3.4.5 Failure Modes & Real-World Practice

Basic Prerequisites: 04 · Advanced Dynamics.

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1. Intuition & Practical Objective

The surface is a fitted object, and fitting is where the money leaks. This page names the failures precisely, in the order a practitioner meets them:

1. Interpolation arbitrage - a smooth fit that is not arbitrage-free (negative densities, decreasing total variance).
2. Forward-skew flattening - the local-vol model matches today’s vanillas but moves wrongly for path-dependent products.
3. Model risk dominating parameter risk - the spread between models is larger than the spread from mis-setting a parameter inside one model.

The objective is not cynicism; it is the discipline of knowing exactly which assumption to distrust, and measuring the residual in money terms. The recurring lesson from Gatheral ch 8–10 and Bergomi ch 2–3: stress the modeling assumptions themselves, not just the parameters.

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2. Mathematical Ground Truth & Derivations

2.1 Static arbitrage conditions

A call surface $C(K,T)$ is arbitrage-free iff (Bergomi §2.2):

• Butterfly/vertical: $\frac{{\partial }^{2}C}{\partial K^2}=e^{-rT}\varphi (K,T)\ge 0$ - equivalently every butterfly spread has non-negative value.
• Calendar/convex order: for $T_1\le T_2$ and fixed moneyness $k=K/F_T$,

$$e^{q{T}_1}C(k{F}_{T_1},T_1)\le e^{q{T}_2}C(k{F}_{T_2},T_2)⟺T_1{\hat{\sigma }}_{k{T}_1}^2\le T_2{\hat{\sigma }}_{k{T}_2}^2,$$

i.e. total implied variance is non-decreasing in maturity at fixed moneyness (Bergomi eq 2.9/2.14/2.15). Violating it is a calendar-spread free lunch.

In the $(y,T)$ coordinates used for interpolation, the conditions become simply: $f(t,y)=(t-t_0){\hat{\sigma }}^2$ must satisfy $f_{i+1}(y)\ge f_i(y)$ (profiles must not cross), and affine interpolation in $T$ between slices preserves convex order (Bergomi eq 2.20).

2.2 Local-vol dynamics failure - the arithmetic

The local-vol forward skew is (Bergomi eq 2.91/2.92)

$${\mathcal{S}}_{\theta }(\tau )={\mathcal{S}}_{\tau +\theta }-\frac{\tau }{\theta }\left(\frac{1}{\theta }{\int }_{\tau }^{\tau +\theta }{\mathcal{S}}_{t}dt-{\mathcal{S}}_{\tau +\theta }\right),{\mathcal{S}}_{\theta }(\tau )\propto {\left(\frac{\theta }{\tau }\right)}^{\gamma }{\mathcal{S}}_{\theta }.$$

For a decreasing skew term structure the bracket is positive (the running average exceeds the endpoint), so the leading minus makes ${\mathcal{S}}_{\theta }(\tau )<{\mathcal{S}}_{\tau +\theta }$: every future skew is weaker than today’s, and dies out like $(\theta /\tau {)}^{\gamma }$. A forward-start option (cliquet) is an option on precisely this forward skew, so LV underprices it. And because LV also cannot lock the forward vol-of-vol (it changes with recalibration), the exotic’s P&L is large and unpredictable.

The vol-of-vol mismatch is equally structural: LV implies $\mathrm{vol}({\hat{\sigma }}_{F_{T}T})=({\mathcal{S}}_T+\frac{1}{T}{\int }_0^T{\mathcal{S}}_{t}dt)\frac{{\hat{\sigma }}_{F_{0}0}}{{\hat{\sigma }}_{F_{T}T}}\to 2{\mathcal{S}}_T$ as $T\to 0$ (Bergomi eq 2.83–2.85), while a time-homogeneous SV model has time-independent vol-of-vol - they cannot agree (Bergomi §2.5).

2.3 The size of the errors (from the corpus)

• Digitals: dropping the skew term misprices a 1-year ATM digital by $\sim 12\%$ of notional (Gatheral §8.4).
• Cliquets: a 5-year 6% digital cliquet can be mispriced by up to $5.76\%$ of notional (Bergomi/Gatheral ch 8/10); a locally-capped-globally-floored cliquet differed 3.53% (Heston) vs 2.55% (LV) - a 2.94% upfront gap (Gatheral §10.1), a large multiple of the provider’s margin.
• Barriers: LV vs SV price gaps can exceed the bid-offer (Gatheral ch 9); a one-touch is worth two European binaries (reflection, zero log-drift) but the ratio is highly model-dependent.

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3. Computational Implementation - arbitrage detection & forward-skew flattening

We (i) exhibit a butterfly arbitrage from an invalid SVI slice (negative Breeden–Litzenberger density) against a valid one, (ii) show a calendar-arbitrage violation numerically, and (iii) quantify the LV forward-skew flattening of §2.2. Stdlib only.

The bad slice produces a negative risk-neutral density ($-6.92$ at $k=-0.04$) - a butterfly free lunch that a naive spline would happily generate. The good slice’s total variance is increasing ($0.0400>0.0200$); the bad one’s is not, so it admits a calendar arbitrage. And the LV forward skew is $0.586\times$ today’s - the measured flattening that underlies every forward-skew mispricing above.

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

1. Interpolation arbitrage (statics). Unconstrained polynomials/splines fit the quotes but violate ${\partial }_K^{2}C\ge 0$ or ${\partial }_{T}w\ge 0$. Fix: parametrize $w(k,T)$ directly with arbitrage-aware objects (SVI with calendar constraints, affine-in-$T$ profiles).
2. LV’s flat forward skew (dynamics). Demonstrated: future skew $=0.586\times$ today’s. Digitally-capped cliquets, barriers, digitals all misprice; LV sellers of forward-skew risk “win the deal and lose money” (Gatheral ch 10).
3. Recalibration risk. LV has no lockable forward vol-of-vol; forward skews/vol-of-vols move with each recalibration ⇒ large unpriced carry P&L when residual gammas are sizeable (Bergomi §2.6).
4. Model risk > parameter risk. The spread between models (LV vs SV vs SVJ) can be several % of notional on cliquets/barriers - far wider than the spread from getting a parameter wrong inside one model. Napoleon-style products hinge on volatility convexity that LV underprices (Gatheral §10.3). Manage by pricing under several models, not by tuning one.
5. Delta convention risk. Without a stated sticky rule, “the delta” is undefined; BS delta ≠ minimum-variance delta (Hull §20.5). Hedging with the wrong convention is a systematic bleed.
6. Extrapolation & wing risk. VS vols and low-strike vols are dominated by the smile outside the traded strikes; a wrong wing extrapolation silently reprices the whole book (Bergomi §5.2). Enforce consistency with liquid VS quotes.

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 04 · Advanced Dynamics · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: BSM · Failure Modes · Heston & SABR