HOFIS 67 - Credit Derivative Valuation and Risk

Question 1

Desirable properties of CDS valuation model

Answer 1

1. Models a credit event as a single event that can occur at any time in the future
2. Must capture the timing of the default
3. Incorporates the recovery rate when the credit event occurs
4. Reprice the term structure of market prices of credit instruments (i.e. investment grade bonds of varying maturities)
5. Be quick to price in order to enable fast calibrations of the model to market prices

Question 2

EPV of Premium leg of CDS in terms of risky annuity

Answer 2

V_Premium = C(T) * A(T)

• C(T) = fixed annual coupon rate for a T-maturity CDS
• A(T) = EPV of a risky annuity with maturity T

Question 3

Formula for CDS upfront cost at time 0 in terms of Par CDS Spread

Answer 3

U(0) = [S(T) - C(T)] * A(T)

Question 4

Importantance of having a model to price a CDS

Answer 4

1. CDS market is OTC; which means we can’t rely on market quotations to determine what prices are.
2. We need to use a model to price an exotic CDS with features that are different from plain vanilla CDS
3. A valuation model is important to use in risk management

Question 5

Calibrating a Term Structure of Probabilities to CDS

Answer 5

• Assuming constant default rate at discrete time intervals, we can calibrate default rates from CDS market quotes of the upfront cost
• Start with one-year CDS which only requires knowledge of Q(1)
  
  • Set model price of 1-year CDS equal to market upfront cost to solve for h₁
    
    • Q(1) = e^-h₁
• ​In a similar manner, we can move on to the 2-year CDS to solve for h₂

Question 6

Framework of no-arbitrage relationship between CDS and floating rate bonds

Answer 6

• Floating rate bond issued by the same company underlying the CDS
• LIBOR + C = coupon paid by the floating rate note (C is the spread)
• Fixed coupon rate of the CDS = spread on the floating rate bond C
• Investor borrows P + U at funding rate Libor and makes quartely payments
• Investor uses P to buy 1 unit of the floating rate bond maturing at T
• Invesrot uses U to buy CDS protection on 1 unit of face balue of the bond
  
  • makes quartely coupon pmts

Question 7

Cashflows of no-arbitrage relationship between CDS and floating rate bonds

Answer 7

there are three sources of cashflows:

1. Floating rate bond
2. CDS
3. LIBOR funding

cashflows occur at the following times:

1. Time 0
2. Quarterly coupon dates of the CDS, floating rate bond, and funding payments
3. Date of a credit event
4. Time T (maturity of CDS and Floating Rate Bond)

In order for no arbitrage to occur from this trade, the net cashflows at each payment time listed above must equal 0.

Question 8

Assumptions of no-arbitrage relationship between CDS and flo

Answer 8

• interest rate on the bond is locked until maturity
• credit event of the reference bond occurs on a coupon payment date
• no tax effects from coupon and principal payments of the floating rate bond.
• Transaction costs are negligible.
• There is no counterparty risk in the CDS.
• The T-maturity floating rate bond that we mentioned above is actually a tradeable instrument;

in reality, however, this is just a theoretical security that does not exist in the marketplace.