4.2.6 Advanced Extensions

Basic Prerequisites: 02 · Parametric and 05 · Failure Modes.

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1. Intuition & Practical Objective

Two practical problems remain after the three VaR methods are understood:

1. Options are nonlinear - delta-normal VaR can’t price their risk. The fix is the delta–gamma approximation: keep the convexity (second-order) term the delta-only method throws away. It is the middle rung between the fast-but-linear delta-normal and the exact-but-slow full revaluation.
2. A VaR model is only a model - you must check it against reality. The fix is backtesting: compare the model’s predicted breach rate with what actually happened. The two canonical statistics are Kupiec’s Proportion-of-Failures (frequency) and Christoffersen’s independence (clustering).

The practical objective of this page: the exact delta–gamma math, a clean demonstration of where it helps, and a runnable Kupiec + Christoffersen backtest - plus the regulatory context (Basel traffic-light zones, FRTB’s shift to ES).

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2. Mathematical Ground Truth & Derivations

2.1 Delta–gamma: the quadratic approximation of the P&L

Delta-normal (linear, Hull Ch 22 eq. 22.6). For $N$ factors, $\Delta P\approx {\sum }_i{S}_i{\delta }_i\Delta x_i={\delta }^T\Delta S$, $a_i=S_i{\delta }_i$. Suitable only when the portfolio is delta-linear in the factors.

Delta–gamma (quadratic, Hull Ch 22 eq. 22.7/22.8; Glasserman Ch 9 eq. 9.2). Add the curvature:

$$\Delta V\approx \frac{\partial V}{\partial t}\Delta t+{\delta }^T\Delta S+\frac{1}{2}\Delta S^T\Gamma \Delta S,$$

${\delta }_i=\partial V/\partial S_i$, ${\Gamma }_{ij}={\partial }^{2}V/\partial S_i\partial S_j$ (the cross-gammas matter). For one factor this is the familiar

$$\Delta V\approx \Theta \Delta t+\delta \Delta S+\frac{1}{2}\gamma (\Delta S{)}^2.$$

The point: $\gamma >0$ (long options) means the P&L bends, so a linear delta-normal VaR systematically misstates the risk - it ignores that the position gains/loses convexly as the move grows.

Diagonalizing to a sum of quadratic forms (Glasserman Ch 9 §9.2). Write $\Delta S=CZ$, $C{C}^T=\Sigma$, and diagonalize $-\frac{1}{2}{C}^T\Gamma C=\Lambda =\text{diag}({\lambda }_1,\dots ,{\lambda }_m)$. Then the loss is a quadratic in independent normals

$$L\approx Q=a+\sum _{j=1}^m(b_j{Z}_j+{\lambda }_j{Z}_j^2),a=-\Theta \Delta t.$$

Its moment-generating function is closed form, which (i) gives a fast VaR via inversion, and (ii) is the exact sampling engine for importance sampling in the tail (Glasserman §9.2; Table 9.1: CV≈2–5×, IS≈7–27×, stratified-IS up to ~173× variance reduction for tail probabilities).

Where it helps / where it doesn’t. For mild convexity (a near-ATM FX put, one day), delta-gamma reproduces full-revaluation VaR almost exactly; delta-only is off a few percent. For delta-neutral, gamma-heavy positions (a long straddle: $\delta \approx 0$ but big $\gamma$), delta-only claims “no risk” while the realized tail risk is real - exactly the case full revaluation and delta–gamma exist to catch (see the failure mode in 05/Reference).

2.2 Backtesting VaR - Kupiec POF and Christoffersen independence

Kupiec (1995) Proportion-of-Failures (POF). Let $x$ be breaches ($L_t>{\text{VaR}}_{\alpha }$) over $T$ days, expected rate $p=1-\alpha$ under $H_0$. The likelihood-ratio statistic

$${\text{LR}}_{\text{POF}}=-2\ln \left[\frac{(1-p{)}^{T-x}{p}^x}{(1-\hat{p}{)}^{T-x}{\hat{p}}^x}\right]\sim {\chi }_1^2,\hat{p}=\frac{x}{T}.$$

Reject at 5% if ${\text{LR}}_{\text{POF}}>3.841$ (the ${\chi }_1^2$ 95% point). This only checks the rate - it is blind to ordering.

Christoffersen (1998) independence. A model can pass Kupiec yet have its breaches clustered (all in one crash week) - the worst failure mode there is. Build the $2\times 2$ transition count matrix $\{n_{ij}\}$ of the violation sequence ($0\to 0,0\to 1,1\to 0,1\to 1$). Let $p_0=n_{01}/(n_{00}+n_{01})$, $p_1=n_{11}/(n_{10}+n_{11})$, and overall $p=(n_{01}+n_{11})/(n_{00}+n_{01}+n_{10}+n_{11})$. The independence LR

$${\text{LR}}_{\text{ind}}=-2[(n_{00}+n_{10})\log (1-p)+(n_{01}+n_{11})\log p-n_{00}\log (1-p_0)-n_{01}\log p_0-n_{10}\log (1-p_1)-n_{11}\log p_1]\sim {\chi }_1^2.$$

Reject at 5% if $>3.841$. Combined (conditional) coverage = ${\text{LR}}_{\text{POF}}+{\text{LR}}_{\text{ind}}\sim {\chi }_2^2$ catches both wrong-rate and clustering.

Regulatory reading (BCBS 1996). Supervisors grade internal-models VaR by breaches: green (≤4 exceptions in 250) to red (≥10) with escalating capital multipliers - the frequency test mapped to a capital add-on. Modern FRTB (2019) moves market-risk capital from 99% VaR to 97.5% Expected Shortfall for that exact reason (VaR’s non-subadditivity and tail blindness) - see VaR & ES.

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3. Computational Implementation - delta–gamma and the backtest, stdlib only

A. Delta–gamma recovers full-revaluation VaR on the FX put (the canonical comparison). Same put as 04:

Delta-only is ~7% off on a mildly convex put; delta–gamma lands within 1% of the full-revaluation benchmark. On a delta-neutral straddle the delta-only estimate collapses toward zero while the true VaR stays real - that gap is why delta–gamma exists and why full revaluation is the standard for options-heavy books.

B. Backtest a 99% VaR model and watch Kupiec catch a bad one (stdlib). A well-calibrated model produces ~10 breaches/1000 days and is accepted; an under-stated-vol model breaches 115× and is rejected by Kupiec (while - instructive - still failing nothing on independence, because its breaches are scattered, not clustered):

The under-stated-vol model breaches ~11× too often - Kupiec rejects it decisively (LR 363 ≫ 3.841). Its breaches are scattered, so the independence test (correctly) sees no clustering - the two tests are complementary: Kupiec is the coverage check, Christoffersen the clustering check, and together the conditional-coverage test is their sum.

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4. Failure Modes & First-Principles Breakdowns

1. Delta–gamma still ignores higher-order risk (vanna, volga, jumps) and cross-gammas if ${\Gamma }_{ij}$ is dropped. The quadratic bends, but a kink (jump) or stochastic vol defeats it - the gateway to stress testing and full MC per scenario.
2. Backtesting has low power at high confidence. At 99% the expected breaches are sparse; a single bad run can under/over-reject. Basel’s traffic-light zones (green ≤4/250 · red ≥10/250) exist precisely because small-sample frequency tests are noisy.
3. Kupiec alone is blind to clustering - a serial-crash model passes on count but is the most dangerous kind. Always pair it with Christoffersen independence (the conditional-coverage test).
4. Backtest = history; the next regime may differ. Passing a backtest on calm history rewards underestimating the next crisis. This is why regulators pair backtesting with stress testing and why FRTB moved to ES at 97.5%.
5. VaR’s subadditivity failure is structural. Backtesting a subadditive measure can reward desks for concentrating risk - one reason ES/CVaR is the coherent successor (VaR & ES).

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5. References

• Glasserman, Monte Carlo Methods in Financial Engineering
• Hull, Options, Futures, and Other Derivatives
• Kupiec, Paul H.: Techniques for Verifying the Accuracy of Risk Measurement Models, Journal of Derivatives 3(2):73–84 (1995)
• Christoffersen, Peter F.: Evaluating Interval Forecasts, International Economic Review 39(4):841–862 (1998)
• BCBS: Supervisory Framework for the Use of Backtesting… (1996)

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub.
• Forward: FRTB successor) · Stress Testing & Scenario Analysis.
• Theory: The Greeks & Delta-Hedging (the source of $\delta ,\gamma$) · Numerical Methods.

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All numbers above (3,396.15 / 3,289.54 / 3,405.97, the Student-$t(4)$ 1.5565%, the ghost 3,465 / 3,196, the delta-gamma 0.0141 vs full 0.0143, and the backtest LR=363.3 rejection) were generated and verified with the standard library only.