4.11.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes & Practice and VaR: Definition & Flaws.

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1. Intuition & Practical Objective

Two things turn a pile of sensitivities into a risk-management system:

1. A number. The delta–gamma (a.k.a. quadratic) approximation puts the second-order term back into VaR, so that a quantile can be computed from sensitivities alone - no full revaluation, no Monte Carlo. This is the RiskMetrics/Hull framework, and it is the reason a bank can compute a portfolio VaR in milliseconds for a trading desk that reprices a million instruments.
2. A constraint. Limit systems convert sensitivities into a governance structure: a maximum delta, a maximum gamma, a vega bucket cap, a VaR limit, an escalation procedure. The sensitivity vector is not an answer; it is the unit of account in which risk appetite is written.

The key insight this page delivers: the sign of the gamma determines the direction of the error in a delta-normal VaR. Long gamma makes the P&L distribution positively skewed, thinning the left tail - so delta-normal VaR overstates. Short gamma does the reverse and delta-normal understates the exact case a limit system exists to catch.

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2. Mathematical Ground Truth & Derivations

The two models (Hull §22.5, eq. 22.6–22.8).

• Linear (delta-normal): with exposure vector $b$, factor covariance $\Sigma$, and horizon $h$,

$$\Delta V=b^{\top }\Delta f,{\mathrm{VaR}}_{\alpha }=z_{\alpha }\sqrt{b^{\top }\Sigma b}(=z_{\alpha }|\delta |\sigma S\sqrt{h}\text{for one equity factor}).$$

Assumes the P&L is normal, i.e. that the portfolio is linear in the factors. It is exactly a sensitivity method: it needs only deltas.

• Quadratic (delta–gamma): with the Hessian $H$ (gamma and cross-gamma),

$$\Delta V=b^{\top }\Delta f+\frac{1}{2}\Delta f^{\top }H\Delta f.$$

For a single equity factor with $\Delta S=\sigma SZ,Z\sim N(0,1)$, this is a quadratic form in a normal:

$$\Delta V=aZ+b{Z}^2,a=\delta \sigma S,b=\frac{1}{2}\gamma {\sigma }^2{S}^2.$$

Exact moments of $aZ+b{Z}^2$ (derived from $\mathbb{E}[Z^2]=1,\mathbb{E}[Z^4]=3,\mathbb{E}[Z^6]=15$):

$$\mathbb{E}[\Delta V]=b,\mathrm{Var}(\Delta V)=a^2+2b^2,{\gamma }_1=\frac{6a^{2}b+8b^3}{(a^2+2b^2{)}^{3/2}},{\gamma }_2^{\text{ex}}=\frac{3a^4+60a^2{b}^2+60b^4}{(a^2+2b^2{)}^2}-3.$$

Note the pure-gamma limit ($a=0$): ${\gamma }_1=2\sqrt{2}\approx 2.83$ and ${\gamma }_2^{\text{ex}}=12$ - the moments of a ${\chi }_1^2$, as they must be. A gamma-only position is a chi-square, not a normal, and calling it normal is a modelling error with a known, computable size.

The Cornish–Fisher quantile correction. The true quantile of a skewed, fat-tailed distribution is approximated from the normal quantile $z$:

$$z_{\alpha }^{\text{CF}}=z+\frac{(z^2-1)}{6}{\gamma }_1+\frac{(z^3-3z)}{24}{\gamma }_2-\frac{(2z^3-5z)}{36}{\gamma }_1^2,{\mathrm{VaR}}_{\alpha }^{\Delta \gamma }=-\mathbb{E}[\Delta V]+\sqrt{\mathrm{Var}}{z}_{\alpha }^{\text{CF}}(-{\gamma }_1,{\gamma }_2),$$

with the skewness sign flipped because VaR is a quantile of the loss $L=-\Delta V$.

Limit systems (the governance layer). Typical structure, in the units established on the index page:

Limit type	Unit	Measures	Failure it prevents
Delta limit	shares / $ notional	directional exposure	concentrated directional bets
Gamma limit	$\Delta$-change per $1\%$ move	convexity / realised-vs-implied	being short the crash convexity
Vega limit	$ per vol point, bucketed by expiry	vol exposure	unhedged surface shape
Theta limit	$ per day	carry bleed	slow insolvency of a long-gamma book
VaR / ES limit	currency, one-day or ten-day	the aggregate	the whole
Stress limit	currency, per scenario	non-linearity beyond the model	the tail the model cannot see

The escalation rule that makes limits real: limits are hard at the desk level (breach $=$ mandatory reduction next session) and soft at the division level (breach $=$ notification and a remediation plan). Without a stated action, a limit is a report. This is where the sensitivity machinery connects to policy - see Basel & Regulation for the same structure at regulatory scale (the FRTB sensitivities-based method prescribes delta, vega and curvature buckets, which is this page’s mathematics written into capital law).

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3. Computational Implementation - delta-normal vs delta-gamma vs full revaluation, and a limit dashboard

One Monte Carlo run prices the truth. Then the two sensitivity models are compared for a long and a short gamma book, showing that the error flips sign with the position - the central practical message of this folder. Stdlib only.

Four readings.

1. The error flips sign with the position, and this is the whole lesson. Delta-normal VaR is $+7.41\%$ too high for the long-gamma book and $-6.19\%$ too low for the short-gamma book - using the identical volatility, delta magnitude and confidence level. The difference is the sign of $b=\frac{1}{2}\gamma {\sigma }^2{S}^2$, i.e. the skewness of the P&L. A delta-only VaR is conservative for the option buyer and dangerously optimistic for the option seller.
2. Delta-gamma is accurate to a fraction of a percent in both cases ($+0.21\%$, $+0.10\%$) at a one-day horizon - which is exactly why it survived as the industry’s pre-Monte Carlo workhorse, and exactly why it fails over ten days or in stress (see 05 · Failure Modes).
3. The chi-square limit is visible in the moments. For a gamma-dominated position ($a\to 0$) the formulas give skewness $2\sqrt{2}$ and excess kurtosis $12$; here $a=75.3$ dwarfs $b=2.17$, so the deviations are small (${\gamma }_1=\pm 0.173$) - but the sign is the sign of the position.
4. Delta-normal alone would have passed this book. The dashboard’s delta utilisation is $59.77\%$ - comfortable - while the gamma limit is in breach at $109.44\%$. A limit system that watches only delta (or only VaR) reports this book as fine. The gamma and vega columns are not decoration; they are the limits that bind.

The FRTB echo. The sensitivities-based method in Basel’s Fundamental Review of the Trading Book requires exactly these three buckets - delta, vega, curvature - risk-weighted and aggregated per risk class. The regulatory capital regime is this page’s mathematics, with the aggregation rules fixed by law. See Basel & Regulation.

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4. Failure Modes & First-Principles Breakdowns

1. Cornish–Fisher is an expansion, not a distribution. It is a truncated cumulant series; for extreme skew/kurtosis (short-dated gamma books, deep tails) the corrected quantile can become non-monotone or even exceed the feasible range. Always sanity-check $z^{\text{CF}}$ against the second model.
2. Delta-gamma assumes the factor is normal. The quadratic form handles the portfolio’s non-linearity but not the factor’s fat tails. Both errors must be handled: non-linearity by the Hessian, fat tails by the factor distribution (EVT / historical simulation).
3. Cross-gamma is where multi-factor books hide. With $N$ factors the Hessian has $N(N-1)/2$ cross terms; most systems report only the diagonal. A book of options on correlated underlyings has a large off-diagonal exposure that is invisible in a per-factor gamma report.
4. Limits without an action are reports. The mechanism that makes a limit a limit is the stated consequence of breach, the escalation path, and the time allowed to cure. A limit system is an operating procedure with arithmetic attached, not the reverse.
5. Limits must be in the same units as the hedge. A vega limit in raw units while the desk hedges per-point is a $100\times$ mismatch; a theta limit per trading day while the report is per calendar day is a $45\%$ mismatch. Unit mismatches are the most common cause of a limit system that appears to work and does not.
6. A VaR limit is not a stress limit. The 99% one-day quantile says nothing about a five-sigma week. Every sensitivity-based VaR limit must be paired with a scenario/stress limit that is not derived from the same local model (see Stress Testing & Scenario Analysis).

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5. References

• Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
• J.P. Morgan / RiskMetrics: Technical Document, 4th ed. (1996)
• Dowd, Kevin: Measuring Market Risk (2nd ed., 2005)
• Fisher, R. A. & Cornish, E. A.: Moments and Cumulants in the Specification of Distributions, Biometrika 30(3–4):262–291 (1938)
• Alexander, Carol: Market Risk Analysis, Vol. IV (Value at Risk Models) (2008)
• BCBS: Minimum Capital Requirements for Market Risk (January 2019, d457
• RiskMetrics / MSCI and Hull, Risk Management and Financial Institutions

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6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · Index Hub
• Sibling: 06 · VaR Extensions (delta–gamma & backtesting) · Expected Shortfall (the coherent measure FRTB substitutes for VaR)
• Forward: Stress Testing & Scenario Analysis · Basel & Regulation (FRTB SBM: delta, vega, curvature)
• Base: Pillar 3 · The Greeks · Numerical Methods