Single Name Credit Derivatives

Transcription

1 Single Name Credit Derivatives Paola Mosconi Banca IMI Bocconi University, 22/02/2016 Paola Mosconi Lecture 3 1 / 40

2 Disclaimer The opinion expressed here are solely those of the author and do not represent in any way those of her employers Paola Mosconi Lecture 3 2 / 40

3 Reference Book D. Brigo and F. Mercurio Interest Rate Models Theory and Practice. With Smile, Inflation and Credit Springer (2006) Paola Mosconi Lecture 3 3 / 40

4 Outline 1 Introduction Credit Derivatives Credit Risk 2 CDS Stylized Facts 3 CDS Pricing General Pricing Framework Model Independent Framework 4 CDS Bootstrapping General Framework Constant Intensity 5 Selected References Paola Mosconi Lecture 3 4 / 40

5 Introduction Outline 1 Introduction Credit Derivatives Credit Risk 2 CDS Stylized Facts 3 CDS Pricing General Pricing Framework Model Independent Framework 4 CDS Bootstrapping General Framework Constant Intensity 5 Selected References Paola Mosconi Lecture 3 5 / 40

6 Introduction Credit Derivatives Credit Derivatives Credit derivatives are bilateral financial contracts that isolate specific aspects of credit risk from an underlying instrument and transfer that risk between two parties. The J.P. Morgan Guide to Credit Derivatives (1999) Go to Credit Risk Paola Mosconi Lecture 3 6 / 40

7 Introduction Credit Derivatives Single Name Credit Derivatives: a List Credit Derivatives can be single name or multi-name instruments. A list of single name credit derivatives includes: Defaultable zero coupon bonds Defaultable coupon bonds Defaultable floaters Credit Default Swaps (CDS) Constant Maturity CDS Equity payoffs with counterparty risk Equity Default Swaps... Go to CDS Paola Mosconi Lecture 3 7 / 40

8 Introduction Credit Risk Credit Risk: Definition Credit risk is associated with an underlying asset issued by a reference entity (government, financial, corporate) and originates from the inability of the reference entity to meet its contractual obligations with its counterparties. A default (or credit) event is determined by the ISDA agreement 1 (2003) and typically includes: 1 bankruptcy, 2 failure to pay, 3 debt restructuring Paola Mosconi Lecture 3 8 / 40

9 Introduction Credit Risk Credit Risk: Definition (Alternate Take) Credit risk is related to the possibility that an unexpected change in a reference entity s creditworthiness may generate a corresponding unexpected change in the market value of the associated credit exposure. Types of Risk: Default Risk It is the risk that the issuer will not be able to pay its obligations of interest and principle. Downgrade (or Migration) Risk Risk of changes in the credit quality of the issuer, as a consequence of a downgrade by the rating agencies. Credit Spread Risk Risk of changes in the credit spreads of the issuer, for example due to market conditions. Paola Mosconi Lecture 3 9 / 40

10 Introduction Credit Risk Credit Risk Measurement Credit risk measurement is based on these fundamental parameters: Probability of Default (PD) The likelihood that the borrower will fail to make full and timely repayment of its financial obligations. Exposure at Default (EAD) The expected value of the loan at the time of default. Loss Given Default (LGD): The amount of the loss if there is a default, expressed as a percentage of the EAD. Recovery Rate (Rec = 1 LGD): The proportion of the EAD the bank recovers. Paola Mosconi Lecture 3 10 / 40

11 Introduction Credit Risk Recovery Rate Recovery rates can vary widely, as they are affected by a number of factors: Type of instrument and its seniority The recovery rate is directly proportional to the instrument s seniority, i.e. an instrument that is more senior in the capital structure will usually have a higher recovery rate than one that is lower down in the capital structure. Corporate issues: capital structure and leverage Debt instruments issued by a company with a relatively lower level of debt in relation to its assets may have higher recovery rates than a company with substantially more debt. Macroeconomic conditions: economic cycle, liquidity conditions... If a large number of companies are defaulting on their debt (e.g. in a deep recession) the recovery rates may be lower than during normal economic times. For a review of recovery rate modeling see Altman et al (2005). In this course, we will assume recovery rates (and LGD) to be constant and given by historical estimates of rating agencies. Paola Mosconi Lecture 3 11 / 40

12 Introduction Credit Risk Probability of Default The default event, being unpredictable, is commonly described by a random variable τ, denoting the default time. The probability of default occurring before time T, given that default has not happened until time t, is given by the so-called cumulative default probability: p(t,t) = Pr(τ T τ > t) (1) Probability of default between two times T 1 and T 2: p(t,t 1,T 2) = Pr(T 1 < τ T 2 τ > t) = p(t,t 2) p(t,t 1) Marginal default probability, i.e. the probability of default in the period (T 1,T 2], conditional on having survived until the beginning of time T 1: p M (t,t 1,T 2) = Pr(T 1 < τ T 2 τ > T 1) = p(t,t2) p(t,t1) 1 p(t,t 1) (2) Paola Mosconi Lecture 3 12 / 40

13 Introduction Credit Risk Probability of Default: Example I We follow Ballotta et al in order to illustrate the relationship between marginal and cumulative default probabilities. Consider t = 0 and T i = 1,2,3,... year. T 1 = 1: T 2 = 2: p(0,2) = p(0,1) }{{} default in 1st year p(0,1) = p M (0,0,1) + }{{} OR (1 p M (0,0,1)) p M (0,1,2) }{{}}{{} survive 1st year default in 2nd year Solving for p M (0,1,2) we obtain a result in agreement with Eq. (2): p M (0,1,2) = p(0,2) p(0,1) 1 p(0,1) Paola Mosconi Lecture 3 13 / 40

14 Introduction Credit Risk Probability of Default: Example II T 3 = 3: p(0,3) = p(0,1) }{{} default in 1st year + }{{} OR (1 p M (0,0,1)) }{{} survive 1st year (1 p M (0,0,1)) p M (0,1,2) }{{}}{{} survive 1st year default in 2nd year (1 p M (0,1,2)) p M (0,2,3) }{{}}{{} survive 2nd year default in 3rd year + }{{} = p(0,2)+(1 p M (0,0,1))(1 p M (0,1,2)) p M (0,2,3) Solving for p M (0,2,3) we get, in accordance with Eq. (2): p M (0,2,3) = p(0,3) p(0,2) 1 p(0,2) OR and so on... Paola Mosconi Lecture 3 14 / 40

15 Introduction Credit Risk Probability of Default: Graphical Illustration Go to Credit Derivatives Paola Mosconi Lecture 3 15 / 40

16 CDS Stylized Facts Outline 1 Introduction Credit Derivatives Credit Risk 2 CDS Stylized Facts 3 CDS Pricing General Pricing Framework Model Independent Framework 4 CDS Bootstrapping General Framework Constant Intensity 5 Selected References Paola Mosconi Lecture 3 16 / 40

17 CDS Stylized Facts Credit Default Swaps (CDS) Credit Default Swaps (CDS) are basic protection contracts that provide the protection buyer an insurance against the default of a reference entity, in exchange for periodic payments to the protection seller. The market for a large number of reference entities is quite liquid and CDS spreads are determined by supply and demand. There is no need to use pricing models to value CDS contracts at inception. However, models are necessary in order to: extract market default probability from CDS quotations through bootstrapping price more complex credit derivatives, by calibrating models to CDS market quotes determine the mark-to-market MTM value of existing CDS positions. Pricing models for default are mainly of two kinds: 1 reduced form (intensity) models 2 structural models Paola Mosconi Lecture 3 17 / 40

18 CDS Stylized Facts CDS Features A CDS contract ensures protection against default of a reference entity. The protection buyerapaystotheprotection seller B agiven rater, attimest a+1,...,t b or until default (the premium leg), in exchange for a single protection payment LGD (the protection leg) at default time τ of a reference entity C, provided that T a < τ T b. Schematically: Protection protection LGD at default τ if T a < τ T b Protection Seller B rate R at T a+1,...,t b or until default τ Buyer A At evaluation time t, the amount R is set at a value R ab (t) that makes the contract fair, i.e. such that the present value of the two exchanged flows is zero. This is how the market quotes CDS. Paola Mosconi Lecture 3 18 / 40

19 CDS Stylized Facts Assumptions and Notations Let us denote: the year fraction between two consecutive dates as α i T i T i 1 the first date among the {T i } i=a+1,...,b following t as T β(t), i.e. t [T β(t) 1,T β(t) ) the stochastic discount factor as D(t,T) = B(t)/B(T), where B(t) = e t 0 rudu represents the bank account numeraire, r being the instantaneous short interest rate. Let us assume the Loss Given Default LGD 1 Rec to be deterministic the default time τ and the interest rate process r to be independent unit notional amount Paola Mosconi Lecture 3 19 / 40

20 CDS Stylized Facts CDS Cash Flows Premium Leg payments The premium leg pays at each time T i, if default has not occurred yet, the periodic fee: Rα i 1 {τ Ti} at default time τ, if τ < T b, the accrued interest: R(τ T β(τ) 1 ) 1 {Ta<τ<T b } where 1 {X} is the indicator function of set X. Protection Leg payments The protection leg pays at default time τ, if τ T b, the Loss Given Default: LGD 1 {τ<tb } Discounted payoff seen from B Π CDSab (t) = Π PremLab (t) Π ProtLab (t) (3) b = D(t,τ)(τ T β(τ) 1 )R 1 {Ta<τ<Tb } + D(t,T i)α i R 1 {τ Ti } D(t,τ)LGD 1 {Ta<τ T b } i=a+1 Paola Mosconi Lecture 3 20 / 40

21 CDS Pricing Outline 1 Introduction Credit Derivatives Credit Risk 2 CDS Stylized Facts 3 CDS Pricing General Pricing Framework Model Independent Framework 4 CDS Bootstrapping General Framework Constant Intensity 5 Selected References Paola Mosconi Lecture 3 21 / 40

22 CDS Pricing General Pricing Framework General Pricing Framework Let use denote by CDS a,b (t,r,lgd) the price at time t of the CDS. At t = 0, the price is given by the expectation of the discounted payoff Π CDSab (0) under the risk neutral measure Q: CDS a,b (0,R,LGD) = E[Π CDSab (0)]. (4) This is equivalent to pricing (analytically) both legs. Paola Mosconi Lecture 3 22 / 40

23 CDS Pricing General Pricing Framework Goal In general, the resulting pricing formulas depend on the choice of the underlying model (intensity based or structural) used to describe the dynamics of: the default event interest rates the loss given default (when stochastic) However, such formulas will not be used to price CDS that are already quoted in the market. Rather, they will be inverted in correspondence of CDS market quotes to calibrate the models to the CDS quotes themselves. Paola Mosconi Lecture 3 23 / 40

24 CDS Pricing Model Independent Framework Model Independent Framework Before tackling the general problem of deriving pricing formulas for CDS under specific model assumptions, which will be dealt with in the next lectures, we show how to derive model independent formulas, given, at time t = 0: the initial zero coupon curve (bonds) observed in the market: P mkt (0,T i ) the survival probabilities Paola Mosconi Lecture 3 24 / 40

25 CDS Pricing Model Independent Framework Survival Function (Li 2000)... Let F(t) denote the distribution function of default time τ: F(t) := Pr(τ t) t 0 and F(0) = 0. If F is differentiable, its derivative f represents the density function and: F(t) = t f(u)du. The probability that default does not occur before time t represents the survival probability and is described the survival function S(t): S(t) := Pr(τ t) = t f(u)du = 1 F(t). Paola Mosconi Lecture 3 25 / 40

26 CDS Pricing Model Independent Framework Premium Leg I Under independence between default τ and interest rates, the expected value of the premium leg, under the risk neutral measure Q, is given by: PremL ab (R) = E D(0,τ)(τ T β(τ) 1 )R 1 {Ta<τ<Tb } + b i=a+1 [ = E D(0,t)(t T β(t) 1 )R 1 {Ta<t<Tb } 1 {τ [t,t+dt)} ]+ 0 [ Tb ] = E D(0,t)(t T β(t) 1 )R 1 {τ [t,t+dt)} + T a Tb = E[D(0,t)](t T β(t) 1 )R E[1 {τ [t,t+dt)} ]+R T a Tb = P(0,t)(t T β(t) 1 )R Q(τ [t,t +dt))+r T a D(0,T i )α i R 1 {τ Ti } b E[D(0,T i )]α i R E[1 {τ Ti }] i=a+1 b P(0,T i )α i R Q(τ T i ) i=a+1 b P(0,T i )α i Q(τ T i ) i=a+1 b P(0,T i )α i Q(τ T i ) i=a+1 Paola Mosconi Lecture 3 26 / 40

27 CDS Pricing Model Independent Framework Premium Leg II Given that dt is infinitesimal, it follows Q(τ [t,t +dt)) = Q(τ < t +dt) Q(τ < t) = dq(τ < t) = d[1 Q(τ t)] = dq(τ t) and [ PremL ab (R) = R Tb T a P(0,t)(t T β(t) 1 )dq(τ t)+ ] b P(0,T i)α i Q(τ T i) i=a+1 Paola Mosconi Lecture 3 27 / 40

28 CDS Pricing Model Independent Framework Protection Leg Under independence between default τ and interest rates, the expected value of the protection leg, under the risk neutral measure Q, is given by: ProtL ab = E [ ] LGDD(0,τ) 1 {Ta<τ Tb } [ ] = LGD E D(0,t) 1 {Ta<τ Tb } 1 {τ [t,t+dt)} = LGD = LGD = LGD and 0 Tb T a E[D(0,t) 1 {τ [t,t+dt)} ] Tb T a E[D(0,t)]E[1 {τ [t,t+dt)} ] Tb ProtL ab = LGD T a P(0,t) Q(τ [t,t +dt)) Tb T a P(0,t)dQ(τ t) Paola Mosconi Lecture 3 28 / 40

29 CDS Pricing Model Independent Framework CDS Model Independent Pricing Formula The CDS price at time t = 0 is given by: CDS ab (0,R,LGD; Q(τ > )) = PremL ab (Q(τ > )) ProtL ab (Q(τ > )) [ ] Tb b = R P(0,t)(t T β(u) 1 )dq(τ t)+ P(0,T i)α i Q(τ T i) T a i=a+1 Tb +LGD P(0,t)dQ(τ t) T a The integrals in the survival probabilities can be approximated numerically by summations through Riemann-Stieltjes sums, considering a low enough discretization time step. Paola Mosconi Lecture 3 29 / 40

30 CDS Pricing Model Independent Framework CDS Market Quotes The market quotes, at time 0, the fair CDS rate mkt MID R = R0,b (0) with initial protection time and final protection time T a = 0 T b {1y,3y,5y,7y,10y}. Paola Mosconi Lecture 3 30 / 40

31 CDS Pricing Model Independent Framework CDS Rates mkt MID R0,b (0) is obtained by solving: mkt MID CDS ab (0,R0,b (0),LGD; Q(τ > )) = 0 which gives: mkt MID R0,b (0) = LGD Tb 0 P(0,t)dQ(τ t) Tb 0 P(0,t)(t T β(t) 1 )dq(τ t) b i=1 P(0,Ti)αi Q(τ Ti) Using the initial zero coupon curve (bonds) observed in the market, P mkt (0,T i), and the mkt MID market quotes of R0,b (0), starting from T b = 1y, we can find the market implied survival {Q(τ t), t 1y} and iteratively bootstrap all the survival probabilities {Q(τ t),t (1y,3y]} and so on up to T b = 10y. This is a way to strip survival probabilities from CDS quotes in a model independent way, with no need to assume an intensity or structural model of default! However, the market in doing the above stripping typically resorts to hazard functions associated with the default time. Paola Mosconi Lecture 3 31 / 40

32 CDS Pricing Model Independent Framework...Hazard Functions (Li 2000) Recalling the survival function introduced above, survival probabilities can be equivalently expressed in terms of the so-called hazard functions. The default rate at time t, conditional on survival until time t or later, defines the hazard rate function: Pr(t τ < t +dt) F(t +dt) F(t) h(t) := lim = = f(t) dt 0 dtpr(τ t) dt S(t) S(t) = (t) S S(t) The cumulative hazard function H(t) is defined as: H(t) := t 0 h(u)du = t 0 (5) d(lns(u)) = lns(t) (6) Then, the survival function S(t), describing the survival probability up to time t, can be expressed in terms of hazard functions as: S(t) = e t 0 h(u)du = e H(t) Paola Mosconi Lecture 3 32 / 40

33 CDS Bootstrapping Outline 1 Introduction Credit Derivatives Credit Risk 2 CDS Stylized Facts 3 CDS Pricing General Pricing Framework Model Independent Framework 4 CDS Bootstrapping General Framework Constant Intensity 5 Selected References Paola Mosconi Lecture 3 33 / 40

34 CDS Bootstrapping General Framework General Framework GOAL: Quoted spread survival probabilities intensity Assume the existence of a deterministic default intensity γ(t), such that γ(t) dt represents the probability rate of defaulting in [t,t +dt) having not defaulted before t: Q(τ [t,t +dt) τ > t) = γ(t)dt Recalling definitions (5) and (6) of hazard functions, the cumulated intensity is nothing but a cumulated hazard rate: Γ(t) := t 0 γ(t) dt The survival probability (under the risk neutral measure) can be written as Q(τ t) = e Γ(t) (7) Also, dq(τ t) = γ(t)e Γ(t) dt Paola Mosconi Lecture 3 34 / 40

35 CDS Bootstrapping General Framework Interest Rate Analogy Usually, the actual model assumes for the default time τ a more complex structure (e.g. stochastic intensity). However, γ mkt are retained as a mere quoting mechanism for CDS rate market quotes. If the intensity is stochastic(λ(t)) the survival probability formula(7) must be generalized as follows: Q(τ t) = E [e ] t 0 λ(u)du This is just the price of a zero coupon bond in an interest rate model with short rate r replaced by λ. In this way, survival probabilities can be interpreted as zero coupon bonds and intensity λ (or γ in the deterministic case) as instantaneous credit spreads. In the following, we will show a very simple quoting mechanism, based on constant intensity. Further examples, based on deterministic and stochastic intensities, will be dealt with extensively, when considering Reduced Form (Intensity) Models. Paola Mosconi Lecture 3 35 / 40

36 CDS Bootstrapping Constant Intensity Constant Intensity Formula The market makes intensive use of a simple formula for calibrating a constant intensity (hazard rate) γ(t) = γ to a single CDS rate, R 0,b : γ = R 0,b LGD Since γ can be interpreted as an instantaneous credit spread, this interpretation can be extended to R as well. Therefore, the CDS premium rate R admits a double interpretation, either as a credit spread or as a default probability. This formula is handy, since it does not require the interest rate curve. However, it does not take into account the term structure of CDS, being based on a single quote for R. It can be used in any situation where one needs a quick calibration of the default intensity to a single CDS quote (e.g. the liquid 5y one). Paola Mosconi Lecture 3 36 / 40

37 CDS Bootstrapping Constant Intensity Constant Intensity: Formula Derivation I Assumptions independence between default time and interest rates the premium leg pays continuously until default the premium rate R of the CDS: i.e. in the interval [t,t +dt) the premium leg pays Rdt. Premium Leg [ T PremL = E = R 0 T 0 ] T D(0,t) 1 {τ>t} Rdt = R E[D(0,t)]E[1 {τ>t} ]dt = R 0 T 0 E[D(0,t) 1 {τ>t} ]dt P(0,t) Q(τ > t)dt Paola Mosconi Lecture 3 37 / 40

38 CDS Bootstrapping Constant Intensity Constant Intensity: Formula Derivation II Protection Leg ProtL = E [ ] T LGDD(0,t) 1 {τ T} = LGD E [ ] D(0,t) 1 {τ [t,t+dt)} 0 T T = LGD P(0,t) Q(τ [t,t +dt)) = LGD P(0,t)dQ(τ > t) 0 0 Given the constant intensity assumption, i.e. Q(τ > t) = e γ t and dq(τ > t) = γe γ t dt = γ Q(τ > t)dt it follows: ProtL = γlgd T 0 P(0,t) Q(τ > t)dt. R T 0 P(0,t) Q(τ > t)dt = γlgd Premium Leg = Protection Leg T 0 P(0,t) Q(τ > t)dt γ = R LGD Paola Mosconi Lecture 3 38 / 40

39 Selected References Outline 1 Introduction Credit Derivatives Credit Risk 2 CDS Stylized Facts 3 CDS Pricing General Pricing Framework Model Independent Framework 4 CDS Bootstrapping General Framework Constant Intensity 5 Selected References Paola Mosconi Lecture 3 39 / 40

40 Selected References Selected References Altman, E., Resti, A. and Sironi, A. (2005), Default Recovery Rates in Credit Risk Modeling: A Review of the Literature and Empirical Evidence, Journal of Finance Literature Ballotta, L., Fusai, G. and Marena, M. (Forthcoming), Introduction to Default Risk and Counterparty Credit Modelling. The J.P. Morgan Guide to Credit Derivatives (1999) to Credit Derivatives.pdf Li, D. X., (2000), On Default Correlation: A Copula Approach, Journal of Fixed Income, 9 Paola Mosconi Lecture 3 40 / 40