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Define a forward on a CDS and an option on a CDS.

FRM® Part II · Credit Risk Measurement and Management · Credit Derivatives · FRM2.CR.13.e

The idea

A forward CDS is an agreement to buy or sell protection over a future period, from T1 to T2, at a spread fixed today. It usually terminates with no payment if the reference entity defaults before T1. Because of that knock-out, it can be replicated by buying T2 protection and selling T1 protection. Here s1 and s2 are the CDS spreads for each maturity, and A1 and A2 are the matching risky annuities. Because the knock-out is replicated by those two trades, the forward spread over the period is about F = (s2 × A2 − s1 × A1) ÷ (A2 − A1). An option on a CDS, a credit default swaption, gives the right but not the obligation to enter a CDS at a strike spread. A payer swaption is the right to buy protection and gains when spreads widen. A receiver swaption is the right to sell protection. At expiry, the payer swaption pays Payoff = max(spread − strike, 0) × annuity × notional.

Formulas

Forward CDS spread

forward spread = (s2 × A2 − s1 × A1) ÷ (A2 − A1)

s1
the CDS spread to T1
A1
the risky annuity to T1
s2
the CDS spread to T2
A2
the risky annuity to T2

Ignores accrued premiums. Divide by the forward period's annuity A2 − A1, not by A2.

Payoff of a payer CDS option at expiry

payoff = max(S − K, 0) × A × notional

S
the CDS spread at expiry
K
the strike spread
A
the risky annuity of the underlying CDS
notional
the notional

A receiver uses max(K − S, 0). Leaving out the annuity gives one year's spread gain rather than its present value.

Black value of a payer CDS option

value = A × [F × N(d1) − K × N(d2)] × notional

A
the forward risky annuity
F
the forward CDS spread
N
the standard normal cumulative distribution function
d1
(ln(F ÷ K) + σ² × T ÷ 2) ÷ (σ × √T)
K
the strike spread
d2
d1 − σ × √T
notional
the notional

σ is the spread volatility and T the time to expiry. Using intrinsic value alone ignores time value.

A practice question

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

Which description defines a forward CDS?

  1. An agreement to buy or sell protection over a future period at a spread fixed today, usually void on an earlier default. - correct
  2. A contract that gives the right but not the obligation to enter a CDS at a future date at a spread fixed today.Why not: Describes a CDS option; a forward commits both sides to the trade.
  3. An agreement to buy protection that starts immediately, with the first premium payment deferred to a later date only.Why not: A forward CDS starts protection at the future date, not now; only deferring premiums is not a forward.
  4. A CDS whose notional automatically increases at a future date if the reference entity's spread has widened by then.Why not: Describes an invented step-up feature; a forward sets the spread and period of a future CDS.

A forward CDS commits the parties to a CDS starting at T1 and ending at T2 at an agreed spread. If the entity defaults before T1, the forward typically terminates without payment.

A harder one

An investor holds a payer option on a five-year CDS with a strike spread of 120 bp and USD 50 million notional. At expiry the five-year CDS spread is 170 bp and the risky annuity is 4.1. The payoff on exercise, closest to, is:

  1. USD 250,000Why not: Omits the risky annuity, so the 50 bp gain is not present-valued over the life of the underlying CDS.
  2. USD 3,485,000Why not: Applies the full market spread instead of the 50 bp excess over the strike.
  3. USD 1,025,000 - correct
  4. USD 0Why not: Treats the option as a receiver, which would pay only if the spread had fallen below the strike.

Payer payoff = max(170 - 120, 0) bp x 4.1 x 50,000,000 = 0.005 x 4.1 x 50,000,000 = 1,025,000.

The hardest tier

A trader wants protection on a name only from year 1 to year 3 and none in year 1. A forward CDS terminates if the name defaults before year 1. Which static position replicates this forward protection?

  1. Sell three-year protection and buy one-year protection on the same name and notional.Why not: Reverses both legs and so creates a short forward protection position, not the long position wanted.
  2. Buy three-year protection and buy one-year protection on the same name and notional.Why not: Doubles the protection in year 1 instead of cancelling it; the one-year leg must be sold.
  3. Buy three-year protection and sell two-year protection on the same name and notional.Why not: Uses the length of the forward period rather than its start date; this leaves protection only in year 3.
  4. Buy three-year protection and sell one-year protection on the same name and notional. - correct

If default occurs before year 1, the bought three-year CDS pays (1 - R) and the sold one-year CDS pays it away, netting to zero, as in a knocked-out forward. After year 1 only the extra two years of protection remain.

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