Everything in Chapters 18–22 priced derivatives by asking: "What is the average payoff across all possible random paths?" This is the probabilistic view — compute an expectation. Chapter 23 asks a completely different question: "How does the option price flow and spread across time and stock price levels, like heat spreading through a room?" This is the engineering view — solve a partial differential equation. The stunning result: both answers are identical.
Same problem, two solution methods. The Feynman-Kac Theorem proves they always give the identical answer.
Before we can build the bridge, we need to understand the starting point: a Stochastic Differential Equation (SDE). An SDE is simply the recipe that describes how a price moves over each tiny instant of time. Every SDE has exactly two ingredients.
The predictable part. Like a car's cruise control — a steady, deterministic push in a specific direction. In GBM, the drift is r·X (the risk-neutral growth at the risk-free rate).
The random part. Like a car hitting patches of ice — unpredictable shaking caused by Brownian Motion. In GBM, the diffusion is σ·X (volatility scaled by price).
The Markov Property is what makes SDEs useful for pricing. Because tomorrow's price only depends on today's price (not the entire history), we can build a PDE that tracks the option value across a two-dimensional grid of (time, price) — without needing to store the full path history. This is the computational efficiency that makes PDE methods orders of magnitude faster than Monte Carlo for standard European options.
This is the heart of Chapter 23. The Feynman-Kac Theorem states that the solution to a PDE — a calculus problem about how prices flow across a grid — is identical to the solution of a risk-neutral expectation — a probability problem about averaging random paths. The two approaches are mathematically equivalent.
Monte Carlo averages paths from the bottom up. PDE solves backwards from the expiry payoff. Both reach the same option price.
The room-temperature analogy makes this intuitive: to find the temperature at a point in a room, you could either release 1,000 tiny thermometer-sensors and average their readings (Monte Carlo) — or you could write down the heat equation and solve it mathematically (PDE). In finance, the payoff at expiry is the "heat source," and the option price today is the temperature at an earlier time. Both methods are correct. You choose based on speed and convenience.
For any market model driven by an SDE, Shreve provides a universal three-step process to convert the stochastic problem into a PDE. This recipe works for stocks, interest rates, exchange rates — any Markovian process.
The BSM PDE is the direct output of applying the three-step recipe to GBM (β = rx, γ = σx). Any other SDE produces a different PDE with the same structure but different coefficients. The recipe is universal — only the ingredients change. The Hull-White interest rate model and the CIR model (below) are both products of this exact same three-step process.
Applying the three-step recipe to interest rate models produces PDEs for bond prices. Two famous models illustrate the trade-offs between mathematical tractability and physical realism.
Mean-reverting drift (rubber band toward a/b). Constant volatility σ. The resulting PDE has a closed-form solution — fast and analytically tractable. Used widely for interest rate derivatives.
Same rubber-band drift. But volatility scales as σ√R — shrinks toward zero as R → 0. The PDE's volatility term vanishes at the boundary, preventing negative rates mathematically.
Hull-White (blue dashed) crosses into negative rates. CIR (gold) bounces off the zero floor as σ√R → 0 — the model self-corrects.
The BSM PDE assumes σ is a constant. But in real markets, options traders observe that different strike prices and maturities imply different volatilities. If BSM were perfectly correct, all implied volatilities would be the same. They are not — and the pattern they form reveals important market structure.
Deep in-the-money and out-of-the-money options consistently show higher implied volatility than at-the-money — the "smile" effect. BSM cannot explain it with a constant σ.
When you track the smile across multiple maturities, you get a Volatility Surface — a 3D landscape where σ is a function σ(t, x) that varies with both time to expiry and the strike price. The CEV model captures part of this by letting σ depend on the current stock level.
σ is a single number. Fast and clean. But implies a flat volatility surface — which no real market ever shows. Systematically misprices out-of-the-money options.
PDE: Vₜ + rxVₓ + ½σ²x²Vₓₓ = rV
σ is a function of time and stock price. Matches any observed smile perfectly by construction. Produces a more complex PDE but still solvable numerically.
PDE: Vₜ + rxVₓ + ½σ(t,x)²x²Vₓₓ = rV
The Dupire formula (1994) is the classic result: given observed market option prices at all strikes and maturities, you can extract σ(t,x) exactly — the unique local volatility surface that makes BSM consistent with every observed price simultaneously. This is the PDE approach running in reverse: instead of solving the PDE to find prices, you use observed prices to recover the PDE coefficient σ(t,x).
The Feynman-Kac Theorem proves both methods give the same answer. The choice between them is entirely a question of computational efficiency for your specific problem.
| Feature | Monte Carlo | PDE (Finite Difference) |
|---|---|---|
| How it works | Simulate thousands of paths. Average the payoffs. | Solve backward on a time-price grid. |
| Best for | Path-dependent options (Asian, lookback, barrier) | European and American options in 1–2 dimensions |
| Speed in 1D | Slow (needs many paths for accuracy) | Very fast — microseconds for standard options |
| Speed in high dimensions | Scales well — "curse of dimensionality" avoided | Exponential blowup — impractical above 3D |
| Handles stochastic vol? | Yes — just add a second random driver | Yes — but adds a dimension to the PDE grid |
| Error convergence | 1/√N — slow (need 4× paths to halve error) | O(Δt², Δx²) — fast with fine grid |
| Strategy type | Complex path-dependent backtests | Real-time options pricing engine |
Compare the Monte Carlo price (averaging random paths) with the BSM formula price (the closed-form PDE solution). Both implement the same Feynman-Kac theorem. Watch the Monte Carlo price converge to the exact BSM answer as you increase the number of paths — and observe how much faster the PDE answer is.
Gold dashed line = exact BSM price (PDE solution) | Blue bars = Monte Carlo path payoff distribution | As paths increase, the MC average converges to the BSM price
Chapter 23 — the Feynman-Kac bridge in five ideas:
What comes next: The multidimensional extension of these PDE methods — pricing options on multiple correlated stocks, interest rate term structure models, and the computational machinery (finite difference grids, ADI schemes) that makes PDE pricing practical at scale.