國立臺灣大學管理學院財務金融學系暨研究所 碩士論文

國立臺灣大學管理學院財務金融學系暨研究所 碩士論文

Department of Finance College of Management National Taiwan University

利用具相關性之二元樹模型做信用組合違約模擬 Credit Portfolio Simulation Using Correlated Binomial

Lattices

黃琮凱

Tsung-Kai Huang

指導教授：呂育道 博士、王之彥 博士 Advisors: Professor Yuh-Dauh Lyuu, Ph.D.

Dr. Jr-Yan Wang, Ph.D.

中華民國 97 年 7 月 July 2008

Abstract

We revisit the models developed in Das and Sundaram (2004) and Bandreddi, et al.

(2007). Bandreddi, et al. (2007) use a simplified version of the model developed by Das and Sundaram for correlated default simulation. We find that in their setting, problematic probabilities may arise which may cause biased results for the purpose of default simulation and the pricing of derivative products. We suggest an alternative model — the D-CEV model, as an alternative to address this problem. The new model is an extension of a popular binomial model and is easy to implement. We further explore the natural characteristics of our alternative method with several numerical experiments. Our proposed model is found to resolve the unpleasant flaws in the model of Bandreddi, et al. (2007) while preserving its desirable properties. We also show how this framework accounts for several empirical features.

TABLE OF CONTENTS

I. INTRODUCTION...4

II. LITERATURE REVIEW ...8

2.1 T^HE DEFAULTABLE CRRM^ODEL...8

2.2 CALIBRATING P^ARAMETERS...9

2.3 S^IMULATING C^ORRELATED D^EFAULT... 11

III. AN ALTERNATIVE MODEL: THE DEFAULTABLE CEV MODEL...14

3.1 PROBLEMS WITH THE D-CRRM^ODEL...14

3.2 T^HE D^EFAUTABLE CEVM^ODEL...15

IV. NUMERICAL EVALUATIONS ...20

4.1 B^ASE C^ASE D^ATA...20

4.2 T^HE I^{MPACT OF} E^QUITY C^ORRELATION...22

4.3 T^HE I^{MPACT OF} E^QUITY V^OLATILITY...23

4.4 T^HE I^{MPACT OF} I^NTENSITY F^UNCTION P^ARAMETERS...28

V. FURTHER RESEARCH AND APPLICATIONS ...33

5.1 B^ASKET D^EFAULT S^WAPS...33

5.2 I^NTENSITY CORRELATION VS.CONDITIONAL C^ORRELATION...33

5.3 RESULTS FOR NTH-^TO-D^EFAULT C^ONTRACTS...37

5.4 T^HE VALUATION OF CDOT^RANCHES...42

5.5 RESULTS FOR CDOT^RANCHES...44

VI. CONCLUSIONS ...47

REFERENCES ...48

I. Introduction

The market in credit derivative products has experienced a tremendous growth in the past years and plays an important role in today’s financial market by facilitating the transfer and trading of credit risk. The International Swaps and Derivatives Association (ISDA) reported in April 2007 that the total notional amount on outstanding credit derivatives was $35.1 trillion with a gross market value of $948 billion. Portfolio credit derivatives such as collateralized debt obligations (CDOs) and basket default swaps account for a significant portion of the market, which have drawn much attention in recent years.¹ A CDO is an asset-backed securitized structure that distributes credit risk to investors by creating “tranches.” Tranches are each responsible, in a sequential order, for credit losses in the reference portfolio backing the CDO. Thus its valuation highly depends on the correlated default behavior of the underlying assets.

Two main quantitative approaches for valuing credit risk and default correlation are the structural-form credit models and the reduced-form credit models.

Structural-form models are based on the original framework developed by Merton (1974) and its extension by Black and Cox (1976), using the principles of option pricing. In this model a default occurs if the value of a company falls below a default barrier before the maturity of its debt liabilities. Zhou (2001b), Hull and White (2001), and Hull, Predescu, and White (2005) propose structural-form models for multi-issuer cases. Their models are dynamic in the sense that the credit qualities of companies evolve through time. Structural-form models possess economic rationale in which they provide a link between the credit quality of a firm and the firm’s financial situation. Still, due to the assumption of complete information about the firm’s assets and liabilities, the default event is not a total surprise. This is often referred to as the

“predictability” of structural-form models (see Giesecke (2004) and Jarrow and Protter (2004)). Since default can be anticipated, the model price of a credit sensitive security converges continuously to its recovery value, in conflict with empirical observation where prices abruptly drop to its recovery value upon default. Moreover, the model implied credit spread for the firm’s debt tends to zero for short time-to-maturity, at odds with positive short-term spreads seen in practice (see

1 According to the Securities Industry and Financial Markets Association, aggregate global CDO issuance totaled US$ 157 billion in 2004, US$ 272 billion in 2005, US$ 552 billion in 2006 and US$

486 billion in 2007. Research firm Celent estimates the size of the CDO global market to close to $2 trillion by the end of 2006.

Giesecke (2004)). Hence, such predictability of default times in this type of models is regarded as a major drawback. This consideration brings us to reduced-form models.

Reduced-form models, first studied by Jarrow and Turnbull (1995) and Duffie and Singleton (1999), overcome this deficiency of predictability by assuming that default occurs without warning at an exogenous default rate (or intensity), characterized by jump processes. The intensity is extracted from the market prices of a firm’s defaultable instruments such as corporate bonds or credit default swaps, which contain a default risk premium demanded from investors of uncertainty about default events. The main shortcoming with reduced-form models is that the arrival of default is not based on any characteristic of the firm’s underlying credit quality, but on market prospects. Nevertheless, they are more commonly used by practitioners for pricing, hedging, and trading purposes (see Jarrow and Protter (2004)). Among reduced-form approaches for multi-issuer cases, copula methods have become popular for pricing correlated default. These types of models were first introduced by Li (2000) and later extended by Gregory and Laurent (2005), which has become the standard market model for portfolio credit derivatives. However, common copula models are static models; they assume constant hazard rates through the whole term of a contract.

But a changing default environment is more realistic.

Nowadays, researchers and practitioners seek to find innovative methods for the valuation of credit portfolio defaults. For example, Longstaff and Rajan (2006) develop a so called top-down approach to model credit portfolio losses using a direct method that does not require modeling individual correlations, whereas Carayannopoulos and Kalimipalli (2003), and Das and Sundaram (2004) suggest another method where equity correlations may be used to drive intensity correlations.

Das and Sundaram (2004) introduce a simple model for pricing securities with equity, interest-rate, and default risk. Their model is a reduced-form model. Default probabilities in their framework are derived endogenously on a binomial lattice calibrated to credit default swap markets. It is claimed that the model captures default information from both equity- and debt-market information rather than just from equity-market information (as in structural-form credit models) or just from debt-market information (as in reduced-form credit models). Bandreddi et al. (2007) use a simplified version of their model to simulate correlated defaults for credit portfolios. We will call it the defaultable CRR model (D-CRR for short). Their framework contains three main components. First, one develops for all reference issuers in a credit portfolio their equity binomial lattices with default risk considered.

Second, one calibrates the lattices to the credit default swap market. Third, one simulates default with the correlated lattices to examine default risk distributions and default correlations.

Unfortunately, the model developed in Bandreddi et al. (2007) gives rise to problematic probabilities on the lattices. In particular, probabilities outside the range of 0 to 1 can arise, leading to biased results when applied to default modeling.

Furthermore, our numerical analyses show other deficiencies of their model in practice, such as its conflict with traditional structural-form models in the relationship between credit risk and equity volatility. To remedy them, we suggest an alternative method to build the lattice. Our method is a generalized extension of the well-known CEV model that takes into account the “leverage effect” and is easy to implement. We call it the defaultable-CEV model (D-CEV for short). The D-CEV model is a simpler version of the model first seen in Das and Sundaram (2007), which was used for the pricing of convertible bonds with equity, interest rate, and credit risk. This thesis will analyze how our proposed model addresses the drawbacks of the D-CRR model without sacrificing their desirable properties. We compare the D-CEV model with the D-CRR model in the pricing of credit derivative products, and find that our model produces higher mean levels of default, which was neglected by the previous D-CRR approach. An examination on how different input parameters affect the results for default simulation in our framework is also performed.

In summary, the framework discussed in this thesis is a new approach that is easy to understand, in which observable equity prices are used along with an intensity-base model to simulate default in an arbitrage-free setting. The model is dynamic in that hazard rates evolve through time, which is an appealing feature for the hedging of credit positions and the valuation of new-generation credit products. The flexibility of the model enables accommodation of several known empirical phenomena. First, empirical research has shown evidence of the joint movements between credit spreads and stock option implied volatilities (see Hull, Nelken and White (2004) and Carr and Wu (2006)). The D-CEV model proposed in our thesis can be simultaneously calibrated to both the term structure of credit default swap spreads and the equity options market. We will also show how in the D-CRR model one may encounter undesirable outcomes when we calibrate to equity volatility. Second, there is evidence indicating that corporate defaults “cluster” in time (see Das, Duffie, Kapadia, and Saita (2005)). The framework discussed in this thesis allows for the accommodation of this fact. We can switch between the assumption of independence of default events, where we consider only intensity correlations, and the assumption of dependence of default events conditioned on intensities, where the default clustering effect is matched.

The remainder of this thesis is organized as follows. In Chapter II we review the previous framework for default modeling. Chapter III points out their disadvantages and describes our improved model. In Chapter IV we perform numerical experiments

to explore the characteristics of our model. Chapter V makes a comparison between the D-CRR model and D-CEV model in the valuation of portfolio credit derivatives.

Chapter VI concludes the thesis.

II. Literature Review

The equity binomial lattice introduced in Das and Sundaram (2004) extends the Cox, Ross, Rubinstein (CRR) model by including a jump-to-default branch for each node.

Bandreddi et al. (2007) apply this model to simulate correlated default. Their framework is summarized in this Chapter.

2.1 The Defaultable CRR Model

In a discrete-time setting with time intervals of length h, the evolution of the equity price S to a stochastic value _t S_{t h+} is assumed to be of the following pattern:

(up move) w/prob (1 ) (down move) w/prob (1 )(1 ) 0 (default) w/prob

t t

t h t t

t

uS q

S dS q

π π π

+

⎧ −

=⎪⎨ − −

⎪⎩

(1)

The respective “up-move” and “down-move” parameters u and d are based on CRR’s settings, where u=e^{σ h} and d =1/u. Intuitively, equity is a security that is assumed to receive zero recovery upon the occurrence of a credit event, thus a third branch to 0 is incorporated. Given the risk-neutral default intensity from time t to

t+h as ξ_t , the default arrival follows a Poisson process, where the default probability π_t over the period t to t+h should be

1 e ^th

t

π = − ⁻ξ .

In a risk-neutral world, the equity price should fulfill the martingale condition

[ ]

rh

t t h

S =e⁻ E S₊ . Hence,

(1 ) (1 ) (1 ) 0

rh

t t t t t t

e S =uS × × −q π +dS × − × −q π + × , π leading to the risk-neutral probability

(r )h

e d

q u d

ξ

+ −

= − . (2)

Notice that the probability of an up-move becomes higher compared to that of CRR’s since the jump to zero of the stock price must be compensated.

The most critical step for building the lattice in this model is to give a characterization of the default intensity. The intensity is endogenously determined with the following equation:

exp( )

t

t

t S^β

ξ = α γ⁺ , (3)

where α β γ, , are entitled the intensity-function parameters. The idea of this function comes from the fact that equity prices tend to reflect the credit risk of a firm.

It is easy to see that in this inverse relationship, as stock prices tend to be low, the default intensity moves higher. We can visualize how such a feature is analogous to structural-form models, where lower equity value leads to lower firm value and higher tendency of hitting the default barrier. The zero equity value upon default is used as a response to the barrier condition. When a firm’s value is below its liabilities, equity holders get nothing. This link to equity values is economically attractive since stock prices in this model are observable whereas firm values in traditional structural models are unobservable. In the D-CRR model, the actual default event is not determined by the absolute equity value, however, but by the intensity in a Poisson arrival, as in reduced-form models. Therefore this model differs from typical structural-form models in the degree of predictability of default. The intensity in this model is an endogenous process characterized by the parameters α β, , and γ . We will show in chapter IV that the three intensity function parameters α β γ, , capture the level, slope, and curvature of the term structure of credit default swap spreads, respectively.

2.2 Calibrating Parameters

The parameters , , α β γ are calibrated to the credit default swap market. A credit default swap is an insurance-like instrument to transfer the credit risk of fixed-income products. It is a contract between two counterparties in which a protection buyer pays a fixed fee periodically to a protection seller that guarantees a contingent payment

Figure 1

Illustration of the Pricing of Credit Default Swaps

(1) (2)

upon a credit event (such as default or failure to pay) happening in the reference entity agreed upon in the contract. The market of credit default swaps represents one of the fastest-growing derivatives markets.² Its high liquidity provides efficient default information, where the availability of term structures of credit default swap spreads allows one to fit the intensity-function parameters mentioned above.

Credit default swaps are quoted as the spread payment per annum made by the protection buyer. How they are priced is much discussed in the literature (for example, see Duffie (1999) or Hull and White (2000) for their no-arbitrage approaches). The main concept behind pricing a credit default swap is that the periodic payment should be the spread that equates the present value of payments made by the protection buyer with the expected loss on default over the life of the default swap contract. This is easy to implement on the lattice since the probabilities of moving on to all states and defaulted nodes are already determined when the lattice is built. As shown in Figure 1, one may begin by putting a spread payment s h× on each non-defaulted node. Then, by calculating the expected present value of these payments through backward recursion, one may get s h PV× × ₁, where PV is the present value of $1 paid at all ₁ non-defaulted nodes. Second, by putting a loss-given-default of 1 RR− per dollar on each defaulted node, where RR is the recovery rate, and zero elsewhere, one can

2 The Bank for International Settlements reported the notional amount on outstanding OTC credit default swaps to be $42.6 trillion in June 2007, up from $28.9 trillion in December 2006 ($13.9 trillion in December 2005).

s h×

s h×

s h×

s h× s h×

s h×

s h×

s h×

s h×

1 RR−

1 RR−

1 RR− 1 RR−

1 RR−

1 RR−

compute the present value of expected losses PV . Finally, equate_L s h PV× × ₁ with PV , and the annual spread payment made from the protection buyer should be: _L

1

10000

(in bps) PVL

s= PV × h (4)

The illustration in Figure 1 is a general treatment that does not specify the timing of defaults and the notional value of the credit default swap. There is much flexibility in considering these details. For example, the notional value of the default swap can be assumed to be a fixed amount for convenience. In our thesis we will assume that (a) in any period in which default occurs, recovery payoffs are realized at the end of the period, (b) default is based on the default intensity at the beginning of the period. In addition, we will adopt the recovery of market value (RMV) condition in our research.

Basically, the RMV condition stipulates that at default of one counterparty, the other counterparty’s claim is a fraction, 1 RR− , of the market value of a non-defaulted, but otherwise equivalent, security (if the value of this security is positive). This condition is briefly described in Appendix A and can be seen in detail in Schönbucher (2003).

As mentioned before, the main purpose of this pricing procedure is to calibrate the parameters , , α β γ. This can be done by extracting different credit default swap spreads of the reference entity from its term structure of credit default swap spreads, and then fitting the parameters to this market data. The three parameters can be solved directly given three market spreads, or a sum-of-least-squares fit can be used for more data provided.

2.3 Simulating Correlated Default

Now, to simulate correlated default, one starts by building the above binomial lattices for all reference entities in a credit portfolio. Assuming N entities (issuers) in a portfolio, one may get N sets of parameters by calibrating to market data, leading to

N intensity functions of the form

exp( )

, 1,...,

i

i i

it

it

t i N

S ^β

ξ = α γ⁺ =

.

This equation implies that if one simulates a path of stock pricesS for an issuer i , it _it is equivalent to simulating a path of default intensities ξ_it for the issuer. Therefore, with the given equity correlation, one can simulate a joint process of stock prices for all reference issuers in the portfolio, which at the same time produces the joint process of ξ_it, i.e., correlated intensity paths. This is how equity correlations drive intensity

correlations.

The simulation starts from the root node of each issuer. At the root nodes, the initial stock prices S_i0,i=1,..., ,N are known, which implies realizing the initial values of the intensities ξ_i0 for all firms. The simulation then proceeds in two steps:

1. One first checks if any of the reference entities has defaulted or not. When the intensity ξ_it at a time t is realized, the default probability π_it at that time is also determined — which is π_it = −1 exp(−ξ_ith). Then, draw a set of uniform random numbers between 0 and 1, denoted by u={ , , ,u u_{1 2} u_N}. If u_i ≤π_it, then issuer i defaults at time t . The random numbers u can be independent uniform random numbers between 0 and 1 drawn with correlation using copula methods (or other techniques).

2. For the firms that have not defaulted in the first step, one further determines whether the stock prices move up or down on the binomial lattices for all the non-defaulted issuers. The probabilities of an up move q and a down move _it 1− are known at each time period. Given the correlation matrix Σ of stock q_it returns, one can decompose Σ using a Cholesky factorization and sample a set of correlated standard normal random variables x { , , ,= x x_{1 2} x_N}. Given Φ ⋅ ( ) the standard normal cumulative distribution function, if ( )Φ x_i ≤q_it, then the stock price goes up for issuer i . On the other hand, if Φ( )x_i >q_it, the stock price goes down for i .

By repeating these two steps, starting from the root node until maturity, N sample paths of stock movements (equivalently, intensity movements) are obtained.

The timing and number of defaults can also be found as a result. Therefore, we are able to model correlated defaults to evaluate portfolio credit derivatives.

Drawing independent random numbers in the first step assumes independence of default events, where we consider only intensity correlations. This way a

“doubly-stochastic assumption” is invoked, similar to most intensity-based models.

The doubly-stochastic assumption means that, conditional on the path of the underlying state process determining default intensities, the respective default times are the first event times of independent Poisson arrivals. Comparatively, drawing random numbers with correlation assumes “conditional dependence,” where additional correlation between default events is injected besides intensity correlation.

We present a brief introduction of the one-factor Gaussian copula model in Appendix B, which is the method we use to draw correlated random variables in our thesis.

These assumptions will be investigated in Chapter V to understand their relative

strengths.

In addition to the original framework, we further suggest a more particular criterion in step 2 to grasp a general concept. Since stock values reduce to zero and remain there after default events, there should actually be no equity correlation between the defaulted firms and those that have survived. Therefore, given the outcomes after the first step in each time period, the portion of those firms that have defaulted should be excluded from the equity correlation matrix. In this way we are ensuring that the random variable draws are not interfered with by irrelevant draws in the simulation algorithm.

III. An Alternative Model: the Defaultable CEV Model 3.1 Problems with the D-CRR Model

The D-CRR model gives rise to problems of negative probabilities. Recall in the model for stock price movements discussed in the previous chapter, the probability of an up move is given by Eq. (2), which is a modification of the transition probabilities in the original CRR model. Furthermore, the stock-intensity relationship is given by Eq. (3). We now examine a set of base parameters as selected in Bandreddi et al.

(2007):

50

S = , σ =0.3, α = −0.5, 1β = , 0.1γ = , r_f =0.03,

and a term of 5 years with monthly time intervals h=1/12. When implementing this model, we encounter problematic probabilities in the lower part of the lattice. For example, when the stock price follows the lowest nodes for 47 time steps, it will become Sd⁴⁷ =50*(e⁻0.3 1/12 47) =0.854 . The intensity is then computed as

( 0.5 0.1*47 /12)

e / 0.854 1.05

ξt = ^{− +} = , and the probability of an up move will be

(

) (

)

q= e ⁺ −e⁻ e −e⁻ = , a dubious number. In fact, the probabilities on the nodes with stock price lower than 0.854 are all out of their valid range in this example. Figure 2 shows that part of the lattice.

One can easily examine how this happens for different parameter values. This is due to the intensity-function setting of Eq. (3) where the relationship between stock and intensity is actually a convex function. When stock prices are low, a slight decrease in stock price will lead to an immense increase in default intensity. In this case where intensity is not bounded, probabilities move out of their valid range.

Although in practice this can be relieved for some cases by choosing an appropriate time interval h, how large the number of time steps should be chosen varies widely for different cases. Therefore, implementation of the model is unstable and computationally inefficient for simulation purposes when a large number of time steps is needed.

The existence of these probabilities also affects the results when we simulate correlated defaults. Recall the simulation method described in section 2.3. The stock price movements are determined by comparing the risk-neutral probability, q, with the cumulative distribution function values of the sample draws. Since the cumulative distribution function values are always in the range of 0 to 1, the stock price will

Figure 2

always move up according to the algorithm when q> . This implies that the stock 1 price can never reach below the stock price that starts yielding invalid probabilities (0.854 in the above example). This portion of the lattice, which contains higher default intensities (and higher default probabilities) than any other parts of the lattice, will never be reached inside during the simulation process. It is hence unused, despite its significant contribution to the calibration of the intensity-function parameters.

Moreover, pricing other derivatives on the lattice through backward induction seems unreasonable with these dubious probabilities.

One main motivation of our thesis is to address this issue. What we need is an alternative model that mitigates the problem and still preserves the desirable properties of the D-CRR model. We come up with the defaultable CEV model (D-CEV for short) in the next section.

3.2 The Defautable CEV Model

A similar version of this model was first seen in Das and Sundaram (2007) to price convertible bonds with default risk. In their paper, stochastic interest rates are also considered, yet it is switched off here in our work for credit modeling objectives.

Prior to introducing our model, we first take a glance of what the “leverage effect” is and how taking account of it benefits the solution of the probability issue.

The “leverage effect” is a phenomenon suggesting that stock price and the volatility of its return are negatively correlated. Previous research shows that the

q=1.21

Time … 46h 47h 48h … … 52h … q=0.97

q=1.02

q=1.08

q=1.14

q=1.28 q=0.93

q=0.98

q=1.03

q=1.08

q=1.15 q=1.21 q=0.99

q=1.04

q=1.37

log-normality of stock prices assumed in the Black-Scholes framework does not hold empirically due to the leverage effect in actual stock price behavior. Therefore, to account for this evidence, different diffusion processes are studied in the literature.

Cox (1975) and Cox and Ross (1976) focused on a general class of stochastic processes known as the constant elasticity of variance (CEV) diffusion class:

dS =μSdt+σS dZc , 0< ≤c 1.

Here the instantaneous variance of dS S/ (percentage price change) is equal to

2/S2 2^c

σ ⁻ and hence an inverse function of the stock price. When the stock price is high, volatility is smaller than when the stock price is low. This matches the leverage effect qualitatively. In Eq. (2), the denominator u−d will become larger with higher volatility when stock prices are low, in accordance with the leverage effect, forming a force to pull back the probabilities to their appropriate range. Therefore, the probability issue can be mitigated with this alternative setting. Actually, taking account of both the leverage effect and the inverse relationship between stock prices and default intensities describes empirical phenomena more faithfully. It is a fact that a firm usually becomes more volatile when its default intensity is high, connecting both our intensity-function setting and the leverage effect in the equity prices.

Nelson and Ramaswamy (1990) developed a recombining binomial lattice that converges weakly to the CEV process, which proves to be computationally simple.

We apply their method for our lattice construction, summarized as follows:

1. First, since the instantaneous volatility of the stock price is stochastic in the CEV process, building a discrete model for the stock process will result in a non-recombining lattice. Therefore, we need to make a transformation from the process of S into another process X, that produces constant instantaneous volatility. Nelson and Ramaswamy (1990) show that the transformation can be computed as

1 1

( ) (1 )

S c S c

X S Z dZ

σ c

σ

− − −

= =

∫

The function X(S) is the said mapping from S to X. Notice that the inverse function of the X(S) is

[ ]

( ) (1 ) ^c

S X = σ −c X ⁻ . (5)

Figure 3

The X-Lattice in the Nelson and Ramaswamy (1990) Framework

After building a lattice for the process with constant volatility, X, we are able to transform from X back to S using this function. Nelson and Ramaswamy (1990) show that the process of S(X) is monotonically increasing in X; therefore a lattice built for S with this transformation inherits the computational simplicity that the lattice built for X demonstrates.

2. The next step is to form the X-lattice correspondent to the stock price, defined as

(up move) w/prob (1 ) (down move) w/prob (1 )(1 ) 0 (default) w/prob

t t

t h t t

t

X h q

X X h q

π π π

+

⎧ + −

=⎪⎪⎨ − − −

⎪⎪⎩

,

with the initial condition X₀ =X S( )₀ . Hence by applying the relationship shown in Eq. (5), this lattice can be mapped to the stock price process

X+ h

2 X+ h

X− h

X X

X−2 h

X+n h ( 2) X+ −n h

. . . . . . .

( 2) X− −n h X−n h

Time 0 h 2h … … nh=T

( ) ( ) (up move) w/prob (1 ) ( ) ( ) (down move) w/prob (1 )(1 ) 0 (default) w/prob

t t t t

t h t t t t

t

S X S X h q

S S X S X h q

π π π

+

− +

⎧ = + −

=⎪⎪⎨ = − − −

⎪⎪⎩

with the same set of probabilities. Here, the probability of an up-move is

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

r h r h

e S X S X e d X

q S X S X u X d X

ξ ξ

+ − +

+ −

− −

= =

− −

by risk-neutral arguments, where ( ) ( ) / ( )

u X =S⁺ X S X and ( )d X =S⁻( ) / ( )X S X .

3. Finally, to build a D-CEV lattice that effectively alleviates the probability issue described in Section 3.1, we need to select an appropriate choice for the CEV leverage coefficient, c, in the CEV diffusion process. We have seen that the instantaneous variance of dS S/ is equal to σ²/S^{2 2− c} from the CEV diffusion process. Thus when c=1 the instantaneous variance is simply σ², which corresponds to the constant volatility assumption in the Black-Scholes framework. As c is varied from 1 to 0, the negative effect of the stock price on the instantaneous variance becomes more profound, where c=0 yields an inverse relationship with quadratic order, σ²/ S². Consequently, c is the parameter that determines the degree of the leverage effect, and can be chosen to build a D-CEV lattice with appropriate leverage effect that relieves the probability issue.

To choose the appropriate c the following inequality should hold:

( ) ( ) ( )

0 1

( ) ( )

r h

e S X S X

q S X S X

ξ

+ −

+ −

≤ = − ≤

− ^, (6) This does not need to be checked for all probabilities q on the lattice. Since we claimed that problematic probabilities happen in the lower part of the lattice due to the convex relationship between the stock price and the intensity in Eq. (3), all we need to do is to assure c is chosen such that the risk-neutral probability for

the lowest node satisfies the inequality, for example, the node with value X−n h in Figure 3. As long as the node with the lowest equity value, and hence highest intensity, is free of the probability issue, then all the other risk-neutral probabilities are likely as well. This provides a criterion to choose the leverage coefficient. Still, note that as long as the leverage effect is considered in the D-CEV lattice, the probability issue can always be eased, as part (or even all) of the problematic probabilities found in the equivalent D-CRR lattice can fall within their appropriate range.

Das and Sundaram (2007) show that a change in the CEV leverage coefficient, c, in the CEV diffusion process has small quantitative effect on the prices of credit default swap spreads derived from the D-CEV lattice. In the remainder of our thesis we set a moderate c=0.5 in the given examples and computations to considerably account for some of the leverage effect. We will see that this is an appropriate choice for our examples to ease the probability issue.

IV. Numerical Evaluations

We now examine how the models perform for default modeling purposes. The numerical experiments allow us to investigate several characteristics and compare across the models. Recall that our experiments are carried out through the following three steps:

1. Build the correlated equity binomial lattices for all reference issuers in the credit portfolio for the D-CRR model and the D-CEV model.

2. Calibrate the lattices to the credit default swap market.

3. Simulate default with the correlated lattices and extract the numbers of default as well as their respective default times.

4.1 Base Case Data

Now, consider a credit portfolio of 10 identical reference entities with the following base level parameters:

50

S = , σ =0.3, α = −0.5, 1β = , 0.1γ = , 0.03r_f = , RR=0.5, and ρ =0.5 The term horizon is 5 years with monthly time steps h=1/12 (year). This is the same data in Bandreddi et al. (2007) used for the D-CRR model. It should be noted that the calibrated values of , , α β γ will actually be different between the D-CRR and D-CEV model even though a same term structure of credit default swap spreads is given. Hence we first assign these values to the D-CRR model and then compute the credit spreads implied from these parameters. The resulting spreads are treated as hypothesized market spreads, which are then used to calibrate our D-CEV model.

This is to ensure we are applying the same set of data to the D-CEV model for our comparison. The fitted parameters are shown in Table 1. We can see that the D-CEV model fits well the hypothesized spreads.

Then we vary different parameters and simulate 10000 paths for each case and each model. In this chapter we will assume that only default intensities are correlated, whereas default events are not. Therefore, we draw independent random numbers from a uniform distribution between 0 and 1 to determine default along the simulated paths. The total numbers of defaults and their respective time of default are derived to form default frequency distributions. We further calculate the moments of the distributions for our investigation. The methodology we use is similar to that in Bandreddi et al. (2007), but instead of adding up all intensities in a simulated path and then drawing only one random number for each path to determine default events, such

as in their work, we make default draws period by period as an extension to their approach. In this way we assure that the findings from our investigation can apply to pricing products that are dependent on the timing of defaults, such as CDOs.

Table 1

Parameters for Different Models and Their Fitted Spreads

Parameters Fitted spreads (bps)

Model

α β γ _{1yr 2yr 3yr 4yr 5yr}

D-CRR −0.50 1.00 0.10 60.39 68.09 74.84 81.68 88.87 D-CEV −0.76 0.93 0.18 60.46 68.08 74.77 81.63 88.93

The base parameters 0.5α = − , β = ,1 γ =0.1 are first given to the D-CRR model, implying market spreads of 1yr=60.39 (bps), 2yr=68.09 (bps), 3yr=74.84 (bps), 4yr=81.68 (bps), and 5yr=88.87 (bps), i.e., assuming these were the spreads observed in the market, we get from the D-CRR model the parameters 0.5α = − , β = , 0.11 γ = . We calibrate the D-CEV model to this data and obtain its intensity function parameters.

4.2 The Impact of Equity Correlation

First, we vary equity correlation. The moments of the default-frequency distributions for different models are shown in Table 2. We find that in both models, the mean and variance of the number of defaults does not vary much under different correlations.

This is consistent with most intensity-based models in which loss distributions are less sensitive to intensity correlations when default events are assumed to be independent than when they are assumed to be correlated, conditional on the correlation between intensities. We will show in Chapter V how imposing conditional correlation of default events besides intensity correlation can make default frequency distributions much more sensitive to correlation assumptions.

A critical finding here is that, the D-CEV has slightly higher mean numbers of defaults than the D-CRR model. This is interesting since we claimed that there exists an unused area where the stock price cannot reach on the D-CRR lattice. The area is associated with lower stock prices and hence higher default probabilities. Therefore, avoiding this area will underestimate the number of defaults during the simulation.

Nevertheless, the D-CEV model is mostly free of the issue, and its resulting higher mean numbers of defaults proves that the probability problem indeed causes significant influence on the simulation results.

Above, the original base case was considered. We will show later how the resulting difference of the two models widens with higher credit levels and different volatility assumptions.

Table 2

Moments of Loss Distributions when Correlation is Varied

This table presents the moments of the number of defaults under different models when equity correlation is varied. Assuming 10 identical firms with the base level parameters: S =50, σ =0.3, and RR=0.5. The term structure of credit default swap spreads is 1yr=60.39 (bps), 2yr=68.09 (bps), 3yr=74.84 (bps), 4yr=81.68 (bps), and 5yr=88.87 (bps), as from Table 1’s D-CRR row. The risk free rate is 0.03.

The term horizon is 5 years with monthly time steps h=1/12 (year). We simulated 10000 sample paths and computed the moments with various equity correlation assumptions for the D-CRR model and the D-CEV model.

D-CRR

Correlation Mean Variance Skewness Kurtosis

0 0.80 0.74 0.64 2.11

0.1 0.80 0.75 0.64 2.06

0.4 0.79 0.75 0.66 2.17

0.8 0.80 0.75 0.68 2.31

1 0.79 0.74 0.72 2.38

D-CEV

Correlation Mean Variance Skewness Kurtosis

0 0.82 0.75 0.61 2.05

0.1 0.81 0.75 0.64 2.12

0.4 0.82 0.76 0.68 2.28

0.8 0.80 0.75 0.68 2.32

1 0.81 0.74 0.61 1.98

4.3 The Impact of Equity Volatility

Second, we fix the equity correlation ρ =0.5, same as in Section 4.1, and vary the equity volatility. This test was done also in Bandreddi, et al. (2007), where they claim that their outcomes are consistent with structural-form models in which the mean number of defaults in a portfolio increases with higher volatility. Nevertheless, they use a wrong methodology to yield this expected result. Indeed, in their work they neglect the true effect of volatility since they vary volatility without simultaneously re-adjusting the intensity-function parameters. But since volatility is determined at the

beginning of the contract and held constant throughout the term in the model, the calibrated values of intensity-function parameters should be affected not only by the term structure of credit default swap spreads, but also by different volatility assumptions.

To rectify their sensitivity analysis, we simultaneously recalibrate the parameters every time the volatility is varied. In this recalibration methodology we generate results that concur with the empirical evidence on the joint movements between credit spreads and stock option implied volatilities. This has been discussed extensively in the literature (for example, Hull, Nelken and White (2004), Carr and Wu (2006)). It is pointed out that the credit default swap market and the stock options market contain overlapping information on the market and credit risk of the company, and moreover suggested that credit spreads are positively correlated with stock options’ implied volatilities. Our goal is to see if the D-CRR model and the D-CEV model capture this empirical fact.

Table 3 shows our results. We stated that the analysis for equity volatility in Bandreddi, et al. (2007) is flawed. Indeed, following our method, D-CRR no longer yields the relationship whereby the mean number of defaults increases with volatility.

Instead, the mean number of defaults drops as volatility increases, contrary to the claim of consistency with structural-form models in Bandreddi, et al. (2007). Even more, when volatility becomes higher, the difference in mean grows larger between that produced by the D-CRR model and by the D-CEV model. We believe this is caused by the rising numbers of problematic probabilities while volatility increases.

We can see that the difference in moments and calibrated parameters between the two models is smaller with lower volatility. It is likely because there are fewer problematic probabilities found in the D-CRR model when volatility is low.

From the results for the D-CEV model, we observe that the fitted parameters and the resulting mean numbers of defaults are almost the same for different volatility assumptions. However, to match the claim of consistency to structural-form models the mean numbers of defaults should rise with volatility. So does the D-CEV fail to corroborate the evidence of higher default numbers given increased volatility as in structural-form models? The reason the simulation results are not sensitive to volatility is that the term structure of credit defaults swap spreads is held constant in our example. In the real world, however, higher equity volatility often comes with higher credit spreads for a firm, and hence higher probability of default. We can check in Table 4 the effect from different term structure of credit default swap spreads on simulation results as we vary the base case term structure of credit default swap spreads given in Section 4.1. It is clear from the table that increased spreads lead to increased mean numbers of defaults. Hence, combining both results from Table 3 and

Table 3

Moments of Loss Distributions when Volatility Is Varied

This table shows the moments of the number of defaults under different models when equity volatility is varied. Assuming 10 identical firms with the base level parameters:

50

S = , ρ =0.5, and 0.5RR= . The term structure of credit default swap spreads is 1yr=60.39 (bps), 2yr=68.09 (bps), 3yr=74.84 (bps), 4yr=81.68 (bps), and 5yr=88.87 (bps), as given in Table 1. Intensity function parameters α , β , and γ are recalibrated to both the credit default swap term structure and equity volatility. The risk free rate is 0.03. The term horizon is 5 years with monthly time steps h=1/12 (year). We simulated 10000 sample paths and computed the moments with various volatility assumptions for the D-CRR model and the D-CEV model.

D-CRR

Moments Calibrated Parameters

Volatility

Mean Variance α β γ

Number of Problematic Probabilities

0.1 0.82 0.76 0.79− 0.92 0.17 0

0.2 0.81 0.76 0.67− 0.96 0.15 1

0.3 0.78 0.77 0.50− 1.00 0.10 60

0.4 0.75 0.76 0.45− 1.01 0.03 107

0.5 0.72 0.78 0.41− 1.03 –0.06 110

D-CEV

Moments Calibrated Parameters

Volatility

Mean Variance α β γ

Number of Problematic Probabilities

0.1 0.83 0.78 −0.76 0.93 0.18 0

0.2 0.82 0.78 −0.76 0.93 0.18 0

0.3 0.82 0.76 −0.76 0.93 0.18 0

0.4 0.82 0.76 −0.76 0.93 0.18 0

0.5 0.82 0.76 −0.76 0.93 0.18 0

Table 4

Moments of Loss Distributions When The Term Structure of Credit Default Swap Spreads Is Varied

This table shows the moments of the number of defaults under different models when the term structure of credit default swap spreads is varied. Assuming 10 identical firms with the base level parameters: S =50, σ =0.3, ρ =0.5, and 0.5RR= . The base case term structure of credit default swap spreads is 1yr=61.09 (bps), 2yr=68.73 (bps), 3yr=75.42 (bps), 4yr=82.26 (bps), and 5yr=89.52 (bps), as given in Table 1.

The base case is then varied by -20 (bps), -10 (bps), +10 (bps), and +20 (bps).

Intensity function parameters α , β, and γ are always recalibrated to both the credit default swap term structure and equity volatility for each case. The risk free rate is 0.03. The term horizon is 5 years with monthly time steps h=1/12 (year). We simulated 10000 sample paths and computed the moments with various term structure of credit default swap spread assumptions for the D-CEV model.

D-CEV

Term Structure of Credit Default Swap Spreads (bps) Moments

1yr 2yr 3yr 4yr 5yr Mean Variance

+20 (bps) 81.09 88.73 95.42 102.26 109.52 0.98 1.10 +10 (bps) 71.09 78.73 85.42 92.26 99.52 0.88 1.01 Base Case 61.09 68.73 75.42 82.26 89.52 0.80 0.94 -10 (bps) 51.09 58.73 65.42 72.26 79.52 0.73 0.85 -20 (bps) 41.09 48.73 55.42 62.26 69.52 0.65 0.79

Table 4, we know that in our D-CEV model (1) higher credit spreads will lead to higher mean numbers of defaults and (2) higher volatility will not have any significant influence on the mean number of defaults. It is thus evident that as both credit spreads and equity volatility jointly increase in line with empirical facts, the mean number of defaults will increase in the D-CEV model. For the D-CRR model this will not always hold. Indeed, in our example the positive effect from higher term structure of credit default swap spreads on the mean numbers of defaults can be offset by a negative effect from higher volatility. Hence, we conclude that, when applied in practice with empirical data, the D-CEV model resembles structural-form models in their relationship between the equity volatility and the mean number of defaults in a better way than the D-CRR model. That is, for the D-CEV model the mean number of defaults can increase with higher volatility as in structural-form models given the condition that credit spreads increase with volatility.

One last thing to mention in Table 3 is that, the number of problematic probabilities is reduced to zero in our example. This is because the leverage coefficient c we chose satisfies the inequality given in Eq. (6). It should be noted that the D-CEV model may not completely eliminate the problem for some cases where the inequality condition does not hold. Still, whether this condition is imposed or not, by accounting for stochastic volatility with the leverage effect and adjusting part of the probabilities, the D-CEV model has this problem to a less extent.

4.4 The Impact of Intensity Function Parameters

We now return to the intensity function parameters and reveal their effect on the term structure of credit default swap spreads. These parameters contain important information about the term structure of credit default swap spreads.

We plot the term structure of credit default swaps spreads for different parameter cases, in Figure 4, Figure 5, and Figure 6. Here we present the results for the D-CEV model. A base case is considered where α = −0.76, 0.93β = , and γ =0.18 within the D-CEV model. These are the parameters from the D-CEV row in Table 1. In Figure 4 we vary α , in Figure 5 we vary β, and in Figure 6 we vary γ . The parameters are individually varied by −0.5, −0.3, −0.1, +0.1, +0.3, +0.5.

We can clearly see in Figure 4 that α primarily captures parallel movements of the curve, where higher values of α lead to higher levels of credit spreads. It is thus regarded as a “level” modulator. Figure 5 shows that β is responsible for part of the level and slope, but holds an inverse relationship with the spreads. Since β lies on the denominator of the intensity function described in Eq. (3), a high value of β leads to lower intensity, as well as lower levels and slopes of the term structure of credit default swap spreads. As for γ , we can see in Figure 6 that the slope and convexity of the curve increases with γ , thus the shape is also captured.

Consequently, with these parameters we have sufficient degrees of freedom to describe the level, slope, and shape of the term structures of credit default swap spreads. This serves a direct link to credit markets and a convenient way to measure how valuation of credit products is affected by changes in overall credit market conditions such as those reflected in the term structure of credit default swaps spreads.

To have a better understanding of the effects of different intensity-function parameters on simulation, Table 5 shows what happens to the moments of the default frequency distributions when the intensity-function parameters are changed. The results are intuitive now that we have looked into the implications of the parameters.

We see that with higher α and γ the level of credit spreads increases, indicating lower credit quality and higher mean numbers of defaults. On the other hand, lower mean numbers of defaults come with higher β’s.

Figure 4

Term Structure of Credit Default Swap Spreads When Alpha Is Varied This figure shows the impact of the intensity function parameter α . The D-CEV model is applied for this examination. We consider a base case where α = −0.76,

β =0.93, and γ =0.18. The parameter α is varied by −0.5, −0.3, −0.1, +0.1, +0.3, +0.5. We plot the respective spread quotes for the credit default swaps of maturities one year to five years, i.e., the term structure of credit default swap spreads.

Term Structure of Credit Default Swap Spreads

0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00 160.00

Time to M aturity

Spread (bps)

Alpha+0.5 101.25 114.50 126.22 138.25 151.01 Alpha+0.3 82.87 93.71 103.33 113.22 123.76 Alpha+0.1 67.83 76.71 84.59 92.73 101.41 Base Case 61.37 69.40 76.54 83.91 91.79 Alpha-0.1 55.52 62.79 69.26 75.93 83.08 Alpha-0.3 45.45 51.40 56.70 62.18 68.06 Alpha-0.5 37.20 42.08 46.42 50.92 55.75

1 2 3 4 5

Figure 5

Term Structure of Credit Default Swap Spreads When Beta Is Varied This figure shows the impact of the intensity function parameter β. The D-CEV model is applied for this examination. We consider a base case where α = −0.76,

β =0.93, and γ =0.18. The parameter β is varied by −0.5, −0.3, −0.1, +0.1, +0.3, +0.5. We plot the respective spread quotes for the credit default swaps of maturities one year to five years, i.e., the term structure of credit default swap spreads.

Term Structure of Credit Default Swap Spreads

0.00 100.00 200.00 300.00 400.00 500.00 600.00 700.00 800.00

Time to Maturity

Spread (bps)

Beta+0.5 8.65 9.75 10.73 11.74 12.83 Beta+0.3 18.92 21.36 23.51 25.72 28.10 Beta+0.1 41.44 46.82 51.58 56.50 61.75 Base Case 61.37 69.40 76.54 83.91 91.79 Beta-0.1 90.93 103.02 113.81 124.97 136.91 Beta-0.3 200.52 228.88 254.75 281.81 310.97 Beta-0.5 447.95 521.55 592.17 668.10 751.53

1 2 3 4 5

Figure 6

Term Structure of Credit Default Swap Spreads When Gamma Is Varied This figure shows the impact of the intensity function parameter γ . The D-CEV model is applied for this examination. We consider a base case where α = −0.76,

β =0.93, and γ =0.18. The parameter γ is varied by −0.5, −0.3, −0.1, +0.1, +0.3, +0.5. We plot the respective spread quotes for the credit default swaps of maturities one year to five years, i.e., the term structure of credit default swap spreads.

Term Structure of Credit Default Swap Spreads

0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00

Time to M aturity

Spread (bps)

Gamma+0.5 78.38 118.97 180.24 275.03 418.12 Gamma+0.3 70.90 95.01 125.57 166.36 220.90 Gamma+0.1 64.34 76.82 89.66 104.25 121.15 Base Case 61.37 69.40 76.54 83.91 91.79 Gamma-0.1 58.58 62.90 65.80 68.36 70.81 Gamma-0.3 53.51 52.19 49.71 47.14 44.68 Gamma-0.5 49.04 43.86 38.65 34.19 30.46

1 2 3 4 5

Table 5

Moments of Loss Distributions When Intensity Function Parameters Are Varied

This table presents the moments of the number of defaults under the D-CEV model varying intensity function parameters. Assuming 10 identical firms with the same parameter values: S =50, ρ=0.4, and RR=0.5. The base case for intensity function parameters is set as: α = −0.5, β = , and 1 γ =0.1. The risk free rate is 0.03. The term horizon is 5 years with monthly time steps h=1/12 (year). We simulated 10000 sample paths and computed the moments for various parameter assumptions.

D-CEV

Alpha Mean Variance Skewness Kurtosis

−0.5 0.66 0.62 0.55 1.56

0 1.03 0.93 0.78 3.17

0.5 1.61 1.34 0.87 5.40

D-CEV

Beta Mean Variance Skewness Kurtosis

0 9.85 0.14 –0.13 0.18

0.5 3.72 2.35 0.64 15.51

1 0.65 0.61 0.55 1.58

2 0.01 0.01 0.01 0.02

D-CEV

Gamma Mean Variance Skewness Kurtosis

−0.1 0.41 0.39 0.35 0.70

0 0.51 0.48 0.43 1.09

0.1 0.66 0.60 0.48 1.35

1 7.09 2.11 –0.95 12.82

V. Further Research and Applications

We mentioned in the previous chapter that loss frequency distributions are less sensitive to intensity correlations when default events are assumed to be independent after conditioning on intensity correlation. Therefore, intensity correlations are insufficient to fully describe the joint movements in credit portfolios. In this chapter, we will first discuss about basket default swaps. Then we will look into different kinds of correlations that should be taken with care, specifically the correlation between intensities and the conditional correlation between default events. We further present some results for the valuation of nth-to-default contracts and CDOs.

5.1 Basket Default Swaps

A basket default swap is similar to a single entity credit default swap except that the underlying is a portfolio of entities rather than one single entity. One popular type of basket default swaps is an nth-to-default swap. For an nth-to-default swap whenever the nth default occurs in the reference portfolio, the buyer stops paying the periodic swap premium and receives the loss-given-default amount, 1 RR− . The premium does not stop until the nth default, even if there are already defaults in the portfolio.

The cost of protection for a basket default swap depends on its probability of being triggered within a specific time, i.e., the probability that the seller has to payout the loss-given-default amount. Correlation assumptions between the reference entities are key elements to the evaluation of such probabilities. They determine how credit risk in the reference portfolio is distributed among different types of default swaps.

In the following section, we will explore the correlation between intensities and the correlation between actual default events.

5.2 Intensity Correlation vs. Conditional Correlation

As we have introduced in Chapter I, Das, Duffie, Kapadia, and Saita (2005) show empirical evidence that corporate defaults cluster in time and that the doubly-stochastic assumption is rejected. The doubly-stochastic assumption says that, conditional on the path of the underlying state process determining default intensities, the respective default times are the first event times of independent Poisson arrivals.

The assumption rules out the presence of contagion or frailty (incompletely observed default covariates not captured by the correlation in intensity processes across firms).

We will test how our model accommodates the default contagion effect observed in