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FRM Exam Part II · VaR Mapping

Mapping Nonlinear Instruments and Options for VaR

Updated 11 October 2026 · Fact-checked

Mapping an option means replacing it with exposure to its underlying risk factor. The delta approximation treats the option as a linear position: option VaR ≈ |Δ| × underlying VaR. The delta-gamma approximation adds a curvature term. Linear mapping understates risk for large moves, especially for long gamma or short gamma positions near the strike.

Understand Mapping Nonlinear Instruments and Options

A linear instrument, such as a stock or a forward, changes in value in proportion to its risk factor. Its VaR is easy: exposure times the factor's VaR. An option is nonlinear. Its value curve bends, so a given move in the underlying changes value by different amounts depending on where you start.

Mapping means expressing a position as exposures to a few risk factors. For an option, the first step is the delta approximation. You take the slope of the price curve at today's price and treat the option as Δ units of the underlying. Then VaR of the option ≈ |Δ| × VaR of the underlying (per unit). This is the basis of delta-normal VaR.

The slope is only right for small moves. The price curve bends, and gamma measures that bending. A delta-gamma approximation uses a second-order Taylor expansion: change in option value ≈ Δ × ΔS + ½ × Γ × (ΔS)². The squared term adds a convexity adjustment.

The sign of gamma matters. A long option (long gamma) gains from large moves in either direction, so delta-normal VaR overstates the loss on the downside tail. A short option (short gamma) loses more than the linear estimate in large moves, so delta-normal VaR understates risk. This is the key limit to remember.

For big moves or long horizons, full revaluation (Monte Carlo or historical simulation with repricing) is more accurate. Delta-gamma sits in between. It is faster than full revaluation but can still be wrong for extreme moves, for strongly nonlinear payoffs, and when volatility or other factors also move.

Key formulas to remember

Delta approximation
ΔV ≈ Δ × ΔS
Δ = ∂V/∂S. Valid for small moves in S. Treats the option as a linear position.
Delta-normal option VaR
VaR(option) ≈ |Δ| × S × σ × z × √t
σ is volatility of returns for the horizon unit, z is the normal quantile (1.645 at 95%, 2.326 at 99%). Use the absolute value because VaR is a loss amount.
Delta-gamma approximation
ΔV ≈ Δ × ΔS + ½ × Γ × (ΔS)²
Γ = ∂²V/∂S². Second-order Taylor expansion of option value.
Worst-case S move at the VaR quantile
ΔS* = ± S × z × σ × √t
Evaluate delta-gamma at the adverse move. For a long call or put (Γ > 0) check both ends; the loss is usually at the end the delta points to.
Delta-gamma loss at the adverse move
Loss ≈ −(Δ × ΔS* + ½ × Γ × (ΔS*)²)
This quantile shortcut is exact only when the loss is monotonic in S. It is an approximation otherwise.
Delta of a position
Position delta = number of options × Δ × contract size
Convert to underlying-equivalent exposure before applying VaR.

How to solve Mapping Nonlinear Instruments and Options questions

Use this sequence for any question on mapping options or the limits of linear VaR.

  1. 1Identify the risk factor the option maps to, usually the underlying price or index.
  2. 2Compute or read the delta and gamma, and scale by the number of options and contract size.
  3. 3Find the adverse move in the underlying at the stated confidence level and horizon: S × z × σ × √t.
  4. 4For delta-normal VaR, multiply the absolute delta-equivalent exposure by the factor's VaR.
  5. 5For delta-gamma, put the adverse move into Δ × ΔS + ½ × Γ × (ΔS)² and take the loss. Use the move in the direction that hurts the position.
  6. 6Compare the two answers and interpret using the sign of gamma: long gamma means delta-normal overstates VaR; short gamma means it understates VaR.
  7. 7State the remaining limits: second-order only, other Greeks (vega, theta) ignored, and full revaluation is the benchmark.

Quickest way: Sign-of-gamma shortcut

When to use it: Use when a question asks only which direction the error goes, or which VaR is larger, with no heavy computation.

  1. Read the gamma sign. Long option positions have Γ > 0; short option positions have Γ < 0.
  2. If Γ > 0, the delta-gamma loss is smaller than the delta-only loss, so delta-normal VaR is conservative.
  3. If Γ < 0, the delta-gamma loss is larger, so delta-normal VaR understates risk.
  4. Check the size of the move. The gamma term grows with (ΔS)², so the gap widens as horizon or confidence level rises.
  5. If a number is needed, compute the delta loss first, then add the gamma correction ½ × Γ × (ΔS)² with the right sign.

Common mistakes in Mapping Nonlinear Instruments and Options

  • Applying delta-normal VaR to a short option position and treating it as safe.

    Delta looks small near the strike, so the position looks low risk.

    Fix: Check gamma. A short gamma position can lose far more than delta suggests in large moves.

  • Forgetting to multiply by contract size and number of options.

    Delta is quoted per unit of underlying.

    Fix: Convert to position delta first, then to a currency exposure.

  • Dropping the square on the move in the gamma term.

    Rushing the Taylor formula.

    Fix: Write ½ × Γ × (ΔS)² explicitly. The move is squared, not Γ.

  • Using the signed delta as VaR and getting a negative number.

    A put has negative delta.

    Fix: Use |Δ| for the exposure size, then work out which direction of the underlying hurts the position.

  • Claiming delta-gamma is exact.

    It feels like a more complete answer.

    Fix: It is still a second-order approximation. It ignores higher orders and other factors such as volatility, and full revaluation is more accurate.

  • Saying long options always make delta-normal VaR too low.

    Confusing the direction of the gamma effect.

    Fix: Long gamma gains from curvature, so delta-normal VaR overstates the loss. It is short gamma that is understated.

Worked examples

Example 1

A trader is long 10,000 call options on a stock priced at USD 100. Each call has delta 0.60 and gamma 0.02. Daily volatility is 2%. Using a 95% confidence level (z = 1.645) and a one-day horizon, find the delta-normal VaR and the delta-gamma VaR. Assume each option covers one share.

Show the solution
  1. Adverse move: the call loses when S falls. ΔS = −100 × 1.645 × 0.02 = −3.29.
  2. Position delta = 10,000 × 0.60 = 6,000 shares.
  3. Delta-normal VaR = 6,000 × 3.29 = USD 19,740.
  4. Gamma term per option: ½ × 0.02 × (3.29)² = 0.01 × 10.8241 = 0.108241. For 10,000 options: USD 1,082.41.
  5. Delta-gamma change in value = −19,740 + 1,082.41 = −18,657.59.
  6. Delta-gamma VaR ≈ USD 18,658.

Answer: Delta-normal VaR is about USD 19,740; delta-gamma VaR is about USD 18,658. The long gamma position makes the linear estimate conservative.

Example 2

A bank is short 5,000 put options on an index at 2,000. Each put has delta −0.40 and gamma 0.001 (per index point), one unit per option. Daily volatility is 1.5%. Using a 99% confidence level (z = 2.33), find the delta-gamma loss. State whether delta-normal VaR over- or understates risk.

Show the solution
  1. Short puts lose when the index falls. Position delta = −5,000 × (−0.40) = +2,000 index units. The bank is effectively long 2,000 units.
  2. Adverse move: ΔS = −(index level 2,000) × 2.33 × 0.015 = −69.9 index points.
  3. Delta loss = 2,000 × 69.9 = 139,800.
  4. Position gamma = −5,000 × 0.001 = −5 (short gamma).
  5. Gamma term = ½ × (−5) × (69.9)² = −2.5 × 4,886.01 = −12,215.03.
  6. Total change in value = (2,000 × −69.9) + (−12,215.03) = −139,800 − 12,215.03 = −152,015.03.
  7. Delta-gamma loss ≈ 152,015 versus delta-normal 139,800.

Answer: Delta-gamma loss is about 152,015 versus 139,800 under delta-normal. Because gamma is negative, delta-normal VaR understates risk.

Exam tips

  • Always decide first whether the position is long or short gamma. Many questions only test the direction of the error.
  • Convert to position delta and gamma before using any formula, and note whether the question gives per-option or per-position Greeks.
  • Work out which direction of the underlying hurts you before choosing the sign of ΔS.
  • Expect interpretation items: when delta-gamma is better, and when full revaluation or Monte Carlo is still needed.
  • Be ready to name other risk factors an option depends on, such as volatility (vega), which delta-gamma on the underlying alone ignores.

Practice questions from VaR Mapping

Mapping Nonlinear Instruments and Options in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Mapping Nonlinear Instruments and Options: frequently asked questions

What is the difference between linear and nonlinear VaR mapping?

Linear mapping assumes value changes in proportion to the risk factor, so one exposure number is enough. Nonlinear mapping, as for options, recognises that the slope changes with the underlying. It adds gamma or uses full revaluation to capture the curvature.

Why does delta-normal VaR fail for options?

It uses only the slope at today's price. Options have curved payoffs, so for large moves the slope changes and the linear estimate is wrong. It is especially misleading for short gamma positions and near the strike.

How do I compute option VaR using delta mapping?

Convert the position to delta-equivalent units of the underlying. Multiply by the underlying price and by z × σ × √t, using the absolute value. The result is the delta-normal VaR.

When is the delta-gamma approximation not enough?

It is not enough for very large moves, long horizons, deep nonlinearity such as barriers, and when volatility or rates also move. In those cases full revaluation by Monte Carlo or historical simulation is more reliable.