Stressed Scenarios and Model Risk

Hard·25 min read

Risk & GreeksScenario AnalysisModel RiskP&L AttributionVaR

Quick Quiz

1. Expected Shortfall (ES) is preferred over Value at Risk (VaR) as a regulatory risk measure primarily because:

ES is easier to compute from historical dataES is a coherent risk measure (satisfies subadditivity), while VaR can violate subadditivity for non-elliptical distributionsES always produces a smaller number than VaR, resulting in lower capital requirementsES requires fewer assumptions about the return distribution

2. A delta-hedged options book shows a systematic negative P&L residual (actual P&L < Greek-attribution P&L) on days when the market falls significantly. What does this suggest?

The book is accidentally delta-long and loses on market fallsThe pricing model is missing a risk factor — likely the smile dynamics (vanna risk) or jump risk not captured by the first-order and second-order GreeksThe theta estimate is too small — the desk is overstating time decayThe gamma hedges are too large, over-hedging the spot move

3. A barrier option is priced at 3.5 under local volatility and 5.2 under stochastic volatility (Heston), both calibrated to the same vanilla implied vol surface. The base model is local vol. What is the model risk reserve?

(5.2 - 3.5)/2 = 0.85

5.2 - 3.5 = 1.7

3.5 \times 0.5\% = 0.0175

(a regulatory percentage)

\max(3.5, 5.2) = 5.2

4. Under FRTB (Basel Fundamental Review of the Trading Book), a trading desk can use its internal risk model for capital calculation as long as its pricing model is independently validated, regardless of P&L attribution test results.

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5. For the second-order P&L decomposition $\delta V \approx \Delta\delta S + \frac{1}{2}\Gamma(\delta S)^2 + \nu\delta\sigma + \text{Vanna}\cdot\delta S\cdot\delta\sigma + \frac{1}{2}\text{Volga}\cdot(\delta\sigma)^2$ , which term dominates for a short-dated near-the-money straddle on a day with a large spot move but stable implied vol?

The vega term

\nu\delta\sigma

The gamma term

\frac{1}{2}\Gamma(\delta S)^2

The delta term

\Delta\delta S

The vanna term

\text{Vanna}\cdot\delta S\cdot\delta\sigma

6. Historical simulation VaR applies the last $N = 250$ days of observed market factor changes to today's portfolio and takes the empirical 1st percentile (99% VaR). What is the main weakness of this approach for an exotic options book?

It requires too many repricings (250 full repricings) to be practicalIt is backward-looking and cannot capture scenarios not present in the 250-day window — crises outside the sample period are missed, and the window may include calm periods that understate tail riskIt always overestimates VaR because historical periods include stress eventsIt requires the portfolio to have zero delta to be applied correctly

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