3.8.5 Failure Modes & Numerical Practice

Basic Prerequisites: 02 · Finite Differences and 03 · Monte Carlo Pricing.

---

1. Intuition & Practical Objective

Every numerical price is wrong, and the only useful question is how. This page quantifies the five failure modes that actually bite in production, each tied to the first-principle assumption it violates:

1. Crank–Nicolson rings at a kink - stability without positivity.
2. Explicit FDM blows up outside its CFL bound - a violated stability condition.
3. Euler discretisation biases every path functional - weak order 1, and worse for extrema.
4. Domain truncation silently prices a different contract - a false boundary condition.
5. Importance sampling makes things worse when mis-tilted - an unbounded likelihood-ratio variance.

The discipline: measure the error against a benchmark you trust (a closed form, a tree, or a refined run) before believing a price.

---

2. Mathematical Ground Truth & Derivations

The discrete maximum principle is what “no ringing” means. For a two-level scheme $U^{n+1}=Q{U}^n$, if $Q$ has non-negative entries and rows summing to at most one, then positivity of the input implies positivity of the output (Duffy Def 9.1, Lemma 11.1). Crank–Nicolson violates this at high frequency: its symbol is

$${\rho }_{\mathrm{CN}}(\xi )=\frac{1-2\lambda {\sin }^2(\xi /2)}{1+2\lambda {\sin }^2(\xi /2)}⟶-1(\text{high frequency}),$$

so the highest modes flip sign every step and decay only slowly. An unconditionally stable scheme can still be a ringing scheme. The Richardson/Cooney cure is the extrapolated implicit-Euler scheme

$$V(t+k)=2U_{k/2}^{(2)}-U_k^{(1)},U^{(1)}=(I+kA{)}^{-1}V,U^{(2)}=(I+\frac{k}{2}A{)}^{-2}V,$$

which is second-order and positive (Duffy eq. 6.36); the industrial variant is Rannacher’s method: two fully implicit steps, then CN.

The explicit stability bound in financial coordinates (Duffy eqs. 12.15–12.18). For the BSM operator with a grid of step $h$ and $S_{\max }$ large,

$$k\le \frac{h^2}{{\sigma }^2{S}_{\max }^2}(\text{equivalently }h\le {\sigma }^2{S}_j/r,k\le 1/({\sigma }^2{j}^2+r)).$$

The bound scales with $h^2$ but inversely with $S_{\max }^2$: the further out you truncate the domain to control failure mode 4, the more time steps failure mode 2 demands. That is the explicit scheme’s trap.

The MC bias budget (Glasserman §6.1–6.3): Euler’s weak order is $1$ - bias $\approx ch$ - while the sampling error is $\sigma /\sqrt{n}$. Balancing them with a work budget $s$ gives $h^{\ast }\propto s^{-1/(2\beta +1)}$ and $\sqrt{\mathrm{MSE}}\propto s^{-\beta /(2\beta +1)}$ (eqs. 6.47–6.48): for Euler ($\beta =1$) the achievable rate is $s^{-1/3}$, strictly worse than the unbiased $s^{-1/2}$. For running extrema/barriers the Euler-on-maximum scheme is only weak order $\le \frac{1}{2}$ - bias removal requires Brownian interpolation:

$${\hat{M}}_i=\frac{{\hat{X}}_{i+1}+{\hat{X}}_i+\sqrt{({\hat{X}}_{i+1}-{\hat{X}}_i{)}^2-2b_i^{2}h\log U_i}}{2},{\hat{p}}_i=\mathbb{P}({\hat{M}}_i\le B\mid {\hat{X}}_i,{\hat{X}}_{i+1})=1-\exp \left(-\frac{2(B-{\hat{X}}_i)(B-{\hat{X}}_{i+1})}{b({\hat{X}}_i{)}^{2}h}\right).$$

Truncation boundary conditions. The call’s far-field condition $V(S_{\max },t)=S_{\max }-K{e}^{-r(T-t)}$ is exact only in the limit. Duffy’s own warning (Ch 4): “specifying boundary conditions for the Black–Scholes equation is somewhat of a black art”; his remedies are far-field truncation at a multiple of $K$, the transformation $x=S/(S+K)$ onto $(0,1)$ (coefficients vanish at the endpoints, so no boundary condition is needed), or the linearity condition ${\partial }^{2}V/\partial S^2=0$ (Hull eq. 21.x, Duffy B3).

---

3. Computational Implementation - the five failures, measured

One script, five experiments. Every number is compared against either the closed form or a refined reference run.

Read the five verdicts off the numbers: CN inflates the true maximum gamma $0.02182$ by $32\times$ at $\Delta t=0.25$ and is still off by $6\times$ at $\Delta t=0.0625$, while implicit and Rannacher stay within $6\%$ throughout; the explicit scheme returns $-1.96\times 10^7$ and $-7.48\times 10^{10}$ on the wrong side of its bound, then $10.41$ once inside it; Euler’s bias is $-0.2430$ at a single step and falls roughly as $1/\text{steps}$, disappearing into the $\pm 0.02$ Monte Carlo noise by 8 steps, while the log-Euler scheme is exact at one step; truncating at $S_{\max }=120$ costs $0.2157$ ($2.1\%$) and nothing improves past $S_{\max }=150$; and a tilt of $\mu =1$ cuts the digital-free call’s standard error $3.2\times$ ($9.99\times$ in variance) while $\mu =2$ and $\mu =4$ make it worse than no tilt at all ($0.0481$ and $1.1247$ against $0.0651$).

---

4. Failure Modes & First-Principles Breakdowns (numbered)

1. Ringing without instability (CN at a kink). CN’s symbol is negative at high frequency, so the payoff kink’s error alternates in sign and decays slowly. It is stable, hence convergent, hence wrong only in the Greek/high-frequency content - which is precisely what hedgers use. Fix: Rannacher (two implicit start-up steps), Richardson-extrapolated implicit Euler, or smooth the initial datum (Duffy Ch 33).
2. Violating the CFL bound (explicit). $k\le h^2/({\sigma }^2{S}_{\max }^2)$ is a hard constraint: values of $-10^7$ are not “approximate”, they are meaningless. Fix: switch to implicit/CN, or use exponential fitting whose stability is independent of $h$ (Duffy Ch 11).
3. Discretisation bias masquerading as noise. Euler’s $-0.2430$ bias at one step is $\approx 8.6\times$ the Monte Carlo standard error, so no path count removes it. Fix: the log transform (exact for GBM), a weak-order-2 scheme, or Richardson extrapolation $2\mathbb{E}[{\hat{X}}^h]-\mathbb{E}[{\hat{X}}^{2h}]$ (Glasserman eq. 6.43) - and for barriers/extrema, Brownian interpolation, since the Euler running-max scheme is only weak order $\le \frac{1}{2}$.
4. Truncation error is a bias, not a mesh error. $S_{\max }=120$ produced a $2.1\%$ error that vanished when $S_{\max }\ge 150$ - refining $h$ inside the truncated domain would never have fixed it. Fix: set $S_{\max }$ from a multiple of $K$ (or use $x=S/(S+K)$ to eliminate the boundary condition entirely) and verify by widening once.
5. A badly tilted importance sampler is worse than none. The variance of the likelihood ratio is the whole game; tilting towards a region the payoff does not reach inflates it. Diagnostic: monitor the second moment of the weights, and reach for stratification on top of IS rather than a bigger tilt.
6. Reporting one number. “The model says 10.42” is the failure mode that hides all five others. Report the price plus its error budget: $\pm$ MC standard error, the mesh/order term, the boundary term, and the scheme (CN vs Rannacher vs implicit).

---

5. References

• Duffy, Finite Difference Methods in Financial Engineering
• Glasserman, Monte Carlo Methods in Financial Engineering
• Hull, Options, Futures, and Other Derivatives
• Haug, Complete Guide to Option Pricing Formulas

---

6. Connected Graph Bridges

• Back: 02 · Finite Differences · 03 · Monte Carlo · 04 · Variance Reduction · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling robustness pages: BSM · Failure Modes · Implied Volatility Surfaces