2.3 Methodology
2.3.2 Framework using CDS spreads
Credit default swaps are one example of credit derivatives, and credit derivative markets
have experienced explosive growth in recent years, which have attracted attentions of deal-
ers, investors, regulators, and lately, the general public. According to the International Swaps
and Derivatives Association (ISDA), on a gross notional outstanding basis, the market has
roughly doubled in size every year from 0.6 trillion dollars in 2001 to 62.2 trillion dollars in
20076. After the breakout of the financial crisis in 2008, the level of CDS notional outstand-
ing has been decreasing in every subsequent year since then, however, this decrease is at-
tributable to portfolio compression, which is a process designed to terminate existing trades
and replace them with a smaller number of trades with substantially smaller notionals that
carry the same risk profile as the initial portfolio. In doing so, portfolio compression reduces
the overall notional size and number of outstanding CDS contracts, and thereby improving
derivatives risk management. ISDA reported 25.1 trillion dollars notional at year end 2012.
Though notionals outstanding is declining, market risk transaction activity which measures
trading volume during a period of time is thought to be a better way to look at the relative dy-
namics of the CDS market. A review of such activity shows that CDS transaction volumes as
5An alternative error minimisation function is, mi nXNc
i=1 wi(C(Xi|θ)−CM(Xi|θ))2+ Nc+Np X i=Nc+1 wi(P(Xi|θ)− PM(Xi|θ))2, the weighted squared error loss function introduced by Bliss and Panigirtzoglou (2002). This loss function is claimed particularly appropriate when the source of option price measurement error predomi- nantly reflects the discreteness of option prices. Bliss and Panigirtzoglou (2002), Liu et al. (2007), and Taylor et al. (2014) applied equal weighting of squared errors to obtain parameters estimates for the mixture of log- normal risk-neutral density. This loss function may provide smallerGvalues and quicker convergence. The convergence of optimisation routine using equation 2.5 performs reasonably well in most estimates through- out our sample. Testing the robustness of the loss function is beyond the scope of this study, thus minimising the sum of the squared errors is retained in this study.
6See Markit report, The CDS Big Bang: Understanding the Changes to the Global CDS Contract and North
measured by notionals increased through 2011 to 2013, and the number of trades executed
increase in 2013 after a slightly fall in 2012. Given the nature of a credit default swap, which
is a contract bought by the protection buyer that protects against default of a certain refer-
ence entity, and the expanding size and engagement activity in the market, it is reasonable
to assume that we can derive a measure of probability of default for a particular reference
entity from CDS market data, and compare it with the estimate obtained from option data.
In a credit default swap, the protection buyer pays a series of periodic fixed payments
known as premium to the protection seller to protect against default of a reference entity.
In return, the protection seller does nothing unless the reference entity gets into financial
difficulty. In that case, the protection seller will be obliged to buy back from the protection
seller the defaulted bond at its face value. In a cash settlement case, the protection seller
will pay the difference between the face value of the defaulted bond and the current value
determined by an auction, to the protection buyer. The percentage of the value of the bond
after default to its face value is known as the recovery rate.
Following the methodology adopted by Bharath and Shumway (2008), letS be the CDS
spread, which is the amount paid per year as a percentage of the notional principal. Notional
amount is assumed 1 dollar. Let T determine the life of the CDS contract. CDS contracts
have maturities range from 6 month, 1, 2, 3...10 years, up to 20, 30 years. 5 year contract is
the most liquid one, and I also examine 1 year contract in this study. Assume the premium
payments are made once every three months, e.g. March, June, September, and December.
Assume the probability of a reference entity defaulting between two consecutive payment
days conditional on no prior default isp, and for simplicity, I assume the default only hap-
pens halfway through the three months. In the event of default, a final accrual payment is
required by the protection buyer, which is equal to 0.5∗0.25S. The risk-free rate is r and
The present value of the expected premium payments made by the protection buyer is then, T
X
t=0.25 h p(1−p)4t−1e−r(t−18)0.5∗0.25S+(1−p)4te−r t0.25S i ,wheret=0.25, 0.5, 0.75, 1 for 1 year contract, andt=0.25, 0.5, 0.75, 1, ..., 5 for 5 year contract.
The first term represents the present value of the expected accrual payments in the event
of a default assuming it happens midway between two consecutive premium payments. The
second term represents the present value of the expected periodic premium payments made
at the end of each period provided that the reference entity survives untilt.
Similarly, the present value of the expected payoff is given by
T
X
t=0.25
p(1−p)4t−1(1−δ)e−r(t−18).
An estimate of the recovery rateδis needed in order to value the payoff. As of The CDS Big Bang in 2009, the new CDS contract has been standardized in a number of ways. The CDS
contracts are grouped based on geological regions, such as North America, Europe, Asia, etc.,
and classed into different types, such as standard corporate, index, sovereign, etc. Different
classes of contracts have different estimates of recovery rates as input into models to price
CDS spreads. As of my sample Standard North American Corporate Senior, I use the market
consensus estimate 40% as recovery rate in the study.
Setting the present value of the expected premium payments equal to the present value
of the expected payoff, so that the value of the contract is zero when both parties enter the
contract, we can solve for the probability of default p from the resulting non-linear equa-
tion. Since the calculations assume no risk-premium associated with default, the resulting
implied probability should be considered a risk-neutral measure. Note thatp is assumed
in reality the instantaneous jump to default rate may change at different point in time based
on changes in the firm’s fundamentals and market conditions. The default probability in this
model represents the average default rate aggregated over the life of the CDS contract.