Financial Engineering
David Oakes
–
IC302
4. Credit Derivatives
Autumn Term 2020/1
Credit Default Swaps
2
Credit Default Swap
• Credit default swap (CDS): a derivative security in which one party makes a payoff to the other when a specified reference entity suffers a credit event.
• Reference entity may be a corporate or sovereign entity (single-name CDS) or a credit index (index CDS).
• Payoff is compensation for credit losses related to the credit event.
• CDS market is a market for protection from credit events.
3
Protection Buyer and Seller
• Protection buyer makes periodic payments (CDS premium) to protection seller over life of contract or until credit event occurs.
• Protection seller compensates buyer if a credit event occurs.
• Credit event payment is based on the recovery rate of a specified reference obligation.
4
CDS Cash Flows
Prior to Credit Event
CDS Premium
Protection Buyer
Following Credit Event
CDS Premium Stops
Credit Event Payment (1 – Recovery Rate)
Protection Seller
Protection Buyer
Protection Seller
5
CDS Par Spread
• CDS premium is the price of protection against credit events that might affect the reference entity.
• The full annualized market value of this protection is the par spread (measured in basis points per year).
• What will happen to the par spread if the credit quality of the reference entity weakens? If it strengthens?
• If you buy protection on a reference entity and its credit quality weakens, will you make a MTM profit or loss?
6
CDS and Credit Risk
• The CDS par spread is a credit spread.
• If the credit quality of the reference entity improves, a credit event is less likely, so the credit event payment is less likely to be made. The CDS spread will tighten.
• This will result in a MTM profit for the protection seller and a MTM loss for the protection buyer.
• If the credit quality of the reference entity gets worse, the CDS spread will widen. The protection buyer makes a MTM profit and the protection seller makes a MTM loss.
7
CDS and Credit Exposure
• The protection seller does better if the credit risk of the reference entity improves and worse if it deteriorates.
• Protection seller has long exposure to credit risk of reference entity (like being long a credit-risky bond).
• The protection buyer does better if the credit risk of the reference entity deteriorates and worse if it improves.
• Protection buyer has short exposure to credit risk of reference entity (like being short a credit-risky bond).
8
How Market Participants Use CDS
• Credit trading (alternative trading venue for credit risk)
• Hedging credit risk in corporate and sovereign bonds
• Hedging credit risk in loans held in the banking book
• CVA/XVA pricing and hedging (counterparty risk management)
9
CDS Documentation
10
CDS Documentation
• CDS are traded under ISDA® master agreements and documentation that establish certain market conventions for CDS cash flows, including:
– Standard maturity and coupon dates – Fixed coupons
– Full first coupon
• Key ISDA® documents:
– 2014 ISDA® Credit Derivatives Definitions
– 2009 ‘Big Bang’ and ‘Small Bang’ protocols – 2003 ISDA® Credit Derivatives Definitions
– 1999 ISDA® Credit Derivatives Definitions
11
Standard Maturity Dates
• CDS trade with standard maturity dates of 20 June and 20 December. All CDS now mature on one of these dates.
• The dates on which trading moves to the next standard maturity date are called roll dates.
• Trading moves to the 20 June maturity date on 20 March and to the 20 December maturity date on 20 September.
12
Standard Coupon Dates
• In the North American and European markets, CDS coupons are paid on a quarterly basis.
• Standard coupon payment dates are 20 March, 20 June, 20 September and 20 December.
• If the coupon payment date is a holiday, the payment is made on the following good business day (the same convention applies to maturity dates).
• Coupon payments are calculated on an Actual/360 basis. 13
Full First Coupon
• A full first coupon is paid on all CDS transactions.
• If we sell protection on 14 October 2020, we will receive a full first coupon on 20 December 2020, calculated on an Actual/360 basis from 20 September 2020.
• We pay the protection buyer accrued interest from the previous coupon date (20 September 2020) to the trade date (14 October 2020).
• This is analogous to accrued interest in the bond market. 14
Fixed Coupons
• The par spread is the annual CDS premium that is equal in value to the protection purchased.
• It is the fair value of the CDS premium.
• Under ISDA® documentation, however, the market convention is for the CDS premium to be paid at a standard annual fixed coupon rate (e.g. 100 bp or 500 bp).
15
Upfront Payments
• Since the fixed coupon will not usually equal the par spread, the protection buyer will typically be either overpaying or underpaying for protection.
• An upfront payment is therefore exchanged between the buyer and seller to ensure that the CDS is fairly valued.
16
Fixed Coupons and Upfront Payments
17
CDS Cash Flows (Par Spread)
• Suppose that we sell 5-year protection on thyssenkrupp AG at 350 bp per year. If the CDS premium were equal to
the par spread and paid in equal quarterly instalments, the cash flows in the CDS would be as shown below:
Sell 5-year protection at 350 bp (no credit event over CDS term)
Quarterly payments of 87.5 bp
Sell 5-year protection at 350 bp
(credit event during Q4 of second year)
Quarterly payments of 87.5 bp
Accrued coupon to credit event date
12345 12345 Credit event payment
18
CDS Cash Flows (Fixed Coupon)
•
If instead we sell 5-year protection on thyssenkrupp AG at a par spread of 350 bp per year, but with a contractual
fixed coupon of either 100 bp or 500 bp per year. The scheduled cash flows in the CDS will be as follows:
Par Spread 350bp
Fixed Coupon 100 bp
Receive upfront payment of 11.25% Quarterly payments of 25 bp
1 2 3 4 5
Fixed Coupon 500 bp
Quarterly payments of 125 bp
12345 Make upfront payment of 6.75%
Quarterly payments of 87.5 bp
12345
Upfront payments are based on assumed risky PV01 of 4.5 for 5-year CDS (see below)
19
Calculating Upfront Payments
• The upfront payments compensate for underpayment or overpayment for protection due to the difference between the contractual fixed coupon and the par spread.
• Size of upfront payment depends on:
– Difference between par spread and fixed coupon
– Risky present value a basis point (risky PV01) for the CDS
• Risky PV01 is present value of 1 bp per year in spread over the remaining life of the CDS based on the survival probabilities of the entity to the coupon payment dates.
20
Buy 5-year senior CDS on reference entity Apple Inc.
Source: Refinitiv Eikon
21
Source: Refinitiv Eikon
22
Fixed coupon 100 bp
Par spread 29.6754 bp
Upfront payment -3.6196% (i.e. protection buyer receives
3.6196% of the notional amount up front from the protection seller)
Source: Refinitiv Eikon
23
Upfront payment (including accrued interest) for notional amount USD 1 million
Source: Refinitiv Eikon
24
Credit Events and CDS Settlement
25
Reference Entity and Obligation
• Reference entity: underlying credit or ‘name’ on which protection is bought or sold (corporate or sovereign).
• Reference entity for index CDS is a portfolio of names.
• Ensure that reference entity is the exact name you wish to
trade (check reference entity database Markit RED).
• Reference obligation: indicates seniority of credit risk
transferred. This affects expected recovery, hence price.
26
Credit Events
• Credit event: event that, if it occurs, will trigger a credit event payment by the protection seller in a CDS.
• Typical credit events for corporate entities: – Bankruptcy
– Failuretopay
– Restructuring(dependingonreferenceentityandwheretraded)
• Typical credit events for sovereign entities: – Failuretopay
– Repudiationormoratorium – Restructuring
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Determinations Committees
• It may not always be clear whether the circumstances affecting a reference entity constitute a credit event.
• ISDA® regional determinations committees decide whether a credit event has occurred.
• Determinations made by these committees are binding on all market participants.
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Cash or Physical Settlement
• CDS may be cash settled or physically settled.
• Cash settlement: protection seller makes cash payment to protection buyer based on post-credit-event value of the cheapest-to-deliver deliverable obligation.
• An auction process is used to establish this value.
• Physical settlement: protection buyer delivers a deliverable obligation to the protection seller in exchange for the CDS notional amount.
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Cash Settlement
• Most CDS are cash settled following a credit event. Following Credit Event
CDS Premium Stops
Credit Event Payment (1 – Recovery Rate)
• Recovery rate is the post-credit-event auction price. For Lehman in 2008, this was 8.625 per 100 nominal, so credit event payment was 91.375 per 100 notional.
• Protection buyer stops paying CDS premium after a credit event but must pay any accrued premium up to the date on which the credit event is deemed to have occurred.
Protection Buyer
Protection Seller
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Credit Trading with CDS
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CDS Market
• CDS dealers quote bid and offer prices for protection.
• Period over which protection is offered is tenor of CDS.
• The most liquid market in any name is typically 5 years,
and most credit trading with CDS is done in this tenor.
• Dealers may quote prices for other tenors in liquid names.
• There is less liquidity in single-name CDS now than before
the financial crisis, but good liquidity in main index CDS.
32
Source: Refinitiv Eikon
33
Credit Trading with CDS
• Outright trades on individual entity or ‘name’
• Relative value trades between entities
• Curve trades (trade in different tenors to bet that curve will steepen or flatten or to create forward exposure)
• Default swap basis trades (relative value trade between bond and CDS of same entity)
34
Outright Trades
• Suppose that you think that the credit quality of Apple Inc. will improve, and you decide to trade this view using CDS.
• Should you buy or sell protection on Apple Inc.?
• Suppose that 5-year CDS on Apple Inc. is trading at a par spread of 30 bp when you enter the trade. If the spread moves to 20 bp, will you make a profit or loss?
35
Relative Value Trades
• CDS can also be used to express views about changes in the relative credit quality of different entities.
• Suppose that you think Apple Inc. is likely to deteriorate in credit quality relative to Samsung Electronics Co. Ltd.
• How could you trade this view using 5-year CDS on the two names? On which entity would you buy protection and on which entity would you sell protection?
36
Relative Value Trades
• Suppose that the 5-year par CDS spreads when you enter the trade are 30 bp for Apple and 22 bp for Samsung.
• If the CDS spreads move to 40 bp for Apple and 25 bp for Samsung, will your trade make a profit or a loss?
• What overall exposure would this trade have to credit risk in the consumer electronics sector?
37
Curve Trades
• Suppose that you think that the 1-year CDS spread on Apple is likely to increase relative to the 5-year spread.
• How could you trade this view using 1-year and 5–year CDS on Apple Inc.?
• Would you trade the same notional in both tenors?
• Why might the credit curve change in this way?
38
Default Swap Basis Trades
• The CDS spread is a kind of credit spread, and the CDS market is an alternative venue for trading credit risk.
• CDS spread for a given entity will be related to credit spreads on bonds issued by that entity through hedging activity of CDS dealers (who use asset swaps to hedge unmatched credit risk on their dealing book) and relative value trades by other market participants.
• This is another example of how hedging costs affect derivatives prices and of how we can create synthetic versions of various financial instruments.
39
Asset Swap as Hedge for CDS
• CDS dealer buys protection on reference entity. Dealer is short the credit risk of the reference entity.
CDS Premium
Credit Event Payment
CDS Dealer
Protection Seller
40
Asset Swap as Hedge for CDS
• Dealer goes long the credit risk of the reference entity going long a bond issued by it.
CDS Premium
Credit Event Payment
Funding (Repo) Rate Fixed Coupon
CDS Dealer
Protection Seller
Repo Dealer (Funding)
Obligation of Reference Entity
41
Asset Swap as Hedge for CDS
• Buying the bond in an asset swap structure in which the dealer pays fixed in an interest rate swap hedges the interest rate risk but leaves the dealer long the credit risk .
Floating
Fixed
CDS Premium
Credit Event Payment
Fixed Coupon
Interest Rate Swap
CDS Dealer
Protection Seller
Funding (Repo) Rate
Repo Dealer (Funding)
Obligation of Reference Entity
42
Asset Swap as Hedge for CDS
• The asset swap acts as a hedge for the CDS position.
CDS Premium
Credit Event Payment
CDS Dealer
Protection Seller
Floating
Fixed
Interest Rate Swap
Funding (Repo) Rate
Fixed Coupon
CDS
Asset Swap (Hedge for CDS)
Repo Dealer (Funding)
Obligation of Reference Entity
43
Asset Swap as Hedge for CDS
• CDS premium the dealer can pay in the CDS will be related to the asset swap margin she earns on the bond.
CDS Premium
Credit Event Payment
CDS Dealer
Protection Seller
Floating
Fixed
Interest Rate Swap
Funding (Repo) Rate
Fixed Coupon
CDS
Asset Swap (Hedge for CDS)
Repo Dealer (Funding)
Obligation of Reference Entity
44
Default Swap Basis Trades
• The default swap basis is the CDS premium minus the asset swap margin:
𝐷𝑒𝑓𝑎𝑢𝑙𝑡 𝑆𝑤𝑎𝑝 𝐵𝑎𝑠𝑖𝑠 = 𝐶𝐷𝑆 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 − 𝐴𝑠𝑠𝑒𝑡 𝑆𝑤𝑎𝑝 𝑀𝑎𝑟𝑔𝑖𝑛
• A significantly positive or negative default swap basis (or one that is trading away from its usual value) may represent a relative value opportunity.
• Investors can profit by trading the bond against the CDS (e.g. by going long the bond and buying protection when the default swap basis is negative).
45
CDS Pricing and Revaluation
Credit Risk and Credit Spreads
46
Credit Risk
• Credit risk is the risk that a promised future payment will not be made.
• Credit risk depends on three factors:
– Exposure at default (EAD)
– Loss given default (LGD) (equivalently, recovery rate R) – Default probability
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Expected Loss and Credit Spreads
• Borrowers must compensate lenders for expected loss due to default.
• This compensation is the source of the risk premium or credit spread on credit risky debt.
• It is also the key factor driving CDS premiums.
48
Credit Spread
• Investors demand a credit spread (enhanced yield) in exchange for accepting credit risk.
• Credit spread is measured relative to a specific benchmark.
• Benchmark may be government bonds or interest rate swaps.
Swap rate
Benchmark yield
Credit spread to swaps
Credit spread over risk- free rate
Company specific default risk
Market risk premium
Swap spread
• Measured credit spread includes market risk premium driven in part by relative liquidity, funding and other issues not related to default.
Government bond yield
49
Credit Spread Measures
• Nominal spread: bond yield minus benchmark government bond yield or swap rate of the same maturity
• Asset swap margin: spread to floating rate that can be earned by buying the bond and hedging the interest rate risk by paying fixed in a swap
• Z-spread: spread that must be added to the benchmark zero-coupon yield curve to give a set of discount rates that reproduce the bond’s market price
50
Credit Spread Measures
• Option-adjusted spread (OAS): calculated like the Z- spread, but after taking into account the effect of embedded optionality on the bond’s cash flows
51
Apple Inc 2.9% of September 2027
Source: Refinitiv Eikon
52
Nominal Spreads
Asset Swap Margin
Source: Refinitiv Eikon
Z-spread OAS
53
Credit Risk of Specific Obligations
• Default probabilities are associated with entities (e.g. companies or other individual obligors).
• Recovery rates are characteristics of specific obligations (so that different obligations of the same entity may have different recovery rates).
• Credit analysis of fixed income securities must therefore take into account the seniority and security of the obligation, since these affect the expected recovery rate.
54
Seniority and Security
•
Reorganization or liquidation following bankruptcy is governed by a rule of priority with respect to security holders and other creditors.
Bank Loans Senior Bonds
Subordinated Bonds Equity
Secured creditors have a prior claim over pledged collateral.
Unsecured creditors are repaid in order of seniority.
Equity holders get anything that remains after all other creditors have been repaid.
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Seniority and CDS Prices
• Senior claims (e.g. bank loans or senior bonds ) are repaid
before subordinated claims (e.g. subordinated bonds).
• Senior claims have a higher expected recovery rate, so CDS for which the reference obligation is a senior bond will trade at a lower price than CDS on the same entity for which the reference obligation is a subordinated bond.
56
CDS Pricing and Revaluation
Basic Concepts
57
Real-World Default Probabilities
• Real-world default probabilities are the actual probabilities that govern the defaults we experience.
• They can be estimated from historical data on defaults or by using models based other relevant information.
• Real-world probabilities are relevant for managing credit portfolio risk and for pricing risk we cannot fully hedge.
• Together with expected recovery rates, real-world default probabilities drive the prices of credit risky bonds.
58
Risk-Neutral Valuation
• Derivatives prices (e.g. futures, swaps, options) are to a
considerable extent determined by hedging costs.
• Market participants with the same hedging opportunities, will agree on the price at which a derivative should trade, regardless of their individual risk preferences.
• We can therefore calculate derivatives prices by imagining that investors have any arbitrary set of risk preferences.
59
Risk-Neutral Valuation
• Risk-neutral investors would discount expected future cash flows on all assets at the risk-free interest rate.
• So we can calculate derivatives prices by assuming that investors are risk-neutral and discounting expected cash flows under this assumption at the risk-free interest rate.
• This is called risk-neutral valuation.
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Risk-Neutral Default Probabilities
• In a world of risk-neutral investors, expected cash flows would not be the same as in the real world, since the expected return on all assets would be the risk-free rate.
• We must also therefore adjust the probabilities associated with possible future payoffs.
• For credit derivatives, this means working in terms of risk- neutral default probabilities.
61
Risk-Neutral Default Probabilities
• Risk-neutral default probabilities are market-implied values derived from observed credit spreads.
• They are the default probabilities that would reproduce observed credit spreads if investors discounted the expected value of risky cash flows at risk-free rates.
• They are not estimates of actual default probabilities.
• They are a calibration tool used to analyze credit derivatives such as CDS using risk-neutral valuation.
62
A Simple Example
• Suppose that a one-year bond with face value 100 and no default risk trades at 90.91:
• At the same time, we observe a one-year credit risky bond with face value 100 that trades at 80. Suppose that the expected recovery rate in event of default is 40.
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Implied Default Probability
• The implied risk-neutral default probability is the probability that reproduces the price of 80 when the expected cash flows are discounted at the risk-free rate:
𝑃𝑟𝑖𝑐𝑒 = 𝑅𝑖𝑠𝑘 𝑁𝑒𝑢𝑡𝑟𝑎𝑙 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑃𝑎𝑦𝑜𝑓𝑓 1+𝑟
1 − 𝑞 × 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝑃𝑎𝑦𝑜𝑓𝑓 + 𝑞 × 𝑅 × 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝑃𝑎𝑦𝑜𝑓𝑓 1+𝑟
= 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝑃𝑎𝑦𝑜𝑓𝑓 − 𝑞 × 1 − 𝑅 × 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝑃𝑎𝑦𝑜𝑓𝑓 1+𝑟
𝑤h𝑒𝑟𝑒 𝑟 = 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
𝑅 = 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑟𝑎𝑡𝑒 𝑖𝑛 𝑒𝑣𝑒𝑛𝑡 𝑜𝑓 𝑑𝑒𝑓𝑎𝑢𝑙𝑡
𝑞 = 𝑟𝑖𝑠𝑘 𝑛𝑒𝑢𝑡𝑟𝑎𝑙 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦
=
64
Implied Default Probability
• Rearranging this gives the risk-neutral default probability:
𝑞=𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑𝑃𝑎𝑦𝑜𝑓𝑓−𝑃𝑟𝑖𝑐𝑒× 1+𝑟 1 − 𝑅 × 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝑃𝑎𝑦𝑜𝑓𝑓
• For our example:
𝑞=𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑𝑃𝑎𝑦𝑜𝑓𝑓−𝑃𝑟𝑖𝑐𝑒× 1+𝑟 =100−80 1+0.10 =0.2
1 − 𝑅 × 𝑃𝑟𝑜𝑚𝑖𝑠𝑒𝑑 𝑃𝑎𝑦𝑜𝑓𝑓 1 − 0.40 100
• This is the implied risk-neutral default probability consistent with the market price of the credit risky bond.
65
Implied Default Probability
• The implied risk-neutral default probability may be very different from the real-world default probability.
• Changing the recovery rate or risk-free interest rate will result in a different risk-neutral default probability.
66
Pricing CDS
• Now suppose that we buy protection in a one-year CDS for which the reference obligation is the credit risky bond.
• What CDS premium (i.e. par spread) will we pay?
67
Pricing CDS: Protection Leg
• Protection leg of CDS has two possible future payoffs.
• Using risk-neutral valuation, its present value is 10.91:
Expected payoff to protection buyer
= 0(1-q)+100(1-R)(q)
=0(0.8)+100(1-0.4)(0.2)=12
Prob = 0.8
0
Next Year (No Default)
60
10.91 Today
Risk-free rate r = 10%
PV of expected payoff to protection buyer
Prob = 0.2
Rate 40%)
= 12 =12=10.91 1+r 1.1
Next Year (Default with Recovery
• Note that we have used the risk-neutral default probability implied by the price of the credit risky bond.
68
Pricing CDS: Premium Leg
• For the CDS to be fairly priced, the premium leg (i.e. the CDS premium payments we make) must have the same present value as the protection leg.
• If the premium is paid in a single instalment one year from now, it should be 12 (i.e. 1200 bp), since this will have present value 10.91 when discounted at the risk-free rate.
10.91 = CDS = CDS 1+r 1.1
Þ CDS = 10.91(1.1) = 12.00 (i.e. 1200 bp)
10.91
Today
12.00
Next Year
• We should expect to pay a CDS premium of 1200 bp.
69
CDS Pricing in Practice
• We priced the CDS using a risk-neutral default probability implied by the market price of a credit risky bond.
• This is sometimes done to calculate ’fair value’ prices.
• More typically, we take market CDS prices as given and
look for trading or hedging opportunities based on them.
• We use risk-neutral valuation and implied default probabilities as tools to analyze risk, relative value and trading opportunities in the CDS market.
70
Credit Triangle
• Credit triangle summarizes relationship between credit spread, expected recovery rate, and default probability:
𝐶𝑟𝑒𝑑𝑖𝑡𝑆𝑝𝑟𝑒𝑎𝑑=𝐷𝑒𝑓𝑎𝑢𝑙𝑡𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦× 1−𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦𝑅𝑎𝑡𝑒
• This approximation works well in many situations (see Hull, ch. 24) and can be a helpful intuitive guide to factors that affect credit spreads in bonds, loans, and CDS.
• Notice that our CDS example satisfies the credit triangle: 𝐶𝐷𝑆𝑆𝑝𝑟𝑒𝑎𝑑=𝑞× 1−𝑅 =0.2 1−0.4 =0.12 (𝑖.𝑒.1200𝑏𝑝)
71
CDS Pricing and Revaluation
Risk-Neutral Valuation Model
72
Term Structure of Default
• Credit risk depends on credit quality of the obligor over the entire life of the relevant transaction.
• Probability of default may vary with horizon.
• There is a term structure of default.
73
Cumulative Default Probability
• Cumulative default probability 𝐹 𝑡 is probability that
default occurs sometime between now and date 𝑡.
• Probability that default occurs between two dates 𝒕𝟏 and 𝒕𝟐 (asviewedfromtoday)is𝑞 𝑡Q,𝑡R =𝐹 𝑡R −𝐹 𝑡Q .
74
Survival Probability
• Survival probability is probability 𝑆 𝑡 = 1 − 𝐹 𝑡 that
the entity will not yet have defaulted by a future date 𝑡.
• Consider an entity with 𝐹 𝑡Q = 1% and 𝐹 𝑡R = 1.99%.
• Probability that the entity defaults between dates 𝑡Q and 𝑡R (asviewedfromtoday)is𝑞 𝑡Q,𝑡R =𝐹 𝑡R −𝐹 𝑡Q = 1.99% − 1% = 0.99%.
• Survivalprobabilitiesare𝑆 𝑡Q =1−𝐹 𝑡Q =1−1%= 99%and𝑆 𝑡R =1−𝐹 𝑡R =1−1.99%=98.01%.
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Application to CDS
• For CDS, we use the probability that default occurs between two future dates (as viewed from today) to find the expected payoff and value of the protection leg.
• We use the survival probabilities to find the value of the premium leg, since the protection buyer will only pay the premium for the period if the entity has not yet defaulted.
• We calibrate our model by choosing probabilities that are consistent with observed market CDS premiums.
76
CDS Pricing and Revaluation
• We show how this can be done in a simplified example.
• Divide the life of the CDS into a number of periods.
• An entity that survives to the start of any period either
defaults during that period or survives to the next period.
• The probability that an entity defaults in any period, conditional on having survived to the start of that period, is its conditional default probability for that period.
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Probability Structure
• The assumed probability structure is therefore:
Note that we have adjusted notation slightly, so that q2from today here is equivalent to
q(t1, t2) in our earlier notation. Similarly, qtcumulative = F(t) and qtsurv= S(t) .
• This is essentially the hazard rate model described in Hull (ch. 24 and 25) and Kosowski and Neftci (ch. 18).
78
Probability Structure Example
No Default = 0.9
1 – q1cond
q1cond = q1cumula