4.8.2 Sources of Model Risk

Basic Prerequisites: 01 · Model Risk from Zero and The European Pricing Formulas (the Greeks used as error sensitivities).

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1. Intuition & Practical Objective

Model risk is not one thing that “goes wrong”; it is a set of independent failure channels, each with a different owner, a different detection method, and a different fix. This page names them precisely and turns them into an error budget: a quantitative decomposition of the gap between a model’s output and the truth, in which each channel contributes a dollar amount.

Two canonical taxonomies anchor the field, and they are complementary:

• SR 11-7 (regulatory, two causes). Model risk arises because (1) a model may have fundamental errors - it produces inaccurate outputs relative to its design objective; or (2) a model may be misused - applied outside its assumptions, or with a misunderstanding of its limitations.
• Derman (1996, seven types). A finer practitioner taxonomy that separates, crucially, being wrong about the world from being right about the model and wrong about the arithmetic. Derman’s warning: “you’re worse off thinking you have a model and relying on it than in simply realizing there isn’t one.”

The objective: given a live model, fill in the budget $\Delta V={\Delta }_{\text{form}}+{\Delta }_{\text{param}}+{\Delta }_{\text{impl}}+{\Delta }_{\text{use}}$ with actual numbers, so “model risk” becomes a number with a sign and an owner rather than a worry.

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2. Mathematical Ground Truth & Derivations

The two taxonomies, mapped. Each SR-11-7 cause expands into Derman’s types, and each type maps to one term of the error budget.

Derman (1996) type	SR-11-7 cause	Error-budget channel	Primary detection
Inapplicability of modeling	fundamental error	form (unbounded)	conceptual soundness - is the phenomenon model-able at all?
Incorrect model (wrong factors / dynamics / dependence / deterministic-vs-stochastic split)	fundamental error	form	conceptual soundness + benchmark comparison
Unstable data	fundamental error	parameter	calibration-window sensitivity, orthogonal data
Correct model, incorrect solution	fundamental error	implementation	analytic vs numerical test on a solvable case
Badly approximated solution	fundamental error	implementation	convergence study (grid/time-step refinement)
Software and hardware bugs	fundamental error	implementation	independent re-implementation, boundary tests
Correct model, inappropriate use	misuse	use	usage inventory, limits, “effective challenge”

The budget, first order. Let $V_{\theta }$ be the model value and $\theta =(\sigma ,r,q,\dots )$ its calibrated parameters. A first-order (delta/gamma) expansion gives

$$\Delta V=\underset{\text{form}}{\underbrace{(V_{{\theta }^{\star }}-V^{\star })}}-\underset{\text{parameter}}{\underbrace{\sum _i\frac{\partial V}{\partial {\theta }_i}\Delta {\theta }_i}}+\underset{\text{convexity}}{\underbrace{\mathcal{O}(\Vert \Delta \theta {\Vert }^2)}}+\underset{\text{implementation}}{\underbrace{{\epsilon }_{\text{impl}}}},$$

where $\Delta {\theta }_i$ is the calibration error in parameter $i$ and ${\epsilon }_{\text{impl}}$ the discretisation/bug term. The parameter channel is dominated by the Greeks: ${\Delta }_{\text{param}}V\approx {\sum }_i{G}_i\Delta {\theta }_i$ with $G_i=\partial V/\partial {\theta }_i$ (e.g. vega for $\sigma$, rho for $r$).

Adding channels. If the channels are approximately independent, the total model-risk standard deviation adds in quadrature:

$${\sigma }_{\text{total}}=\sqrt{{\sigma }_{\text{form}}^2+{\sigma }_{\text{param}}^2+{\sigma }_{\text{impl}}^2+{\sigma }_{\text{use}}^2}.$$

Two consequences worth stating: (i) a channel you have not measured is not zero - the budget is a lower bound on ignorance; (ii) the largest channel dominates the sum, so effort belongs on the biggest term, not on the smallest “bug”.

Error sensitivity as a scale. The parameter channel scales linearly in the calibration error (vega × 1 vol-point), while form error is a model-level jump that no amount of calibration removes. Empirically, for a vanilla option the ordering is often form > parameter > implementation - which is why “we recalibrated daily” answers the wrong question.

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3. Computational Implementation - building the error budget

Take an at-the-money 1-year call, $S=X=100$, $r=5\%$, $\sigma =20\%$. We price the same contract under five lenses: the analytic BSM value; a $\pm 1$ vol-point parameter error (vega); a $\pm 1$bp rate error (rho); a 50-step CRR tree (discretisation); a 360-vs-365 day-count bug; and a Merton jump-diffusion model (form error). Stdlib only.

Reading the budget: the 1-vol-point calibration error alone moves the price by $3.59\%$ - larger than the day-count bug ($0.85\%$) and the discretisation error ($0.38\%$) combined. The jump-diffusion form error ($5.05\%$) dominates all parameter channels. The quadrature total is $6.26\%$ of price. Note the vega number $37.524$ is exactly the raw vega from the Pillar-3 Greek table - the budget is built from quantities you already compute.

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4. Failure Modes & First-Principles Breakdowns

1. Calibrating to the wrong observable. Fitting $\sigma$ to a price that was itself produced by a different model makes $\Delta \theta$ absorb (and hide) the form error - the parameter channel looks small because it is doing the form channel’s job. This is the “unstable data” channel in disguise.
2. The budget’s missing terms. Derman’s “inapplicability” and SR-11-7’s “misuse” have no Greek - they cannot be measured by perturbation. Omitted the most dangerous channels from your quadrature sum, and the total is a floor, not an estimate.
3. Correlated channels. Quadrature assumes independence; calibration error and form error are often positively correlated (a bad model needs a distorted $\sigma$ to fit the smile). The independent sum then understates total error.
4. Convergence theatre. A discretisation study that refines a grid while the form is wrong certifies the wrong number to more decimals - “correct model, incorrect solution” read backwards.
5. Day-count and convention bugs are model risk. Derman’s point: the most avoidable large errors hide in the simple part (counting coupons, day-count, settlement dates) while attention sits on the exotic part.

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5. References

• Derman, E., Model Risk, Goldman Sachs QSR Notes (1996)
• Federal Reserve / OCC, SR 11-7 (2011)
• Morini, M., Understanding and Managing Model Risk (2011)
• Hull, J. C., Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 01 · Model Risk from Zero
• Forward: 03 · Validation & Backtesting · Index Hub
• Base: European Pricing Formulas · Greeks & Hedging