4.11 Risk-Factor Sensitivities

Basic Prerequisites: Multivariable Calculus (only the idea of a partial derivative - no options knowledge required at this entry point; the Pillar-3 Greeks are developed/utilised from 02 · Delta, Gamma & Vega onward). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

A risk manager never sees the future. What she can see is how her book’s value responds to a small change in each thing that can move. That response is a sensitivity: a derivative of portfolio value with respect to a risk factor. This folder is about turning a portfolio into a vector of sensitivities, and using that vector as the primary instrument of risk control.

The pivot from Pillar 3 is deliberate and sharp:

Pillar 3 derives the Greeks - it asks what is $\partial V/\partial S$? - because it needs them to replicate an option and thereby price it. Pillar 4 uses the Greeks - it asks given these sensitivities, how much can the book lose, and where must I stop? - because risk management needs a map from positions to P&L, not a price.

The practical objective of this page is the lookup table (job #1 of this pillar): every sensitivity a desk or a risk system quotes, its definition, its units convention, and a numerically verified value. Six sub-pages then walk from raw intuition through option sensitivities, rate sensitivities, factor decomposition, failure modes, and the delta–gamma/limit extensions.

The one-sentence essence. “A portfolio is not a list of trades, it is a vector of exposures to risk factors; sensitivities are the linear map from factor moves to P&L, and every approximation failure in risk management is the second-order term you dropped.”

2. Mathematical Ground Truth & the Sensitivity Lookup

Quick-Reference Lookup (job #1). All formulas are transcribed from Hull Ch 19 (Greeks, verified) and Ch 22 (delta-normal, delta–gamma, duration mapping, verified), cross-checked against Haug §2 (the formula-authoritative Greeks) and RiskMetrics (1996). The numbers in the check column were re-executed and reproduced exactly from the reference implementation in §3 on the shared test book (Haug option $S=98,X=100,T=.25,r=10\%,b=5\%,\sigma =30\%$; book $=$ long 100 calls $S=100,X=100,T=.5,r=b=5\%,\sigma =20\%$ $+$ short 50 puts $S=100,X=95,T=.25,\sigma =25\%$; bond book $=$ a 5y and a 10y 5%-coupon bond on a flat 4% zero curve).

Notation: $V$ portfolio value, $S$ spot, $\sigma$ volatility, $T$ time to expiry, $r$ rate, $y_i$ the zero rate at curve node $i$, $w$ position vector, $\beta$ factor-loading matrix, $z_{\alpha }={\Phi }^{-1}(\alpha )$.

Critical scaling caveat (inherited from Pillar 3). Raw derivatives are per unit; screen values quote Vega/Rho per 1 point ($=$ raw$/100$), Theta per day ($=$ raw$/365$). A risk system that mixes raw and per-point units mis-sizes every limit by $100\times$ or $365\times$. Every table in this folder states its convention in the header.

The risk-factor map (position $\to$ risk factors $\to$ P&L). This is the object a risk system actually stores:

Delta-normal vs delta-gamma (Hull eq. 22.6–22.8). Mapping the book onto factors $f$ with exposures $b$ and factor covariance $\Sigma$:

The quadratic term $H$ carries gamma (own-second derivative) and cross-gamma ${\gamma }_{ij}={\partial }^{2}V/\partial f_i\partial f_j$. With one equity factor and $\Delta S=\sigma SZ$, the quadratic form becomes $aZ+b{Z}^2$ with $a=\delta \sigma S$, $b=\frac{1}{2}\gamma {\sigma }^2{S}^2$, giving exact closed-form moments

which feed the Cornish–Fisher quantile adjustment for non-normal VaR

3. Computational Implementation - the sensitivity engine

This runs on the standard library only (math.erf gives the exact normal CDF and density). It reproduces every verified number in §2: the Haug Greeks, the aggregated book sensitivities, the DV01/key-rate ladder, the factor-exposure decomposition, and the delta-normal/delta-gamma VaR comparison against full-revaluation Monte Carlo.

The single most important structural fact. Sensitivities are additive across positions but not across factors: the book’s delta is ${\sum }_i{q}_i{\Delta }_i$, but the book’s VaR depends on $\sqrt{b^{\top }\Sigma b}$, where the covariance $\Sigma$ couples the factors. Aggregation without $\Sigma$ is a sum of risks that never diversifies.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Gamma risk is the second-order term you dropped. Any linear (delta-only) risk measure is wrong by $\frac{1}{2}\Gamma (\Delta S{)}^2$; on a $40\%$ down move of the shared test option the delta-only P&L is $-23.91$ against a true $-6.89$ - a $247\%$ error, and the delta-gamma error itself is $-4.87$ ($+71\%$).
2. Cross-greek interaction dominates in a crash. Price and vol move together: at $\Delta S=-20$, $\Delta \sigma =+20$ vol points, delta-gamma misses by $+3.14$ and adding vega still misses by $-2.33$ - the residual is vanna/volga, which no two-Greek system can hold.
3. Beyond second order there is no limit. A third-order residual, a jump, or a regime change breaks any polynomial. Sensitivities are local; stress tests and full revaluation exist precisely because the local map is not global.

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)
• J.P. Morgan / RiskMetrics: *RiskMetrics
• Alexander, Carol: Market Risk Analysis, Vol. III (Pricing, Hedging and Trading Financial Instruments) and Vol. IV (Value at Risk Models) (2008, Wiley)
• 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)

6. Connected Graph Bridges

• Foundational base: Calculus & Optimization · Linear Algebra & Covariance · Pillar 3 · Greeks & Hedging (where the Greeks are derived)
• Sub-pages (in-folder): 01 From Zero · 02 Delta, Gamma, Vega · 03 Rates & Key-Rate Duration · 04 Factor Exposures · 05 Failure Modes · 06 Advanced Extensions (delta–gamma VaR & limits)
• Sibling topics: VaR & Expected Shortfall · Parametric, Historical & Monte Carlo VaR · Stress Testing & Scenario Analysis

Beginner: start at 01 · Practitioner: start at 05