Structural Models. Paola Mosconi. Bocconi University, 9/3/2015. Banca IMI. Paola Mosconi Lecture 3 1 / 65

Transcription

1 Structural Models Paola Mosconi Banca IMI Bocconi University, 9/3/2015 Paola Mosconi Lecture 3 1 / 65

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 / 65

3 Main References Merton Model Hull, J.C., Nelken, I., and White, A. (2004), Merton s Model, Credit Risk and Volatility Skews, Journal of Credit Risk, 1, 3-28 KMV Model Crosbie, P., and Bohn, J. (2002), Modeling Default Risk, KMV First Passage Time Models Brigo, D. and Tarenghi, M. (2004), Credit Default Swap Calibration and Equity Swap Valuation under Counterparty Risk with a Tractable Structural Model, id= Paola Mosconi Lecture 3 3 / 65

4 Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 4 / 65

5 Introduction Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 5 / 65

6 Introduction Introduction Structural models provide a way of relating credit risk of a firm to its capital structure (assets and liabilities). The key idea is that if the value of the company s assets goes below a given safety level, the firm is not able to repay its debts and is subject to default. In basic structural models, default is induced by observable market information. Modeling features, common to all structural models, are: the stochastic process for the asset value of the firm the default barrier modeling the safety level the default time Paola Mosconi Lecture 3 6 / 65

7 Merton Model Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 7 / 65

8 Merton Model Assumptions Assumptions Merton (1974), based on the work of Black and Scholes (1973), proposes a simple model of the firm that provides a way of relating credit risk to the capital structure of the firm. Assumptions: the firm has issued two classes of securities: equity and debt. the debt is a pure discount bond where the payment of D is promised at maturity T the value of the firm is assumed to be a tradable asset and obey a lognormal diffusion process with constant volatility and interest rate (equity shareholders receive no dividend). Under the risk neutral measure: da(t) = r A(t)dt +σ A A(t)dW(t) Paola Mosconi Lecture 3 8 / 65

9 Merton Model Basic Idea Basic Idea I The limited liability feature of equity means that the equity holders have the right, butnottheobligation,topayoffthedebtholdersandtakeovertheremainingassets of the firm. At time T two situations can occur: if the firm s asset value exceeds the promised payment D, the lenders are paid the promised amount and the shareholders receive the residual asset value; if the asset value is less than the promised payment D, the firms defaults, the lenders receive a payment equal to the asset value and the shareholders get nothing. Equity is the same as a call option on the firm s assets with a strike price equal to the book value of the firm s liabilities. Paola Mosconi Lecture 3 9 / 65

10 Merton Model Basic Idea Basic Idea II Merton (1974) calculates: the equity value the probability of default the debt value the implied credit spread Paola Mosconi Lecture 3 10 / 65

11 Merton Model Basic Idea Equity Value Define: E = value of the firm s equity (E T at time T and E 0 at t = 0) A = value of the firm s assets (A T at time T and A 0 at t = 0) Since equity is a call option on the assets of the firm with strike equal to the promised debt payment, the payment to the shareholders at time T is given by: E T = max[a T D,0]. In the Merton (Black Scholes) framework, the current price of equity is: E 0 = A 0N(d 1) De rt N(d 2) (1) where: d 1 = ln(a0ert / D) + 1 σ A T 2 σ A T d2 = d 1 σ A T and σ A = volatility of the asset value, and r = risk free rate of interest are both assumed to be constant. Paola Mosconi Lecture 3 11 / 65

12 Merton Model Basic Idea Equity Value in Terms of the Leverage Defining D 0 = De rt as the present value of the promised debt payment, represents a measure of leverage. L. = D 0 A 0 The current price of equity (1) can be expressed in terms of the leverage as: where E 0 = A 0[N(d 1) LN(d 2)] (2) d 1 = ln(l) + 1 σ A T 2 σ A T d2 = d 1 σ A T The equity value depends on the leverage L, the asset volatility σ A and the time of repayment T. Paola Mosconi Lecture 3 12 / 65

13 Merton Model Basic Idea Probability of Default The risk neutral probability that the company defaults by time T is the probability that shareholders will not exercise their call option to buy the assets of the company for D at T: Q(A T < D) = Q(ln(A T ) < ln( D)) ( = Q W ln D T A 0 r σ2 A 2 )T < T σ A T ( ln A 0 D = N + r σ2 A 2 )T = N( d 2) σ A T ( Recalling d 2 as a function of the leverage L, i.e. d 2 = ln(l) σ A T + 1 σ ) 2 A T the probability of default depends on the leverage L, the asset volatility σ A and the time of repayment T. Paola Mosconi Lecture 3 13 / 65

14 Merton Model Basic Idea Debt Value Let D 0 be the market price of debt at time t = 0. The value of the asset at any time is given by: D t = A t E t. Therefore, at t = 0 using eq. (2) D 0 becomes: D 0 = A 0[N( d 1)+LN(d 2)] The debt value depends on the leverage L, the asset volatility σ A and the time of repayment T. Recall that: D 0 = De rt present value of promised debt payment D D 0 = De yt market value of debt at t = 0 (where y is the yield to maturity on the debt) Paola Mosconi Lecture 3 14 / 65

15 Merton Model Basic Idea Implied Credit Spread of Risky Debt The yield to maturity on the debt is defined implicitly by: D 0 = De yt = D 0e (r y)t and therefore, recalling the definition of leverage (A 0 = D 0/L): [ ] ln N(d 2)+ N( d 1) L y = r. T Credit spread implied by the Merton model [ ] s =. ln N(d 2)+ N( d 1) L y r = T depends only on the leverage L, the asset volatility σ A and the time to repayment T. Paola Mosconi Lecture 3 15 / 65

16 Merton Model Asset Volatility Estimation Asset Volatility Estimation We have seen that the equity value, the debt value, the implied credit spread and the probability of default all depends on the leverage L, the time to repayment T and the asset volatility σ A. How to estimate the asset volatility σ A? Linking asset volatility to equity volatility. Possible approaches include: 1 Jones et al (1984) use Itô formula 2 Geske ( ) uses compound options (see Hull, Nelken and White (2004)) Paola Mosconi Lecture 3 16 / 65

17 Merton Model Asset Volatility Estimation Jones Formula Let σ A = asset volatility and σ E = equity volatility. Equity is interpreted as a call on the asset value: E t = call(t,a t). By differentiating and using Itô formula: de t = dcall(t,a t) = (...)dt + call A σ AA t dw t. Asset volatility as a function of equity volatility Comparing with a hypothetical dynamics: de t = (...)E tdt +σ E E tdw t we find the important relation: A σ E = σ A call E where call = N(d 1) is the Delta greek of the call or equivalently: (3) σ E = σ A N(d1) N(d 1) LN(d 2) Equity volatility can be estimated historically or implied from the market. Paola Mosconi Lecture 3 17 / 65

18 Merton Model Asset Volatility Estimation Hull Nelken White (2004) Approach I Idea Infer asset volatility σ A from implied volatility of equity options. If equity is an option on the asset value A, then equity options are options on options on A (i.e. compound options), which have been analytically priced by Geske ( ). The price of an equity put with strike K and expiry T < T is given at t = 0 by: T T PutGeske(K) = De rt N 2 a 2,d 2 ; A 0 N 2 a 1,d 1 ; +Ke r T N( a 2 ) T T where ln A 0 A T ( ) + r σ2 A 2 T a 1 = a 2 = a 1 σ A T σ A T N 2(.,.;.) is the bivariate normal cumulative distribution and A T the critical asset value for which the equity value at time T equals K (i.e. it is the asset value below which the put on the equity will be exercised). Paola Mosconi Lecture 3 18 / 65

19 Merton Model Asset Volatility Estimation Hull Nelken White (2004) Approach II An equity put can also be computed by using the Black Scholes formula: PutBS(IVol(K),K) = Ke r T N( d 2) E 0N( d 1) where IVol(K) is the equity volatility smile and ( ) ln E 0 K d + r IVol2 (K) 2 T 1 = IVol(K) T d 2 = d 1 IVol(K) T. If we equate this expression with the Geske one for two quoted strikes and using A 0 = D 0/L in Geske formula we can solve for σ A and L. PutGeske(σ A,L;K 1) = PutBS(IVol(K 1),K 1) PutGeske(σ A,L;K 2) = PutBS(IVol(K 2),K 2) Paola Mosconi Lecture 3 19 / 65

20 Merton Model Asset Volatility Estimation Hull Nelken White (2004) Approach III Hull Nelken White use: T = 2 months K 1 = 25 delta and K 2 = 50 delta put options The two months 25 delta (50 delta) put implied volatility is an estimate of the implied volatility of a two-month put option that has a delta of 0.25 ( 0.50), obtained by interpolating between the implied volatilities of the two-month options whose delta are closest to 0.25 ( 0.50). Paola Mosconi Lecture 3 20 / 65

21 Merton Model Limits and Extensions Limits of Merton Model 1 A company can only default at its debt maturity date T. This implies that the probability of default is zero, for any t < T, and credit spreads are given accordingly by 1 : { 0, t < T s(t) = s(t), t T. 2 All debts are mapped into a single zero-coupon bond. 3 Interest rates are assumed to be constant. 4 The value of the firm is assumed to be a tradable asset, but its parameters are not even directly observable. 1 The rigidity in modeling credit spreads is common to all basic structural models. Paola Mosconi Lecture 3 21 / 65

22 Merton Model Limits and Extensions Extensions of Merton Model I 1 Early defaults can be modeled, by specifying a threshold level such that default occurs when the firm s asset value falls below that critical level. Extensions in this direction include: the Black and Cox model (1976), and other First Passage Time models; the KMV model, where the threshold level is determined by the default point (Crosbie and Bohn (2002)). 2 Multiple classes of liabilities (short-term, long-term, convertible debt etc.) have been modeled according to different approaches: Geske s compound option model represents the first structural model of this category (Geske ( )); the KMV model incorporates different liabilities into the default point (Crosbie and Bohn (2002)). Paola Mosconi Lecture 3 22 / 65

23 Merton Model Limits and Extensions Extensions of Merton Model II 3 A stochastic interest rate model can be incorporated into Merton model or its extended versions. In this case, correlation between asset and interest rate processes can be introduced. 4 More sophisticated structural models, tackling the problem of possibly unreliable balance sheet information (e.g. stochastic volatility, jump diffusion etc.) have been proposed (see e.g. Brigo et al (2004, 2006 and 2009) for the Scenario Barrier Time-varying Volatility (SBTV) model). Paola Mosconi Lecture 3 23 / 65

24 Merton Model Limits and Extensions Applications of Merton Model KMV model (Kealhofer McQuown and Vasicek) CreditMetrics (Gupton et al (1997)) Basel 2 (2.5, 3): the single-factor, infinitely granular model based on the works by Finger and Gordy CreditGrades (Finkelstein et al (2002)) Paola Mosconi Lecture 3 24 / 65

25 KMV Model Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 25 / 65

26 KMV Model KMV Model I KMV provides a commercial solution that, extending Black-Scholes-Merton framework, produce a model of default probability. Default occurs when a certain threshold (default point) is breached and multiple classes of liabilities are taken into account (for a review, see Crosbie and Bohn (2002)). Rather than using the default probability à la Merton, KMV, relying on a proprietary database, estimates the frequency with which firms default, based on the fraction of those firms, sharing similar features, that actually did default in previous years. Paola Mosconi Lecture 3 26 / 65

27 KMV Model KMV Model II The default probability of a firm is determined through the following steps: Estimate asset value and volatility: Asset value and asset volatility of the firm are estimated from the market value and volatility of equity and the book value of liabilities. Calculate the distance-to-default: The distance-to-default (DD) is calculated from the asset value and asset volatility and the book value of liabilities. Calculate the default probability: A default database is used to derive an empirical distribution relating the distance to-default to a default probability. Paola Mosconi Lecture 3 27 / 65

28 KMV Model Estimate of Asset Value and Volatility Estimate of Asset Value and Volatility KMV estimates asset value and asset volatility by solving simultaneously the following equations: { E = f(a,σa,t,capital structure) σ E = g(a,σ A,T,capital structure) where E and σ E are input determined from the market. The first equation is given explicitly by (1) while the second one is a (proprietary) generalization of eq. (3). Given that eq. (3) holds only instantaneously, KMV devised a more complex iterative procedure to solve for the asset volatility (see Crosbie and Bohn(2002)). Paola Mosconi Lecture 3 28 / 65

29 KMV Model Distance to Default (DD) Distance to Default (DD) I Firms do not necessarily default when their asset value reaches the book value of their total liabilities. The long-term nature of some of their liabilities provides these firms with some breathing space. KMV defines the default point (DP) as the asset value at which the firm defaults and find that it lies somewhere between total liabilities (LTD) and current, or short-term, liabilities (STD). A popular choice is: DP = STD+ 1 2 LTD A firm will default when its market net worth, i.e. firm s assets minus the firm s default point, reaches zero. Paola Mosconi Lecture 3 29 / 65

30 KMV Model Distance to Default (DD) Distance to Default (DD) II A single measure of default risk is given by the distance to default (DD), which represents the number of standard deviations the asset value is away from default: (Bohn and Crosbie (2002)) DD t = At DP A t σ A Figure: Mean Distance to Default (DD). Paola Mosconi Lecture 3 30 / 65

31 KMV Model Expected Default Frequency (EDF) Expected Default Frequency (EDF) I KMV defines the Expected Default Frequency (EDF) as the probability that the asset value falls below the default point. If the future distribution of the distance-to-default were known, the default probability would simply be the likelihood that the final asset value stayed below the default point. In practice, the distribution of the distance-to-default is difficult to measure. However, the usual assumptions of normal distribution (like in the original Merton model) is a very poor choice. Figure: Expected default frequency (EDF). Paola Mosconi Lecture 3 31 / 65

32 KMV Model Expected Default Frequency (EDF) Expected Default Frequency (EDF) II KMV obtains the relationship between distance-to-default and default probability from data on historical default and bankruptcy frequencies, taken from a proprietary database. The resulting empirical distribution of default rates has much wider tails than the normal distribution. Figure: Mapping from DD to EDF. Paola Mosconi Lecture 3 32 / 65

33 First Passage Time Models Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 33 / 65

34 First Passage Time Models Introduction I Default can now occur at any time and not only at the maturity of the debt T. Figure: First Passage (1P) vs Merton model. Paola Mosconi Lecture 3 34 / 65

35 First Passage Time Models Introduction II We will consider two versions of First Passage Time Models: 1 Black and Cox model (1976) 2 Analytically Tractable First Passage models (AT1P) (Brigo et al ) Goal Find the survival probabilities that, entered in the pricing formula for CDS, allow to calibrate models to the market. Paola Mosconi Lecture 3 35 / 65

36 First Passage Time Models Black and Cox Model Black and Cox Model (1976) I Black and Cox assume a barrier representing safety covenants for the firm. Safety covenants provide the firm s bondholders with the right to force the firm to bankruptcy or reorganization if the firm is doing poorly according to a set standard. Default occurs if the firm s asset value A hits the barrier from above. Paola Mosconi Lecture 3 36 / 65

37 First Passage Time Models Black and Cox Model Black and Cox Model (1976) II In formulas: Under the risk neutral measure the asset dynamics is assumed to follow a geometric Brownian motion: da(t) = (r q)a(t)dt +σ A A(t)dW(t) and the barrier is assumed to have a time dependent, exponential behavior: { D, t = T H(t) = K e γ(t t), t < T where γ and K are positive constants. Moreover, K e γ(t t) < De r(t t) i.e. safety covenants never exceeds the face value of debt, discounted at the risk-free rate. Default time is defined as: τ = inf{t [0,T] : A t H(t)} (inf = ) Paola Mosconi Lecture 3 37 / 65

38 First Passage Time Models Black and Cox Model Black Cox: Barrier Options In the equity market, analytical formulas have been developed to price barrier options (e.g. knock-in, knock-out and digital options). A special case is the down and out digital option or down and out bond (DOB), a contract paying 1 at T if, during the life of the contract, the underlying never hits the barrier H(t) from above: DOB(0,T) = E[D(0,T) 1 {τ T} ] Since the model assumes deterministic interest rates, D(0,T) = P(0,T) and DOB(0,T) = P(0,T)E[1 {τ>t} ] = P(0,T) Q(τ > T) (4) where Q(τ > T) is the (survival) probability of never touching the barrier before T. Paola Mosconi Lecture 3 38 / 65

39 First Passage Time Models Black and Cox Model Black Cox: from Barrier Options to CDS DOB(0, T) can alternatively be priced through an analytical formula developed for barrier options (see Bielecki and Rutkowski (2001) and Brigo and Tarenghi (2004)): DOB(0,T) = P(0,T) [ N ( ln A 0 H(0) +νt σ A T where ν = r q γ σ2 A 2 and a = ν/σ 2 A. Comparing with eq. (4), we derive the survival probability: Q(τ > T) = [ N ( ln A 0 H(0) +νt σ A T which enters the CDS pricing formula: CDS 0,b (0,R,LGD) = accrual R b i=1 ) ( ) ( )] 2a H(0) ln H(0) A N 0 +νt A 0 σ A T ) ( ) ( )] 2a H(0) ln H(0) A N 0 +νt, A 0 σ A T Tb P(0,T i)α i Q(τ > T i) LGD P(0,t)dQ(τ > t) 0 Paola Mosconi Lecture 3 39 / 65

40 First Passage Time Models Black and Cox Model Black Cox: CDS Calibration I Black Cox Model Typically the market quotes 5 to 10 CDS spreads. Is it possible to reproduce liquid CDS data? R Mkt 0,1y R Mkt 0,2y. R Mkt 0,10y da(t) = { (r q)a(t)dt +σ A A(t)dW(t) D, t = T H(t) = K e γ(t t), t < T model parameters: σ A, D, K, γ 4 parameters are not enough to allow for a flexible calibration! Extension (local volatility): R0,1y Mkt R0,2y Mkt da(t) = (r q)a(t)dt +σ A (t)a(t)dw(t) H(t) =.... model parameters: t σ A (t), D, K, γ R0,10y Mkt Now we have infinite parameters. Paola Mosconi Lecture 3 40 / 65

41 First Passage Time Models Black and Cox Model Black Cox: CDS Calibration II Is it possible to have time dependency in both the volatility and the barrier, still preserving closed form pricing formulas? In general, barrier option problems do not allow for an analytically tractable solution for time-dependent volatility of the underlying and general curved barriers. However, for a particular shape of the barrier a closed form solution has been found by Lo et al (2003) and Rapisarda (2003). Brigo et al ( ), starting from this result, have built the first extension of the Black Cox model, called Analytically Tractable 1st Passage model (AT1P), which is able to calibrate the term structure of CDS market quotes, as satisfactorily as in a reduced form model. Paola Mosconi Lecture 3 41 / 65

42 First Passage Time Models AT1P Models AT1P Models Analytically Tractable First Passage (AT1P) models have been introduced and extensively studied by Brigo et al ( ). Risk neutral asset dynamics: Time dependent default barrier { Ĥ(t) = H exp da(t) = (r(t) q(t))a(t)dt +σ A (t)a(t)dw(t) where H and B are free parameters. t 0 [ ] } q(s) r(s)+bσa(s) 2 ds Survival probability: Q(τ > T) = N ln A 0 +β T ( ) 2β H 0 σ2 A(s)ds H N ln H A 0 +β T 0 σ2 A(s)ds T 0 σ2 A (s)ds A 0 T 0 σ2 A (s)ds where β = 2B 1 2 Paola Mosconi Lecture 3 42 / 65

43 First Passage Time Models AT1P Models AT1P Models: Default Barrier The default barrier varies in time, following the firm and market conditions: { t [ ] } Ĥ(t) = H exp q(s) r(s)+bσa(s) 2 ds = 0 H E[A t] A } 0 {{} exp Backbone of the barrier ( B t 0 ) σa(s)ds 2 }{{} Cutting some slack in high volatility conditions H and A 0 always appear in formulas in ratios like H A 0. Therefore, it is possible to rescale the initial value of the firm s assets A 0 = 1 and express the (free) barrier parameter H as a fraction of it. In this case, it is not necessary to know the real value of the firm. Paola Mosconi Lecture 3 43 / 65

44 First Passage Time Models AT1P Models AT1P Models: CDS Calibration The survival probability, and hence the CDS pricing formula, is a function of (σ A (t),b,h). Choosing a choose a piece-wise constant shape for the volatility, i.e.: R Mkt 0,1y,...R Mkt 0,10y σ 1y,...σ 10y,H,B where σ A (t) = σ i for T i 1 t < T i (T 0 = 0), the number of parameters to calibrate is equal to n+2 (with n the number of market quotes). Given that the number of model parameters exceeds by 2 the number of market quotes, Brigo et al ( ) proposes two alternative options: 1 Fix the parameters B and σ 1 (e.g. using an exogenous estimate of (the first) volatility following Hull, Nelken and White (2004)) and calibrate H and the remaining n 1 volatility parameters. 2 Fix B and H (e.g. H/A 0 = 40%, in analogy with the CDS recovery) exogenously and calibrate the n volatility parameters. Paola Mosconi Lecture 3 44 / 65

45 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration I Parmalat default history September 12, 2003: Parmalat dropped plans for a e300 million debt sale. November 14, 2003: The Chief Financial Officer resigned after questions were raised about Parmalat financial transactions. December 9, 2003: Parmalat missed a e150 million bond payment, while the management claimed this was due to a customer not paying its bills. December 19, 2003: a claimed $ 3.9 billion liquidity was revealed not to exist. December 24, 2003: Parmalat filed bankruptcy. (Source: Brigo and Morini (2006)) Paola Mosconi Lecture 3 45 / 65

46 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration II Parmalat credit situation is analyzed in four different days: September 10, 2003: just before the beginning of the final Parmalat default history. November 28, 2003: after the story of Parmalat crisis began to unfold but before its peak. December 8-10, 2003: one day before and after Parmalat missed a e150 million bond payment: the fraud was not yet clear but the company was suspected to be on the verge of bankruptcy. Paola Mosconi Lecture 3 46 / 65

47 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (September 10, 2003) Recovery Rate = 40%, H = 0.4 and B = 0.7 Figure: Maturity dates and corresponding CDS quotes in bps (left Table). Calibration with piece-wise constant volatility (right Table). Source: Brigo (2009). Paola Mosconi Lecture 3 47 / 65

48 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (September 10, 2003) Figure: Volatility term structure. Source: Brigo (2009). Paola Mosconi Lecture 3 48 / 65

49 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (November 28, 2003) Recovery Rate = 40%, H = 0.4 and B = 0.7 Figure: Maturity dates and corresponding CDS quotes in bps (left Table). Calibration with piece-wise constant volatility (right Table). Source: Brigo (2009). Paola Mosconi Lecture 3 49 / 65

50 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (November 28, 2003) Figure: Volatility term structure. Source: Brigo (2009). Paola Mosconi Lecture 3 50 / 65

51 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (December 8, 2003) Recovery Rate = 25%, H = 0.4 and B = 0.7 Figure: Maturity dates and corresponding CDS quotes in bps (left Table). Calibration with piece-wise constant volatility (right Table). Source: Brigo (2009). Paola Mosconi Lecture 3 51 / 65

52 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (December 8, 2003) Figure: Volatility term structure. Source: Brigo (2009). Paola Mosconi Lecture 3 52 / 65

53 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (December 10, 2003) Recovery Rate = 15%, H = 0.4 and B = 0.7 Figure: Maturity dates and corresponding CDS quotes in bps (left Table). Calibration with piece-wise constant volatility (right Table). Source: Brigo (2009). Paola Mosconi Lecture 3 53 / 65

54 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (December 10, 2003) Figure: Volatility term structure. Source: Brigo (2009). Paola Mosconi Lecture 3 54 / 65

55 First Passage Time Models AT1P Models AT1P Models: Parmalat CDS Calibration (December 10, 2003) Figure: Survival probability exp( Γ): intensity models (left) and AT1P model (right). Source: Brigo and Mercurio (2006) and Brigo (2009). Paola Mosconi Lecture 3 55 / 65

56 First Passage Time Models AT1P Models AT1P Models: Remarks AT1P models can calibrate exactly CDS market quotes. Under stress conditions, calibration seems to perform better with AT1P models rather than with intensity models, where some intensity pillars assume negative values. The value of the first volatility parameter, under some circumstances, seems inconsistent with the following ones. This reflects the difficulty of structural models to achieve short term spreads. Due to the lognormal dynamics of the underlying, the likelihood of default within a short time horizon is extremely small, because the process has continuous paths and it takes time to cross the default boundary (Duffie and Singleton (2003)). In the presence of a deterministic barrier, a considerable probability of default can only be reached in a very short horizon at the price of a very high (short term) volatility. Paola Mosconi Lecture 3 56 / 65

57 First Passage Time Models AT1P Models AT1P Models: Extensions AT1P models introduced so far rely on the assumption of a deterministic barrier. This assumption poses two problems: it implies that accounting data are fully reliable and transparent. This condition has been violated many times under stress circumstances. it makes hard the correct determination of short term credit spreads. AT1P models extensions: the Scenario Barrier Time-varying Volatility model (SBTV) by Brigo and Morini (2006), where the barrier value H is assumed to be a discrete random variable, assuming different values in different scenarios; the CreditGrades model by Finkelstein et al (2002), where the barrier is modeled as a continuous stochastic variable; Jump-diffusion (Zhou (1997)): a jump component is added to the underlying continuous process, but the model is somehow untractable; Uncertainty on the firm value (Duffie and Lando (1997)). Paola Mosconi Lecture 3 57 / 65

58 Link to Equity Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 58 / 65

59 Link to Equity Link to Equity We have seen that there is a dependence between CDS spreads and the equity volatility σ E of the reference entity. In the past, data on equity have been used to deduce credit spreads, but now that CDS are quite liquid the procedure has lost attractiveness. Reversing the logic, we use liquid CDS data to infer estimates of the equity volatility σ E, according to the line of reasoning: CDS quotes structural model σ A σ E = function(σ A ) This is useful when evaluating contracts where no quoted implied equity volatilities for the underlying are present (e.g. due to the level of strike, the long maturity etc.) Paola Mosconi Lecture 3 59 / 65

60 Summary Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 60 / 65

61 Summary Summary I 1 We have introduced the basic structural models: Merton model (1974): lognormal assumption on A(t), flat and static barrier, default possible at (a single) debt maturity. KMV model: empirical distribution of asset value, multiple liabilities, early defaults. Black and Cox model (1976): lognormal assumption on A(t), exponential barrier, early defaults but rigid calibration of credit spreads. 2 Analytically Tractable First Passage (AT1P) models (Brigo et al ( )) extend the Black Cox model, giving the barrier and the asset volatility a time dependent structure which allows for an exact calibration of CDS data, though highlighting the problem of very short term credit spreads. Paola Mosconi Lecture 3 61 / 65

62 Summary Summary II 3 Hints are given at how to solve the problem of short term credit spreads, which is related to the inability of a continuous diffusive process for the asset and a deterministic barrier to capture the uncertainty related to unreliable accounting information (stochastic, continuous and discrete, extensions of the AT1P models) 4 Link to equity: liquid CDS data may be used to infer estimates of the equity volatility σ E, in cases where market quotes of equity are unavailable (e.g. credit-equity hybrid products). Paola Mosconi Lecture 3 62 / 65

63 Selected References Outline 1 Introduction 2 Merton Model Assumptions Basic Idea Asset Volatility Estimation Limits and Extensions 3 KMV Model Estimate of Asset Value and Volatility Distance to Default (DD) Expected Default Frequency (EDF) 4 First Passage Time Models Black and Cox Model AT1P Models 5 Link to Equity 6 Summary 7 Selected References Paola Mosconi Lecture 3 63 / 65

64 Selected References Selected References I Bielecki, T., and Rutkowski, M. (2001): Credit risk: Modeling, Valuation and Hedging. Springer Verlag Brigo, D. (2009), Essex Lecture Notes, Unit 3 Brigo, D., and Morini, M. (2006), Structural Credit Calibration, Risk Magazine, April issue Brigo, D., Morini, M., and Tarenghi M. (2009), Credit Calibration with Structural Models: The Lehman case and Equity Swaps under Counterparty Risk, Bielecki, Brigo and Patras (Editors) Recent advancements in theory and practice of credit derivatives, Bloomberg Press Duffie, D., and Lando, D. (1997), The term structure of credit spreads with incomplete accounting information, Econometrica, 69, Gupton, G.M., Finger, C.C., and Bathia, M. (1997), CreditMetrics, Technical Document Finkelstein, V., Lardy, J.P., Pan, G., Ta, T., and Tierney, J. (2002), CreditGrades Technical Document Paola Mosconi Lecture 3 64 / 65

65 Selected References Selected References II Geske, R. (1977), The Valuation of Corporate Liabilities as Compound Options, Journal of Financial and Quantitative Analysis, 5, Geske, R. (1979), The Valuation of Compound Options, Journal of Financial Economics, 7, Jones, E.P., Mason, S.P., and Rosenfeld, E. (1984), Contingent Claims Analysis of Corporate Capital Structure: An Empirical Investigation, Journal of Finance 39, Lo C.F., Lee H.C., and Hui, C.H. (2003): A Simple Approach for Pricing Barrier Options with Time-dependent Parameters, Quant. Fin., 3 Merton, R. (1974), On the pricing of corporate debt: The risk structure of interest rates. J. of Finance 29, Rapisarda F. (2003): Pricing Barriers on Underlyings with Time-dependent Parameters, Working Paper. id= Zhou, C. (1997), A Jump-Diffusion Approach to Modeling Credit Risk and Valuing Defaultable Securities Paola Mosconi Lecture 3 65 / 65