Charta is on Google Play: practise on the go. Get the app
Charta

Explain how options, forwards, forward rate agreements (FRAs) and interest rate swaps are each handled when they are mapped.

FRM® Part II · Market Risk Measurement and Management · VaR Mapping · FRM2.MR.5.g

The idea

Derivatives are mapped by decomposing them into positions in the primitive risk factors. An option is replaced by delta times the underlying position, which is the delta-normal approach. It is accurate for small moves but misses gamma. Delta-gamma or full revaluation is needed for large moves and for positions with near-zero net delta, such as a straddle. A forward to buy foreign currency is long the foreign spot exposure, long the foreign zero-coupon bond and short the domestic zero-coupon bond. A forward rate agreement 6x12 that pays fixed and receives floating is long a 6-month zero and short a 12-month zero of equal present value. A pay-fixed swap is short a fixed-rate bond and long a floating-rate note. The note carries only the rate risk up to its next reset date. A receive-fixed swap is the reverse. Both pay-fixed positions gain when rates rise and lose when they fall, because the zero or the fixed bond they are short falls in value as rates rise. Exam point: the sign on each leg decides whether the risks offset.

Formulas

Delta-normal VaR of an option position

VaR = z × σ × Δ × n × S

z
the standard normal quantile
σ
the volatility of the underlying's return over the horizon
Δ
the option delta, net across legs
n
the number of options
S
the underlying price

Small moves only. The exposure is delta times underlying value, not the premium; near-zero net delta, as in a straddle, badly understates risk.

A practice question

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

When an option position is mapped using the delta approach, it is represented as:

  1. a position in the underlying equal to the option's premium.Why not: Uses market value as the exposure instead of delta times the underlying notional.
  2. a position in the underlying asset equal to delta times the underlying notional. - correct
  3. a zero-coupon bond maturing on the option's expiry date.Why not: Treats the option as a bond; delta mapping works through the underlying risk factor.
  4. a position equal to the option's gamma times the underlying notional.Why not: Gamma is a second-order sensitivity; first-order mapping uses delta.

Delta-normal mapping replaces the option by delta x notional of the underlying, so the option's VaR is delta times the underlying's VaR. It is linear and so accurate only for small moves.

A harder one

A trader holds 20,000 call options on a stock priced at USD 80. Each option is worth USD 4.20 and has a delta of 0.55. The stock's daily volatility is 1.8 percent. The one-day 95 percent delta-normal VaR of the options is closest to:

  1. USD 47,400Why not: Treats the options as 20,000 shares, ignoring delta.
  2. USD 2,500Why not: Uses the option premium as the exposure instead of delta times the underlying value.
  3. USD 26,100 - correct
  4. USD 36,800Why not: Uses the 99 percent quantile 2.326 instead of the 95 percent quantile 1.645.

Delta-equivalent exposure = 0.55 x 20,000 x 80 = 880,000. VaR = 1.645 x 1.8% x 880,000 = 26,100. The option price is not the exposure; delta times underlying value is.

The hardest tier

A firm buys a 6x12 forward rate agreement on USD 10 million, which starts in 6 months and covers a further 6 months, paying the fixed rate and receiving floating. Rates use semiannual compounding and the 6-month zero rate is 4.0 percent. The FRA maps to a long 6-month zero and a short 12-month zero, each with a present value of USD 10 million / 1.02 = USD 9.80 million. One-day 95 percent VaR rates for the zeros are 0.15 percent (6-month) and 0.30 percent (12-month), with a correlation of 0.80. The VaR of the FRA is closest to:

  1. USD 19,700 - correct
  2. USD 44,100Why not: Adds the two stand-alone VaRs, ignoring that the long and short legs offset because they are highly correlated.
  3. USD 32,900Why not: Treats the two vertices as uncorrelated.
  4. USD 42,100Why not: Treats both legs as long exposures, missing that the 12-month leg is short.

Stand-alone VaRs: 9.804m x 0.15% = 14,706 and 9.804m x 0.30% = 29,412. With opposite signs, VaR = sqrt(14,706² + 29,412² - 2 x 0.8 x 14,706 x 29,412) = 19,700. An FRA is a spread position, so its risk is much lower than that of either leg.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in VaR Mapping