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Point out where the delta-normal method breaks down.

FRM® Part I · Valuation and Risk Models · Calculating and Applying VaR · FRM1.VR.2.e

The idea

The delta-normal method fails whenever its two assumptions fail: that value changes are linear in the risk factors and that the risk factors are jointly normal. The linear assumption breaks down for portfolios with large gamma, such as at-the-money options near expiry, for large market moves, for long horizons, and above all for a short option book where gamma is negative and actual losses exceed the delta estimate. A long option has positive gamma, so delta overstates its loss. A delta-neutral book can show a near-zero delta-normal VaR while still carrying substantial risk, because the second-order term is ignored. Gamma term = 0.5 × gamma × (ΔS)². Here ΔS is the change in the underlying. Fat tails and skewness add further error. Remember that delta only gives the slope at today's price, so the error grows with the size of the move.

Formulas

Delta-gamma approximation of an option's value change

ΔV = Δ × ΔS + 0.5 × Γ × (ΔS)²

Δ
the option delta
ΔS
the change in the underlying price
Γ
the option gamma

Delta-normal keeps only the first term. For a short option the gamma term adds to the loss, so dropping it understates risk; forgetting the 0.5 overstates the correction.

A practice question

From the question bank, with the reason each wrong option is wrong.

The delta-normal method is LEAST reliable for which kind of portfolio?

  1. A book of short at-the-money options with high gamma. - correct
  2. A portfolio of equity index futures and forward contracts.Why not: These are linear instruments, for which the delta-normal approach is exact if returns are normal.
  3. A book of deep in-the-money options over a one-day horizon.Why not: Delta is close to one and gamma is small, so the linear approximation works well.
  4. A diversified portfolio of cash equities.Why not: Cash equities are linear, so the approximation causes no error from non-linearity.

Delta-normal ignores curvature. Where gamma is large and moves are big, the neglected second-order term matters; it hurts most for short options because their losses accelerate.

A harder one

For which position is a delta-normal VaR most likely to understate the true loss from a large adverse stock move?

  1. A long position in at-the-money options with positive gamma.Why not: Positive gamma makes actual losses smaller than the delta estimate, so delta-normal tends to overstate risk.
  2. A long position in stock index futures.Why not: A futures position is linear, so delta-normal captures its loss exactly.
  3. A short at-the-money option position with negative gamma. - correct
  4. A long position in deep in-the-money calls with delta close to one.Why not: Such calls behave almost like stock, so the linear approximation is accurate.

A short option loses more than delta predicts as the move grows, because negative gamma adds a loss of 0.5 x |gamma| x (move)². A long option has the opposite convexity, so its delta-based loss estimate is too large.

The hardest tier

A trading book has a net delta close to zero and large negative gamma. Its delta-normal VaR is reported as very small. Which conclusion is best?

  1. The VaR is reliable, because a net delta of zero means the portfolio is not really exposed to the market.Why not: Treats delta as the whole exposure; the second-order term still produces losses for large moves.
  2. The VaR overstates risk, because negative gamma positions actually profit from large market moves.Why not: Confuses signs: negative gamma loses on large moves in either direction; positive gamma gains.
  3. The VaR probably understates risk, because the gamma loss grows with the square of the move and delta-normal ignores it. - correct
  4. The VaR is reliable as long as the confidence level is raised to 99.9%.Why not: A higher quantile only scales a figure that is near zero; it cannot capture an exposure that the formula leaves out.

With delta near zero the linear model shows almost no sensitivity, but the loss from negative gamma is 0.5 x |gamma| x (move)², which is large for large moves in either direction. A delta-normal figure misses this entirely.

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