Credit default swaps

This page follows QuantLib's CDS example, its first two cases: a hazard curve bootstrapped from quoted spreads, then ISDA's standard model. Its values match that example's output. The sample, CreditDefaultSwaps.cs, carries QuantLib's notice.

A hazard curve from spreads

Four CDS quoted at 150 bp, from three months to two years, with a recovery of 50% and a flat 1% discount curve. A SpreadCdsHelper per quote, and a piecewise curve of flat hazard rates bootstrapped through them:

Calendar calendar = new TARGET();
Date today = calendar.Adjust(new Date(15, Month.May, 2007));
var context = new EvaluationContext { EvaluationDate = today };
var discount = new FlatForward(today, 0.01, new Actual365Fixed());

int settlementDays = 1;
double recoveryRate = 0.5;
(Period Tenor, double Spread)[] quotes =
[
    (new Period(3, TimeUnit.Months), 0.0150), (new Period(6, TimeUnit.Months), 0.0150),
    (new Period(1, TimeUnit.Years), 0.0150), (new Period(2, TimeUnit.Years), 0.0150),
];

var helpers = new List<BootstrapHelper<DefaultProbabilityTermStructure>>();
foreach (var (tenor, spread) in quotes)
{
    helpers.Add(new SpreadCdsHelper(spread, tenor, settlementDays, calendar, Frequency.Quarterly,
        BusinessDayConvention.Following, DateGenerationRule.TwentiethIMM, new Actual365Fixed(), recoveryRate,
        discount, context));
}

var curve = new PiecewiseDefaultCurve<HazardRate, BackwardFlat, IterativeBootstrap>(
    today, helpers, new Actual365Fixed(), context);

The curve's nodes and its survival probabilities:

Node	Hazard rate
2007-05-15	0.0299685
2007-09-20	0.0299685
2007-12-20	0.0299613
2008-06-20	0.0299619
2009-06-22	0.0299607

Horizon	Survival probability
1Y	97.040077%
2Y	94.175796%

Each quoted CDS, priced on the curve by the mid-point engine:

Calendar calendar = new TARGET();
Date settlement = calendar.Advance(h.Today, 1, TimeUnit.Days);
foreach (var (tenor, spread) in h.Spreads)
{
    Date maturity = calendar.Adjust(settlement + tenor, BusinessDayConvention.Following);
    Schedule schedule = new MakeSchedule().From(settlement).To(maturity).WithFrequency(Frequency.Quarterly)
        .WithCalendar(calendar).WithTerminationDateConvention(BusinessDayConvention.Unadjusted)
        .WithRule(DateGenerationRule.TwentiethIMM).ToSchedule();
    var cds = new CreditDefaultSwap(ProtectionSide.Seller, 1e6, spread, schedule, BusinessDayConvention.Following,
        new Actual365Fixed());
    CreditDefaultSwapResults result = MidPointCdsEngine.Calculate(cds, h.Context, h.Curve, 0.5, h.Discount);
    results.Add((Name(tenor), result));
}

Quote	Fair spread	Default leg	Coupon leg
3M	1.500000%	-5177.05	5177.05
6M	1.500000%	-8841.72	8841.72
1Y	1.500000%	-16101.8	16101.8
2Y	1.500000%	-30154.5	30154.5

Each is worth nothing at its quoted spread, the bootstrap's condition, and its default and coupon legs offset.

ISDA's standard model

ISDA's rate curve from deposits and swaps, a flat hazard rate, and a reference trade, five years of protection bought on 100 million at 100 bp, priced by IsdaCdsEngine:

// QuantLib's example sets IborCoupon::Settings::createAtParCoupons(), the context's UseAtParCoupons.
var context = new EvaluationContext { EvaluationDate = new Date(21, Month.October, 2014), UseAtParCoupons = true };
var helpers = new List<BootstrapHelper<YieldTermStructure>>();
foreach (var (months, rate) in new[]
         {
             (1, 0.000060), (2, 0.000450), (3, 0.000810), (6, 0.001840), (9, 0.002560), (12, 0.003370),
         })
{
    helpers.Add(new DepositRateHelper(rate, new Period(months, TimeUnit.Months), 2, new TARGET(),
        BusinessDayConvention.ModifiedFollowing, false, new Actual360(), context));
}

var euribor6M = new Euribor(new Period(6, TimeUnit.Months));
foreach (var (years, rate) in new[]
         {
             (2, 0.002230), (3, 0.002760), (4, 0.003530), (5, 0.004520), (6, 0.005720), (7, 0.007050),
             (8, 0.008420), (9, 0.009720), (10, 0.010900), (12, 0.012870), (15, 0.014970), (20, 0.017000),
             (30, 0.018210),
         })
{
    helpers.Add(new SwapRateHelper(rate, new Period(years, TimeUnit.Years), new TARGET(), Frequency.Annual,
        BusinessDayConvention.ModifiedFollowing, new Thirty360(Thirty360Convention.BondBasis), euribor6M, context));
}

var rateCurve = PiecewiseYieldCurve<Discount, LogLinear, IterativeBootstrap>.Create(
    0, new WeekendsOnly(), helpers, new Actual365Fixed(), context);

FlatHazardRate defaultCurve = FlatHazardRate.Create(0, new WeekendsOnly(), 0.016739207493630, new Actual365Fixed(),
    context);
var schedule = new Schedule(new Date(22, Month.September, 2014), new Date(20, Month.December, 2019),
    new Period(3, TimeUnit.Months), new WeekendsOnly(), BusinessDayConvention.Following,
    BusinessDayConvention.Unadjusted, DateGenerationRule.CDS, false, context);
var trade = new CreditDefaultSwap(ProtectionSide.Buyer, 100e6, 0.01, schedule, BusinessDayConvention.Following,
    new Actual360(), true, true, new Date(22, Month.October, 2014), null, new Actual360(true), true);
CreditDefaultSwapResults result = IsdaCdsEngine.Calculate(trade, context, defaultCurve, 0.4, rateCurve, false,
    IsdaCdsEngineNumericalFix.Taylor, IsdaCdsEngineAccrualBias.NoBias, IsdaCdsEngineForwardsInCouponPeriod.Piecewise);

The rate curve at its pillars:

Pillar	Zero rate	Discount
2014-11-24	6.08332e-05	0.999994
2014-12-23	0.00044368	0.999923
2015-01-23	0.000804988	0.999793
2015-04-23	0.00184508	0.99907
2015-07-23	0.00257462	0.998062
2015-10-23	0.00339272	0.996594
2016-10-24	0.00221731	0.995551
2017-10-23	0.00274915	0.991764
2018-10-23	0.00352107	0.985986
2019-10-23	0.00451599	0.977637
2020-10-23	0.0057249	0.966173
2021-10-25	0.00707627	0.951562
2022-10-24	0.00847959	0.934305
2023-10-23	0.009823	0.91529
2024-10-23	0.0110503	0.895248
2026-10-23	0.0131302	0.85407
2029-10-23	0.0153845	0.793724
2034-10-23	0.0176039	0.702988
2044-10-24	0.0188487	0.567776

Reference trade	Value
First period's accrual on 24 October 2014	88888.9
NPV	-40991.9

Table of Contents

Credit default swaps

A hazard curve from spreads

ISDA's standard model