Financial PDEs: Turning Contract Terms Into a Valuation Problem

A partial differential equation links a financial value to local changes in its state variables and time. It does not determine a unique contract value on its own. Payoffs, boundary conditions, model assumptions and the direction of time complete the problem.

This reference explains that full specification, starting from elementary PDE structure and ending with richer valuation models. It is a reference on pricing equations and their interpretation, not a forecast about derivative markets.

What a financial PDE describes

An ordinary differential equation depends on one independent variable. A PDE relates derivatives with respect to several variables. An option value $V(S,t)$, for example, depends on underlying price $S$ and time $t$. A stochastic-volatility value $V(S,v,t)$ adds variance $v$ as another state.

The order of a PDE is the highest derivative order. A linear PDE is linear in the unknown function and its derivatives, although its coefficients may depend on state and time. Nonlinear control problems can contain maxima or nonlinear combinations of the derivatives.

Equation families guide different mathematical problems

These labels guide the conditions and numerical methods required, but financial boundary behaviour still needs explicit analysis.

A terminal payoff starts backward valuation

For a European call with strike $K$ and maturity $T$, the terminal condition is

$$V(S,T)=\max(S-K,0).$$

The known cash flow occurs at maturity; valuation works backward to an earlier time. Setting a future payoff is not the same as declaring today’s unknown value to be an initial condition.

Introducing time to maturity $\tau=T-t$ converts backward calendar-time valuation into forward evolution in $\tau$. A codebase should choose one convention and apply derivative signs consistently. Many apparent numerical instabilities are actually a mixture of the two conventions.

Boundary conditions specify behaviour at the edges of the state domain. A Dirichlet condition fixes value; a Neumann condition fixes a derivative. A mixed condition can combine them. Boundaries may represent a genuine contractual event, such as knockout, or an artificial truncation of an unbounded mathematical domain.

The basic Black–Scholes equation

For a non-dividend-paying asset with constant volatility $\sigma$ and constant rate $r$, the standard equation is

$$V_t+\frac12\sigma^2S^2V_{SS}+rSV_S-rV=0.$$

The first term is passage of time; the second captures curvature under diffusion; the third reflects the pricing-measure drift; the final term discounts the value. The equation assumes the familiar frictionless replication setting. Jumps, financing asymmetries or incomplete hedging markets require additional structure.

For a call, the lower boundary is zero at $S=0$. At a sufficiently high underlying level,

$$V(S,t)\approx S-Ke^{-r(T-t)}.$$

For a put, value at $S=0$ is $Ke^{-r(T-t)}$, while its high-price value tends to zero. These conditions follow from the payoff and discounting assumptions, rather than being arbitrary numerical conveniences.

Dividends change the drift and asymptotic conditions. Discrete dividends can require explicit jump conditions in the valuation timeline. A model that reproduces a no-dividend analytical price can still be wrong for a dividend-paying contract.

Heat-equation transformations reveal the structure

Using log price and time to maturity transforms the variable-coefficient structure of the basic equation. Further rescaling removes drift and discount terms, leaving a heat-equation form. The transformation explains why Gaussian kernels appear in the analytical solution.

Three ways to represent compatible valuation problems

Each method works best when the coefficients, domain and boundary conditions have a compatible structure.

Their value is not limited to closed-form answers. Analytical special cases provide benchmarks for numerical solvers. A solver that cannot recover a simple known solution should not be trusted merely because its more complicated outputs look smooth.

The expectation and the PDE are two views of one model

Let the pricing-measure state follow $dX_t=a(X_t,t)dt+b(X_t,t)dW_t$, with discount rate $r(X_t,t)$ and terminal payoff $g(X_T)$. Under suitable regularity, the valuation

$$V(x,t)=\mathbb E_{\mathbb Q}\left[ e^{-\int_t^T r(X_s,s)ds}g(X_T)\mid X_t=x\right]$$

In words: The value is the expected terminal payoff discounted along each future state path, given the current state and pricing probabilities.

corresponds to

$$V_t+aV_x+\frac12b^2V_{xx}-rV=0, \qquad V(x,T)=g(x).$$

This is the central Feynman–Kac connection. A Monte Carlo calculation approximates the expectation; a PDE scheme approximates the differential equation. Agreement between independently implemented versions is powerful evidence when both use the same conventions and assumptions.

More state variables change the equation and the hedge

In a stochastic-volatility model, asset price and variance evolve jointly. A common square-root variance specification uses mean-reversion speed $\kappa$, long-run variance $\theta$, volatility of variance $\xi$ and correlation $\rho$. The valuation equation includes both variance derivatives and a mixed price-variance derivative.

The correlation contribution is proportional to $\rho\xi vS V_{Sv}$. It affects the shape of the implied-volatility surface and the response to simultaneous price and variance changes. Adding a variance state therefore adds both numerical dimension and model risk.

A local-volatility model retains one price state but replaces constant volatility with $\sigma_{\mathrm{loc}}(S,t)$. It can fit a suitable smooth arbitrage-consistent vanilla surface within its assumptions. That fit does not guarantee realistic future smile dynamics or accurate prices for every path-dependent product.

The local-volatility derivation also connects call-price curvature across strikes to a pricing density. For smooth European call prices $C(K,T)$, deterministic constant rate $r$, no dividends and suitable boundary behaviour, the forward relation gives

$$\sigma_{\mathrm{loc}}^2(K,T)= \frac{2[C_T(K,T)+rK C_K(K,T)]}{K^2C_{KK}(K,T)}.$$

In words: Double the sum of the time change and interest adjustment, then divide by strike squared times the change in slope across strikes.

Here subscripts denote derivatives with respect to strike $K$ and maturity $T$. The denominator involves strike curvature, so noisy or sparse option quotes can produce unstable inferred local volatility. Integration by parts in the density-based derivation requires the relevant boundary terms to vanish. Arbitrage-consistent smoothing and explicit boundary assumptions are therefore substantive parts of the inversion, not cosmetic preprocessing.

A jump-diffusion model adds discontinuous price changes. Its valuation equation becomes a partial integro-differential equation because a jump can move the state across a range of prices in one event. A diffusion-only grid approximation that ignores that integral changes the model.

Early exercise is an obstacle problem

An American-style option must be worth at least its immediate exercise payoff $g(S)$. Its holder chooses between exercising and continuing. In a basic formulation, the valuation satisfies a complementarity condition of the form

$$\max\{g(S)-V,\;V_t+\mathcal LV-rV\}=0,$$

where $\mathcal L$ is the spatial diffusion operator. The exercise boundary is part of the solution. A European PDE solved with only the terminal payoff cannot capture this choice.

At discrete exercise dates, backward induction compares continuation value with exercise value. Continuous exercise requires an appropriate obstacle or free-boundary treatment. Smooth-pasting conditions can apply under suitable regularity but should not be assumed unchanged for every payoff or jump model.

Path dependence requires sufficient state information

For an arithmetic-average option, the current underlying level alone does not determine the remaining payoff distribution. A running sum or integral must be added to the state, unless a valid dimension-reduction argument is available.

For a continuously monitored barrier option, the barrier can be a domain boundary with the contractual rebate or knockout value. Discrete monitoring is different: the barrier condition applies at monitoring dates, not continuously between them. Replacing one convention with the other changes the contract price.

Lookback products require information about a running maximum or minimum. Each added state increases computational cost. This is why Monte Carlo may become attractive for high-dimensional path dependence even when PDE methods remain effective for low-dimensional exercise problems.

A simple heat-equation solution shows why conditions matter

Consider the illustrative heat equation $u_\tau=a u_{xx}$ on $0<x<L$, with zero values at both endpoints and initial profile $u(x,0)=\sin(\pi x/L)$. Separation of variables yields

$$u(x,\tau)=e^{-a\pi^2\tau/L^2}\sin(\pi x/L).$$

Here $a>0$ is diffusion strength and $L$ the domain length. The spatial shape remains a sine wave while its amplitude decays. Increasing diffusion or reducing the domain length accelerates that decay.

The same differential equation with a different initial profile has a different solution. A nonzero boundary value also changes the evolution. This simple example demonstrates why writing down a pricing PDE without its terminal and boundary conditions does not yet identify a price.

More general profiles can be represented as sums of compatible modes. Each mode decays at its own rate, with higher-frequency components damping faster. This helps explain both the smoothing effect of diffusion and why nonsmooth payoff features create demanding short-time numerical behaviour.

Forward density and backward value answer complementary questions

The backward equation evolves a conditional value as the current state changes. The forward equation evolves the probability density of the future state. They use related but different differential operators.

For a diffusion with drift $a(x,t)$, diffusion $b(x,t)$ and density $p(x,t)$, the forward equation takes the form

$$p_t=-\partial_x(ap)+\frac12\partial_{xx}(b^2p).$$

The derivatives act on the products containing the density. Replacing this with the backward operator applied directly to $p$ can be wrong when coefficients depend on state. Boundary flux must also be consistent with probability conservation or the intended absorbing state.

Integrating the terminal payoff against a correctly evolved pricing density produces the same value as backward valuation under matching assumptions. The two approaches offer different computational advantages and an opportunity for independent checks.

Default introduces a contractual loss term

For a pre-default value $V$, default intensity $\lambda$ and recovery value $R$, a simplified valuation equation can contain the term $-\lambda(V-R)$. This removes value at the rate of default arrival while adding the specified recovery contribution.

The recovery variable must have the same value units as $V$. It may represent recovery of par, recovery of market value or a closeout amount; those are different conventions. Calling the entire resulting equation a universal “CVA PDE” would hide that it prices a risky claim under a particular default and recovery setup.

Counterparty adjustments can require joint modelling of exposure and credit quality. Netting, collateral, closeout timing and wrong-way risk change the relevant payoff. The equation should be derived from those contractual definitions rather than selected because it contains a familiar intensity term.

Complete the valuation specification

Questions that complete a financial valuation specification

A Green-function representation also needs the initial or terminal and boundary contributions appropriate to the problem. Writing only an integral against an internal source term does not reproduce a nonzero terminal payoff by itself. In financial applications, that payoff is often the principal input rather than a minor addition. Keeping all contributions explicit prevents a formally correct kernel calculation from omitting the cash flow that is actually being valued.

A well-posed PDE specification makes numerical work meaningful. A fast solution of an equation with the wrong payoff, missing state or inappropriate boundary is an accurate answer to the wrong question. The practical aim is a valuation problem whose economic and mathematical components can be checked together.