3.8 Numerical Methods

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Multivariable Calculus. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Closed forms exist for a handful of contracts. Everything else - American exercise, path-dependent payoffs, multi-factor models, barriers with monitoring dates - has no formula, and must be priced by discretising the object it came from. There are exactly two objects to discretise:

• The PDE (Feynman–Kac’s differential form): replace the space and time derivatives by divided differences, and solve the resulting linear system on a grid. → Finite-Difference Methods.
• The expectation (Feynman–Kac’s integral form): replace the integral over $\mathbb{Q}$-paths by an average over simulated paths. → Monte Carlo Pricing.

Trees sit between the two: a binomial/trinomial tree is a backward-induction PDE solver on a log-price grid, and (Hull Ch 21) the explicit finite-difference scheme is a trinomial tree.

This folder is the hub: it gives the fast scheme and error lookup below and routes you to six sub-pages that build the two families from first principles, quantify their errors, and show what breaks.

The one-sentence essence. “There are two ways to solve a pricing problem numerically - march a grid or average paths - and each has exactly one error knob (the mesh $h,k$, or the path count $n$); knowing the order of each error is the whole discipline.”

2. Mathematical Ground Truth & Derivations

Lookup 1 - the three families. All verified numerically in §3 against the same ATM call ($S=X=100$, $T=1$, $r=5\%$, $\sigma =20\%$, exact $=10.4506$).

Lookup 2 - the finite-difference schemes (Duffy eqs. 6.17–6.19). Writing the one-factor parabolic operator as $\mathcal{L}u$ and $\theta$ as the weight on the old time level (the new level carries $1-\theta$),

Von Neumann (Duffy eqs. 8.34–8.39, printed with a typographical $4{\lambda }^2$ - the correct symbol has a single $\lambda$, consistent with the printed condition $\lambda \le \frac{1}{2}$):

The triangle that governs everything (Definitions 8.1/8.3/8.4 + Theorem 8.1): consistency (truncation error $\to 0$) + stability ($\Vert Q^n\Vert \le K$) $⟺$ convergence - Lax equivalence. Order $(p,q)$ means $\Vert {\tau }^n\Vert =O(h^p)+O(k^q)$.

Lookup 3 - the Monte Carlo estimator (Glasserman eqs. 1.1–1.8, 1.39, 3.20).

Error $O(n^{-1/2})$ in every dimension - versus the trapezoidal rule’s $O(n^{-2})$ in one dimension and $O(n^{-2/d})$ in $d$, which is the entire reason Monte Carlo exists for exotics. Halving the error costs $4\times$ the paths; one extra decimal costs $100\times$.

Lookup 4 - which scheme for which problem (Duffy’s own guidance, Ch 30 + Ch 19–29):

3. Computational Implementation - the three families on one contract

One contract, three engines, stdlib only. Run it and watch the two discretisation families land on the same number.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the full analysis is in 05 · Failure Modes & Practice. In one line each:

1. Stability is a property of $(h,k)$ jointly, not of the formula. The explicit scheme is correct and useless outside its CFL bound: at $k>h^2/({\sigma }^2{S}_{\max }^2)$ it produced $10.41\to -7.5\times 10^{10}$ on the same contract.
2. Second-order accuracy is not the same as no ringing. Crank–Nicolson is unconditionally stable, yet at $\Delta t=0.25$ it inflates the strike-region gamma by $32\times$ (Duffy Ch 33: “spurious oscillations near the strike price”).
3. Monte Carlo’s $n^{-1/2}$ is slow and its bias is separate. Discretising an SDE adds an $O(h^{\beta })$ bias on top of the sampling error; the fix is a better scheme (log-Euler is exact for GBM) or MSE balancing, not more paths.

5. References

• Duffy, Daniel J.: Finite Difference Methods in Financial Engineering (Wiley, 2006)
• Glasserman, Paul: Monte Carlo Methods in Financial Engineering (Springer, 2004)
• Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
• Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed.)

6. Connected Graph Bridges

• Foundational base: Stochastic Calculus & Itô · Calculus
• Upstream theory: Black–Scholes–Merton Hub (the PDE being discretised) · No-Arbitrage & Binomial Trees (the discrete seed)
• Sub-pages (in-folder): 01 From Zero · 02 Finite Differences · 03 Monte Carlo · 04 Variance Reduction · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05