Adding the gamma term to parametric VaR, why a delta-hedged book needs the two-tailed quantile, and why gamma losses do not scale with the square root of time.
Delta-gamma VaR adds the second-order term to the linear estimate: the loss at the quantile move is delta × ΔS plus one half of gamma × ΔS squared. On an illustrative short position in 10,000 three-month at-the-money calls, delta-normal VaR at 99 per cent over one day is $24,458, delta-gamma VaR is $27,003, and full revaluation gives $26,897. Hedge the delta away and delta-normal VaR falls to zero, while the position can still lose $2,567 on a bad day.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Parametric VaR treats every position as a straight line in its risk factor: a 1 per cent move in the underlying produces delta times that move, whichever way the market goes. Options are curves. A short call loses more on the second dollar of a rally than on the first, because its delta climbs as the market rises. Gamma measures that curvature, and a book that is short gamma has a loss distribution with a fatter left tail than any linear model will show. Delta-gamma VaR is the cheapest fix: it keeps the speed of a formula while capturing most of the curvature, and it shows exactly where the linear number goes wrong.
| Input | Value |
|---|---|
| Underlying price / strike | 100 / 100 |
| Implied volatility, annual | 30% |
| Risk-free rate | 4% |
| Time to expiry | 3 months |
| Position | short 10,000 calls |
| Call price / premium received | 6.46 / $64,595 |
| Delta per call / gamma per call | 0.556 / 0.0263 |
| Confidence, horizon | 99%, 1 day (252-day year) |
Position delta is −10,000 × 0.556 = −5,563 shares; position gamma is −10,000 × 0.0263 = −263 per dollar. A one-day standard deviation is 100 × 30% / √252 = 1.89, and the 99 per cent quantile move is 2.326 × 1.89 = 4.40.
Delta-normal VaR = |position delta| × σΔS × z = 5,563 × 1.89 × 2.326 = $24,458
Delta-gamma loss at the adverse move (a rally of 4.40, since the book is short calls):
ΔP = Δ × ΔS + ½ × Γ × ΔS2 = −5,563 × 4.40 + ½ × (−263) × 4.402
= −$24,458 − $2,545 = −$27,003, so delta-gamma VaR = $27,003
Full revaluation: reprice every call at each possible one-day price and one day less to expiry, weight by probability, read off the 1 per cent quantile: $26,897
Excel, with delta in B1, gamma in B2 and the quantile move in B3: =-(B1*B3+0.5*B2*B3^2), checking both B3 and -B3 and keeping the larger loss.
| Method | VaR | Gap to full revaluation |
|---|---|---|
| Delta-normal | 24,458 | −9.1% |
| Delta-gamma | 27,003 | +0.4% |
| Full revaluation | 26,897 | benchmark |
The gamma term adds $2,545, and it takes the estimate from 9.1 per cent short of the true figure to within 0.4 per cent of it. The small residual comes mainly from two sources: the move is lognormal rather than normal, and full revaluation also earns a day of time decay, which delta-gamma ignores.
Now buy 5,563 shares against the calls. The book is delta-neutral, so delta-normal VaR is exactly $0. The book is still short 263 of gamma, and it loses money on a large move in either direction: ΔP = ½ × Γ × ΔS2, which is negative whatever the sign of ΔS.
Losses come from both tails, so the 1 per cent is split: 0.5 per cent each side, z = 2.576, not 2.326
Quantile move = 2.576 × 1.89 = 4.87
Delta-gamma VaR = ½ × 263 × 4.872 = $3,120
Full revaluation, including one day of theta worth $550 to a short option book: $2,567
Two lessons sit in those numbers. First, using the one-tailed 2.326 on a delta-neutral book gives $2,545, which understates the delta-gamma figure, because a short straddle-like profile loses at both ends. Second, on a hedged book theta is no longer a rounding error: it is worth 18 per cent of the delta-gamma loss, and full revaluation, which collects it, comes in at $2,567, close to $3,120 less $550.
| Horizon | Naked: delta-normal | Naked: full | Linear shortfall | Hedged: delta-gamma | Hedged: full |
|---|---|---|---|---|---|
| 1 day | 24,458 | 26,897 | 9.1% | 3,120 | 2,567 |
| 5 days | 54,690 | 66,902 | 18.3% | 15,598 | 12,732 |
| 10 days | 77,344 | 101,792 | 24.0% | 31,197 | 25,172 |
The linear error grows with the horizon because the gamma term grows with the square of the move. On the hedged book the effect is starker: the delta-gamma loss scales with the horizon itself, not with its square root. Applying the square-root-of-time rule to the one-day $3,120 gives $9,865 for ten days, against $31,197 delta-gamma and $25,172 on full revaluation. The rule in how to scale a one-day VaR to ten days assumes a linear book, and an options book is not one.
Delta-gamma VaR is delta times the quantile move plus half gamma times its square, here $27,003 against a full revaluation of $26,897, while delta-normal misses 9.1 per cent. On a delta-hedged book it is the only parametric figure that is not zero. The book's own value at risk, expected shortfall and backtest workbook is in the free companion files.
Delta-normal VaR treats the option as a straight line in the underlying and multiplies position delta by the quantile move. Delta-gamma adds half of gamma times the move squared, capturing the curvature. On an illustrative short call position, the gamma term adds $2,545 to a one-day 99 per cent VaR of $24,458, bringing it within 0.4 per cent of full revaluation.
Because position delta is zero, the linear model sees no exposure. A short gamma book still loses on a large move either way: half of gamma times the move squared. For the illustrative book short 263 of gamma, the 99 per cent loss at a 4.87 move is $3,120 delta-gamma and $2,567 after a day of theta on full revaluation.
No. The gamma loss grows with the variance of the move, which grows with the horizon itself. For the illustrative delta-hedged book, scaling the one-day $3,120 by the square root of ten gives $9,865, while the ten-day delta-gamma figure is $31,197 and full revaluation $25,172.
This article is one calculation from Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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