1)}(t,x) + \cdots, \ u^{(0)}(t, x) &= \sum_q A_q^{(0)} e^{\lambda_q^{(0)} t} \Psi_q^{(0)}(x), \ u^{(1)}(t, x) &= \sum_q \left[A_q^{(1)} e^{\lambda_q^{(0)} t} \Psi_q^{(0)}(x) + A_q^{(0)} \lambda_q^{(1)} t e^{\lambda_q^{(0)} t} \Psi_q^{(0)}(x) + A_q^{(0)} e^{\lambda_q^{(0)} t} \Psi_q^{(1)}(x)\right]. \end{align*}

• Initial Condition Expansion:

$A_q^{(0)} = (\Psi_q^{(0)}, h)_s,$

where $(\cdot, \cdot)_s$ is an appropriately weighted inner product.

This mathematical machinery systematically reduces the pricing problem to determining functionals and derivatives with respect to averaged (fast mean-reverting) coefficients, which are then used to reconstitute the solution.

6. Relation to Previously Established Asymptotic Methods

A noteworthy result is the rigorous equivalence, at leading and $\mathcal{O}(\sqrt{\epsilon})$ orders, with the singular perturbation technique of Fouque, Papanicolaou, and Sircar (FPS), as established in Theorems 4.1 and 4.2 of (Fouque et al., 2010). This harmonization confirms that spectral decomposition methods offer no loss of accuracy relative to more traditional singular perturbation techniques for fast mean-reverting settings—even when applied to complex contracts such as rebates and knock-outs. The spectral approach, however, routinely delivers advantages in implementation and generalizability to more complicated domains and boundary conditions.

7. Broader Impact and Extensions

The spectral decomposition technique in this setting:

• Systematically incorporates and enforces boundary conditions for a broad range of payoff profiles, facilitating the pricing of path-dependent or barrier options.
• Extends naturally to higher dimensions and to other fast–slow stochastic systems where spectral decomposability of the operator is tractable.
• Reduces the analytical and computational complexity by separating variables and leveraging Fourier- or Sturm-Liouville–type eigenfunction expansions.

A plausible implication is that for other classes of singularly perturbed stochastic differential equations, spectral decomposition may similarly yield both analytical clarity and computational tractability, provided that the generator is amenable to such expansion and that efficient numerical evaluation of eigenvalues/eigenfunctions is feasible.

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In summary, the spectral decomposition technique transforms a high-dimensional, fast–slow PDE problem into an eigenfunction expansion, enabling accurate and efficiently computable option price approximations in fast mean-reverting stochastic volatility models. This approach maintains parity with established asymptotic results while providing improved flexibility—particularly in handling boundary conditions for exotic options—and forms the mathematical basis for further extensions in stochastic analysis and quantitative finance.