4.11.1 Risk-Factor Sensitivities from Zero

Basic Prerequisites: Multivariable Calculus (only the idea of a partial derivative).

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1. Intuition & Practical Objective

You own a portfolio. Something in the world moves - a share price, a yield, an implied vol, a spread. How much money did you just make or lose? There are only two honest ways to answer:

1. Full revaluation: re-price the whole book under the new world and subtract. Exact, slow, and - if you want a distribution of outcomes - expensive.
2. Sensitivities: measure once how the book responds to a small nudge in each factor, then multiply. Approximate, instant, and the thing every risk limit is written in terms of.

This page builds the second answer from nothing, and shows precisely when it breaks. No options knowledge required.

The mental model to hold: a portfolio is a machine with a handful of input dials. The dials are the risk factors ($S$, $y$, $\sigma$, spreads, FX…). The sensitivities are the gear ratios from dial turns to money. Risk management is the art of knowing the gear ratios, and of knowing that gear ratios change when the dial turns far.

Three steps:

1. A sensitivity is a derivative, and a derivative is a local exchange rate. “Delta $=0.6$” means: for a small move in the share price, I gain $60$ cents per $1 of move. It is a rate, not a prediction - it says nothing about which way the price will go.
2. Sensitivities add across positions but not across factors. If desk A has delta $100$ and desk B has delta $-40$, the firm has delta $60$ - simple addition. But the risk is not the sum of the parts, because factors move together and correlate. That coupling lives in a covariance matrix, not in the sensitivities.
3. Every approximation has a remainder, and the remainder is the risk. A first-order (delta) answer misses the curvature term $\frac{1}{2}\Gamma (\Delta S{)}^2$. For small moves that is noise. For a crash it is the entire loss.

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2. Mathematical Ground Truth & Derivations

The Taylor map. Let $V(f_1,\dots ,f_n)$ be portfolio value as a function of risk factors $f$. For a move $\Delta f$,

$$V(f+\Delta f)-V(f)=\underset{\text{sensitivities }\times \text{ moves}}{\underbrace{\sum _i\frac{\partial V}{\partial f_i}\Delta f_i}}+\underset{\text{curvature}}{\underbrace{\frac{1}{2}\sum _{i,j}\frac{{\partial }^{2}V}{\partial f_i\partial f_j}\Delta f_i\Delta f_j}}+O(\Vert \Delta f{\Vert }^3).$$

The first-order vector $b_i=\partial V/\partial f_i$ is the exposure vector (the “Greeks” when the factors are option inputs). The second-order matrix $H_{ij}={\partial }^{2}V/\partial f_i\partial f_j$ is the gamma/cross-gamma matrix. Everything in sensitivity-based risk management is a decision about how many Taylor terms to keep.

Bump-and-revalue (Hull §21.8 for Greeks from a grid; the universal production method). When no closed form exists, measure the derivative numerically by central differences:

$$\frac{\partial V}{\partial f_i}\approx \frac{V(f_i+h)-V(f_i-h)}{2h},\frac{{\partial }^{2}V}{\partial f_i^2}\approx \frac{V(f_i+h)-2V(f_i)+V(f_i-h)}{h^2}.$$

This is identical in spirit to what a trading system does: shock each risk factor by a small amount, re-price the book, difference. It works for any instrument and any model, which is why it is the industry default even when formulas exist.

The risk-factor map table (position $\to$ factors $\to$ P&L). The first modelling judgement is which dials to use:

Position	Risk factors	First-order P&L
Equity	$S$	$\Delta \Delta S$
Bond / swap	$y_1,\dots ,y_n$ (curve nodes)	${\sum }_{i}KR{D}_i\Delta y_i$
Vanilla option	$S,\sigma ,r$	$\Delta \Delta S+\nu \Delta \sigma +\rho \Delta r$
Cross-currency swap	2 curves, FX spot	$\sum KRDs+\text{FX delta}$

Rule of thumb. Map to primary factors you can observe and whose covariance you can estimate - not to the prices of your own instruments. A correlation between two bespoke swaps is unmeasurable; a correlation between two government-bond yields is a time series.

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3. Computational Implementation - bump-and-revalue, and where the first order fails

Re-produce the sensitivities of a European call without using a single Greek formula, then watch the delta-only and delta-gamma approximations lose the plot as the shock grows. Stdlib only.

Read the table as the whole subject in miniature. At $\Delta S=1$ the delta-only answer is off by $2.2\%$ and delta-gamma is essentially exact ($0.611414$ vs $0.611284$). At $\Delta S=40$ the delta-only answer is off by $33\%$, and even delta-gamma - which over-shoots ($+45.80$ vs $+35.61$) - is off by $-10.19$, i.e. $29\%$ of the truth. The curvatures that are invisible locally dominate globally. This is why a VaR number built on deltas alone is not a conservative number; it is a wrong number, and the sign of the error depends on the position.

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4. Failure Modes & First-Principles Breakdowns

1. Linearisation is not a safety margin. A delta-only P&L can be too optimistic (long a hedge that decays) or too pessimistic. It has no directional bias - which is worse than a known bias, because you cannot correct for it.
2. The shock size is a modelling choice, and it is doing the work. Central differences with $h=0.01$ gave delta and gamma to six decimals here, but a $h$ that is too small drowns in floating-point cancellation and one that is too large straddles curvature. A sensitivity is only meaningful together with the shock that produced it.
3. Wrong factor map, wrong risk. If you hedge a 30-year bond with a 10-year future on a “parallel shift” assumption, your delta is right for a parallel curve move and wrong for every actual curve move (see 03 · Rates & Key-Rate Duration).
4. Sensitivities are local, positions are not. A book of 10,000 different options has 10,000 different local maps; aggregating them into one firm-level delta throws away the distribution of curvature. This is the structural argument for factor decomposition (04 · Factor Exposures).

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5. References

• Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
• RiskMetrics (J.P. Morgan): Technical Document, 4th ed. (1996)
• Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed., 2006)
• Alexander, Carol: Market Risk Analysis, Vol. III (2008)

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6. Connected Graph Bridges

• Base: Calculus & Optimization · Pillar 3 · The Greeks (the same objects, derived for pricing)
• Continue: 02 · Delta, Gamma, Vega · Index Hub
• Sibling: Monte Carlo VaR (the alternative to sensitivities: full revaluation)