Converting from Running to Upfront

A CDS market quote is given in terms of a standard spread (usually 100 bp or 500 bp) and an upfront payment, or in terms of an equivalent running or breakeven spread, with no upfront payment. The functions cdsbootstrap, cdsspread, and cdsprice perform upfront to running or running to upfront conversions.

For example, to convert the market quotes to upfront quotes for a standard spread of 100 bp:

Settle = '17-Jul-2009';  % valuation date for the CDS
MarketDates = datenum({'20-Sep-10','20-Sep-11','20-Sep-12','20-Sep-14',...
'20-Sep-16'});
MarketSpreads = [140 175 210 265 310]';
MarketData = [MarketDates MarketSpreads];

ZeroDates = datenum({'17-Jan-10','17-Jul-10','17-Jul-11','17-Jul-12',...
'17-Jul-13','17-Jul-14'});
ZeroRates = [1.35 1.43 1.9 2.47 2.936 3.311]'/100;
ZeroData = [ZeroDates ZeroRates];

[ProbData,HazData] = cdsbootstrap(ZeroData,MarketData,Settle);

Maturity3 = MarketData(:,1);
Spread3Run = MarketData(:,2);
Spread3Std = 100*ones(size(Maturity3));
Price3 = cdsprice(ZeroData,ProbData,Settle,Maturity3,Spread3Std);
Upfront3 = Price3/10000000; % Standard notional of 10MM
display(Upfront3);

This resulting value is:

Upfront3 =

    0.0047
    0.0158
    0.0327
    0.0737
    0.1182

The conversion can be reversed to convert upfront quotes to market quotes:

ProbData3Upf = cdsbootstrap(ZeroData,[Maturity3 Upfront3 Spread3Std],Settle);
Spread3RunFromUpf = cdsspread(ZeroData,ProbData3Upf,Settle,Maturity3);
display([Spread3Run Spread3RunFromUpf]);

Comparing the results of this conversion to the original market spread demonstrates the reversal:

ans =

  140.0000  140.0000
  175.0000  175.0000
  210.0000  210.0000
  265.0000  265.0000
  310.0000  310.0000

Under the flat-hazard rate (FHR) quoting convention, a single market quote is used to calibrate a probability curve. This convention yields a single point in the probability curve, and a single hazard rate value. For example, assume a four-year (standard dates) CDS contract with a current FHR-based running spread of 550 bp needs conversion to a CDS contract with a standard spread of 500 bp:

Maturity4 = datenum('20-Sep-13');
Spread4Run = 550;
ProbData4Run = cdsbootstrap(ZeroData,[Maturity4 Spread4Run],Settle);
Spread4Std = 500;
Price4 = cdsprice(ZeroData,ProbData4Run,Settle,Maturity4,Spread4Std);
Upfront4 = Price4/10000000;
fprintf('A running spread of %5.2f is equivalent to\n',Spread4Run);
fprintf('   a standard spread of %5.2f with an upfront of %8.7f\n',...
   Spread4Std,Upfront4);

A running spread of 550.00 is equivalent to
   a standard spread of 500.00 with an upfront of 0.0167576

To reverse the conversion:

ProbData4Upf = cdsbootstrap(ZeroData,[Maturity4 Upfront4 Spread4Std],Settle);
Spread4RunFromUpf = cdsspread(ZeroData,ProbData4Upf,Settle,Maturity4);
fprintf('A standard spread of %5.2f with an upfront of %8.7f\n',...
   Spread4Std,Upfront4);
fprintf('    is equivalent to a running spread of %5.2f\n',Spread4RunFromUpf);

A standard spread of 500.00 with an upfront of 0.0167576
    is equivalent to a running spread of 550.00

As discussed in Beumee et. al., 2009 (see Credit Derivatives), the FHR approach is a quoting convention only, and leads to quotes inconsistent with market data. For example, calculating the upfront for the three-year (standard dates) CDS contract with a standard spread of 100 bp using the FHR approach and comparing the results to the upfront amounts previously calculated, demonstrates that the FHR-based approach yields a different upfront amount:

Maturity5 = MarketData(3,1);
Spread5Run = MarketData(3,2);
ProbData5Run = cdsbootstrap(ZeroData,[Maturity5 Spread5Run],Settle);
Spread5Std = 100;
Price5 = cdsprice(ZeroData,ProbData5Run,Settle,Maturity5,Spread5Std);
Upfront5 = Price5/10000000;
fprintf('Relative error of FHR-based upfront amount: %3.1f%%\n',...
   ((Upfront5-Upfront3(3))/Upfront3(3))*100);

Relative error of FHR-based upfront amount: -0.8%