FRM Exam Part I · Calculating and Applying VaR

Delta, Gamma and Full Revaluation for VaR on Options

Updated 11 October 2026 · Fact-checked

Nonlinear positions such as options do not move one-for-one with their risk factors. You estimate VaR by shocking the risk factor at the chosen quantile, then repricing with delta only, delta plus gamma, or full revaluation. Full revaluation is most accurate; delta is fastest but misses curvature.

Understand Delta, Gamma and Full Revaluation for Nonlinear Positions

VaR needs a loss for a given move in the market. For a stock or a futures position, loss is linear in the price move. For an option or a bond, it is not. The price curve bends.

The first step is risk factor mapping. You break each position into exposures to a small set of risk factors, such as an equity index, an FX rate, or points on the yield curve. An option maps to its underlying price and implied volatility. A bond maps to zero rates at key maturities. This keeps the model small enough to run.

The second step is to choose how to reprice. The delta approximation uses the slope only: ΔV ≈ δ × ΔS. It turns the option into a linear position, so you can use delta-normal VaR. It is accurate for small moves and badly wrong for large moves or for options with high gamma, such as near-the-money options close to expiry.

The delta-gamma approximation adds curvature: ΔV ≈ δΔS + ½ΓΔS². A long option has positive gamma, so delta alone overstates the loss. A short option has negative gamma, so delta alone understates the loss. For bonds the same idea is duration and convexity.

Full revaluation reprices the instrument with its pricing model at each new risk factor value. It is exact within the model, and it is what historical simulation and Monte Carlo normally use. It costs more computing time. In a portfolio, the approximations are quick but full revaluation is the benchmark.

Key formulas to remember

Delta approximation: ΔV ≈ δ × ΔS; Linear in the underlying move. Use for small moves. Portfolio delta is the sum of position deltas on the same underlying.
Delta-gamma approximation: ΔV ≈ δ × ΔS + ½ × Γ × (ΔS)²; Second-order Taylor expansion. Gamma is positive for long options and negative for short options.
Delta-normal VaR for an option: VaR = |δ| × S × z × σ (σ is the volatility of the return over the horizon); Worst-case move in S at the confidence level, times |δ|. z is 1.645 at 95% and 2.326 at 99% (one-tailed).
Bond duration-convexity approximation: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)²; Δy in decimals. Same logic as delta-gamma.
Full revaluation loss: Loss = V(S₀) − V(S*); S* is the shocked risk factor. Reprice with the full model.

How to solve Delta, Gamma and Full Revaluation for Nonlinear Positions questions

Use this order for any question on VaR of an option or bond position.

1Identify the position and its risk factors (underlying price, rate, volatility).
2Find the shock for the confidence level and horizon: ΔS = ±z × σ × S. For a long call or long stock use the down move; for a put or short call use the move that hurts.
3Pick the method the question asks for: delta, delta-gamma, or full revaluation.
4For delta: compute δ × ΔS. For delta-gamma: add ½ΓΔS², keeping the sign of Γ.
5For full revaluation: reprice at S₀ + ΔS with the model and subtract from the current value.
6Convert the loss to a positive VaR and scale by the number of contracts or units.
7Compare answers if asked: long gamma makes delta-only overstate the loss; short gamma makes it understate.

Quickest way: Sign-of-gamma shortcut

When to use it: When options ask which method gives a higher or lower VaR, or when you need the answer before computing everything.

Check the sign of gamma for the whole position, not each leg.
Positive net gamma: delta-only VaR is too high. Negative net gamma: delta-only VaR is too low.
Compute the shock once, then the delta loss, then adjust by ½ΓΔS².
With a loss, the gamma term for a long option is positive and reduces it. Subtract it from the delta loss.
Eliminate options that break the direction rule before doing any other arithmetic.

Common mistakes in Delta, Gamma and Full Revaluation for Nonlinear Positions

Forgetting to square ΔS in the gamma term.
Candidates remember δΔS and add Γ × ΔS by habit.
Fix: Write ½ × Γ × ΔS² every time and compute ΔS² separately.
Dropping the ½ in the gamma term.
The factor comes from the Taylor expansion and is easy to forget under time pressure.
Fix: Tie it to the bond version, ½ × C × Δy², so you remember it in both.
Using the wrong direction of shock.
Candidates always take the down move, but a put or short call loses when the price rises.
Fix: Ask which move hurts. Then apply the signed delta to that move.
Saying delta VaR always overstates risk.
Textbook examples use long options.
Fix: Overstates for net positive gamma, understates for net negative gamma.
Mixing percent and dollar shocks.
The σ given is a return, but the formula needs a price change.
Fix: Convert first: ΔS = z × σ × S. Then use it in dollars per share.
Applying delta-normal VaR with the delta-gamma term still included as if the result were normal.
Adding gamma makes the payoff nonlinear, so a normal underlying does not give a normal option P&L.
Fix: Treat delta-gamma as a quadratic approximation. Use it to estimate the loss at a chosen shock, not as a normal quantile.

Worked examples

Example 1

A trader holds 1,000 long call options on a stock priced at USD 100. Each call has delta 0.60 and gamma 0.04. Daily volatility is 2%. Estimate the one-day 99% VaR using (a) delta only and (b) delta-gamma. Use z = 2.33.

Show the solution

Shock: ΔS = −2.33 × 0.02 × 100 = −4.66.
Delta only, per option: 0.60 × (−4.66) = −2.796.
Delta only, position: −2.796 × 1,000 = −2,796. VaR = USD 2,796.
Gamma term, per option: ½ × 0.04 × 4.66² = 0.02 × 21.7156 = 0.4343.
Delta-gamma change per option: −2.796 + 0.4343 = −2.3617.
Position: −2,361.7. VaR ≈ USD 2,362.

Answer: Delta VaR ≈ USD 2,796; delta-gamma VaR ≈ USD 2,362. The long option has positive gamma, so delta alone overstates the loss.

Example 2

A portfolio is short 500 call options on a stock priced at USD 50. Each call has delta 0.50 and gamma 0.05. The 10-day 95% move in the stock is 8% (up or down). Use delta-gamma to estimate the loss for the adverse move, and say whether delta only understates or overstates it.

Show the solution

Short calls lose when the stock rises. Adverse move: ΔS = +0.08 × 50 = +4.
Position delta = −500 × 0.50 = −250. Position gamma = −500 × 0.05 = −25.
Delta term: −250 × 4 = −1,000.
Gamma term: ½ × (−25) × 4² = −12.5 × 16 = −200.
Total change = −1,000 − 200 = −1,200. Loss = USD 1,200.
Delta only gives USD 1,000, which is lower than 1,200.

Answer: Delta-gamma loss ≈ USD 1,200. Delta only gives USD 1,000, so it understates the loss because net gamma is negative.

Exam tips

Check the sign of gamma before computing. It tells you the direction of the error and often removes two options at once.
Questions give σ as a return. Convert it to a price shock with S and z before anything else.
For a short option or a put, find the adverse direction first. The sign of δ × ΔS must be a loss.
Know when full revaluation matters: large moves, high gamma, and options near expiry. Expect conceptual questions on that.
Keep ΔS² and ½Γ separate in your working. Most lost marks come from arithmetic slips, not concepts.

Practice questions from Calculating and Applying VaR

Delta, Gamma and Full Revaluation for Nonlinear Positions in other exams

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

FRM Part IIMapping Nonlinear Instruments and Options

Delta, Gamma and Full Revaluation for Nonlinear Positions: frequently asked questions

What is the difference between delta-gamma and full revaluation VaR?

Delta-gamma uses a second-order Taylor expansion of the price around today's value. Full revaluation reprices the instrument with its model at the shocked risk factor values. Full revaluation is more accurate for large moves but slower to run.

Why does delta-only VaR fail for options?

Delta is a slope at one point, but the option price curve bends. The error grows with the size of the move and with gamma. For a short option, the loss grows faster than delta predicts.

What is risk factor mapping in VaR?

It means expressing each position as exposure to a limited set of market variables, such as an index, an FX rate or zero rates at key maturities. The VaR model then works with those factors instead of every instrument.

Which VaR methods use full revaluation?

Historical simulation and Monte Carlo normally reprice the portfolio at each scenario, so they handle nonlinearity directly. Delta-normal VaR uses only the linear approximation.