4.11.5 Failure Modes & Practice

Basic Prerequisites: 02 · Delta, Gamma, Vega and 04 · Factor Exposures.

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1. Intuition & Practical Objective

Every sensitivity is a local statement. This page is about the ways the local map stops describing the book, in order of how much money they cost:

1. Gamma risk - the second-order term. Large, systematic, and directionally signed.
2. Cross-greek interaction - the off-diagonal terms of the Hessian. Price and vol do not move independently in a crisis; they move together.
3. Non-linearity beyond second order - the third-order residual, which no two-Greek system can hold, plus the genuinely non-analytic events: jumps, gaps, and regime change.

The discipline. A sensitivity-based risk system is a quadratic approximation of a non-polynomial function. It is useful exactly insofar as the shock is small relative to the curvature. The professional rule is therefore not “trust the Greeks” but: know the size of the shock at which your approximation dies, and have a different tool (stress test, full revaluation) for beyond it.

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2. Mathematical Ground Truth & Derivations

The full Taylor map, and what each block means.

$$V(f+\Delta f)-V(f)=\underset{\text{delta / DV01}}{\underbrace{b^{\top }\Delta f}}+\underset{\text{gamma, cross-gamma, convexity}}{\underbrace{\frac{1}{2}\Delta f^{\top }H\Delta f}}+\underset{\text{third order / "speed"}}{\underbrace{\frac{1}{6}\sum _{ijk}{\partial }_{ijk}V\Delta f_i\Delta f_j\Delta f_k}}+\cdots$$

For a single equity factor with $\Delta S$ and $\Delta \sigma$, writing $\Gamma ={\partial }^{2}V/\partial S^2,\nu =\partial V/\partial \sigma ,\mathcal{V}={\partial }^{2}V/\partial {\sigma }^2$ (volga) and $\text{Vanna}={\partial }^{2}V/\partial S\partial \sigma$:

$$\Delta V\approx \Delta \Delta S+\nu \Delta \sigma +\frac{1}{2}\Gamma (\Delta S{)}^2+\text{Vanna}\Delta S\Delta \sigma +\frac{1}{2}\mathcal{V}(\Delta \sigma {)}^2+\Theta \Delta t.$$

The five terms are not equally important, and the ranking is empirical:

Term	Magnitude in a normal day	Magnitude in a crash	Who is short it
$\Delta \Delta S$	dominant	dominant	delta hedgers
$\Theta \Delta t$	small, certain	small	long-gamma books
$\frac{1}{2}\Gamma (\Delta S{)}^2$	small	large	short-gamma books (option sellers)
$\text{Vanna}\Delta S\Delta \sigma$	tiny	large - spot and vol fall together	skew/smile-hedged books
$\frac{1}{2}\mathcal{V}(\Delta \sigma {)}^2$	tiny	large - vol-of-vol	short-volga books

Why vanna and volga awaken together with gamma. In an equity selloff, spot falls and implied vol rises (the leverage effect). So $\Delta S<0$ and $\Delta \sigma >0$ simultaneously, and every off-diagonal term is active at once. This is the structural reason a “delta-hedged, vega-hedged” book can still lose: it is short the $S\times \sigma$ cross term, which neither hedge touches.

The gamma–theta identity restated as a P&L decomposition. For a delta-hedged position over a step $\Delta t$,

$$\text{P\&L}\approx \frac{1}{2}\Gamma S^2\left[{\left(\frac{\Delta S}{S}\right)}^2-{\sigma }^2\Delta t\right]$$

• the realised-variance minus implied-variance trade. Its expectation under $\mathbb{Q}$ is zero; its variance is $\frac{1}{2}{\Gamma }^2{S}^4{\sigma }^4\Delta t^2$ per step (accumulating to $\frac{1}{2}{\Gamma }^2{S}^4{\sigma }^{4}T\Delta t$ over horizon $T$), which is why the risk of a delta-hedged book is proportional to ${\Gamma }^2$, and why short-gamma positions (selling options) have a P&L distribution with a fat left tail and a thin right one: many small gains, rare large losses.

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3. Computational Implementation - where the approximation dies

Two experiments on a long ATM call ($S=X=100$, $T=0.5$, $r=b=5\%$, $\sigma =20\%$): first an equity-only shock, then a joint spot-and-vol shock of the kind seen in March 2020. Stdlib only.

Experiment 1 - gamma risk, and its signature sign. At $\Delta S=-5$ the delta-only estimate is $13\%$ too pessimistic and delta-gamma is nearly exact. At $\Delta S=-40$ the delta-only estimate is $-23.91$ against a true $-6.89$ - a $247\%$ overstatement of the loss - because a long call floors at zero intrinsic. Note the residual does not grow monotonically ($0.0125,0.0701,0.0504$, then $-4.87$): once the option is far out of the money the third and higher derivatives, which are not in the model at all, do the work. Delta-gamma is not conservative; it is wrong in both directions.

Experiment 2 - cross-greek interaction. At $\Delta S=-20$ with a $20$-point vol spike, delta-gamma says $-6.48$ while the truth is $-3.34$: an error of $+3.14$. Adding the vega term brings the estimate to $-1.01$, which reduces the error to $-2.33$ - but only at this shock size. Pushing further, the same correction makes things worse: at $\Delta S=-30$ the residual moves from $+1.44$ (delta-gamma) to $-6.77$ (delta-gamma-vega), and at $\Delta S=-40$ from $-2.86$ to $-13.80$. The reason is structural: vega is itself a first-order approximation of a non-linear dependence on vol, so bolting a linear vega term onto a quadratic spot term does not produce a second-order model in $(S,\sigma )$ - it produces a model with one second-order term and one first-order term, which is accurate only in a thin neighbourhood. The missing pieces are vanna and volga: delta itself moves as vol moves, and vega itself grows as vol grows. A system that tracks only delta, gamma and vega cannot represent that, no matter how carefully each is computed.

The gamma–theta identity holds exactly to the analytic formulas ($5.471732$ both ways) - a reminder that these are not approximations of each other, they are two views of one quantity.

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

1. Gamma risk is asymmetric and it is the P&L of the business. Long gamma: bounded loss (premium), unbounded gain, and a certain theta cost. Short gamma: bounded small gain, unbounded loss. The sensitivity report shows $+\Gamma$ or $-\Gamma$; it never shows the asymmetry, which is the actual risk.
2. Cross-greek interaction is where options books actually die. Vanna and volga are invisible to a delta-gamma-vega report. In a crash, spot down and vol up activate both simultaneously, and the “hedged” book loses on a term that was never on the risk page (see Pillar 3 · vanna and volga).
3. Beyond second order there is no convergence rate to rely on. A shock of $40\%$ is not a small perturbation; the Taylor series residual is comparable to the answer. Sensitivities are for limits and hedging, stress tests are for losses.
4. Jumps are not in the Taylor expansion at all. A gap through a strike is a discontinuous $\Delta S$, so no finite-order approximation applies. The BSM hedge cannot remove jump risk (incomplete market) and sensitivity-based VaR cannot measure it (see 05 · BSM Failure Modes and Stress Testing).
5. Pin risk: gamma blows up at expiry. As $T\to 0$ at the money, $\Gamma \to \infty$ and delta flips between $0$ and $1$; the position becomes a digital bet on a coin flip and the hedger is whipsawed. Mitigation is operational, not mathematical: close or roll before expiry.
6. Sensitivities are model sensitivities. They are derivatives of your model’s price. If the model is wrong (a flat vol, a single curve), the sensitivities are wrong in the same direction - model risk enters the risk system through the front door.
7. Aggregation destroys curvature information. Summing gammas across strikes produces one number that describes no actual scenario. Two books with identical net gamma can have opposite third-order behaviour; net a book’s gamma at your peril.

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5. References

• Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
• Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed., 2006)
• Taleb, Nassim Nicholas: Dynamic Hedging (Wiley, 1997)
• Derman, Emanuel: Model Risk (Goldman Sachs QSR Notes, 1996)
• Alexander, Carol: Market Risk Analysis, Vol. IV (2008)

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6. Connected Graph Bridges

• Back: 02 · Delta, Gamma, Vega · 04 · Factor Exposures
• Continue: 06 · Advanced Extensions (delta–gamma VaR, limit systems) · Index Hub
• Sibling: Pillar 3 · BSM Failure Modes (the same breakdowns, viewed from pricing) · Extreme Value Theory & Fat Tails
• Forward: Reverse Stress Testing (the tool for everything beyond the Taylor expansion)