Forward PDE and Fokker-Planck Equation¶

The backward Kolmogorov equation propagates option prices backward from the terminal payoff to the present. Its dual, the forward Kolmogorov equation (or Fokker-Planck equation), propagates the transition density forward in time. In the local volatility setting, the forward equation provides the theoretical foundation for Dupire's formula: inverting the Fokker-Planck equation for the local volatility function yields a direct formula in terms of observable call prices. This section develops the forward PDE, derives its key properties, and shows how it connects the local volatility surface to the evolution of the risk-neutral density.

Learning Objectives

After completing this section, you should be able to:

Derive the Fokker-Planck equation for the local volatility diffusion from first principles
Explain the duality between forward and backward Kolmogorov equations
Show how the Fokker-Planck equation leads to the Dupire formula
State the conditions under which the forward PDE has a unique smooth solution
Interpret the forward equation as a conservation law for probability

The Forward Kolmogorov Equation¶

Derivation from the SDE¶

Under the risk-neutral measure \(\mathbb{Q}\), the asset price evolves as:

\[ dS_t = (r - q)S_t \, dt + \sigma_{\text{loc}}(S_t, t) S_t \, dW_t^{\mathbb{Q}} \]

Let \(p(S, t \mid S_0, 0)\) denote the transition density: the probability density that \(S_t = S\) given \(S_0\) at time \(0\). For any smooth test function \(\varphi(S)\) with compact support, Ito's lemma gives:

\[ d\varphi(S_t) = \left[(r - q)S_t \varphi'(S_t) + \frac{1}{2}\sigma_{\text{loc}}^2(S_t, t) S_t^2 \varphi''(S_t)\right] dt + \sigma_{\text{loc}}(S_t, t) S_t \varphi'(S_t) \, dW_t^{\mathbb{Q}} \]

Taking expectations and using \(\mathbb{E}^{\mathbb{Q}}[\varphi(S_t)] = \int_0^{\infty} \varphi(S) p(S, t) \, dS\):

\[ \frac{d}{dt} \int_0^{\infty} \varphi(S) p(S, t) \, dS = \int_0^{\infty} \left[(r - q)S \varphi'(S) + \frac{1}{2}\sigma_{\text{loc}}^2(S, t) S^2 \varphi''(S)\right] p(S, t) \, dS \]

Integrating by parts twice on the right-hand side (assuming \(p\) and its derivatives vanish sufficiently fast at \(S = 0\) and \(S = \infty\)):

\[ \int_0^{\infty} (r - q)S \varphi'(S) p(S, t) \, dS = -\int_0^{\infty} \varphi(S) \frac{\partial}{\partial S}\bigl[(r - q)S \, p(S, t)\bigr] \, dS \]

\[ \int_0^{\infty} \frac{1}{2}\sigma_{\text{loc}}^2 S^2 \varphi''(S) p(S, t) \, dS = \int_0^{\infty} \varphi(S) \frac{1}{2}\frac{\partial^2}{\partial S^2}\bigl[\sigma_{\text{loc}}^2(S, t) S^2 p(S, t)\bigr] \, dS \]

Since this holds for all test functions \(\varphi\), the density \(p(S, t)\) satisfies:

Theorem 13.3.1 (Fokker-Planck Equation for Local Volatility).

\[ \frac{\partial p}{\partial t} = -\frac{\partial}{\partial S}\bigl[(r - q)S \, p\bigr] + \frac{1}{2}\frac{\partial^2}{\partial S^2}\bigl[\sigma_{\text{loc}}^2(S, t) S^2 \, p\bigr] \]

with initial condition \(p(S, 0) = \delta(S - S_0)\). \(\square\)

General Form¶

For a general Ito diffusion \(dX_t = \mu(X_t, t)\,dt + \sigma(X_t, t)\,dW_t\), the Fokker-Planck equation takes the form:

\[ \frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}\bigl[\mu(x, t) p(x, t)\bigr] + \frac{1}{2}\frac{\partial^2}{\partial x^2}\bigl[\sigma^2(x, t) p(x, t)\bigr] \]

For the local volatility model with \(\mu(S, t) = (r-q)S\) and \(\sigma(S, t) = \sigma_{\text{loc}}(S, t) S\), the diffusion coefficient in the Fokker-Planck equation is \(a(S, t) = \sigma_{\text{loc}}^2(S, t) S^2\).

Duality: Forward vs Backward¶

The Backward Kolmogorov Equation¶

The backward Kolmogorov equation governs the evolution of conditional expectations. For \(u(S, t) = \mathbb{E}^{\mathbb{Q}}[\varphi(S_T) \mid S_t = S]\):

\[ \frac{\partial u}{\partial t} + (r - q)S \frac{\partial u}{\partial S} + \frac{1}{2}\sigma_{\text{loc}}^2(S, t) S^2 \frac{\partial^2 u}{\partial S^2} = 0 \]

with terminal condition \(u(S, T) = \varphi(S)\). The backward equation runs backward in time from \(T\) to \(t\) and uses derivatives with respect to the initial spot \(S\).

Adjoint Relationship¶

The forward and backward operators are formal adjoints. Define:

\[ \mathcal{L} u = (r - q)S \frac{\partial u}{\partial S} + \frac{1}{2}\sigma_{\text{loc}}^2 S^2 \frac{\partial^2 u}{\partial S^2} \]

\[ \mathcal{L}^* p = -\frac{\partial}{\partial S}\bigl[(r - q)S \, p\bigr] + \frac{1}{2}\frac{\partial^2}{\partial S^2}\bigl[\sigma_{\text{loc}}^2 S^2 \, p\bigr] \]

Then for suitable functions \(u, p\) vanishing at the boundary:

\[ \int_0^{\infty} u \, \mathcal{L}^* p \, dS = \int_0^{\infty} p \, \mathcal{L} u \, dS \]

This adjoint relationship is the mathematical expression of the duality between:

Backward equation: Given a fixed terminal payoff, how does the expected value depend on the initial state?
Forward equation: Given a fixed initial state, how does the probability distribution evolve forward?

Proposition 13.3.1 (Duality Identity). For the transition density \(p(S, T \mid S_0, 0)\) and any smooth function \(\varphi\) with compact support:

\[ \int_0^{\infty} \varphi(S) p(S, T \mid S_0, 0) \, dS = u(S_0, 0) \]

where \(u\) solves the backward equation with terminal condition \(u(S, T) = \varphi(S)\). \(\square\)

This identity confirms that both equations encode the same information: one in terms of densities (forward), the other in terms of expected values (backward).

From Fokker-Planck to Dupire's Formula¶

Derivation via Density Integration¶

The European call price is:

\[ C(K, T) = e^{-rT} \int_K^{\infty} (S - K) p(S, T) \, dS \]

Differentiating with respect to maturity \(T\):

\[ \frac{\partial C}{\partial T} = -rC + e^{-rT} \int_K^{\infty} (S - K) \frac{\partial p}{\partial T}(S, T) \, dS \]

Substituting the Fokker-Planck equation and integrating by parts (as detailed in the Dupire Formula Derivation section):

\[ \frac{\partial C}{\partial T} = -rC + (r - q)e^{-rT}\int_K^{\infty} S \, p(S, T) \, dS + \frac{1}{2}\sigma_{\text{loc}}^2(K, T) K^2 e^{-rT} p(K, T) \]

Using the identities:

\[ e^{-rT}\int_K^{\infty} S \, p \, dS = C - KC_K, \quad e^{-rT} p(K, T) = C_{KK} \]

we obtain:

\[ \frac{\partial C}{\partial T} = -qC - (r - q)KC_K + \frac{1}{2}\sigma_{\text{loc}}^2(K, T) K^2 C_{KK} \]

Solving for \(\sigma_{\text{loc}}^2\):

\[ \sigma_{\text{loc}}^2(K, T) = \frac{\partial_T C + (r - q)K \partial_K C + qC}{\frac{1}{2}K^2 \partial_{KK} C} \]

This is Dupire's formula. The Fokker-Planck equation is the theoretical engine behind it: the forward evolution of the density determines how call prices change with maturity, and inverting this relationship yields the local volatility.

Forward Equation in Call Price Space¶

An equivalent and often more useful formulation writes the forward PDE directly in terms of call prices. Defining \(U(K, T) = e^{rT} C(K, T)\) (the undiscounted call price), Dupire's forward PDE becomes:

\[ \frac{\partial U}{\partial T} = \frac{1}{2}\sigma_{\text{loc}}^2(K, T) K^2 \frac{\partial^2 U}{\partial K^2} - (r - q)K \frac{\partial U}{\partial K} - qU \]

with initial condition \(U(K, 0) = (S_0 - K)^+\).

This PDE is forward in the maturity variable \(T\) and has the strike \(K\) as its spatial variable. Its advantage over the backward PDE is computational: a single solve sweeps forward through all maturities simultaneously, producing the entire call price surface from a fixed initial spot \(S_0\).

Computational Advantage

The backward PDE must be solved separately for each option payoff (each strike \(K\)). The forward PDE solves for all strikes simultaneously with a single forward sweep, making it the natural tool for calibrating the entire local volatility surface.

Probability Conservation and Flux Interpretation¶

Conservation Law¶

The Fokker-Planck equation can be written in conservation form:

\[ \frac{\partial p}{\partial t} + \frac{\partial J}{\partial S} = 0 \]

where the probability flux is:

\[ J(S, t) = (r - q)S \, p(S, t) - \frac{1}{2}\frac{\partial}{\partial S}\bigl[\sigma_{\text{loc}}^2(S, t) S^2 \, p(S, t)\bigr] \]

The conservation form expresses the fact that probability is neither created nor destroyed: the total probability \(\int_0^{\infty} p(S, t) \, dS = 1\) is preserved for all \(t\).

Proposition 13.3.2 (Conservation of Probability). If \(p(S, t)\) solves the Fokker-Planck equation with \(p(S, 0) = \delta(S - S_0)\) and \(J(S, t) \to 0\) as \(S \to 0^+\) and \(S \to \infty\), then:

\[ \int_0^{\infty} p(S, t) \, dS = 1 \quad \text{for all } t \geq 0 \]

Proof. Integrate the conservation law over \((0, \infty)\):

\[ \frac{d}{dt}\int_0^{\infty} p(S, t) \, dS = -\bigl[J(S, t)\bigr]_0^{\infty} = 0 \]

by the boundary conditions. Since \(\int_0^{\infty} p(S, 0) \, dS = 1\), the result follows. \(\square\)

Physical Interpretation of the Flux¶

The probability flux \(J\) has two components:

Drift flux \((r - q)S \, p\): Probability is transported in the direction of the risk-neutral drift. For \(r > q\), probability flows toward higher values of \(S\).
Diffusion flux \(-\frac{1}{2}\partial_S[\sigma_{\text{loc}}^2 S^2 p]\): Probability spreads out due to volatility. The diffusion flux moves probability from regions of high concentration to regions of low concentration, but the rate depends on the local volatility.

Where \(\sigma_{\text{loc}}(S, t)\) is large, the diffusion flux is strong and the density spreads rapidly. Where \(\sigma_{\text{loc}}\) is small, the density remains concentrated. The local volatility surface therefore controls the shape of the density at each instant.

Well-Posedness and Regularity¶

Existence and Uniqueness¶

Theorem 13.3.2 (Well-Posedness of the Forward PDE). Suppose \(\sigma_{\text{loc}}(S, t)\) satisfies:

Uniform ellipticity: There exist constants \(0 < \lambda \leq \Lambda < \infty\) such that \(\lambda \leq \sigma_{\text{loc}}(S, t) \leq \Lambda\) for all \((S, t)\)
Holder continuity: \(\sigma_{\text{loc}}\) is Holder continuous in both variables

Then the Fokker-Planck equation with initial condition \(p(S, 0) = \delta(S - S_0)\) admits a unique classical solution \(p \in C^{2,1}((0, \infty) \times (0, T^*])\) satisfying \(p(S, t) > 0\) for all \(S > 0\), \(t > 0\).

The proof relies on the theory of parabolic PDEs (Friedman, 1964) and the parametrix method for constructing the fundamental solution. The strict positivity of \(p\) follows from the strong maximum principle for parabolic equations.

Smoothness of the Density¶

Under the conditions of Theorem 13.3.2, the transition density \(p(S, t)\) has several important properties:

Smoothness: \(p\) is infinitely differentiable in \(S\) for \(t > 0\), even though the initial condition is a Dirac delta
Gaussian bounds: There exist constants \(c, C > 0\) such that

\[ \frac{c}{S\sqrt{t}} \exp\left(-\frac{C(\log S/S_0)^2}{t}\right) \leq p(S, t) \leq \frac{C}{S\sqrt{t}} \exp\left(-\frac{c(\log S/S_0)^2}{t}\right) \]
Short-time behavior: As \(t \to 0^+\), the density concentrates around \(S_0\) and converges to \(\delta(S - S_0)\) in the distributional sense

These bounds ensure that the Dupire formula is well-defined: the denominator \(C_{KK} = e^{-rT} p(K, T) > 0\) for all \(K > 0\) and \(T > 0\).

Forward Equation in Log Coordinates¶

Change of Variable¶

The transformation \(x = \log S\) simplifies the Fokker-Planck equation. Under the SDE for \(x_t = \log S_t\):

\[ dx_t = \left(r - q - \frac{1}{2}\sigma_{\text{loc}}^2(e^{x_t}, t)\right) dt + \sigma_{\text{loc}}(e^{x_t}, t) \, dW_t^{\mathbb{Q}} \]

the transition density \(\tilde{p}(x, t) = p(e^x, t) e^x\) satisfies:

\[ \frac{\partial \tilde{p}}{\partial t} = -\frac{\partial}{\partial x}\left[\left(r - q - \frac{1}{2}\tilde{\sigma}^2(x, t)\right)\tilde{p}\right] + \frac{1}{2}\frac{\partial^2}{\partial x^2}\bigl[\tilde{\sigma}^2(x, t) \tilde{p}\bigr] \]

where \(\tilde{\sigma}(x, t) = \sigma_{\text{loc}}(e^x, t)\). In log coordinates, the diffusion coefficient no longer depends on the price level through \(S^2\), which improves the numerical conditioning of the PDE.

Constant Volatility Case¶

When \(\sigma_{\text{loc}}(S, t) = \sigma\) is constant, the density in log coordinates is Gaussian:

\[ \tilde{p}(x, t) = \frac{1}{\sigma\sqrt{2\pi t}} \exp\left(-\frac{\bigl(x - x_0 - (r - q - \frac{1}{2}\sigma^2)t\bigr)^2}{2\sigma^2 t}\right) \]

where \(x_0 = \log S_0\). This recovers the Black-Scholes lognormal distribution. The forward PDE reduces to the heat equation with drift, confirming the Gaussian transition density.

Worked Example¶

Density Evolution with Piecewise Constant Local Volatility

Setup. Consider a simple local volatility function that depends only on the price level:

\[ \sigma_{\text{loc}}(S, t) = \begin{cases} 0.30 & \text{if } S \leq 100 \\ 0.15 & \text{if } S > 100 \end{cases} \]

with \(r = 0.03\), \(q = 0\), \(S_0 = 100\).

The Fokker-Planck equation has a higher diffusion coefficient below \(S = 100\) and a lower coefficient above. This means:

Probability mass below \(S_0 = 100\) spreads out faster (higher vol region)
Probability mass above \(S_0 = 100\) remains more concentrated (lower vol region)

The resulting density \(p(S, T)\) is negatively skewed: the left tail is fatter than the right tail. This produces an implied volatility surface with a downward-sloping skew, consistent with the empirical equity skew.

The Dupire formula applied to call prices generated by this model recovers the piecewise constant \(\sigma_{\text{loc}}\):

For \(K < 100\): \(\sigma_{\text{loc}}^2(K, T) = 0.09\), yielding \(\sigma_{\text{loc}} = 0.30\)
For \(K > 100\): \(\sigma_{\text{loc}}^2(K, T) = 0.0225\), yielding \(\sigma_{\text{loc}} = 0.15\)

This confirms the consistency of the forward PDE and Dupire inversion.

Connection to Other Results¶

Gyongy's Theorem¶

The Fokker-Planck equation plays a central role in Gyongy's theorem. For any stochastic volatility model, the marginal density \(p(S, t)\) satisfies a Fokker-Planck equation with effective diffusion coefficient \(\mathbb{E}[\sigma_t^2 S_t^2 \mid S_t = S]\). The Markovian projection produces a local volatility model whose Fokker-Planck equation has the same solution -- hence the same marginal densities.

Dupire Formula as Inversion¶

The Dupire formula is the algebraic inversion of the Fokker-Planck equation: given the density \(p(S, T) = e^{rT} C_{KK}(S, T)\) and its time evolution \(\partial_T p\) (encoded in \(C_T\)), one solves for the diffusion coefficient \(\sigma_{\text{loc}}^2 S^2\). The forward PDE determines \(p\); inverting for \(\sigma_{\text{loc}}^2\) recovers the Dupire formula.

Transition to Numerical Methods¶

In practice, the forward PDE is solved numerically using finite difference methods. The PDE is discretized on a \((S, T)\) grid, and the density (or equivalently, call prices) is propagated forward in time using Crank-Nicolson or implicit Euler schemes. The forward formulation is especially efficient for calibration, since a single forward sweep produces call prices at all strikes simultaneously.

Summary¶

The Fokker-Planck equation is the forward counterpart of the backward pricing PDE:

Forward evolution: The Fokker-Planck equation \(\partial_t p = -\partial_S[\mu p] + \frac{1}{2}\partial_{SS}[a \, p]\) propagates the transition density forward from the initial Dirac delta.
Adjoint duality: The forward operator \(\mathcal{L}^*\) is the formal adjoint of the backward operator \(\mathcal{L}\), encoding the same information in dual form.
Foundation for Dupire: Integrating the Fokker-Planck equation against the call payoff and inverting yields the Dupire formula for local volatility.
Conservation law: The forward PDE preserves total probability and admits a flux interpretation with drift and diffusion components.
Computational efficiency: The forward PDE solves for all strikes simultaneously, making it the natural tool for calibrating the local volatility surface.
Well-posedness: Under uniform ellipticity and regularity conditions on \(\sigma_{\text{loc}}\), the forward PDE has a unique, strictly positive, smooth solution.

Exercises¶

Exercise 1. Starting from the SDE \(dS_t = (r - q)S_t \, dt + \sigma_{\text{loc}}(S_t, t) S_t \, dW_t\), carry out the integration-by-parts steps to derive the Fokker-Planck equation. Specifically, show that:

\[ \int_0^{\infty} \frac{1}{2}\sigma_{\text{loc}}^2 S^2 \varphi''(S) p(S, t) \, dS = \int_0^{\infty} \varphi(S) \frac{1}{2}\frac{\partial^2}{\partial S^2}[\sigma_{\text{loc}}^2 S^2 p] \, dS \]

State the boundary conditions on \(p\) required for the boundary terms to vanish.

Solution to Exercise 1

We must show that the second-derivative integral can be transferred from \(\varphi\) onto \(\sigma_{\text{loc}}^2 S^2 p\) via two integrations by parts.

First integration by parts. Let \(u = \varphi'(S)\) and \(dv = \frac{1}{2}\sigma_{\text{loc}}^2 S^2 p \, dS\). Then:

\[ \int_0^{\infty} \frac{1}{2}\sigma_{\text{loc}}^2 S^2 \varphi''(S) p \, dS = \left[\varphi'(S) \cdot \frac{1}{2}\sigma_{\text{loc}}^2 S^2 p\right]_0^{\infty} - \int_0^{\infty} \varphi'(S) \frac{\partial}{\partial S}\left[\frac{1}{2}\sigma_{\text{loc}}^2 S^2 p\right] dS \]

The boundary term vanishes provided \(\sigma_{\text{loc}}^2 S^2 p \to 0\) as \(S \to 0^+\) and \(S \to \infty\).

Second integration by parts. Apply integration by parts to the remaining integral with \(u = \varphi(S)\):

\[ -\int_0^{\infty} \varphi'(S) \frac{\partial}{\partial S}\left[\frac{1}{2}\sigma_{\text{loc}}^2 S^2 p\right] dS = -\left[\varphi(S) \frac{\partial}{\partial S}\left(\frac{1}{2}\sigma_{\text{loc}}^2 S^2 p\right)\right]_0^{\infty} + \int_0^{\infty} \varphi(S) \frac{\partial^2}{\partial S^2}\left[\frac{1}{2}\sigma_{\text{loc}}^2 S^2 p\right] dS \]

The boundary term again vanishes provided \(\frac{\partial}{\partial S}[\sigma_{\text{loc}}^2 S^2 p] \to 0\) as \(S \to 0^+\) and \(S \to \infty\).

Combining both steps:

\[ \int_0^{\infty} \frac{1}{2}\sigma_{\text{loc}}^2 S^2 \varphi''(S) p \, dS = \int_0^{\infty} \varphi(S) \frac{1}{2}\frac{\partial^2}{\partial S^2}[\sigma_{\text{loc}}^2 S^2 p] \, dS \]

Required boundary conditions: The density \(p(S, t)\) and the diffusion coefficient \(\sigma_{\text{loc}}^2(S, t) S^2\) must decay fast enough at \(S = 0^+\) and \(S = \infty\) so that all boundary terms vanish. Specifically:

\(\sigma_{\text{loc}}^2 S^2 p(S, t) \to 0\) as \(S \to 0^+\) and \(S \to \infty\)
\(\frac{\partial}{\partial S}[\sigma_{\text{loc}}^2 S^2 p(S, t)] \to 0\) as \(S \to 0^+\) and \(S \to \infty\)

These conditions are satisfied for the lognormal-type densities arising in local volatility models.

Exercise 2. Write the Fokker-Planck equation for the log-coordinate \(x = \log S\), with density \(\tilde{p}(x, t) = p(e^x, t)e^x\). Verify that when \(\sigma_{\text{loc}}\) is constant, the solution is a Gaussian density with mean \(x_0 + (r - q - \sigma^2/2)t\) and variance \(\sigma^2 t\).

Solution to Exercise 2

Under \(x = \log S\), the SDE becomes:

\[ dx_t = \left(r - q - \frac{1}{2}\sigma_{\text{loc}}^2(e^{x_t}, t)\right) dt + \sigma_{\text{loc}}(e^{x_t}, t) \, dW_t^{\mathbb{Q}} \]

Writing \(\tilde{\sigma}(x, t) = \sigma_{\text{loc}}(e^x, t)\), the drift is \(\tilde{\mu}(x, t) = r - q - \frac{1}{2}\tilde{\sigma}^2(x, t)\) and the diffusion is \(\tilde{\sigma}(x, t)\). The Fokker-Planck equation for \(\tilde{p}(x, t) = p(e^x, t)e^x\) is:

\[ \frac{\partial \tilde{p}}{\partial t} = -\frac{\partial}{\partial x}\left[\left(r - q - \frac{1}{2}\tilde{\sigma}^2(x, t)\right)\tilde{p}\right] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[\tilde{\sigma}^2(x, t)\tilde{p}] \]

Verification for constant \(\sigma_{\text{loc}} = \sigma\). When \(\tilde{\sigma}(x, t) = \sigma\) is constant, the Fokker-Planck equation becomes:

\[ \frac{\partial \tilde{p}}{\partial t} = -\left(r - q - \frac{1}{2}\sigma^2\right)\frac{\partial \tilde{p}}{\partial x} + \frac{1}{2}\sigma^2 \frac{\partial^2 \tilde{p}}{\partial x^2} \]

This is the advection-diffusion (heat) equation with constant drift \(\mu_x = r - q - \frac{1}{2}\sigma^2\) and constant diffusion \(\frac{1}{2}\sigma^2\). The fundamental solution with initial condition \(\tilde{p}(x, 0) = \delta(x - x_0)\) is:

\[ \tilde{p}(x, t) = \frac{1}{\sigma\sqrt{2\pi t}} \exp\left(-\frac{(x - x_0 - (r - q - \frac{1}{2}\sigma^2)t)^2}{2\sigma^2 t}\right) \]

This is a Gaussian with mean \(x_0 + (r - q - \sigma^2/2)t\) and variance \(\sigma^2 t\), confirming the Black-Scholes lognormal distribution.

Exercise 3. The probability flux is \(J(S, t) = (r - q)S p - \frac{1}{2}\partial_S[\sigma_{\text{loc}}^2 S^2 p]\). Consider a piecewise constant local volatility: \(\sigma_{\text{loc}} = 0.30\) for \(S < 100\) and \(\sigma_{\text{loc}} = 0.15\) for \(S > 100\). Qualitatively describe the shape of the density \(p(S, T)\) for \(T = 1\) year. Which region has a fatter tail, and why?

Solution to Exercise 3

The piecewise constant local volatility has \(\sigma_{\text{loc}} = 0.30\) for \(S < 100\) and \(\sigma_{\text{loc}} = 0.15\) for \(S > 100\), with \(S_0 = 100\).

The left tail (\(S < 100\)) is fatter. The diffusion coefficient below \(S = 100\) is \(a(S, t) = \sigma_{\text{loc}}^2 S^2 = 0.09 S^2\), which is four times larger than the coefficient above \(S = 100\) where \(a(S, t) = 0.0225 S^2\) (at the same price level \(S = 100\)). The higher volatility region produces stronger diffusion, causing probability mass to spread out more rapidly on the downside.

Qualitative shape of \(p(S, T)\) at \(T = 1\):

The density has a negative skew (left-skewed distribution)
The left tail is fatter than the right tail because the diffusion flux is stronger for \(S < 100\)
The mode of the density is shifted slightly to the right of where it would be under constant volatility, because probability that diffuses below 100 encounters high volatility and spreads further, while probability above 100 encounters low volatility and stays concentrated
The risk-neutral drift \((r - q)S = 0.03S\) pushes the distribution slightly to the right, but this is a secondary effect compared to the asymmetric diffusion

This density shape produces a downward-sloping implied volatility skew: OTM puts (low \(K\)) have higher implied volatility than OTM calls (high \(K\)), consistent with the empirical equity volatility skew.

Exercise 4. Prove the conservation of probability: starting from \(\partial_t p + \partial_S J = 0\), show that \(\int_0^{\infty} p(S, t) \, dS = 1\) for all \(t \geq 0\). What physical assumptions are needed at the boundaries \(S = 0\) and \(S = \infty\)?

Solution to Exercise 4

Starting from the conservation form of the Fokker-Planck equation:

\[ \frac{\partial p}{\partial t} + \frac{\partial J}{\partial S} = 0 \]

Integrate over \(S \in (0, \infty)\):

\[ \frac{d}{dt}\int_0^{\infty} p(S, t) \, dS = -\int_0^{\infty} \frac{\partial J}{\partial S} \, dS = -\bigl[J(S, t)\bigr]_{S=0}^{S=\infty} = -(J(\infty, t) - J(0, t)) \]

Physical assumptions at boundaries:

At \(S = \infty\): The density \(p(S, t) \to 0\) fast enough that both the drift flux \((r-q)Sp\) and the diffusion flux \(\frac{1}{2}\partial_S[\sigma_{\text{loc}}^2 S^2 p]\) vanish. This requires \(p\) to decay faster than any polynomial, which is guaranteed by the Gaussian upper bounds.
At \(S = 0^+\): The flux \(J(0, t) = 0\) because \(S = 0\) is a natural boundary for the lognormal-type diffusion. The drift term \((r-q) \cdot 0 \cdot p = 0\), and the diffusion coefficient \(\sigma_{\text{loc}}^2 \cdot 0^2 = 0\) at \(S = 0\).

Therefore \(J(\infty, t) = J(0, t) = 0\), which gives:

\[ \frac{d}{dt}\int_0^{\infty} p(S, t) \, dS = 0 \]

Since \(p(S, 0) = \delta(S - S_0)\) satisfies \(\int_0^{\infty} p(S, 0) \, dS = 1\), the total probability is conserved:

\[ \int_0^{\infty} p(S, t) \, dS = 1 \quad \text{for all } t \geq 0 \]

Exercise 5. Explain the duality between the forward and backward Kolmogorov equations. If \(u(S, t)\) solves the backward equation with terminal condition \(u(S, T) = \varphi(S)\) and \(p(S, T \mid S_0, 0)\) solves the forward equation with initial condition \(p(S, 0) = \delta(S - S_0)\), show that:

\[ \int_0^{\infty} \varphi(S) p(S, T \mid S_0, 0) \, dS = u(S_0, 0) \]

Solution to Exercise 5

We must show that the duality identity holds by connecting the forward and backward equations through their adjoint relationship.

Let \(u(S, t)\) solve the backward equation:

\[ \frac{\partial u}{\partial t} + (r-q)S\frac{\partial u}{\partial S} + \frac{1}{2}\sigma_{\text{loc}}^2 S^2 \frac{\partial^2 u}{\partial S^2} = 0, \quad u(S, T) = \varphi(S) \]

Let \(p(S, t \mid S_0, 0)\) solve the forward equation:

\[ \frac{\partial p}{\partial t} = -\frac{\partial}{\partial S}[(r-q)Sp] + \frac{1}{2}\frac{\partial^2}{\partial S^2}[\sigma_{\text{loc}}^2 S^2 p], \quad p(S, 0) = \delta(S - S_0) \]

Define \(I(t) = \int_0^{\infty} u(S, t) p(S, t \mid S_0, 0) \, dS\). Differentiate with respect to \(t\):

\[ \frac{dI}{dt} = \int_0^{\infty} \frac{\partial u}{\partial t} p \, dS + \int_0^{\infty} u \frac{\partial p}{\partial t} \, dS \]

The first integral gives \(\int_0^{\infty} (-\mathcal{L}u) p \, dS\) where \(\mathcal{L}\) is the backward operator. The second integral gives \(\int_0^{\infty} u \, \mathcal{L}^* p \, dS\) where \(\mathcal{L}^*\) is the forward operator. By the adjoint relationship:

\[ \int_0^{\infty} u \, \mathcal{L}^* p \, dS = \int_0^{\infty} p \, \mathcal{L} u \, dS \]

Therefore:

\[ \frac{dI}{dt} = -\int_0^{\infty} (\mathcal{L}u) p \, dS + \int_0^{\infty} (\mathcal{L}u) p \, dS = 0 \]

Since \(I(t)\) is constant, \(I(0) = I(T)\). Evaluating at \(t = T\): \(I(T) = \int_0^{\infty} \varphi(S) p(S, T \mid S_0, 0) \, dS\). Evaluating at \(t = 0\) using \(p(S, 0) = \delta(S - S_0)\): \(I(0) = u(S_0, 0)\). Therefore:

\[ \int_0^{\infty} \varphi(S) p(S, T \mid S_0, 0) \, dS = u(S_0, 0) \]

Exercise 6. The forward PDE in call price space is:

\[ \frac{\partial U}{\partial T} = \frac{1}{2}\sigma_{\text{loc}}^2(K, T) K^2 \frac{\partial^2 U}{\partial K^2} - (r - q)K \frac{\partial U}{\partial K} - qU \]

with \(U(K, 0) = (S_0 - K)^+\). Explain why this formulation computes call prices at all strikes simultaneously in a single forward sweep. Compare the computational cost with the backward PDE approach for pricing vanilla options at \(N\) different strikes.

Solution to Exercise 6

Forward PDE approach. The forward PDE in call price space:

\[ \frac{\partial U}{\partial T} = \frac{1}{2}\sigma_{\text{loc}}^2(K, T)K^2\frac{\partial^2 U}{\partial K^2} - (r-q)K\frac{\partial U}{\partial K} - qU \]

starts from the initial condition \(U(K, 0) = (S_0 - K)^+\), which is defined for all strikes \(K\) simultaneously. A single forward sweep in \(T\) (from \(T = 0\) to the final maturity) produces \(U(K, T)\) at all grid points \((K_j, T_k)\) in one pass. This means call prices at all \(N\) strikes are obtained simultaneously at computational cost \(O(M \cdot N_K)\), where \(M\) is the number of time steps and \(N_K\) is the number of strike grid points.

Backward PDE approach. The backward PDE:

\[ \frac{\partial V}{\partial t} + (r-q)S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma_{\text{loc}}^2(S, t)S^2\frac{\partial^2 V}{\partial S^2} - rV = 0 \]

must be solved with terminal condition \(V(S, T) = (S - K)^+\) for each strike separately. Pricing \(N\) different strikes requires \(N\) separate backward PDE solves, each with cost \(O(M \cdot N_S)\) where \(N_S\) is the number of spot grid points. The total cost is \(O(N \cdot M \cdot N_S)\).

Comparison: For \(N\) strikes, the forward PDE is \(N\) times faster. If \(N_K \approx N_S\) and \(N = 100\) strikes, the forward PDE provides a speedup of roughly two orders of magnitude. This makes the forward PDE the natural tool for calibration, where one must evaluate call prices at hundreds of strike-maturity pairs to match the market surface.

Exercise 7. Under the Gaussian bounds for the transition density (Theorem 13.3.2), show that \(C_{KK} = e^{-rT} p(K, T) > 0\) for all \(K > 0\) and \(T > 0\). Explain why this strict positivity is essential for the well-definedness of Dupire's formula.

Solution to Exercise 7

By Theorem 13.3.2, under uniform ellipticity (\(0 < \lambda \leq \sigma_{\text{loc}} \leq \Lambda\)) and Holder continuity of \(\sigma_{\text{loc}}\), the transition density satisfies the Gaussian lower bound:

\[ p(K, T) \geq \frac{c}{K\sqrt{T}} \exp\left(-\frac{C(\log K/S_0)^2}{T}\right) \]

for some constants \(c, C > 0\), valid for all \(K > 0\) and \(T > 0\).

Since \(c > 0\), \(K > 0\), and \(T > 0\), the prefactor \(\frac{c}{K\sqrt{T}} > 0\). The exponential factor is always strictly positive for any finite argument. Therefore \(p(K, T) > 0\) for all \(K > 0\) and \(T > 0\).

Since \(e^{-rT} > 0\) for finite \(T\), it follows that:

\[ C_{KK}(K, T) = e^{-rT} p(K, T) > 0 \]

Why this is essential for Dupire's formula. The Dupire formula is:

\[ \sigma_{\text{loc}}^2(K, T) = \frac{\partial_T C + (r-q)K\partial_K C + qC}{\frac{1}{2}K^2 \partial_{KK} C} \]

The denominator is \(\frac{1}{2}K^2 C_{KK} = \frac{1}{2}K^2 e^{-rT} p(K, T)\). If \(C_{KK} = 0\) at some point, the Dupire formula has a division by zero and the local volatility is undefined. The strict positivity \(C_{KK} > 0\) guarantees that the denominator never vanishes, so the local volatility is well-defined and finite at every point \((K, T)\) with \(K > 0\) and \(T > 0\). Moreover, the positivity of \(C_{KK}\) is equivalent to the call price being strictly convex in the strike, which is a necessary condition for absence of butterfly arbitrage.