3.6.5 Failure Modes & Real-World Practice

Basic Prerequisites: 03 · Analytic Approximations and 04 · Free Boundary.

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1. Intuition & Practical Objective

American pricing fails in specific, measurable ways: the analytic approximations are maturity-dependent, real contracts allow exercise only at discrete dates, and the free boundary is sensitive to inputs we estimate poorly. This page names the failures precisely and quantifies each - so a practitioner knows which number to distrust and by how much.

The three failures, in one line each:

1. The approximation degrades with maturity. BAW is a short-maturity expansion; error grows from $+0.45\%$ to $+2\%$ between $T=0.5$ and $T=3$.
2. Continuous exercise is an idealisation. A Bermudan (exercise a few times a year) is worth strictly less than the continuous American - $0.13$ less at 4 dates/year for the standard case.
3. The boundary is an input-sensitive object. $L^{\ast }$ depends on $r$ and $\sigma$ through $\gamma =2r/{\sigma }^2$; our uncertainty about $r,\sigma$ is amplified into the exercise decision and hence into the price.

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2. Mathematical Ground Truth & Derivations

Where the assumptions live. Every American result rests on the same skeleton (Shreve II Ch 8; Björk Ch 21):

• (A1) Continuous exercise data - you may stop at any $t$; the free boundary $S^{\ast }(t)$ is a continuum. Real listed options are American (any trading day) but many OTC/structure products are explicitly Bermudan (a fixed exercise-date set $t_1<\cdots <t_m$). The continuous value is an upper bound.
• (A2) Complete, frictionless market with a single Brownian driver - needed for the Snell envelope to be the price. Jumps or stochastic vol make it a range (06 · Advanced Extensions).
• (A3) Constant $r,\sigma$ - the boundary $\gamma =2r/{\sigma }^2$ inherits the uncertainty in $r$ and $\sigma$; the perpetual boundary moves like $1/{\sigma }^2$.

The two error terms, made precise.

Approximation error. BAW and BS-1993 replace the true boundary by an analytic surrogate; their error is small for short maturity (numerically $O(10^{-3})$ on the verified cases) and grows with $T$ (the surrogate boundary drifts from the true curve as $T$ increases).

Discretisation error of exercise. Replacing continuous exercise by $m$ dates removes value. Writing $V_m$ for the Bermudan value with $m$ exercise dates, $V_m\uparrow V^{\text{Am}}$ as $m\to \infty$ (Björk §21.4, the discrete Snell envelope converges to the continuous one). The gap is a lower bound on what a real contract loses.

The gamma spike at the boundary. Because $V_{SS}$ jumps at $S^{\ast }$, the delta $V_S$ is continuous but the second derivative is not; a delta-hedged American short position has a discontinuous gamma across the boundary, which is a first-principles source of hedging error independent of all discretisation (BSM · Failure Modes).

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3. Computational Implementation - the failures in numbers

Two experiments. (A) BAW error versus a 4000-step tree, across maturities. (B) Bermudan versus continuous exercise, across exercise frequencies. Stdlib only.

Reading it. BAW’s error is one-directional and grows steadily - a $+2\%$ bias at $T=3$ is a real P&L leak on a long-dated book. The Bermudan gap is large and practically decisive: an annual-exercise right is worth exactly the European value (its only exercise date is maturity), quarterly exercise recovers $\approx 74\%$ of the continuous premium, monthly $\approx 91\%$. The exercise-frequency clause of a contract is worth more than the choice of approximation.

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4. Failure Modes & First-Principles Breakdowns

1. Using BAW for long-dated options. Error grows with maturity; use BS-2002 or a tree when $T\gtrsim 1$y, and always benchmark once.
2. Pricing a Bermudan as an American. The continuous value overstates by the discretisation gap ($0.13$ at 4 dates/year here). For structured notes with explicit call schedules this is a systematic mark-to-model error.
3. Trusting “exercise when deep ITM” heuristics. Any fixed rule is suboptimal and low-biases the price (the LSM low bound in 06 · Advanced Extensions makes this quantitative).
4. Sensitivity of the boundary to $r,\sigma$. Since $\gamma =2r/{\sigma }^2$, a 10% vol error moves the perpetual boundary by $\approx 20\%$. The exercise decision - and therefore the value and the hedge - inherits that.
5. Gamma at the boundary. $V_{SS}$ jumps, so delta-hedging an American short has a gamma discontinuity; near-the-boundary hedges need finer rebalancing than a smooth gamma profile would suggest, and the discretisation ringing of naive schemes sits right there.
6. Ignoring the early-exercise premium in risk. An American put’s delta at the boundary is exactly $-1$ (exercise ⇒ linear). A European-model delta ignores this and overstates the hedge ratio deep in the money.

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5. References

• Haug, The Complete Guide to Option Pricing Formulas
• Björk, Arbitrage Theory in Continuous Time
• Shreve, Stochastic Calculus for Finance II
• Hull, Options, Futures, and Other Derivatives
• Duffy, Finite Difference Methods in Financial Engineering

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6. Connected Graph Bridges

• Back: 04 · Free Boundary · Index Hub
• Forward: 06 · Advanced Extensions
• Practical branches: Numerical Methods · Failure Modes · BSM · Failure Modes · Implied Volatility Surfaces