3.2.5 Failure Modes & Practice

Basic Prerequisites: 03 · Trees & Convergence.

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1. Intuition & Practical Objective

A binomial engine is thirty lines of code and it is easy to make it produce a number that is wrong in a specific, nameable way. This page names those ways, with the numbers attached, so a practitioner can tell a real price from an artefact.

Five failures, in one line each:

1. A step too coarse breaks the no-arbitrage bracket - the risk-neutral probability leaves $[0,1]$ and the tree is an arbitrage machine.
2. Applying a European formula to an American put understates value by the early-exercise premium.
3. Expecting (or fearing) early exercise on a dividend-free American call - it never happens, and a tree “confirming” it is just showing discretization error.
4. Trusting a single $n$ - CRR error oscillates and does not shrink monotonically.
5. Discounting under the real-world measure - the most common conceptual error, and the one that produces a plausible-looking wrong number.

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2. Mathematical Ground Truth & Derivations

2.1 The bracket is a constraint, not a convention

CRR requires $d<e^{r\Delta t}<u$, i.e. $e^{-\sigma \sqrt{\Delta t}}<e^{r\Delta t}<e^{\sigma \sqrt{\Delta t}}$, i.e.

$$\sigma \sqrt{\Delta t}>r\Delta t⟺\boxed{\sigma >r\sqrt{\Delta t}}$$

Violate it and $p=\frac{e^{r\Delta t}-d}{u-d}$ leaves $[0,1]$: the “probability measure” is not a measure, the state prices $\zeta =\tilde{\mathbb{P}}/(1+r)$ go negative, and a claim’s value can be below its arbitrage bound. This is not a numerical bug; it is FT1 failing for the discretized market.

2.2 Early exercise: the two asymmetric facts

American put (and dividend-paying call). The value satisfies the backward recursion

$$v_k(x)=\max \{\underset{\text{continuation}}{\underbrace{\frac{1}{1+r}[\tilde{p}{v}_{k+1}(ux)+\tilde{q}{v}_{k+1}(dx)]}},\underset{\text{intrinsic}}{\underbrace{g(x)}}\},$$

so $V^{\text{Am}}\ge V^{\text{Eu}}$ always, with a strictly positive early-exercise premium whenever the exercise region is non-empty. Applying the European formula is therefore a lower bound, not a price.

American call, no dividend. For a convex payoff with $g(0)=0$ and $r\ge 0$ (Shreve Ch 7, Cor. 2.25) the discounted payoff $(1+r{)}^{-k}g(S_k)$ is a $\tilde{\mathbb{P}}$-submartingale, so by optional sampling

$$\tilde{\mathbb{E}}\left[(1+r{)}^{-n}g(S_n)\right]=\underset{\tau }{\max }\tilde{\mathbb{E}}\left[(1+r{)}^{-\tau }g(S_{\tau })\right],$$

hence $C^{\text{Am}}=c^{\text{BSM}}$ - never exercise early. A tree that appears to show otherwise is showing tree error, not economics.

2.3 The American value as a supermartingale

The structural statement (Shreve Def 6.1): the discounted American value $\{(1+r{)}^{-k}{V}_k\}$ is the smallest $\tilde{\mathbb{P}}$-supermartingale dominating the intrinsic process $\{G_k\}$; the optimal exercise time is ${\tau }^{\ast }=\min \{k:V_k=G_k\}$, the first node where value equals intrinsic. Hedging then needs consumption $C_k=V_k-\frac{1}{1+r}\tilde{\mathbb{E}}[V_{k+1}\mid F_k]\ge 0$ (Shreve Lemma 2.21) - the seller who loses an exercise race has already banked the difference.

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3. Computational Implementation - the failures, in numbers

Stdlib only.

Experiment 1 - breaking the bracket. A long-dated single-step tree with $20\%$ volatility and a $5\%$ rate.

At $n=1$, $T=20$: $\sigma \sqrt{\Delta t}=0.894<r\Delta t=1.0$, so $u<e^{r\Delta t}$ and $p=1.134>1$. The same parameters with a fine step are perfectly well-behaved - the failure is entirely one of step size.

Experiment 2 - early-exercise premium, and its absence for a call.

The put carries a $+5.2\%$ early-exercise premium that no European formula can see. The call’s “premium” is negative and shrinking like $1/n$ - that gap is pure discretization error (the tree values are below the European price and converging upward). The economic prediction $C^{\text{Am}}=c^{\text{BSM}}$ is confirmed in the limit; a coarse tree merely looks like it disagrees.

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4. Failure Modes & First-Principles Breakdowns

1. The bracket, unguarded. Always assert $d<e^{b\Delta t}<u$ inside a tree routine and raise on violation. The condition $\sigma >r\sqrt{\Delta t}$ will be met comfortably for equity-like inputs but fails for low vol, long maturity, or an accidental step size.
2. European formula on an American put. Understates by $0.2427$ ($5.2\%$) on the running example. The premium is not small when the put is in-the-money and rates are positive - it is the whole point of the contract.
3. Hunting for an early-exercise boundary on a dividend-free call. There is none ($C^{\text{Am}}=c^{\text{BSM}}$, Shreve Ch 7); an observed “boundary” at coarse $n$ is the discretization error of Experiment 2. With a discrete dividend, early exercise immediately before the ex-date can be optimal - a completely different mechanism (Hull §15.12, §13.5).
4. Real-measure discounting. $V_0={\mathbb{E}}^{\mathbb{P}}[f]/(1+r)$ is wrong whenever $\tilde{p}\ne p$. Use ${\mathbb{E}}^{\tilde{\mathbb{P}}}$ or, equivalently, ${\mathbb{E}}^{\mathbb{P}}[\zeta f]$ with $\zeta =\frac{d\tilde{\mathbb{P}}/d\mathbb{P}}{1+r}$ (02).
5. Single-$n$ reporting. Error oscillation (03) means a favourite $n$ can be a lucky or unlucky one. Report an $n$-study or an averaged value.
6. Wrong carry $b$. Index/FX/commodity trees need $e^{(r-q)\Delta t}$ (or $e^{(r-r_f)\Delta t}$) as the growth factor, not $e^{r\Delta t}$ (Hull §13.11).

Practice checklist. Assert the bracket → choose $b$ from the instrument → average or smooth over $n$ → price European-and-tree to confirm the engine against the closed form → only then apply the American branch.

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5. References

• Shreve, Stochastic Calculus for Finance I
• Hull, Options, Futures, and Other Derivatives
• Haug, The Complete Guide to Option Pricing Formulas
• Björk, Arbitrage Theory in Continuous Time

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6. Connected Graph Bridges

• Back: 03 · Trees & Convergence · 04 · Fundamental Theorems · Index Hub
• Forward: 06 · Advanced Extensions
• Siblings: BSM · 05 Failure Modes · Implied Volatility Surfaces