PDE Finite Difference - Numerical Options Pricing
Last reviewed: by Options Analysis Suite Research.
What Is PDE Pricing?
PDE (Partial Differential Equation) pricing solves the option's pricing equation directly on a discretized grid, typically using finite differences. Black-Scholes, Heston, and Local Volatility all admit PDE formulations: the Black-Scholes PDE is one-dimensional (spot only), Heston is two-dimensional (spot + variance), and Local Volatility is also one-dimensional but with a state-dependent volatility coefficient. Solving the PDE backward from the payoff at expiration gives the option price at all grid nodes simultaneously.
PDE methods have one critical advantage over FFT and Monte Carlo: they handle American exercise and barriers natively. At every grid node, comparing the PDE solution to the immediate exercise value implements the optimal stopping rule. Barriers are imposed as Dirichlet boundary conditions on the relevant grid edges. For exotic equity products with early exercise or knockouts, PDE is often the cleanest tool.
Common PDE Schemes
- Explicit Euler: simple but conditionally stable; requires small time steps relative to the spatial grid.
- Implicit Euler / Crank-Nicolson: unconditionally stable. Crank-Nicolson is second-order accurate in time and the workhorse for production PDE pricing.
- ADI (Alternating Direction Implicit): required for multi-dimensional PDEs (Heston is 2D). Splits each time step into directional sub-steps for tractable linear-system solves.
What Does PDE Capture?
- American exercise: natural via the linear-complementarity formulation at each node.
- Barriers: Dirichlet conditions impose knockout/knock-in cleanly.
- Local volatility surfaces: PDE handles state-dependent σ(S, t) directly.
- Greeks: the spatial derivatives on the grid give delta, gamma, and theta as byproducts of the solve, with no Monte Carlo noise.
What Doesn't PDE Easily Capture?
- Path-dependent payoffs that aren't expressible as boundary conditions: Asians and lookbacks need additional state variables, multiplying grid dimension and cost.
- High-dimensional baskets: PDE scales exponentially in spatial dimension; for d > 3 Monte Carlo is generally faster.
- Models with jumps: adding jumps to a PDE requires PIDE (Partial Integro-Differential Equation) solvers, which are heavier than vanilla PDE.
Boundary Conditions and Grid Construction
A PDE solver requires boundary conditions at the spatial extremes (very small and very
large spot, in 1D; corresponding boundaries in 2D Heston) and an initial condition at
expiration (the option payoff). For a vanilla European call: payoff at expiration is
max(S − K, 0), the lower spot boundary value is 0 (the call is worthless
if spot is near zero), and the upper boundary value approaches S − K·e−r(T−t)
(the call price approaches its intrinsic-value plus discount as spot grows large).
For barrier options, knock-out boundaries are imposed as V = 0 at the
barrier strike at all times. The grid is typically log-uniform in spot (so that
multiplicative-spacing makes the spatial grid well-suited to lognormal dynamics) with
concentrated nodes near strikes and barriers where the price function has the most
curvature.
Time-Stepping Stability
Different time-stepping schemes have different stability properties:
- Explicit Euler: simple but conditionally stable; requires Δt ≤ Δx²/(2σ²) for the Black-Scholes PDE, which means very small time steps on fine spatial grids. Cheap per-step but expensive in total.
- Implicit Euler / Crank-Nicolson: unconditionally stable. Crank-Nicolson is second-order accurate in time and is the workhorse for production PDE pricing. Each step requires solving a tridiagonal linear system, but the time-step size is unconstrained.
- ADI (Alternating Direction Implicit): required for multi-dimensional PDEs (Heston is 2D in spot and variance). Splits each time step into directional sub-steps for tractable linear-system solves rather than dense 2D matrices. Hundsdorfer-Verwer and Craig-Sneyd are common ADI variants for Heston PDE.
- Operator-splitting: for jump models, the diffusion and jump parts of the PIDE can be split and solved sequentially per time step, simplifying the implementation at some cost in convergence order.
What Does PDE Capture?
- American exercise: natural via the linear-complementarity formulation at each grid node, where the option value is the maximum of the discounted continuation value and the immediate exercise value.
- Barriers: Dirichlet conditions impose knockout/knock-in cleanly at the barrier strike, with continuous monitoring naturally captured by the continuous-time formulation.
- Local volatility surfaces: PDE handles state-dependent σ(S, t) directly because the volatility appears as a coefficient on the diffusion term that the solver evaluates pointwise.
- Greeks: the spatial derivatives on the grid give delta, gamma, and theta as byproducts of the solve, with no Monte Carlo noise. This is one of PDE's strongest advantages over MC for Greek-heavy analytics.
What Doesn't PDE Easily Capture?
- Path-dependent payoffs that aren't expressible as boundary conditions: Asians and lookbacks need additional state variables, multiplying grid dimension and cost in ways that quickly become impractical.
- High-dimensional baskets: PDE scales exponentially in spatial dimension; for d > 3 Monte Carlo is generally faster.
- Models with jumps: adding jumps to a PDE requires PIDE (Partial Integro-Differential Equation) solvers, which include an integral term for the jump component and are heavier than vanilla PDE.
How OAS Uses PDE
The browser PDE calculator computes European and American Black-Scholes prices and Greeks. Its WebGPU path evolves batches of price surfaces using compensated arithmetic, with tridiagonal solves for European options. For American options, Howard policy iteration uses tridiagonal Thomas solves to enforce early exercise; consistently bumped price surfaces supply the sensitivities.
Available Solvers
Both CPU and WebGPU solvers are available for European and American Black-Scholes PDE pricing. Choose a backend under Computation Method in the PDE model parameters. If WebGPU is unavailable or a GPU solve fails, the calculation is retried on CPU. Cash-dividend schedules use the CPU solver.
Accuracy and Refinement
Targeted validation uses 1% for first derivatives, 1.5% for second derivatives, and 2% for third derivatives, plus a currency-scaled absolute allowance near zero. These budgets apply to the validated refinement settings and sampled scenarios; they are not a runtime comparison of every GPU result against CPU. Exact expiry, the exercise boundary, and cash-dividend event Greeks are outside that certification.
Richardson Extrapolation is enabled by default. It combines nested grid resolutions to reduce discretization error. Turning it off reduces work at the cost of accuracy; the published Greek budgets are certified with it enabled. Grid layout, spatial order, startup damping, and smoothing are selected automatically by the solver.
GPU speedups depend on the workload and device; software-adapter validation does not establish physical GPU performance or cross-vendor numerical parity.
Use PDE pricing in the calculator
Related Concepts
Binomial (vs) · Local Volatility (via PDE) · Heston (2D PDE) · Black-Scholes · Monte Carlo · FFT Pricing · Greeks Reference · Calibration · Model Divergence · Model Landscape
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