VaR, ES and the stress table all answer the same question, which is how much money is at stake, differing only in how they define the word "much". A second family of risk numbers answers a different question entirely: how much does this book's value change when one named thing changes by one unit. Both families are needed, and a desk that runs only one of them has a specific blind spot rather than a general shortfall of rigour.
What does each sensitivity measure?
| Measure | The one thing that moves | What it gives you | What it cannot see |
|---|---|---|---|
| Delta | Underlying price | Change in value per 1 unit of price | Its own change; anything beyond a small move |
| Gamma | Underlying price | Change in delta per 1 unit of price | Everything other than price |
| Vega | Implied volatility | Change in value per 1 point of vol | Price, time, rates |
| Duration | Yield | Change in value per 1 unit of yield | Curve shape; credit; convexity, unless added |
| Beta | The market factor | Change in value per 1 unit of index move | Anything the index does not span |
Every row shares one limitation and it is worth stating once. A sensitivity is a local, one-factor statement. It is the first term of a Taylor expansion around today's price, and it is exactly right for an infinitesimal move and progressively wrong for a real one. VaR and ES, by contrast, are portfolio-level and probabilistic: they aggregate across factors and attach a likelihood, at the cost of telling you nothing about which exposure produced the number.
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Worked example: why delta alone fails on a call option
Make the local-approximation point with an arithmetic that cannot be argued with. Take a three-month at-the-money call, spot and strike both 100, risk-free rate 5%, implied volatility 30%. It is worth 6.583084, with delta 0.562903, gamma 0.0262649 and vega 0.196986 per volatility point.
| Move in spot | Full reprice | Delta only | Delta plus gamma |
|---|---|---|---|
| −20% | −6.099760 | −11.258059 | −6.005087 |
| −10% | −4.274424 | −5.629029 | −4.315786 |
| +10% | +6.819445 | +5.629029 | +6.942272 |
| +20% | +15.403010 | +11.258059 | +16.511030 |
Read the −20% row and the reason gamma exists is on the page. Delta alone predicts a loss of 11.258059 on a position that can lose at most its 6.583084 premium, an impossible answer produced by an entirely correct formula asked a question outside its range. Adding the gamma term brings the estimate to −6.005087, wrong by 0.094673 rather than by 5.158299. Then read the +20% row and notice that the second-order estimate now overshoots, by 1.108020, because a two-term expansion is still an expansion. Sensitivities are for small moves and for hedging; stress tests are for large ones; and a risk report that offers only one of the two has decided which crises it will be surprised by.