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Calculate VaR for non-linear derivatives by means of the delta-normal approach, after first explaining how that approach operates.

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

The idea

The delta-normal approach handles an option by replacing it with its linear approximation. The change in option value is approximated as delta × the change in the underlying, so the option becomes an equivalent position. Exposure = delta × number of units × underlying price. Risk-factor changes are assumed jointly normal. VaR = z × σ × |delta| × position value. This holds for one underlying, with z = 2.326 at 99 percent and σ the volatility of the underlying's return. For several underlyings, the equivalent exposures are combined using the covariance matrix. A put has a negative delta but its VaR is still a positive loss, so use the absolute value. Remember to scale the daily σ by the square root of the horizon in days. The exam trap is forgetting delta and applying the formula to the full underlying notional.

Formulas

Delta-normal VaR of an option position

VaR = z × σ × |Δ| × N × S

z
the standard normal quantile, 2.326 at 99%
σ
the volatility of the underlying's return over the horizon
Δ
the option delta
N
the number of units of the underlying
S
the underlying price

One underlying, linear approximation and normal returns. Use the absolute delta for puts, and never apply the formula to the full notional without delta.

A practice question

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

How does the delta-normal approach compute VaR for a portfolio of options?

  1. It revalues the options over many random paths of the underlying and takes a percentile of the outcomes.Why not: This is Monte Carlo simulation with full revaluation, not delta-normal.
  2. It applies actual past changes in the underlying to the option pricing formula and ranks the resulting losses.Why not: This is historical simulation, which makes no normality assumption.
  3. It replaces each option by delta times the underlying and applies the normal linear VaR formula. - correct
  4. It adds a gamma term to the delta term to capture the curvature of the option value.Why not: This is the delta-gamma refinement; plain delta-normal uses only the first-order term.

Delta-normal linearises: change in option value is approximately delta x change in underlying. The option becomes an equivalent underlying position and normal returns give VaR = z x sigma x exposure.

A harder one

A trader is long 100,000 call options on a stock priced at USD 50. The option delta is 0.60 and the stock's daily volatility is 2.0%. Using delta-normal, the one-day 99% VaR (z = 2.326) is closest to:

  1. USD 139,600 - correct
  2. USD 232,600Why not: Ignores the delta and treats the position as 100,000 shares.
  3. USD 98,700Why not: Uses the 95% quantile of 1.645 instead of 2.326 for the 99% level.
  4. USD 93,100Why not: Uses (1 - delta) = 0.40 as the delta, confusing the delta with its complement.

Exposure = 0.60 x 100,000 x 50 = USD 3.0 million. VaR = 2.326 x 2.0% x 3.0 million = USD 139,600.

The hardest tier

A portfolio holds call options on USD 12 million of stock A (delta 0.50) and call options on USD 10 million of stock B (delta 0.40). Daily volatilities are 1.2% for A and 2.0% for B, with a correlation of 0.50. The one-day 99% delta-normal VaR (z = 2.326) is closest to:

  1. USD 306,000 - correct
  2. USD 696,000Why not: Applies the formula to the full notionals and ignores the deltas.
  3. USD 354,000Why not: Adds the two stand-alone VaRs, which assumes a correlation of one.
  4. USD 250,000Why not: Drops the correlation term, assuming the two stocks are uncorrelated.

Delta-equivalent exposures are 6.0 and 4.0 million. Dollar volatilities are 72,000 and 80,000. Portfolio sigma = sqrt(72,000² + 80,000² + 2 x 0.5 x 72,000 x 80,000) = 131,700. VaR = 2.326 x 131,700 = USD 306,000.

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