Stochastic models and hybrid derivatives

Claudio Albanese Department of Mathematics / Imperial College London ??? Presented at Bloomberg and at the Courant Institute, New York University

New York, September 22nd, 2005 Co-authors.

• Oliver Chen (National University of Singapore)

• Antonio Dalessandro (Imperial College London)

• Manlio Trovato (Merrill Lynch London) available at: www.imperial.ac.uk/mathfin

1 Contents.

• Part I. A term structure model.

• Part II. Credit barrier models with functional lattices.

• Part III. Estimations under P and under Q.

• Part IV. Pricing credit-equity hybrids.

• PART V. Credit correlation modeling and synthetic CDOs.

• Part VI. Pricing credit-interest rate hybrids.

2 PART I. A stochastic volatility term structure model

It is widely recognized that fixed income exotics should be priced by means of a stochastic volatility model. Callable constant swaps (CMS) are a particularly interesting case due to the sensitivity of rates to implied volatilities for very deep out of the money strikes. In this paper we introduce a stochastic volatility term structure model based on a continuous time lattice which allows for a numerically stable and quite efficient methodology to price fixed income exotics and evaluate hedge ratios.

3 Introduction

The history of interest rate models is characterized by a long series of turns. The Black formula for caplets and was designed to take as underlying a single forward rate under the appropriate forward measure, see (Joshi & Rebonato 2003). This has the advantage to lead to a simple pricing formula for European options but also the limitation of not being extendable to callable contracts. To have a more consistent model, short rate models where introduced in (Cox, Ingersoll & Ross 1985), (Vasicek 1977), (Black & Karasinski 1991) and (Hull & White 1993). These models are distinguished by the exact specification of the spot rate dynamics through time, in particular the form of the diffusion process, and hence the underlying distribution of the spot rate.

4 LMM models

Next came LIBOR market models, also known as correlation models. First introduced in (Brace, Gatarek & Musiela 1996) and (Jamshidian 1997), forward LIBOR models affirmed themselves as a mainstream methodology and are now discussed in textbooks such as for instance (Brigo & Mercurio 2001). Various extensions of forward LIBOR mod- els that attempt to incorporate volatility smiles of interest rates have been proposed. type extensions were pioneered in (Andersen & Andreasen 2000). A stochastic volatility extension is proposed in (Andersen & Brotherton-Ratcliffe 2001), and is further extended in (Andersen & Andreasen 2002). A different approach to stochastic volatility forward LIBOR models is described in (Joshi & Rebonato 2003). Jump-diffusion forward LIBOR models are treated in (Glasserman & Merener 2001), (Glasserman & Kou 1999). A cali- bration framework is proposed in (Piterbarg 2003).

5 Stochastic volatility models

Modeling stochastic volatility within LIBOR market models is a chal- lenging task from an implementation viewpoint. In fact, Monte Carlo methods tend to be slow and inefficient in the presence of a large num- ber of factors. In a strive to achieve a more reliable pricing framework, in recent years, we witnessed a move away from correlation models and the emergence and recognition as market standard of the SABR model by (S.Hagan, Kumar, S.Lesniewski & E.Woodward 2002) in the fixed income domain.

6 SABR

SABR however is unlikely to be the definitive solution and is perhaps rather just yet another stepping stone in a long chain. In fact, like- wise to the Black formula approach, SABR takes up a single forward rate under the corresponding forward measure as underlier. As a con- sequence, within this framework it is not possible to price callable swaps and Bermuda swaptions. There are also calibration inconsis- tencies. Implied swaption volatilities with very large strikes are probed by constant maturity swaps (CMS), structures which receive fixed, or LIBOR plus a spread, and pay the equilibrium swap rate of a given maturity. The asymptotic behavior of implied volatilities for very large strikes turns out to flatten out to a constant, as opposed to diverging rapidly as SABR would predict. Finally, some pricing inconsistencies may emerge with SABR due to the fact that the model is solved by means of asymptotic expansions with a finite, irreducible error.

7 Stochastic volatility term structure models

In this article we attempt to go beyond SABR by introducing a stochas- tic volatility short rate model which has the correct asymptotic behav- ior for implied swaption volatilities and can be used for callable swaps and Bermuda swaptions. Our model is solved efficiently by means of numerical linear algebra routines and is based on continuous time lat- tices of a new type. No calculation requires the use of Montecarlo or asymptotic methods and prices and hedge ratios are very stable, even for extreme values of .

8 Nearly stationary calibration

Our model is made consistent with the term structure of interest rates and the term structure of implied at-the-money volatilities by means of a calibration procedure that greatly reduces the degree of time dependency of coefficients. As a consequence, the model is nearly stationary over time horizons in excess of 30 years.

9 Applications

In this presentation, I discuss the model by reviewing in detail an implementation example. While leaving it up to the interested reader to pursue the many conceivable variations and extensions, we describe the salient features of our modeling by focusing in detail to the problem of pricing and finding hedge ratios for Bermuda swaptions and callable CMS swaps.

10 The model

Our model is built upon a specification of a short rate process rt which combines local volatility, stochastic volatility and jumps. We make our best effort to calibrate the model to a and introduce the least possible degree of explicit time dependence in such a way to refine fits of the term structure of interest rates and of selected at-the-money swaption volatilites. The model is specified in a largely non-parametric fashion within a functional analysis formalism and expressed in terms of continuous time lattice models.

A sequence of several steps is required to specify the short rate process rt.

11 The conditional local volatility processes

We introduce M states of volatility. The process conditioned to stay in one of such states α ∈ {1, ...M} is related to the solution rαt of the following equation:

βα drαt = κα(θα − rαt)dt + σαrαt dW. (1)

12 Short rate volatilities for each of the four volatility states

13 The functional analysis formalism

In the functional analysis formalism we use, these SDEs are associated to the Markov generators

2 2βα 2 r ∂ σαrαt ∂ L = κα(θα − rαt) + . (2) α ∂r 2 ∂r2

14 Functional lattices

To build a continuous time lattice (also called functional lattice), we discretize the short rate variable and constrain it to belong to a finite lattice Ω containing N + 1 points r(x) ≥ 0, where x = 0, ...N, r(0) = 0 and the following ones are in increasing order. The discretized Markov r generator LΩα is defined as the operator represented by a tridiagonal matrix such that

X r LΩα(x, y) = 0 y X r LΩα(x, y)(y − x) = κα(θα − r(x)) y X r 2 2 2βα LΩα(x, y)(y − x) = σαr(x) . y

15 Model parameters

In our example, we select an inhomogeneous grid of N = 70 points spanning short rates from 0% to 50%. We also choose to work with M = 4 states for volatility and make the following parameter choices:

α σα βα θα κα 0 31% 30% 2.10% .17 1 46% 40% 5.50% .18 2 75% 50% 8.50% .23 3 100% 60% 12.00% .24

16 How to solve in the special case of a local volatility model (M=1) and without jumps

Although our model is more complex than a simple local volatility process, it is convenient to describe our resolution method in the r specific case of the operator LΩα with constant α. This method can then be generalized and is at the basis of other extensions such as the introduction of jumps (see below).

17 Spectral analysis

We start by considering the following pair of eigenvalue problems:

r rT LΩαun = λnun LΩαvn = λnvn T where the superscript denotes matrix transposition, un and vn are r the right and left eigenvectors of LΩα, respectively, whereas λn are the corresponding eigenvalues. Except for the simplest cases, the Markov r generator LΩα is not a symmetric matrix, hence un and vn are different.

18 Spectral analysis

Also, in general the eigenvalues are not real. We are only guaranteed that their real part is non-positive Reλn ≤ 0 and that complex eigen- values occur in complex conjugate pairs, in the sense that if λn is an eigenvalue then also λ¯n is an eigenvalue. We set boundary conditions in such a way that there is absorption at the endpoints, and in par- ticular at the point corresponding to zero rates r(0) = 0. With this choice, we are also guaranteed that there exists a zero eigenvalue.

19 Spectral analysis

There is no guarantee, in the most general case, that there exists a complete set of eigenfunctions. However, the chance that such a complete set does not exist for a Markov generator specified non- parametrically is zero, so we can safely assume that this is the case. In the unlikely case that this assumption is not correct, the numerical linear algebra routines needed to solve our will identify the problem and a small perturbation of the given operator will suffice to rectify the situation. Assuming completeness, the diagonalization problem can be rewritten in the following matrix form:

r −1 LΩα = UΛU (3) where U is the matrix having as columns the right eigenvectors and Λ is the diagonal matrix having the eigenvalues λi as elements.

20 Functional calculus

Key to our constructions is the remark that, if the Markov generator is diagonalisable, we can apply an arbitrary function F to it by means of the following formula:

r r −1 F (LΩα) = UF (∗Ωα)U (4) This formula is at the basis of the so-called ”functional calculus”.

As Ito’s formula regarding functions of stochastic processes is central in the mathematical for diffusion processes, functional cal- culus for Markov generators plays a pivotal role in our framework for stochastic volatility models.

21 Functional calculus

This formula has several applications. An immediate one allows us to express the pricing kernel u(r(x), t; r(y),T ) of the process as follows:

(T −t)Lr X λ (T −t) u(r(x), t; r(y),T ) = (e Ωα)(x, y) = e n un(x)vn(y). (5) n

22 Introducing jumps

At this stage of the construction one has the to also add jumps. Although in the example discussed in this paper we are mostly focused on long dated callable swaps and swaptions for which we find that the impact of jumps can be safely ignored, adding jumps involves negligible additional complexities and is thus worth considering and implement- ing in other situations. To add jumps, one can follow the following procedure which accounts for the need to assign different intensities to up-jumps and down-jumps. Jump processes are associated to a special class of stochastic time changes given by monotonously non- decreasing processes Tt with independent increments.

23 Stochastic time changes

The time changes characterizing Levy processes with symmetric jumps are known as Bochner subordinators and are characterized by a Bochner function φ(λ) such that

h −λT i −φ(λ)t E0 e t = e (6)

24 The -gamma model

For example, the case of the variance which received much attention in Finance corresponds to the function µ2  µ φ(λ) = log 1 + λ (7) ν ν where µ is the mean rate and ν is the variance rate of the variance gamma process.

25 The generator of the

The generator of the jump process can be expressed using functional r calculus as the operator −φ(−LΩα). To produce asymmetric jumps, we specify the two parameters differently for the up and down jumps and compute separately two Markov generators

r L± = −φ±(−LΩα) = −U±φ±(Λ)V± (8) where: 2 µ± µ± φ±(λ) = log(1 + λ ) (9) ν± ν±

26 The generator of the process with asymmetric jumps

The new generator for our process with asymmetric jumps is obtained by combining the two generators above   0 ········· 0    L−(2, 1) d(2, 2) L+(2, 3) ···L+(2, n)  r  . . . .  L =  . . .. ··· .  jΩα    L (n − 1, 1) L (n − 1, 2) ··· d(n − 1, n − 1) L (n − 1, n)   − − +  0 0 ······ 0

27 Probability conservation and boundary conditions

Here the element of the diagonal are chosen in such a way to satisfy probability conservation:

X r d(x, x) = − LjΩα(x, y) (10) y6=x Also notice that we have zeroed out the elements in the matrix at the upper and lower boundary: this ensures that there is no probability leakage in the process.

28 Drift condition

In our setting, we choose a short rate as a modeling primitive and we thus do not need to impose a martingale condition. Otherwise, were we working with a forward rate instead, the appropriate method of restoring the martingale condition would be to modify the matrix elements of the resulting generator on the first sub-diagonal and the first super-diagonal.

29 The Local Levy generator

At this stage of the construction, we have therefore obtained a gener- r ator LjΩα for the short rate process, whose dynamics is characterized by a combination of state dependent local volatility and asymmetric jumps. We note that the addition of jumps has not increased the di- mensionality of the problem and is therefore computationally efficient.

30 Modeling the dynamics of stochastic volatility

As a third step, we define a dynamics for stochastic volatility by as- signing a Markov generator to the volatility state variable α which depends on the rate coordinate x. Namely, conditioned to the rate variable being x, the generator has the following form

sv sv sv Lx = (x)L+ + L− (11) where −0.7 0.7 0 0   0 0 0 0   0 1 1 0 8 0 3 1 4 1 4 0 0  sv  − . . .  sv  . − .  L+ =   , L− =   .  0 0 −1.5 1.5  0 3 −3 0  0 0 0 0 0 0 5 −5 (12)

31 Excitability function (x)

32 The total generator

Out of the two generators we just defined, we form a Markov generator L acting on functions of both the rate variable x and the volatility variable α. This generator has matrix elements given as follows:

r sv L(x, α; y, β) = LjΩα(x, y)δα,β + Lx (α, β)δxy. (13)

33 Numerical analysis

In our working example, the matrix L has total dimension MN = 280. For matrices of this size, diagonalization routines such as dgeev in LAPACK are very efficient. Since our underlier is a short rate though, we are not interested in the pricing kernel but rather in the discounted transition probabilities given by  R T  − rsds p(x, t; y, T ) = E e t , |rt = r(x), rT = r(y) . (14)

34 Numerical analysis

This kernel satisfies the following backward equation ∂ p(x, t; y, T ) + (Lp)(x, t; y, T ) = r(x)p(x, t; y, T ). (15) ∂t In functional calculus notations, the solution is given by

G(T −t) p(x, t; y, T ) = e (x, y) where G(x, y) ≡ L(x, y) − r(x)δxy. (16)

35 Numerical analysis with stochastic volatility

The same diagonalization method illustrated above for the local volatil- ity case applies also in this situation. By representing the matrix G in the form G = UΛU−1 (17) where Λ is diagonal, and writing the matrix of discounted transition probabilities as follows

eG(T −t) = UeΛ(T −t)U−1. (18)

36 Calibration and Pricing

In our example, to calibrate our model we aim at matching forward swap rates and at-the-money swaption volatilities, both referring to swaps of 5 year tenor. We start from the following data

1y 2y 3y 4y 5y 7y 10y 15y 20y 30y forward 2.999% 3.311% 3.587% 3.800% 3.984% 4.226% 4.393% 4.477% 4.301% 4.114% ATM vol 21.506% 19.443% 17.962% 16.967% 16.189% 14.897% 13.801% 12.460% 12.665% 11.728%

37 Nearly stationary calibration

The calibration procedure has two steps. In a first step we search for a best fit using the model above without introducing any explicit time dependency. In a second step, we then introduce time dependency to achieve a perfect fit. As a consequence of this procedure, the degree of time variability of model parameters is kept to a bare minimum. To introduce time dependence we combine two operations: a shift of the short rate by a time varying, deterministic function of time and a deterministic time change, i.e. ˙ rt → r˜t = b(t)rb(t) + a(t). (19)

38 Nearly stationary calibration

Here b(t) is monotonously increasing and b˙(t) denotes its time deriva- tive. Using the new process, discounted transition probabilities can be computed as follows:  R T  R T − r˜sds − a(s)ds E e t , |r˜t =r ˜(x), r˜T =r ˜(y) = e t G(x, b(t); y, b(T )) (20) where G is the kernel for the stationary process defined above.

Our choice in the working example is b(t) = 1.095t + 0.008t2. The function a(t) is then defined in such a way to match the term structure of forward swap rates. This adjustment is given in the next slide.

39 Deterministic yield adjustment (EUR)

40 Degree of time dependence

As one can see from this picture, the yield adjustment is less than 20 basis points in absolute value. This ensures that the probability of the modified short rate processr ˜t to attain negative values is small. In a typical implementation of the Hull-White model along similar lines, the short rate adjustment is typically of a few percent. The discrepancy is linked to the fact that the richer stochastic volatility model we construct is capable of explaining most of the salient features of the zero curve even with the constraint that the process be stationary.

41 Advantages of nearly stationary model calibration

The advantage of having a nearly stationary model is that the shapes of yield curves that one obtains depend on the short rate and the volatility state but are largely independent of time. The figure in the next slide shows the yield curves corresponding to different initial volatility states and different starting values for the short rate. As the graphs indicate, yield curves are sensitive to the initial volatility state as they raise if initial volatilities raise. Moreover graphs show that curves invert for high values of the short rate. In our model, this behavior is consistent over all time frames except for corrections of the order of 10 basis points.

42 Yield curves for different values of the initial volatility state and of the short rate

43 Pricing swaptions and callable constant maturity swaps

Implied volatilities for European swaptions are given in the next slide. Here we graph extreme out of the money strikes of up to 15% for swaptions of varying maturities where the deliverable is a 5Y swap. One can notice that implied volatilities naturally flatten out at long maturities, a behavior consistent with what observed in the CMS mar- ket where such extreme strike levels are probed.

44 for European swaptions (EUR)

45 Implied volatility for European swaptions (JPY)

46 Term structure of implied 5Y swaption volatilities (JPY)

47 Bermuda swaptions

Exercise boundaries for 10Y Bermuda swaptions are given in the next slides. The first graph refers to payer swaptions and the second to receiver swaptions.

48 boundaries for payer Bermuda options

49 Exercise boundaries for receiver Bermuda options

The corresponding graphs for callable CMSs are given below. Notice that the exercise boundaries depend on the volatility state.

50 Exercise boundaries for callable payer CMSs

51 Exercise boundaries for callable receiver CMSs

52 Sensitivities for Bermuda swaptions

Sensitivities for Bermuda swaptions are given in the next slides. These sensitivities are computed by holding the volatility state variable fixed and are defined as the of the price for a 10Y payer Bermuda swaption with respect to the rate of the 10Y swap.

53 Delta of a 10Y Bermuda swaption, with semi-annual exercise schedule, with respect to the 10Y swap rate. This is computed while holding fixed the volatility state variable.

54 Gamma of a 10Y Bermuda swaption, with semi-annual exercise schedule, with respect to the 10Y swap rate. This is computed while holding fixed the volatility state variable.

55 Sensitivities for Constant Maturity Swaps

Sensitivities of callable constant maturity swaps are given in the next slides. The delta and gamma are computed similarly to what done for Bermuda swaptions, while the vega is calculated instead with respect to the 10Y into 5Y European swaption.

56 Delta of a 10Y callable CMS swap, paying the 5Y swap rate with semi-annual exercise schedule, with respect to the 15Y swap rate . This is computed while holding fixed the volatility state variable.

57 Gamma of a 10Y callable CMS swap, paying the 5Y swap rate with semi-annual exercise schedule, with respect to the 15Y swap rate. This is computed while holding fixed the volatility state variable.

58 Vega of a 10Y callable CMS swap, paying the 5Y swap rate with semi-annual exercise schedule, with respect to the 10Y into 5Y European swaption price. This is computed while holding fixed the short rate.

59 Conclusions

We present a stochastic volatility term structure model, providing a consistent framework for pricing European and Bermuda options, as well as callable CMS swaps. The model is built upon a specification of a short rate process, which combines local volatility, stochastic volatility and jumps. The richness of the model allows to keep the degree of time variability of model parameters to a bare minimum, and obtain a nearly stationary behaviour. The solution methodology is based on a new type of continuous time lattices, which allow for a numerically stable and quite efficient technique to price fixed income exotics and evaluate hedge ratios.

60 PART II. Credit Barrier Models

Statistical data that we would like a credit model to fit includes:

• historical default probabilities given an initial credit rating

• historical transition probabilities between credit ratings

• interest rate spreads due to credit quality

61 Credit-rating based models

The early models of this class considered the credit-rating migration and default process as a discrete, time-homogenous and took the historical transition probability matrix as the Markov transition matrix.

Deficiencies:

• difficult to correlate

• risk-neutralization leads to unintuitive results

62 Analytic closed form solutions versus numerical linear algebra methods

The former framework for credit barrier models leveraged on solvable models.

In the newer version recently developed we have a flexible non-parametric framework, whereby tractability comes from the use of numerical lin- ear algebra as opposed to coming from the analytical tractability of special functions.

63 The underling diffusion process

The first building block of our construction is a Markov chain process xt on the lattice Ω = {0, h, ..., hN} ⊂ [0, 1] where N is a positive integer and h = 1/N. In the case of a discretized diffusion with state depen- dent drift and volatility, the infinitesimal generator L, of the process xt is a tridiagonal matrix and can be expressed as follows in terms of finite difference operators:

Lx = a(x)∆ + [b(x) − a(x)]∇+ where x ∈ Ω and

(∆f)(x) = f(x+1)+f(x−1)−2f(x), and (∇+f)(x) = f(x+1)−f(x). (21)

64 Continuous space limit

To ensure the existence of a continuous space limit, we derive the functions a(x), b(x) from a drift function µ(ξ) and a volatility function σ(ξ), where ξ ∈ [0, 1], which is identifiable as the credit quality process and can be expressed in terms of its infinitesimal by imposing the following conditions: P y L(x, y)(y − x) = µ(hx) P 2 2 y L(x, y)(y − x) = σ(hx) P y L(x, y) = 0

65 The P and the Q measure

In our model, we actually use two drift functions: µP (ξ) and µQ(ξ), one defining the P or statistical measure and the latter modeling the Q or pricing measure. We postulate that the only difference between the P and the Q measure lies in the specification of these two drift func- tions. Correspondingly, we use the subscripts P and Q to identify the Markov generator and transition probabilities under the corresponding measure. Whenever the subscripts are omitted as here below, formulas apply to both the P and the Q measure.

66 Eigenvalue problems and functional calculus

To manipulate the Markov generator by means of functional calculus, the first step is to diagonalize it. Let λn be the eigenvalues of the operator L and let un(x) and vn(x) be the right eigenvectors, so that

Lun = λnun.

67 Numerical methods for eigenvalue problems

In most cases, Markov generators admit a complete set of eigenvec- tors. Although there are exceptions where diagonalization is not pos- sible and one can reduce the operator at most to a non trivial Jordan form with non-zero off-diagonal elements, these exceptional situations occur very rarely both in a measure theoretic sense, as exceptions span a set of zero measure, and in a topological sense as their complement is dense in the space of all generators. In practical terms, this implies that non-diagonalizable operators arise very rarely if at all in practice and whenever they do, a professional numerical diagonalization algo- rithm would detect the problem and a small perturbation of the model parameters would rectify. To carry out numerical diagonalization, we find that the function dgeev in the public domain package LAPACK is quite suitable.

68 Diagonalizing the Markov generator

We just assume that the operator L admits a complete set of eigen- vectors. In this case, we can form the matrix U whose columns are given by the eigenvectors un(x) and write

L = UΛU−1. (22) −1 We denote with V the operator U and with vn(x) its row vectors.

69 Functional calculus

Key to our constructions is the remark that if the matrix operator L is diagonalizable we can apply an arbitrary function F to it by means of the following formula:

F (L) = UF (Λ)U−1 (23) This formula is at the basis of the so-called ”functional calculus”. As Ito’s formula regarding functions of stochastic processes is central in the stochastic analysis for diffusion processes, functional calculus for Markov generators plays a pivotal role in our framework for stochastic volatility models. This formula has several applications. An immediate one allows us to express the pricing kernel u(x, t; y, t0) of the process as follows:

0 (t0−t)L X λ (t0−t) u(x, t; y, t ) = (e )(x, y) = e n un(x)vn(y). (24) n

70 Introducing jumps

At this stage of the construction we add jumps. Jumps are ubiquitous in credit model and we find that a jump component is necessary in order to reconcile observed default probabilities with credit migration probabilities. Within our framework, adding jumps involves marginal additional complexities from the numerical viewpoint.

71 Asymmetric jumps

To reflect asymmetries in the jump intensities, we model separately up and down jumps. A particularly interesting class of jump processes is associated to stochastic time changes given by monotonously non- decreasing processes Tt with independent increments. These time changes are known as Bochner subordinators and are characterized by a Bochner function φ(λ) such that

h −λT i −φ(λ)t E0 e t = e (25) For example, the case of the variance gamma process which received much attention in Finance corresponds to the function µ2  µ φ(λ) = log 1 + λ (26) ν ν where µ is the mean rate and ν is the variance rate of the variance gamma process.

72 Functional calculus with subordinated generators

The generator of the jump process corresponding to the subordination of a process of generator L can be expressed using functional calculus as the operator −φ(−L). To produce asymmetric jumps, we specify the two parameters differently for the up and down jumps and compute separately two Markov generators

L± = −φ±(−L) = −U±φ(−Λ±)V± (27) where: 2 ! µ± µ± φ±(λ) = log 1 + λ (28) ν± ν±

73 Generators with asymmetric jumps

The new generator for our process with asymmetric jumps is obtained by combining the two generators above   0 ········· 0    L−(2, 1) d(2, 2) L+(2, 3) ···L+(2, n)   . . . .  L =  . . .. ··· .     L (n − 1, n) L (n − 1, 2) ··· d(n − 1, n − 1) L (n − 1, n)   − − +  0 0 ······ 0 Here the element of the diagonal are chosen in such a way to satisfy the condition of probability conservation

d(i, i) = − X L(i, j) (29) j6=i Also notice that we have zeroed out the elements in the matrix at the upper and lower boundary: this ensures that there is no probability leakage in the process.

74 Adding jumps

At this stage of the construction, we have therefore obtained a gen- erator Lj for the process of distance to default, whose dynamics is characterized by a combination of state dependent local volatility and asymmetric jumps. We note that the addition of jumps has not in- creased the dimensionality of the problem and is therefore computa- tionally efficient.

75 PART III: Estimation and calibration: P measure

We first estimate the process for distance to default xt with respect to the statistical measure P by matching transition probabilities over one year and default probabilities over time horizons of 1, 3 and 5 years.

A credit rating system consists of a number K of different classes. In the case of the extended system by Moody’s, K = 18 and the ratings are:

{0, 1,..., 17} ↔ {Default, Caa, B3, Ba3, Ba2,..., Aa3, Aa2, Aa1, Aaa}

76 Introducing barriers

We subdivide the nodes of the lattice Ω into K subintervals of adjacent nodes:

Ii = [xi−1, ...xi] (30) N where 0 = x0 < x1 < ... < xK = N and #(xi − xi−1) = K , for i = 1, ..., K. The interval Ii corresponds to the i-th rating class. If a process is in Ii at time t, then is said to have a credit rating of i. ∀i, x¯i ∈ Ii denotes the initial node. The conditional transition probability p˜ij(t) that an obligor with a given initial rating i at time 0 will have a rating j at a later time t > 0 can be estimated by matching it with historical averages provided by credit assessment institutions.

77 Introducing barriers

For our purposes, we model this quantity as follows:

aj−1 X p˜ij(t) = uP (0, x¯i; t, y). y=aj−1 wherex ¯i is a point in the interval Ii which represents the barycenter of the population density in that credit class and is part of the model specification. For simplicity’s sake, we takex ¯i to be the midpoint of the interval Ii.

78 The state of default

Absorption into the state x = 0 is interpreted as the occurrence of default. The probability that starting from the initial rating i and reaching a state of default by time t is given by

D p˜i (t) = uP (0, x¯i; t, 0).

The model under P is characterized by a drift function µP (ξ), a volatility function σ(ξ) and jump intensities. The first two func- tions are graphed below, while the variance rates we estimated are ν+ = 7.5, ν− = 4.

79 Local volatility σ(ξ) vs. distance to default

80 Local drifts µP (ξ) and µQ(ξ) vs distance to default under the P and the Q measure, respectively.

81 Comparison of discrete model (lines) and historical (dots) one year transition probabilities.

82 Comparison of discrete model (lines) and historical (dots) de- fault probabilities.

83 Estimation and calibration: Q measure

Risk neutralization is defined by changing the drift function µP (ξ) into µQ(ξ), while leaving everything else unaltered.

The new drift is chosen in such a way to fit spread curves. Term structures of probability of default for each rating class are given by

D q˜i (t) = uQ(0, x¯i; t, 0). (31)

84 CDX index spreads

In our example, we use CDS spreads for 125 names in the Dow Jones CDX index. We looked at 5 datasets by Mark-it Partners correspond- ing to the last business days of the months of January, February, March, April and May 2005. The datasets provide CDS spreads at maturities: 6m, 1y, 2y, 3y, 5y, 7y, 10y and tentative recovery rates for each name. We insist that the CDS spreads be matched by our model and take the liberty of adjusting the term structure of recovery rate for each name. Besides having to estimate the drift under Q we also estimate the current distance to default for each name. The objective is to ensure that the term structure of recovery rates be as flat as possible and as close as possible to the tentative input value.

85 Comparison of discrete model (lines) and market (dots) for spread curves of investment grade bonds (Data taken 02/10/2003)

86 Comparison of discrete model (lines) and market (dots) for spread curves of speculative grade bonds (Data taken 02/10/2003)

87 Liquidity spreads

From these pictures one can notice a systematic bias in spreads. Our model appears to systematically underestimate BB spreads and over- estimate BBB spreads.

This can be interpreted in terms of the differential liquidity in the two market sectors.

88 CDS Spreads: Investment Grades (Data from March 2005)

89 CDS Spreads: Non-Investment Grades (Data from March 2005)

90 Implied term structure of recovery rates

We observe that the implied term structures of recovery rates are highly correlated across names and to the general spread level. This is not surprising as recovery levels are known to be linked to the economic cycle. Hence implied recoveries reflect the market perception of the future economic cycle. As we compare the implied recovery cycles on the last business day of January, March and May 2005, we notice that the implied recovery cycle appears equally pronounced in the three months. However, the ones in January and March show a much greater degree of coherence across names, perhaps a signature of the fact that in January and March markets were rather tranquil and efficient, while in May 2005 dislocations occurred.

91 Implied recovery cycles for the CDX components on January 31st, 2005.

92 Implied recovery cycles for the CDX components on March 31st, 2005.

93 Implied recovery cycles for the CDX components on May 31st, 2005.

94 Risk-neutral transition probabilities

In the risk-neutral setting we can also calculate risk-neutral transition probabilities. These are necessary to price credit-rating dependent options.

How do we expect risk-neutral transition probabilities to behave? In- dependent of the model, since risk-neutral default probabilities are greater than historical default probabilities, one would expect down- grades in credit-rating to be more probable in the risk-neutral setting than historically.

95 96 97 98 PART IV. Equity default swaps (EDS)

Equity default swaps are a new class of instruments that several dealers started marketing this year. They are defined as out of the money American digital puts struck at 30% of the spot price. Typical maturity is 5 years and the premium is paid in installments by means of a semi- annual coupon stream.

In this example, I will compare CEV prices with the prices one obtains from credit barrier models. The latter, are models estimated to ag- gregate data, namely the credit transition matrix, default probabilities and curves. The credit equity mapping is obtained by fitting at-the-money implied volatilities as a function of the ratings.

99 Main Finding

It appears that the market is currently pricing EDSs by means of diffusion local volatility models and that this is not entirely consistent with data. The marked differences in prices are due to the fact that the credit barrier model accounts for the phenomenon of ”fallen angels” by introducing and calibrating a jump component in the process.

100 Credit-Equity mapping

101 Mapping to equity

The credit quality is mapped to equity prices via a deterministic, mono- tonic function Φ at some horizon date T :

rT ST (ζ) = e Φ(ζ)

For ti < T , we take the discounted expectation of Φ:

rti Sti(ζ) = e E[Φ(ζ)|ζti]

102 A snapshot of market EDS spreads

103 Credit quality versus stock price (the credit equity mapping)

104 Local volatility

105 At-the-money implied vols as a function of credit quality

106 EDS spreads as a function of rating

107 CDS to EDS spread ratio as a function of rating

108 CDS to EDS spread ratio based on the CEV model

109 PART V. Credit correlation modeling and lattice models for synthetic CDOs.

Having characterized the process for credit quality xt and identified starting points for each individual procsess, the next step is to in- troduce correlations by conditioning to economic cycle scenarios, thus introducing a correlation structure among the credit quality processes.

The economic cycle is modeled by means of a non-recombining lat- tice of the structure sketched below. The underlying index variable is allowed to take up two values on each period ∆t. An upturn corre- sponds to a ”good” period while a downturn to a ”bad” period for the economy. In our example, we chose the time step to be ∆t = 1y and find that this choice is sufficient to provide great flexibility in the tuning of the correlation structure.

110 Conditioning the Lattice to a Market Index

To explain our methodology to introduce correlations, we consider first a simple case whereby the model is characterized by a pair of complementary transition probabilities w, (1 − w) ∈ [0, 1] at each node, which we assume constant. In order to condition the continuous time lattice corresponding to a given credit quality process to the economic index variable we introduce the notion of local beta given by function β(ξ) which provides the corresponding sensitivity. The limiting cases of β(ξ) = 0 and β(ξ) = 1 correspond to zero and full correlation between a name with a given credit quality hx ∈ [0, 1] and the cycle variable.

Along the path of each given scenario on the tree, the unconditional kernel of the credit quality process is replaced by conditional transition probabilities defined as follows:

± ± uw,β(t, x; t + ∆t, y) = (1 − β(hx))u0(t, x; t + ∆t, y) + β(hx)u1 (t, x; t + ∆t, y) 111 Here u0 = u is the unconditional kernel and corresponds to a zero ± β(hx). In the opposite case of β(hx) = 1, conditional kernel u1 (x, y) has the following form:  u(x, y) if y > m(w, x)  + 1  P  u1 (x, y) = w − y>m(w,x) u(x, a(w, x)) if y = m(w, x) 1 − w  u(x, y) = 0 if y < m(w, x) (32) and  0 if y > m(w, x)  − 1  + u1 (x, y) = u(x, m(w, x)) − u1 (x, m(w, x)) if y = m(w, x) (33) w  u(x, y) if y < a(w, x). where ( ) m(w, x) = inf m = 0, ...N| X u(x, y) ≤ w . (34) y

− + u(x, y) = wuβ (x, y) + (1 − w)uβ (x, y). (35)

Conditioning is achieved by forming a weighted sum over all paths in − the event tree. On a given path, we use uβ for a bad period scenario + and uβ for a good one. The weight of a path is the product of a number of factors w equal to the number of bad periods and a number of factors (1 − w) for each one of the good periods. With this method, marginal probabilities are kept unchanged while correlations are induced on the single name processes. More specifically, one can price all credit sensitive instruments specified with the given names one can first evaluate the conditional prices PΓ by means of the following multiperiod kernel:

(t −t )L (t −t )L e i i−1 γi · ... · e n n−1 γi. where Γ = {γ1, . . . , γn} runs over the sets of conditional paths due to the scenario of the index. The (unconditional) price is then given by:

X n−(Γ) n (Γ) P = w (1 − w) + PΓ. Γ

This construction can be generalized. Consider a number M > 1 of percentile levels 0 < w1 < ... < wM < 1 and let qi ∈ [0, 1], i = 1...M be P a corresponding set of probabilities summing up to one, i.e. i qi = 1. Then we can set M ± X ± u−→ (t, x; t + ∆t, y) = q u (t, x; t + ∆t, y). (36) w ,β i wi,β i=1 The formulas above extend also to this case as long as one replaces P the weight w with the average weight i qiwi.

The choices we make for the March and June datasets are graphed below. Here one can observe that the levels we were led to choose in June are lower and the probabilities more uneven than in March. This can be interpreted as saying that the model is detecting a higher level of implied correlation between jumps in the June data than in March. Specifications for the weights qi and percentile levels wi.

112 The local beta function

Modeling correlation is key to pricing basket credit derivatives. Buyers and sellers of basket credit derivatives have a wide range of arbitrage- free prices to choose from, and it is the market that determines, both in principle and in practice, a definite price. In our framework, tranches of varying seniority are priced by calibrating the local beta function β(ξ).

113 Specifications for the function β(x) in March and June 2005.

114 Decoupling of correlation

Notice that as an effect of GM and Ford being downgraded, the local beta function responded by lowering on the side of low quality grades while rising on the high qualities. This resulted in a simultaneous fall of equity prices and widening of senior spreads.

115 Contagion skew

A useful graph to assess the impact of the specification of the local β(ξ) function on our correlation model is the contagion skew in the next slide. This graph is constructed as follows: we first compute the unconditional default probabilities as a function of credit quality. Next, for each value of credit quality, assuming that a name of that quality defaults within a time horizon of 5 years, we compute the conditional probability of defaults for all other name. Finally, we take an average over all values of credit quality of the ratio between the conditional and unconditional probabilities. As the graph shows, the higher is the credit quality of a defaulted name, the larger is the impact on all other names. The steepness of this curve controls precisely the discrepancy between prices for senior tranches as compared to junior ones.

116 Contagion Skew: ratio between conditional and unconditional probability of defaults, where conditioning is with respect to the default of a name whose credit quality is on the X axis.

117 Pricing CDOs

Although CDX index tranches are written on 125 underlying names, we observe that our lattice model performs quite efficiently. We separate the numerical analysis in two different steps. In the first we go through all names and generate conditional lattices. We choose ∆t = 1y and a time horizon of 5y, so that we obtain a total of 32 scenarios. This is a pre-processing step which is independent of the CDO structure. This step typically takes a few minutes for a hundred names and could be carried out periodically and offline for the universe of all traded names. The pricing step instead takes only a few seconds and requires generating the probability distribution function for CDO portfolios over the given time horizon.

118 Expected Loss Distribution for CDX index tranches in March

2005

119 Expected Loss Distribution for CDX index tranches in June 2005

120 Pricing CDOs

The model can be calibrated by adjusting the function β(ξ), ξ ∈ [0, 1], the thresholds wi, i = 1, ...M and the corresponding probabilities qi.

121 Tranche prices for March 20th 2005

attachment detachment spread mktspread 0% 3% 499.6 bp (+32% uff) 500 bp(+32% uff) 3% 6% 187.4 bp 189bp 6% 9% 108.6 bp 64bp 9% 12% 56.5 bp 22bp 12% 22% 6.7 bp 8bp where ”uff” stands for ”upfront fee”.

122 Calibration

Notice that a good agreement can be reached with the equity, junior mezzanine and senior tranche. On the other hand, the model appears to over-estimate the price of the two senior mezzanine tranches 6-9 and 9-12 by a factor 2-3. This might be in relation to the high degree of liquidity of these tranches and appetite for this risk profile.

123 Tranche prices for June 20th 2005

attachment detachment spread mktspread 0% 3% 499.7 bp (+49% uff) 500 bp(+49% uff) 3% 6% 170.1 bp 177bp 6% 9% 30.4 bp 54bp 9% 12% 27.5 bp 24bp 12% 22% 10.0 bp 12bp

124 Hedge ratios of the various CDO tranches for March 2005 plot- ted against 5Y CDS spreads.

125 Hedge ratios.

One can notice that the hedge curves for the equity and the junior mezzanine are fairly different and as a consequence it does not appear as appropriate to use the mezzanine as a proxy to hedge credit expo- sure at the equity tranche level. The differentiation among the two profiles is a direct consequence of the steep aspect of the local beta function.

126 Conclusions

We propose a novel approach to dynamic credit correlation modeling that is based on continuous time lattice models correlated by con- ditioning to a non-recombining tree. The model describes not only default events but also rating transitions and spread dynamics, while single name marginal processes are preserved.

127 PART VI. Credit-interest rate hybrids

Functional lattices for the dynamic CDO model and for the term struc- ture model covered above can be combined and correlated while pre- serving the specification of marginal processes. This opens the possi- bility of pricing credit - interest rate hybrid instruments.

As an example of these applications, in the following, we consider cancellable interest rate swaps which are linked to the default of either one name in the CDX index, or to the first default event of a name in a given basket, or to the default of the CDX equity tranche. We also consider interest floors that cancel upon the default of the equity tranche. In all cases, we evaluate also hedge ratios.

128 Single name, credit linked cancelable swaps

129 First to default cancelable

130 Basis of a CDO subordinated interest rate swap

131 Hedge ratios for a CDO subordinated interest rate swap

132 Price of a CDO subordinated interest rate floor

133 Hedge ratios for a CDO subordinated interest rate floor

134 Conclusions

We find that our model is well suited for interest rate hybrids. It is numerical efficient and since it does not involve a Montecarlo step, hedge ratios have no simulation noise.

We find that, within a local beta model for credit correlations, hedge profiles tend to be relatively higher for the better quality ratings which are more correlated to the business cycle.

135