Delta-Gamma VaR for Option Books

Variance-covariance VaR assumes a portfolio's profit and loss moves in a straight line with the underlying market factor. A stock position genuinely does; an option position does not, its payoff curves, which is precisely why anyone bothers holding it. Using the straight-line assumption on an option book badly misjudges risk near the strike, where the curve bends hardest. Full Monte Carlo revaluation fixes this exactly but costs computation. Delta-gamma VaR is the middle path: keep the fast quadratic formula, but add one more term that captures the curvature a straight line misses.

The analogy before any symbols

Estimating how far a car travels in the next second using only its current speed works fine if the car is cruising at constant velocity. It fails badly if the car is accelerating hard, because speed alone (a straight-line, first-order fact) misses the curvature in the distance-versus-time relationship that acceleration creates. Adding one more term, half the acceleration times time squared, captures most of that curvature cheaply, without needing to simulate the car's exact motion second by second. Delta is an option's "speed" with respect to the underlying; gamma is its "acceleration." Delta-gamma VaR is exactly this second-order correction applied to option risk.

The mechanics

For a small move $\Delta S$ in the underlying, a Taylor expansion of the option's value $V$ gives:

$\Delta V \approx \Delta \cdot \Delta S + \tfrac{1}{2}\Gamma \cdot (\Delta S)^2,$

where $\Delta = \partial V/\partial S$ (delta, the straight-line sensitivity) and $\Gamma = \partial^2 V/\partial S^2$ (gamma, the curvature). In words: the option's profit or loss is approximately delta times the underlying's move, plus a correction of half the gamma times the move squared, and that squared term is always positive for a long option position, which is exactly why buying options tends to help on big moves in either direction and plain delta-hedging misses that benefit entirely.

Because $(\Delta S)^2$ enters the expression, $\Delta V$ is no longer normally distributed even when $\Delta S$ is, it becomes skewed, so the plain variance-covariance formula cannot be reused unchanged. Two common fixes: (1) a Cornish-Fisher style adjustment, computing the skewness the gamma term induces and feeding it into the quantile adjustment from Cornish-Fisher VaR, or (2) a quadratic Monte Carlo, simulating $\Delta S$ many times cheaply and applying the formula above directly rather than fully repricing the option, which is much faster than true full revaluation while still capturing curvature.

Compare the true curved payoff shown here to a straight tangent line drawn at today's spot: near the strike, the two diverge fast, which is exactly the region where a delta-only VaR estimate is most wrong and a delta-gamma correction matters most.

Worked example: an at-the-money option

A book holds 100 at-the-money call contracts (100 shares each) on a $50 stock, delta 0.55 per share, gamma 0.08 per share, so portfolio delta $= 100\times100\times0.55 = 5{,}500$ shares-equivalent and portfolio gamma $= 100\times100\times0.08 = 800$. Daily volatility is 2%, so a 99% move in the stock price is $z_{0.99}\times\sigma\times S = 2.326\times0.02\times50 = 2.326$, i.e. $2.326 per share. Delta-only VaR: $5{,}500\times2.326 = 12{,}793$, i.e. $12,793. Adding the gamma term for that same move: $\tfrac12\times800\times(2.326)^2 = \tfrac12\times800\times5.41 = 2{,}164$, i.e. $2,164. Because gamma benefits a long option position on a large move in either direction, the true 99% loss estimate is actually reduced by this term for a long position, giving a delta-gamma VaR of $12{,}793 - 2{,}164 = 10{,}629$, i.e. $10,629, roughly 17% lower than the delta-only number suggested. A pure delta approximation would have overstated this long option book's downside risk.

Worked example: a short-gamma book

Now the same position is a short (written) option book: delta $-5{,}500$, gamma $-800$. Delta-only VaR on a down move is the same magnitude, $12,793, but the gamma term now works against the position on large moves in either direction, since a short option loses convexity: $\tfrac12\times(-800)\times(2.326)^2 = -2{,}164$, i.e. minus $2,164, which subtracted becomes an addition to the loss, giving delta-gamma VaR of $12{,}793 + 2{,}164 = 14{,}957$, i.e. $14,957, about 17% higher than delta alone suggests. This is the sign flip that makes delta-gamma VaR essential for option-selling desks: gamma always hurts short-option positions on big moves, and a delta-only VaR would understate exactly the risk that matters most for them.

What this means in practice

Delta-gamma VaR is the standard fast approximation for option and convertible-bond desks that need intraday risk numbers without full revaluation, and it is a meaningful improvement over delta-only VaR whenever gamma is large relative to delta, near expiry or near the strike. It degrades for large moves or for portfolios with significant higher-order sensitivities (vega, vanna, volga), where the second-order Taylor expansion itself stops being a good approximation to the true payoff curve.