Separation of Variables¶

Separation of variables is not the natural method for vanilla Black–Scholes on the infinite line — there the Fourier transform (see § Fourier Transform) dominates. Separation becomes essential when the pricing domain is bounded: barrier options impose Dirichlet conditions at one or both ends of the log-price interval, and these boundaries quantize the spectrum of the spatial operator into a discrete sequence of eigenmodes. The whole subsection is built around this single idea — boundaries quantize the spectrum — and works it out first on a toy heat equation, then on the transformed Black–Scholes PDE, and finally on a barrier option.

1. The Toy Problem: Heat Equation on an Interval¶

Before touching Black–Scholes, we work the simplest possible bounded problem. Everything that follows — eigenmodes, exponential decay, the Fourier limit, the barrier picture — is already visible here.

1.1 Setup¶

Consider

with homogeneous Dirichlet boundary conditions

and initial data \(u(x, 0) = f(x)\).

Read this physically: heat in a thin rod of length \(L\), both ends held at temperature \(0\). The rod cannot store an arbitrary temperature profile — the boundaries select a privileged family of shapes.

1.2 Separation Ansatz¶

Look for a product solution \(u(x, t) = X(x)\, T(t)\). Substituting and dividing by \(X T\):

Both sides equal a single constant \(-\lambda\) because the left depends only on \(t\) and the right only on \(x\). This gives two ODEs:

1.3 The Spatial Eigenvalue Problem¶

The spatial ODE with \(X(0) = X(L) = 0\) admits non-trivial solutions only for discrete values

with corresponding eigenfunctions

These are the standing waves of the rod. The pair \((L, X_n)\) encodes everything geometric:

• \(X_1\) has no interior zero — the smoothest mode.
• \(X_n\) has \(n - 1\) interior zeros — successively more oscillatory.
• All modes vanish at the ends, as required by the boundary conditions.

flowchart LR
    A["Bounded domain
[0,L]"] --> B["Dirichlet BCs
u(0)=u(L)=0"]
    B --> C["Discrete eigenvalues
λn = (nπ/L)2"]
    C --> D["Standing-wave basis
sin(nπx/L)"]

Why boundaries quantize: a sinusoid on \([0,L]\) vanishing at both ends must fit a half-integer number of wavelengths, forcing \(kL = n\pi\).

1.4 Temporal Decay¶

The time ODE \(T'_n = -\kappa\lambda_n T_n\) gives

Higher modes decay dramatically faster than lower ones. Doubling \(n\) quadruples the decay rate.

This single fact governs the long-time behavior of every bounded-domain heat or pricing problem:

1.5 General Solution and Coefficient Formula¶

Linear superposition and orthogonality of the sines give

with coefficients fixed by the initial data:

Decomposing \(f\) into the eigenmode basis is the bounded-domain analogue of Fourier analysis: the orthogonality \(\int_0^L \sin(m\pi x/L)\sin(n\pi x/L)\,dx = (L/2)\delta_{mn}\) plays exactly the role of \(\int e^{i(\omega-\omega')x}dx = 2\pi\delta(\omega - \omega')\) for the transform on \(\mathbb{R}\).

2. Geometric Interpretation: What the Eigenmodes Show¶

The toy problem rewards a careful look at what the modes mean. Three observations carry directly to the barrier-option setting.

2.1 Initial Condition as a Sum of Modes¶

Any continuous \(f\) on \([0,L]\) with \(f(0) = f(L) = 0\) decomposes uniquely into sines:

A localized bump near \(x = L/2\) requires many modes; a smooth half-sine requires essentially only \(X_1\). Sharpness in space ↔ slow decay of \(|c_n|\) in \(n\).

2.2 First Mode = Long-Time Shape¶

For any payoff, after enough time the surviving profile is a multiple of \(X_1\). This is the principal eigenmode of the system — the smoothest function compatible with the boundary conditions, and the one that dies slowest. Everything else is a transient.

2.3 Higher Modes = Short-Time Corrections¶

The modes with \(n \geq 2\) control how the solution adjusts from the initial profile to the principal eigenmode. They are oscillatory in space and decay rapidly in time:

is already \(e^{-3\kappa\pi^2 t / L^2}\) for \(n = 2\). Short-time, high-frequency, localized features are exactly the regime where many modes contribute and the spectral picture is most useful.

3. The Infinite-Domain Limit: How Fourier Emerges¶

Push the boundaries apart, \(L \to \infty\). The natural continuous parameter is the wavenumber \(k_n = n\pi/L\), with spacing \(\Delta k = \pi / L\). As \(L \to \infty\), the spacing \(\Delta k\) collapses to zero, and the discrete index \(n\) relaxes into a continuous variable \(k \in \mathbb{R}\); the eigenvalue itself becomes \(\lambda(k) = \tfrac{1}{2}\sigma^2 k^2\) (or \(\kappa k^2\) for the heat equation), so

also collapses to zero. The sum becomes an integral, the sine basis becomes complex exponentials, and the discrete Fourier series of the bounded interval becomes the Fourier transform of the line:

flowchart LR
    A["Bounded [0,L]
discrete λn"] -->|"L → ∞"| B["Spacing Δλ → 0"]
    B --> C["Continuous spectrum"]
    C --> D["Fourier transform
on ℝ"]

Bounded-domain separation and the Fourier transform are two regimes of the same spectral construction. The discrete spectrum is genuine new mathematics precisely because the bounded domain produces it; on the line, the construction degenerates into the integral representation of § Fourier Transform.

4. Barrier Options: The Natural Bounded Domain¶

Now the payoff. A down-and-out call with barrier \(B\) pays \((S_T - K)^+\) at maturity provided the underlying never touches \(B\) on \([0, T]\). If it does, the option is killed.

In log-price \(x = \ln S\), the surviving paths live on the half-line \(x > x_B := \ln B\). Truncating at a large upper level \(x_{\max}\) gives the finite interval

with the boundary conditions

• \(V(x_B, \tau) = 0\) — knocked out at the barrier,
• \(V(x_{\max}, \tau) =\) deep-ITM asymptotic (e.g.\ \(e^{x_{\max}} - K e^{-r\tau}\)).

This is exactly the bounded setup of §1, only with the Black–Scholes operator replacing the simple Laplacian. Barrier options are the canonical reason separation of variables earns its place in option pricing.

flowchart LR
    A["Stock price St"] --> B{"Hits barrier B?"}
    B -->|"Yes"| C["Knocked out
V = 0"]
    B -->|"No"| D["Pays (ST - K)+"]
    C -.absorbing wall.-> E["Dirichlet BC
at x = ln B"]

The barrier acts as an absorbing wall: paths that touch it are removed from the ensemble, exactly as a Dirichlet condition removes amplitude from the eigenfunction expansion.

5. What the Barrier Eigenmodes Will Look Like¶

Before doing the BS-specific eigenvalue calculation in §6, we can already predict the qualitative shape of the barrier-option solution by lifting the toy heat-equation observations of §§1–2 directly to the bounded barrier interval. Sturm–Liouville theory in §6 will confirm what the analogy already strongly suggests; getting the intuition first makes the formal machinery feel inevitable rather than imposed.

The toy heat equation on \([0, L]\) produced eigenmodes \(X_n^{\text{toy}}(x) = \sin(n\pi x/L)\) with eigenvalues \(\lambda_n^{\text{toy}} = (n\pi/L)^2\). The barrier-option problem replaces the toy Laplacian with the BS spatial operator on \([x_B, x_{\max}]\), but the qualitative features of the eigenmode sequence — first mode smoothest, higher modes oscillatory, exponential decay of mode \(n\) at rate \(\lambda_n\) — are governed by the boundary geometry alone, not by the specific operator. We can read off three observations now and verify them in §6.

5.1 First Mode: Long-Maturity Behavior¶

For large \(\tau\), only the slowest-decaying mode survives. Writing the eventual eigenfunction expansion as \(V(x, \tau) = \sum_n c_n X_n(x) e^{-\lambda_n \tau}\):

A single shape, exponentially attenuated. The eigenvalue \(\lambda_1\) is the survival rate of the option value: long-maturity barrier prices decay like \(e^{-\lambda_1 \tau}\). The first mode is the smoothest profile compatible with vanishing at the barrier — supported in the interior, away from both boundaries — by direct analogy with the half-sine \(\sin(\pi x/L)\) of §1.

5.2 Higher Modes: Short-Time Corrections¶

The high-\(n\) modes carry the response to sharp features in the payoff — the kink at \(S = K\) for a call, jumps in a digital, etc. By analogy with §1's sine modes, they oscillate in space with \(n - 1\) interior nodes and decay in time as \(e^{-\lambda_n \tau}\). Near \(\tau = 0\), many modes contribute and the partial sum gives the Gibbs-type behavior expected near a non-smooth payoff; for \(\tau \gtrsim 1/\lambda_2\) they have effectively vanished.

5.3 Mode Hierarchy and Time-to-Forget¶

In the toy problem \(\lambda_n / \lambda_1 = n^2\); for the BS barrier operator the ratio still grows like \(n^2\) (with a constant-coefficient perturbation). Mode \(n\) is irrelevant once

This is a quantitative version of "the bounded domain forgets sharp data quickly." Section 6 below makes the eigenvalue sequence \(\{\lambda_n\}\) precise for the BS operator.

6. Separation on the Transformed Black–Scholes PDE¶

We now do the explicit construction the previous subsection presumed. The log-price change of variables (§ Heat Equation) turns the Black–Scholes PDE into the constant-coefficient form

on \(x \in [x_B, x_{\max}]\) with the boundary conditions of §4 and terminal \(V(x, 0) = \Phi(e^x)\) (after time reversal \(\tau = T - t\)).

6.1 Ansatz and Two ODEs¶

Setting \(V(x, \tau) = X(x)\, T(\tau)\):

So \(T(\tau) = e^{-\lambda\tau}\) and

6.2 Sturm–Liouville Form¶

The spatial operator carries both a second-derivative and a first-derivative term, so it is not symmetric in the unweighted \(L^2\) inner product. The standard remedy is a gauge transformation \(X(x) = e^{-\alpha x} Y(x)\) for a suitable constant \(\alpha\), chosen to eliminate the first-derivative term. This reduces the eigenvalue problem to a self-adjoint second-order operator on \([x_B, x_{\max}]\) with Dirichlet boundary conditions — the canonical setting for Sturm–Liouville theory. Self-adjointness is what guarantees real eigenvalues, orthogonal eigenfunctions, and completeness — the structural properties that make spectral expansions possible.

Sturm–Liouville theory then guarantees exactly the eigenvalue picture used qualitatively in §5:

• a real, discrete, positive, increasing spectrum \(0 < \lambda_1 < \lambda_2 < \cdots \to \infty\);
• orthogonal eigenfunctions \(\{X_n\}\) complete in the appropriate weighted \(L^2\);
• exponential damping \(e^{-\lambda_n \tau}\) of the \(n\)-th component.

We do not need the full functional-analytic machinery — only the qualitative statement above, which underwrites §5's intuition and gives the picture of §1.4 in the present setting.

6.3 Eigenfunction Expansion¶

The full pricing formula is

with the inner product appropriate to the self-adjoint form of §6.2. Truncating at \(N\) modes yields an error bounded by

so convergence is exponential in \(\tau\) — fast for moderate-to-long maturities, slow for \(\tau\) near \(0\). The qualitative observations of §5 are now confirmed quantitatively: first-mode dominance for \(\tau \gg 1/(\lambda_2 - \lambda_1)\), Gibbs-type partial-sum behavior near \(\tau = 0\), and the time-to-forget formula \(\tau \gtrsim 1/(\lambda_n - \lambda_1)\) for mode \(n\) to be negligible.

7. Method of Images: A Spectral Alternative¶

For a down-and-out call on the half-line \([B, \infty)\) — no upper truncation — separation produces a continuous spectrum (no second boundary) and the eigenfunction sum collapses to a closed form. Schematically, that closed form is the method of images:

with a power \(p\) determined by the drift of the log-price dynamics. The reflected term cancels the original on \(S = B\), enforcing the Dirichlet condition.

Conceptually:

For a double-barrier problem the closed form via images becomes an infinite sum of reflections — equivalent to, and no simpler than, the eigenfunction expansion.

8. Two Supporting Cases¶

8.1 Box Spread on a Bounded Interval¶

For \(S \in [S_L, S_U]\) with constant boundary values \(V(S_L, t) = A\) and \(V(S_U, t) = B\), subtract the linear interpolant

so \(W\) satisfies homogeneous Dirichlet conditions and admits the standard eigenfunction expansion. The constant piece carries the "boundary data"; the eigenfunctions handle the dynamics.

8.2 Multi-Asset Options¶

For two assets \((S_1, S_2)\) the cross-derivative term in the Black–Scholes PDE prevents separation unless the correlation \(\rho = 0\). In the uncorrelated case,

For \(\rho \neq 0\), diagonalize the covariance matrix first (principal components) and then separate in the rotated coordinates.

9. Computation¶

The pieces of a numerical scheme are:

• Eigenvalues — either by the shooting method (guess \(\lambda\), integrate the ODE, match the right boundary) or by finite-difference discretization, yielding a matrix eigenvalue problem.
• Coefficients — projection integrals \(c_n = \langle \Phi, X_n\rangle / \langle X_n, X_n\rangle\) evaluated by Gauss quadrature.
• Truncation — keep \(N\) modes; error bound \(\sim e^{-\lambda_{N+1}\tau}\). For \(\tau\) near zero, increase \(N\) or fall back to a finite-difference solver near maturity.

The mode count \(N\) needed for a given accuracy scales like \(\tau^{-1/2}\) near \(\tau = 0\) and is essentially \(1\)–\(5\) for \(\tau\) at the year scale.

10. Where This Sits Among the Solution Representations¶

The pricing semigroup \(e^{\tau\mathcal{L}}\) from § Introduction admits four equivalent representations, and separation of variables provides one of them — the spectral form:

The kernel form \(\int G(x, y, \tau) V_0(y)\, dy\) is the subject of § Heat Equation; the probabilistic form \(\mathbb{E}[V_0(X_\tau)]\) is the subject of § Feynman–Kac. Separation of variables is the representation that turns barrier-option pricing into a spectral problem on a bounded domain — and on that domain it is genuinely the right tool.

11. Summary¶

The structural lesson is the one stated at the top of the subsection: boundaries quantize the spectrum. Vanilla options live on \(\mathbb{R}\) and call for a continuous spectrum (Fourier). Barrier options live on a finite interval and call for a discrete one (separation of variables). The two are the same construction, separated only by the geometry of the state space \(\square\)

Appendix: Figure-Generation Script¶

All six figures above were produced by a single script. Captions and discussion are in the body of the subsection; the script is collected here so the chapter narrative is not interrupted by code.

Exercises¶

Exercise 1. For the heat equation \(\frac{\partial F}{\partial \tau} = \frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2}\) on the domain \(x \in (a, b)\) with \(F(a, \tau) = F(b, \tau) = 0\) (a double-barrier knock-out option), apply separation of variables to show that the separated solutions are

for \(n = 1, 2, \ldots\) (Use the positive decay convention \(T_n' = -\mu_n T_n\) so that \(e^{-\mu_n \tau}\) damps in \(\tau\).)

Exercise 2. Using the eigenvalue decomposition from Exercise 1, write the solution for a European call option knocked out at barriers \(S = B_l\) and \(S = B_u\) (with \(B_l < K < B_u\)) as a Fourier sine series. Discuss the convergence rate of this series for smooth vs. non-smooth initial conditions.

Exercise 3. Explain why separation of variables with a discrete spectrum arises naturally for barrier options (bounded domain) but not for vanilla European options (unbounded domain). What replaces the discrete eigenvalue sum in the unbounded case?

Exercise 4. Consider the Black–Scholes PDE after the change of variables to the heat equation. Substitute \(F(x,\tau) = X(x)T(\tau)\) and derive the two ODEs. Solve each ODE explicitly, identifying the separation constant \(\lambda\).

Exercise 5. The four equivalent representations of the solution operator — eigenfunction expansion, Fourier integral, Green's function, and probabilistic expectation — are listed in the summary. For a European call on the domain \(x \in (-\infty, \infty)\), verify that the Green's function representation (convolution with Gaussian) and the Fourier transform representation yield the same answer.

Exercise 6. For a down-and-out call with barrier at \(S = B\) (where \(B < K\)), describe how the method of images can be used instead of separation of variables to satisfy the boundary condition \(V(B, t) = 0\). What is the "image" solution, and how does the final formula relate to the standard Black–Scholes price?