Credit default swap term structures and structural models

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Credit Default Swap Term

Structures and Structural Models

Panagiotis Pavlou

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Author: Panagiotis Pavlou

Student nr: 10604723

E-mail: pavlou.panos@gmail.com Date: August 15, 2016

Supervisor: dr. Daniël Linders

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Statement of Originality

This document is written by Student Panagiotis Pavlou who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating

it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

The purpose of this paper is to explore the capability of structural models to price credit derivatives and replicate their market behavior. The credit derivative of choice is Sovereign Credit Default Swaps. We employ three structural models for further research. We start with the most simple one, the prototype structural model of Merton (1974) in which the reference entity’s asset value is described by a Brownian motion and default can happen only at maturity if the asset value is lower than a fixed barrier. The second model of choice is a Cox type structural model in which asset value assumptions are the same as in Merton’s model, but default is defined as the first passage of a fixed default barrier. Our last model is a Lévy Variance Gamma structural model where the asset value follows a Lévy process. The models are discussed in terms of their level of realisticity. We confirm that all three models can accurately price individual single name CDSs, but they fail to replicate some features of the CDS term structure.

Keywords Credit risk, Credit Derivatives, Structural models, Credit default swaps, Variance gamma, Lévy, Brownian motion, Monte Carlo sim-ulation.

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1 Introduction.

Credit risk is present in everyones life as long as exchanging products and money exists. Most of the asset transactions happen between a borrower and a lender accompanied with a set of agreed terms. An example of such a transaction is a simple loan from the bank. Suppose that a bank grants a loan to a third party (borrower), by doing so the bank and the borrower enter to the agreement that the borrower will return the money in a specified amount of time. Though, there is always a probability that the borrower will not be able to honor the agreement and this is where the concept of credit risk is entailed. In real terms, the “bank” can be any kind of entity and the “borrower” as well, moreover the agreements and the terms of the agreements can be more complicated.

The financial crisis of 2008, though not a new phenomenon, altered once more the world’s perception and made clear that no financial entity is immune to credit events. World wide panic of investors and national even bailouts un-derlined the importance of mitigating Credit Risk. Credit risk incorporates the inability or the unwillingness of a financial entity to honor its obligations and therefore a credit event is defined as the manifestation of this event. Credit derivative instruments became popular as a way to protect portfolios from this kind of risk. Moreover, Basel II regulating framework allows for financial in-stitutions to use their own credit risk models to price credit derivatives with benefits on their capital requirements, therefore there is an evident need for credit risk models.

There are several different approaches in modeling Credit Risk. The liter-ature is divided between methods that use accounting information to estimate or forecast credit risk, and models that use market prices of assets to assess credit risk. Examples of accounting information methods are the Altman’s Z score(Altman 1968) and Credit rating models. From the market price methods side, the two main streams of models are Structural models and Intensity based models. All these models aim to determine a credit event and then, to asses the probability that this event will happen, in other words; the probability of default.

Structural models connect the event of default with the capital structure of the reference entity, assuming that its equity is traded as an option on its asset value in the market. Therefore, they focus on modeling the asset value of the reference entity and they assume that default is related to wether the level of the asset value is above(survival) or under(default) the level of a default barrier. An appropriate Credit risk model should be intuitive and able to replicate market features. This thesis studies the impact of incorporating jumps in the modeling of the asset value, by answering the following research question:

“What is the impact of incorporating jumps in the estimation of the prob-ability of default in Structural modeling and to what extend this proposes an added value in replicating the market features of the Credit Default Swap term structures.”

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To answer this question, we will follow a comparative approach by employing three structural models for credit risk to explore their performance in estimating the probability of default and their capability to price and replicate the Sovereign CDS term structures presented in the market. The models under consideration will be, the Merton model, the Black - Cox model with constant barrier and the Lévy Variance Gamma Default model. Merton’s model is the origin of structural models, with the other two to be considered as extensions of it. While Merton and Black-Cox models base their modeling assumptions on the Black-Scholes framework and its normality, the Lévy Variance Gamma model is still based on Merton’s intuition, but relies on a Lévy framework, that incorporates jumps, to account for shocks and unexpected events. It is interesting to see the impact of the different modeling assumptions on the output of each model.

In chapter 2, we briefly review the basic definitions and state our assumptions regarding the environment that this paper will be based on. In chapter 3, we are going to analyze Credit Default Swaps, and present the way they are priced. In chapter 4, we present the two fundamental credit risk methods on the market. In chapter 5, we will present the Black-Cox and our main Lévy Variance Gamma default model. In chapter 6, our data set will be presented and visualized. In chapter 7, we discuss the calibration process of our models. And, finally in chapter 8 we present our results.

2 Financial Background and Definitions.

A big part of this paper is dedicated in derivative pricing and more specifically on CDS pricing. Though, it is important, before delving into our models and pricing formulas, to build up on the derivative pricing framework by presenting the basic concepts of mathematics and financial theory behind it. Our sources for this section are Hull and White (2003), Schoutens (2003) , Schoutens W., Cariboni J. (2004) and Tarashev N. (2005).

2.1 Definitions.

We begin by providing the definitions of probability space and random variables which are the core elements of a stochastic process. A specific case of a stochastic process is a Brownian motion which is a key assumption in our models of choice.

2.1.1 Sigma Algebra.

A σ-Algebra is collection of E of subsets of the d−dimensional real space E which

• contains an empty subset: ∅∈ E

• is stable under unions: if {An, n ≥ 1} is a sequence of disjoint elements of

E, then

[

n≥1

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• contains the complementary of any element: ∀A ∈ E = AC ∈ E. 2.1.2 Probability measure and Measurable function. Probability space (E, E , µ) , is a finite measure with mass equal to 1.

A measurable function: Suppose that (E, E ) and (D, D) are two measurable spaces. A function f : E → D is measurable if for any measurable set A∈ D,

f−1(A) = {x ∈ D|f (x) ∈ A}

is a measurable subset of E.

2.1.3 Random Variable.

A random variable is a variable of which the value changes according to a random pattern. A random variable Z : Ω → E defined on a probability space (Ω, F , P) is a measurable function from Ω to some set E.

2.1.4 Stochastic process.

A stochastic process X = {Xt, 0 ≤ t ≤ T } is a collection of random variables

defined on (Ω, F , P), evolving in time t according to a system. 2.1.5 Brownian Motion.

Brownian motion or Wiener process W = {Wt, 0 ≤ t ≤ T } is a fundamental

real-valued stochastic process on the probability space (Ω, F , P). It is closely linked to the normal distribution and satisfies the following conditions :

• W0= 0, starts at 0

• has stationary increments, for example, if FW

t is the filtration that is

generated by W in times 0 ≤ s ≤ t, then the increment Wt− Ws is

independent of the filtration FW s

• has independent increments, for all times l ≤ s ≤ t ≤ u , W (u) − W (t) is independent from W (s) − W (l)

• every increment W (t) − W (s) with s < t , follows a normal distribution N (0, (t − s)).

2.2 Modeling Assumptions-Theory of option pricing.

Later on we will see that our models rely on stochastic processes to describe the behavior of financial assets in the future. To price derivatives on these financial assets, we will follow the rules suggested by the theory of option pricing. Before presenting the concepts of the theory in mathematical notation, we provide a gentle introduction of the intuition of the Fundamental Theorem of option pricing.

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A common human behavioral characteristic is risk aversion. Risk aversion addresses to the fact that people tend to reduce their exposure to uncertainty. This behavior intervenes in their decision making process and makes more prob-able for them to prefer a deal that will result in lower expected payoff but with less uncertainty than a more risky one, with higher expected payoff and higher uncertainty. This behavior affects the pricing process of assets as well, since the asset prices depend on their embedded risk and as a result, the price of risky assets is different than their expected value, usually lower than the ex-pected value, rewarding those who actually bear the risk. Under this context, there cannot exist a universal price of a specific asset since people have different risk preferences and calculate the expected value differently by using their own discount factors.

To overcome this unfortunate convenience-wise event, The Fundamental Theorem of Option Pricing suggests that in a complete market with no arbi-trage opportunities, there exists an alternative approach. The pricing of assets can be done by incorporating all investors premia into the probabilities of the future payoffs and then calculate the expectation under this new probability. This probability is called the risk neutral measure. Under the risk neutral mea-sure, the price of any asset equals its discounted, by the risk free rate, expected payoff. All the notions will be defined in mathematical notation below.

2.2.1 Risk free asset.

The price dynamics of a risk free asset follow the pattern

dBt= rtBtdt

with an explicit solution, when r is constant in time,

Bt= ert

where B = {Bt, 0 ≤ t ≤ T } is the price process, r = {rt, 0 ≤ t ≤ T } is risk free

rate. Hereby, the ratio

D(t, T ) = E B_Bt

T

is defined as the discount factor and will be used to estimate the present value of any cashflow or derivative at time t < T .

In our study, risk free rates are considered flat and the risk free asset is considered as “safe” deposit in a bank account. Thus, the discount factor in our case is defined as:

D(t, T ) = e(−r(T −t)).

2.2.2 Contingent Claim.

Consider a stochastic process X = {Xt, 0 ≤ t ≤ T }. A contingent claim CTX

of an underlying process X and maturity T is any positive FX

T - measurable

stochastic variable. The value of CX

T at time T is determined by the underlying

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2.2.3 Martingale.

A stochastic process X = {Xt, 0 ≤ t ≤ T } is a Ft martingale if:

• X is adapted to the filtration {Ft}t≥0.

• E [|Xt|] <∝ for each t .

• E [Xt|Fs] = Xsholds for all s and t iff s ≤ t.

The first point says that the value of X is observable at every point in time t. The third point states with all the information available today, the expected value of X in a future time is equal to its present value.

2.2.4 Arbitrage and Arbitrage-free Market.

Consider a self- financing investment strategy ϕ with a price process in time Aϕ _{= {A}ϕ

t, 0 ≤ t ≤ T }. The self-financing strategy ˜ϕ that has positive

prob-ability for profit PAϕt˜≥ 0

= 1 with no initial investment at time t = 0, Aϕ₀˜= 0, is defined as arbitrage.

In our study we assume an arbitrage free environment. An arbitrage free environment suggests that investment strategies ˜ϕ as defined above are do not exist. That means that with no initial capital no profit can be made.

2.2.5 Complete market.

A market is characterized as complete in case self financing strategies ϕ exist so that their value can replicate accurately any contingent claim CX

T .

Aϕ_t (T ) = C_TX.

A complete market suggests that, the no-arbitrage consistent price of a con-tingent claim CX

T is given by:

ΠCt = A ϕ t.

2.2.6 Risk neutral measure.

In a given probability space (Ω, F , P) we define Q as the risk neutral measure of P when:

• Q and P have the same set of null events.

• The price process Π = {Πt, 0 ≤ t ≤ T } of any security discounted by the

factor D(t, T ) as defined above, is a martingale under Q.

Therefore,the notation P will refer to real world probabilities, while the notation Q will refer to risk neutral probability measures.

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2.2.7 Risk neutral pricing.

Suppose a stochastic process X = {Xt, 0 ≤ t ≤ T } and a contingent claim

CX

T = Y({Xt, 0 ≤ t ≤ T }) with a price ΠC = {ΠCt, 0 ≤ t ≤ T }. In an arbitrage

free market, the price ΠC _{is given by:}

ΠC_t_{= E}Q_{[D(t, T )Y(X} T)|Ft]

= e(−r(T −t))_EQ_[Y(X

T)|Ft] , 0 ≤ t ≤ T

The risk neutral world suggests that the price of any security is the result of the risk neutral expectation of the discounted cashflows by using the discount factor as mentioned in section 2.2.1 .

3 Credit Default Swaps.

3.1 Credit Default Swaps.

Credit Default Swaps are a class of financial derivatives, similar to insurance products, protecting the buyer from credit risk. The difference between CDSs and insurance contracts is that CDSs are issued by third parties, and there is no necessity for the CDS buyer to have any financial stake in the reference entity. CDS contracts were introduced in the mid - 1990s in the private sector, where commercial banks used them for hedging purposes against credit risk embedded in large corporate loans. The popularity of the product rose, moving forward from private to public sector as well. CDS contracts are used for hedging against credit risk as an insurance-type offsetting instrument, for relative-value trading (having a short position in one country and a long one in another), and arbitrage trading (buy/sell government bonds vs sell/buy sovereign CDS).

On the one end of the agreement we have the CDS buyer. A CDS buyer enters in a contract with the CDS seller, agreeing to pay a fixed coupon, in exchange of the right to be compensated in the case of a predefined credit event of the reference entity. On the other end of the agreement, the CDS seller enters in a contract with the CDS buyer, agreeing to compensate in case a credit event happens. The compensation depends on the notional amount, the recovery rate and estimated probabilities of default, which along with other characteristics such as settlement are predefined in the contract. Since credit events are rare, the evaluation of the spread, and consequently the probability of default, from the seller’s side is a difficult task.

Because of those particular contracts’ nature, they are liquid and are highly traded in the international markets, therefore, it is believed that they contain credible information about the probability of a credit event of the reference entity. Assuming that other influencing factors like recovery rate and interest are flat, higher spreads (fixed coupons) of the CDS contract imply higher probability of a credit event.

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Figure 3.1 Schematic of the cashflow transactions of a CDS contract.[19]

The fee that the protection buyer has to pay is called the CDS spread and is defined as the rate that the protection buyer has to pay to enter in a CDS contract that will cover a predetermined amount of the loss in case of the default of the reference entity. The higher the CDS spread, the higher the estimated default risk for the reference entity.

Figure 3.2 The figure above depicts the cashflows, from the CDS buyer point of view, of a CDS contract with maturity T = 10 years, in the case that the reference defaulted on the 8th_year.[19]

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3.2 Credit Default Swap pricing formula.

In this section we will present the general pricing formula of a credit default swap under the risk neutral pricing assumptions of section 2.2.7. The pricing formula of a Credit Default Swap is a general translation into a mathematical notation of the cashflows depicted in the example of figure 3.2.

In figure 3.2 we can see that the price of a CDS contract is composed by two opposing cashflows. On one hand, we have the fixed periodic payments that the CDS buyer has to pay (negative cashflows), and, on the other hand we have the payment that the CDS seller has to pay in case of default (positive cashflow). To compute the price of the CDS contract today, we first need to find the present value of these two cashflow streams. Therefore, in the CDS pricing formula, we introduce two “legs”, the fixed leg, which refers summation of the present values of periodic payments the protection buyer has to honor multiplied by their respective risk neutral probabilities of taking place, times the payment amount, and the default leg, which is the summation of the expected value of the default payments multiplied by their risk neutral probability of taking place. We define N as the notional amount of the CDS contract, S the CDS spread, SN is the amount of the periodic payments. Since we work under the risk neutral framework the present value of each cashflow can be calculated by using the risk free discount factor D(0, t), R is the recovery rate and ∆t the time interval between the payments. We work under the risk neutral setting because we attempt to find a universal price or else “fair price” for our CDS.

Therefore, the fixed leg is defined as:

P Vf ixed= SN n

X

i=1

D(0, ti)Psurvival(ti)∆ti

The default leg is defined as:

P Vdef ault= (1 − R)N n

X

i=1

D(0, ti)(Psurv(ti−1) − Psurv(ti))

The price of the CDS contract at the time t = 0 , as in all swap agreements, should be equal to 0, this means that:

P Vf ixed= P Vdef ault

Consequently, the par spread S on a Notional amount N = 1 is defined as:

S =(1 − R) Pn

i=1D(0, ti)(Psurv(ti−1) − Psurv(ti))

Pn

i=1D(0, ti)Psurv(ti)∆ti

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and in continuous time:

S = (1 − R)−´₀TD(0, s)dPsurv(s) ´T 0 D(0, s)Psurv(s)ds .

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The par spread in continuous time means that the spread is paid continu-ously. It is not a realistic assumption since it is impossible in real life to pay a fee continuously, but it will be a convenient in our Variance Gamma model assumptions. For the rest of our models we assume a discrete time environment.

The main component in the CDS pricing formula is the probability of survival and the probability of default, and they are defined as:

• Psurv(t) =Probability of no default event in the time interval 0 ≤ t ≤ T .

• Pdef ault(t) =Probability of default event in the time interval 0 ≤ t ≤ T .

• Pdef ault(t) = 1 − Psurv(t).

• Pdef ault(ti−1, ti) = Psurv(ti−1) − Psurv(ti)

Therefore, from now on we are going to focus on and explore the methods and models which are able to estimate these probabilities.

4 Fundamental methods of assessing credit risk.

The two main approaches for credit risk are Structural Models and Intensity Based Models. The main difference is that Intensity based models focus directly on modeling the default intensity of the reference entity. Structural models on the other hand, base their intuition on an option analogy, under the assumption that the reference entity’s asset value is traded as a stock in the market and focus on modeling the reference entity’s asset value.

4.1 Structural models.

In 1973, Fisher Black, Myron Scholes and Robert Merton discovered a mathe-matical formula that would change the course of money markets. The equation of the three economists was the first widely known attempt to describe by a simple and efficient way the stochastic behavior of the stock prices in the fu-ture. Their theory proved to be so important that they were awarded with a Nobel prize in 1997.

The Black - Scholes model changed the options market from marginal to the epicenter of financial decision making. The difference of this model from former attempts was that it is based in variables that are measurable, such as the current stock price, the strike price, interest rates and maturity. The only variable that the model can not calculate is the risk of the investment. Therefore, they connected the variability of the stock price with its risk. In other words, the higher the variability of the stock price in the future, the higher the risk the investors would undertake, and the higher the prices of the options written on the stock. In contrast, the lower the variation of the stock price, the lower the uncertainty therefore lower prices for the options on the stock.

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Merton in 1974, based on the theory above introduced the first structural model for credit risk. He made use of the Black - Scholes formula to estimate the probability of survival of a reference entity. He theorized that the asset value of a financial entity V = {Vt, 0 ≤ t ≤ T } is the summation of it’s equity value,

E = {Et, 0 ≤ t ≤ T } and the value of a zero coupon bond zT = {ztT, 0 ≤ t ≤ T },

with maturity T and face value L:

Vt= Et+ ztT.

If at maturity T , the asset value is not at a sufficient level to meet the obligations (L), the bond holders take over the financial entity and the reference entity goes bankrupt. This outcome in the Merton model is defined as a default event. These assumptions allowed Merton to treat the equity of the financial entity as a European call option with payoff:

ET = max(VT−L, 0) = {VT−L}+= {VT−Lif VT ≥ L} and 0 otherwise. (2)

The equation above shows that the bond holders are short a put option with maturity T and strike price , and the share holders are long a call option, with the same characteristics, on the financial entity’s asset value.

The dynamics of the asset value V , follow a Geometric Brownian motion, consequent to Black- Scholes assumptions.

dVt= Vt(µdt + σVdWt) V0> 0, (3)

were µ is the drift parameter σV is the asset volatility and W is a standard

wiener process. The asset value under this assumption is, at time t, log normally distributed,

log(Vt) − log(V0) ∼ N (t(µ −

1 2σ

2_{), σ}2_t).

In the context of Merton, default can happen only at maturity, no matter if the equity value drops below the face value L at a time t earlier than maturity T . Under the risk neutral measure Q the dynamics of the equity is given by:

Et= VtΦ(d1) − exp(−r(T − t))Φ(d2) (4)

Φ() is the standard normal distribution function and,

d1= ln(Vt L) + (r + σ 2 V/2)(T − t) σV √ T − t . d2= d1− σV √ T − t.

The probability that the call option finishes in the money, held by the share holders, constitutes the conditional probability that default will not occur in (t, T ]. So in the Black Scholes world the probability of survival is defined as:

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In 1976, Black and Cox proposed the first extension to Merton’s model. Their model is considered as a first passage model addressing the fact that default can occur, not only at maturity T, but also at an earlier time t. The default barrier is assumed to be time dependent and exogenous, in contrast to the original Merton’s model, where default is considered indigenous.

These two models lay the basis for the structural approach wing of assessing credit risk . Numerous extensions of the two models have been published. One of them , Credit Grades model is a generalization of the previous two and is widely accepted by practitioners. The model’s main difference compared to its predecessors is that it allows for instantaneous default by incorporating the barrier L as stochastic, resulting into higher short term spreads compared to implementing a deterministic barrier.

Nelson in 1993 and Longstaff and Schwartz in 1995, attempted to avoid some of the shortcomings of the model of Merton. They approach the reference entity’s equity structure in a different way and default can happen at any point in time. In case of a default event, the entity’s debt is paid partly or in fixed amount. Their models implement interest rates as stochastic, accounting for interest rate risk as well. As we have seen, the equity value of a financial entity depends on the stock prices, while the interest rates depend on the level of the interest on government bonds.

In real market terms, however, the behavior of assets are not reflected by the assumption of normality or log normality implied by the Black- Scholes model. The underlying distribution of stock returns may exhibit fat tails, be skewed and may have positive excess kurtosis. In their paper, Schoutens and Cariboni (2004) based on the intuition of the Merton’s model as well, attempt to overcome the flaws of Black - Scholes normality assumptions and suggest a model in which, the stock price process is described by en exponential of a Lévy process driven entirely by jumps. The underlying distributions of the Lévy processes are more flexible and capable of better reflecting the variations of the underlying. This specification allows for instantaneous default without the need for a stochastic barrier L.

4.2 Intensity based models.

An other popular approach of credit risk modeling is Intensity based models or Reduced form models. In this approach, it is assumed that the time to default is not dependent to the equity value of the reference financial entity but default is triggered by exogenous factors. In this way, estimating the reference entity’s asset value is not needed to define default. The approach of intensity based models in credit risk is based on the idea that default is a random event and it is manifested at the time of the first jump of a counting poisson process Nt= {Nt, 0 ≤ t ≤ T } with intensity λ = {λt, 0 ≤ t ≤ T }. The default intensity

can be deterministic or stochastic and is actually the determinant of the credit risk price as it models the default rate of the reference entity. Default intensity is defined as:

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λt= lim h→0

P [t < τ ≤ t + h|τ > t]

h . (6)

where τ is the time of default . The above equation means that, approxi-mately speaking, for a small time interval ∆t > 0 :

P [τ ≤ t + ∆t|τ > t] ≈ λt∆t.

The most representative model in this stream is the Jarrow - Turnbull de-fault model (1995) and it is divided in two cases,(a) homogeneous case and (b) inhomogeneous case. In its homogeneous case, Nt is a homogenous poisson

process with constant default intensity λ in time, with a probability of survival between time (0, t] :

P_SurvivalJ T (t) = e−λt (7)

and the expected time to default τ = 1/λ.

Figure 4.1. Survival and default probability in time in the HJT model with λ = 0.1 .[19]

Though it is credible to assume that default intensities vary over time. This assumption results to the inhomogeneous of Jarrow - Turnbull model. In this case default intensities are a deterministic function of time, λ = {λt, 0 ≤ t ≤ T }

and the probability of survival in this case, in the time fraction of (0, t] is defined as:

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As as special simplified case, in the inhomogeneous Jarrow - Turnbull model, we consider a piecewise constant default intensities, meaning that default inten-sities do change in time but they are constant between payment dates. Thus we define the piecewise default intensity as:

λt= Kj, Tj−1≤ t ≤ Tj, j = 1, 2, ...., 5. (9)

and, as an example, the resulting default probabilities are:

P_survivalIJ T (t) =                exp(−K1t) 0 ≤ t < T1 exp(−K1T1− K2(t − T1)) T1≤ t < T2 exp(−K1T1− K2T2− K3(t − T2)) T2≤ t < T3 exp(−P3 j=1KjTj− K4(t − T3)) T3≤ t < T4 exp(−P4 j=1KjTj− K5(t − T4)) T4≤ t ≤ T5 (10)

In the figures 3 and 4 T = 10 years and λtis chosen to be:

λ(t) =                T1= 1 K1= 0.02 T2= 3 K2= 0.05 T3= 5 K3= 0.07 T4= 7 K4= 0.10 T5= 10 K5= 0.13 (11)

Figure 4.2. Probability of survival and the probability of default in time for the inhomogeneous case of Jarrow - Turnbull model.[19]

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Figure 4.3. Piecewise constant hazard rates Jarrow - Turnbull model. [19] In real world terms, there is uncertainty in the future about the creditwor-thiness of the reference entity. This fact is addressed by the implementation of stochastic intensities of default λ = {λt, 0 ≤ t ≤ T }. The stream of this

type of models are called Cox-type Intensity based models. Worth mentioning studies on this type of intensity based models are Cox et al (1985), Duffie and Singleton (1999) who build up on the Cox model but introduce jumps for the default intensities’ dynamics.

4.3 Which method to choose?

We choose to go further with the Structural methods because they set an intu-itive link between the event of default and the financial markets. Even though Intensity based models are easier to implement, they focus on modeling default intensity as a statistical process with no intuitive connection to the real market.

5 Credit Default Swap Pricing models.

The main credit risk models of our consideration are: the fundamental Merton model as presented in Chapter 4, the Black-Cox model with constant barrier and the Lévy Variance Gamma Default Model. The choice of these specific models is made because of the differences in their modeling assumptions. It is interesting to analyze the effect of these modeling differences on the output of each model in regard to the probabilities of default, CDS spread and lastly the fit of each model to the Sovereign CDS term structures.

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Since the Merton model has already been examined, in this chapter we will present the two latter models. For each model, we proceed firstly by defining the modeling assumptions on the way they describe the asset value of the reference entity and how they define the event of default. Secondly, we will focus on estimating the probabilities of survival and default and derive the formulas for calculating the CDS par spread. Finally, we perform a comparative analysis of the results of the models.

5.1 Black-Cox model with Constant Barrier.

As mentioned in Chapter 4, Black and Cox presented the first extension to Merton’s model. In this paper and our analysis, we are going to employ a modification of their model, and characterize it as a first passage model with with constant barrier.

The asset value of the reference entity in the Black - Cox type of models is modeled via a Brownian motion. Though, in contrast to the original Merton model, this model will address the fact that default can happen earlier than the time of maturity T specified on the contract as well. More specifically, the first time the asset value of the firm V = {Vt, 0 ≤ t ≤ T } falls below a given

threshold L, in the time interval (0 ≤ t ≤ T ] , the reference entity will be considered defaulted. Therefore, time of default is defined :

τ = inf{t > 0|Vt< L}

Thanks to the attributes of the brownian motion, the risk neutral dynamics of the reference entity’s asset value are

Vt= V0exp r − σ 2 2 t + σWt

where r is the risk free rate and Wta wiener process, we can easily estimate

the probability of survival Psurvival , under the risk-neutral measure Q, if the

asset value did not cross the barrier L up to time t, the conditional probability of survival is given by1_:

Psurv(T |Ft) = PQ[τ > T |τ > t] = Φ(d3) −

L(2r/σ2₎₋₁

Vt

Φ(d4)

Φ is the standard normal distribution function and d3, d4are defined as:

d3= ln Vt L + r−σ2 2 (T − t) σ√T − t d4= d3− σ √ T − t. 1_{Appendix A.1.}

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The probability of default under the risk neutral measure is defined as :

PQ

def ault(T |Ft) = 1 − Psurv(T |Ft) = 1 − Φ(d3) +

L Vt

(2r σ2)−1

Φ(d4).

Since probabilities of default and probabilities of survival have been esti-mated we can adjust the general pricing formula of a CDS under the Black-Cox model. The general pricing formula of the CDS spread for a CDS contract at time 0 and maturity T , where i = 1, 2, .., n and t0= 0, ..., tn = T , in discrete

time is :

S(0,T )=

(1 − R)Pn

i=1D(ti)(Psurv(ti−1) − Psurv(ti))

Pn

i=1D(ti)Psurv(ti)∆ti

By combining the two expressions we have as a result :

S(0,T )= (1 − R)Pn i=1D(ti)(Φti−1(d3) − L V0 2r_σ2−1 Φti−1(d4) − Φti(d3) + L V0 _σ22r−1 Φti(d4)) Pn i=1D(ti)(Φti(d3) − L V0 _σ22r−1 Φti(d4))∆ti (12)

5.2 The Lévy Variance Gamma Default Model.

In reality, defaults can be caused by shocks and unexpected events such as natural disasters, fraud or war and we believe that the continuity of the brownian motion in describing the behavior of the asset value of the reference entity is not appropriate in capturing these events. Thus, we will focus on a Lévy Variance Gamma model that models the asset value process to incorporate jumps, addressing to possibility of these shocks. In this section, we build up on a Variance Gamma Lévy model by following W. Schoutens, J. Cariboni (2004). Firstly, we define the traits of a Lévy process and we show that a Variance Gamma process is a pure jump Lévy process. Finally, we set the assumptions of the Lévy default model and we price the CDS spread by connecting its value to a Binary Down and Out Barrier Option.

5.2.1 Lévy Process.

Suppose the characteristic function $(z) of a distribution Λ. We define as an infinitely divisible distribution, the distribution that for every positive integer n, $(z) is also the nth power of a characteristic function. In this case, a stochas-tic process X = {Xt, t ≥ 0} can be defined to every such infinitely divisible

distribution. This stochastic process then is called a Lévy process and has the following traits:

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• has as a starting value 0

X0= 0

• has independent increments , if l < s ≤ t ≤ z , X(z)−X(t) and X(s)−X(l) are independent random variables.

• has stationary increments , if FX

t denotes the filtration generated by X

and if 0 ≤ s < t ,Xt− Xsis independent of FXs

• the distribution of every increment has a characteristic function ($(z))t_.

The function Ψ(z) = log $(z) is the characteristic exponent and satisfies the the following:

Ψ(z) = iγz −ς₂2z2+´_−∝+∝(exp(izx) − 1 − izx1{|x|<1})v(dx)

This formula is the Lévy-Khintchine formula where γ ∈ R , ς2_{≥ 0 and ν is}

a measure on R{0} with´_−∝+∝(1 ∧ x2_{)v(dx) <∝.}

From the Lévy-Khintchine formula we can see that in general a Lévy process consists of a Lévy triplet withγ, ς2, ν(dx) characteristics and has three inde-pendent parts, the first is the deterministic part, the second the Brownian part and the third , a pure jump part. ν(dx) is the Lévy measure that dictates how the jumps occur, jump sizes are governed by a Poisson process with parameter ´

Aν(dx) in the set of A.

In fact, Brownian motions are Lévy processes and are the only type of Lévy process that do not incorporate jumps. Lévy processes are used in various fields of science such as Physics, Engineering, and Economics. They are widely used in option pricing, see Carr and Madan (1999) for example, as a way to overcome the deficiencies of the Black- Scholes model, which most of the times fails to capture the market features. Other Lévy processes are Kou, Generalized Hyperbolic, Normal Inverse Gaussian, and Variance Gamma which is our Lévy process of choice and we will explore further in the next section.

5.2.2 The Variance Gamma Lévy Process.

For our model of interest, we will assume a Variance Gamma process for the behavior of the asset value. The characteristic function of the Variance Gamma distribution with parameters (σ, ν, θ) is infinitely divisible and is defined as :

$V G(u; σ, ν, θ) = (1 − iuθν + σ2νu2/2)−1/ν (13)

Therefore, we can define the Variance Gamma process XV G_{= {X}V G t , t ≥ 0}

, with a Lévy triplet [γ, 0, νV G(dx)], where:

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• it’s increments are independent and stationary where in the time interval [s, t + s]

Xs+tV G− XtV G∼ V G(σ, ν/t, tθ)

σ is the dispersion parameter, ν governs kurtosis and θ governs skewness. A Variance gamma process can be replicated as a Gamma time-changed Brownian motion with drift, or as a the difference of two independent Gamma processes of which one accounts for the positive jumps and the other one for the negative jumps. In our case, we will simulate our Variance Gamma process according to the latter method, following D.B. Madan, P. Carr and E.C. Chang, who proposed a different parametrization.

In their study, instead of a V G(σ, ν, θ) they parametrize the Variance Gamma distribution as V G(C, G, M ) where: C = 1/ν > 0 G = q θ2_ν2 4 + σ2_ν 2 − θν 2 −1 > 0 M = q θ2_ν2 4 + σ2ν 2 + θν 2 −1 > 0 (14)

in this case the Lévy triplet is [γ, 0, ν(dx)] with :

νV G(dx) = ( C exp(Gx)|x|−1dx x < 0 C exp(−M x)x−1dx x > 0 and γ = −C(G(exp(−M ) − 1) − M (exp(−G) − 1)) M G .

Thus, in mathematical notation under the (C, G, M ) parametrization, the increments XV G

s+t− XsV G over the time interval [s, t + s] follow a V G(Ct, G, M )

rule and according to the properties of the V G(C, G, M ) distribution we can replicate the XV G

t process as:

X_tV G= G(1)_t − G(2)_t

where G(1)t is a Gamma process with parameters a = C and b = M and

G(2)_t is a Gamma process with parameters a = C and b = G. A XtV Gprocess is

characterized by finite variation paths and infinitely many jumps in any finite time interval.

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Figure 5.1 A Variance Gamma sample path with C = 20, G = 30, M = 60.[19]

5.2.3 The Lévy Variance Gamma Default Model.

In the previous sections we stated the building blocks that our Variance Gamma Default Model is based on. In this model, the asset value of our reference entity V = {Vt, 0 ≤ t ≤ T } follows an exponential of Variance Gamma stochastic

process X = {Xt, 0 ≤ t ≤ T }. Our asset value process is characterized as a

pure jump process which allows for an instantaneous default. Default happens on the first time the asset value crosses a predetermined barrier L .

Ensuring risk neutrality

We assume an economy that consists of a risk neutral asset B = {Bt, 0 ≤ t ≤ T }

and an asset V = {Vt, 0 ≤ t ≤ T } that is risky , their value processes are

considered, respectively, as:

Bt= B0ert

and

Vt= V0e(r+Xt+ω)t

where r is the risk free rate and it is considered flat. For Vt, ω is the mean

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neutral probability Q, E [Vt] = V0ert holds. Therefore, we need to choose ω

such that:

E[V0e(r+Xt+ω)t] = V0ert

for the Variance Gamma case,

E[exp(Xt)] = 1 − θν − 1 2σ 2_ν −νt

hence, we insure risk neutrality by choosing ω as

ω = 1 ν log 1 −1 2σ 2_{ν − θν} . Defining survival

Under the laws of our model, our reference entity’s asset value follows an expo-nential of a V G(σ, ν, θ) process and for a starting value of V0 at time t default

is defined as Vt= V0eXt > L, V0> 0 or equally, Xt< log L V0 .

The risk neutral probability of survival Psurv(t) between the time interval

(0, t] is : Psurv(t) = PQ Xs> log L V0 , 0 ≤ s ≤ t; = PQ min 0≤s≤tXs> log L V0 ; = EQ 1 min 0≤s≤tXs> log L V0 = EQ 1 min 0≤s≤tVs> L Psurv(t) = EQ 1 min 0≤s≤tVs> L (15)

The indicator function 1(E) is :

1(E) = (

1 , if E is T rue 0 , otherwise

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To continue with our CDS pricing formula we first need to introduce the concept of barrier options and more specifically the Binary Down and Out option or BDOB and the Binary Down and In option or BDIB . The payoff of this kind of option is either 1 unit of currency or nothing depending on the price level of the underlying and a barrier L. Hence, firstly we can present the payoff of a BDOB with a barrier level L and maturity T , which is :

P ayof fBDOB(T ,L)=

(

1 , if Vt> L f or all t, 0 ≤ t ≤ T

0 , otherwise

and the price of BDOB option under the risk neutral measure is:

BDOB(T, L) = D(0, T )EQ 1 min 0≤s≤TVs> L = D(0, T )PQ surv(T ) (16)

On the other hand, the BDIB option comes into existence and pays 1 unit of currency if the value of the underlying crosses the barrier L. The payoff of a BDIB option with barrier L and maturity T is the following:

P ayof fBDIB(T ,L)=

(

1 , if Vt< L f or all t, 0 ≤ t ≤ T

0 , otherwise

equivalently to the price of BDOB option, the price of BDIB option under the risk neutral measure is :

BDIB(T, L) = D(0, T )EQ 1 min 0≤s≤TVs≤ L = D(0, T )PQ def ault(T ). (17)

Under the risk neutrality assumptions, the relationship between BDOB and BDIB options with the same characteristics is:

BDOB(T, L) + BDIB(T, L) = D(0, T )

Under the above relationship , the fact that Pdef ault(0, t) + Psurvival(0, t) = 1

and formula (17) we can calculate the probability of default as:

BDOB(T, L) + BDIB(T, L) = D(0, T ) BDOB(T, L) D(0, T ) + BDIB(T, L) D(0, T ) = 1 PQ def ault(T ) = 1 − BDOB(T, L) D(0, T )

Thus, we can calculate the probability of default through the price of a Binary Down and Out Barrier option.

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Credit Default Swap pricing.

As mentioned in Chapter 3, the price of a CDS with maturity T and par spread S and recovery rate R in continuous time is :

CDS = (1 − R)  − T ˆ 0 D(0, s)dPsurv(s)  − S T ˆ 0 D(0, s)Psurv(s)ds

the par spread S is,

S = (1 − R)−´₀TD(0, s)dPsurv(s) ´T 0 D(0, s)Psurv(s)ds = (1 − R)1 − D(0, T )Psurv(T ) − r ´T 0 D(0, s)Psurv(s)ds ´T 0 D(0, s)Psurv(s)ds (18)

hence, we can connect the price of the par spread S with the price of a BDOB and the resulting formula is :

S =

(1 − R)1 − BDOB(T, L) − r´₀TBDOB(s, L)ds ´T

0 DBOB(s, L)ds

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The problem of the par spread estimation comes down to the pricing of a BDOB at all time points till maturity T .

5.2.4 Monte Carlo CDS pricing. Monte Carlo simulation.

A Monte Carlo method is a stochastic process that uses random numbers and statistics to solve a problem. In a Monte Carlo experiment we use a random number generator to simulate, in our case, random asset paths according to a system. By performing an adequate iterations of simulations/asset paths we can receive a statistically significant estimate of our asset value in the future.

Steps to estimate the CDS price.

It is quite easy to estimate the CDS par spread through Monte Carlo simulations. The steps to do so are listed below:

1. Firstly, following the procedure in the section 6.2.2, we simulate a Variance Gamma process X = [Xt, 0 ≤ t ≤ T } with parameters (σ, ν, θ), in discrete

points in time, we divide the time to maturity T by N time intervals ∆t such that tn= n∆t, n = 1, 2, .., N and we receive values of our VG process

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2. The asset values V in at each point in time are obtained by the risk neutral process of section 6.2.3, and we receive as a result a path of our asset values {Vt1, ..., VtN} for each Monte Carlo iteration.

3. Then we can price our BDOB option at every point in time by checking if the asset value Vtn is higher than the barrier L and by using the formula

(16).

4. Lastly, we receive the BDOB estimate price by averaging the BDOB values at each time tn.

Then, the par spread S can be priced according to the formula (19).

5.3 Model comparison.

In this section we perform a comparison exercise between the models. The parameters used are chosen arbitrarily.

Asset price paths.

In this section we are going to present graphically the differences between the models under consideration. Figure 5.2, shows an asset price path derived from the Black- Cox model, the asset price movement is a brownian motion, which is continuous and does not model for an unexpected instant crossing of the barrier. In this example, the barrier level is set to 40 and we can see that a default event is triggered around the third quarter. In figure 5.3, we present an asset price path generated by our Variance Gamma model. We can see that in in the VG model the asset price incorporates jumps and does allow for instantaneous defaults.

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Figure 5.3. Variance Gamma Asset value path.

Probability of default.

The next graph presents the probabilities of default as derived from our models, by matching the value of the “common” volatility parameter σ. We can see that the Merton model presents the lowest probabilities, close to zero for short maturities. The Black-Cox model results in higher probabilities of default than the Merton model and the Variance Gamma model presents the highest. It seems that the level of the probability of default is proportional to the realisticity of each model. In the merton model, default can only happen at the end of the period; the scenario that the asset price falls below the predetermined barrier at a time t < T but ends above it, is not considered as a default event. But in reality, as mentioned in the previous chapters, this is not the case. The Black -Cox model addresses to this fact by allowing for early default, this fact is indeed reflected to the resulting probabilities which are higher. This is logical since scenarios that in the Merton model are not considered as defaults are indeed counted as defaults. The Variance Gamma model, accounts for early defaults like in the Black-Cox model, and also incorporates instantaneous default events. In reality, defaults can be triggered unexpectedly by events such as war, natural disasters, fraud or terrorist attacks. These kind of events can potentially have a serious impact on the financial health of the reference entity. By incorporating the possibility of such events in the modeling process, we would expect higher probabilities of default and this is the case with the Variance Gamma model. Therefore, the results presented in figure 8 are in line with our intuition. The Variance gamma model seems to be more pragmatic than the other two models, consequently there is evidence that it is the preferred model.

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Figure 5.4 Probability of default. r = 2%, σ = 0.2 for merton and Black-Cox model.VG(0.2, 0.1, 0).

Figure 5.5. Effect of ν parameter on the Probability of default between (0, T = 1year].

In Figure 5.5 we perform a sensitivity analysis, for the VG model, by differing the parameter ν who governs kurtosis and by keeping the rest of the parameters fixed. The results are again in line with our intuition, higher kurtosis results in higher probabilities of default. Again, Black - Cox model presents lower Probability. It seems that the Variance Gamma distribution is indeed more flexible and shows potential on capturing high risk even at short maturities. In

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the next section we test and compare the capability of our models on pricing CDSs.

CDS pricing.

In this section we are going test our models’ capability to price a Credit Default Swap contract with maturity T = 1 in three cases. The risk free rate is taken at 2%. The asset value at time 0 is V0 = 100. The barrier level L = 40 and

recovery rate R = 40%.

Table 1. Experiment with varying parameters on the price of a CDS with maturity 1 year.

From the table above, we can see that the Merton and Black-Cox models present very low CDS prices for our CDS under consideration. The results are in line with the result of figures 5.4 and 5.5 , where for low maturities the derived probabilities of default from Merton and Black- Cox models are close to zero. On the other hand our Variance Gamma model presents higher flexibility in the CDS pricing for low maturity contracts. Therefore, there is evidence that the variance Gamma model is to be preferred since it proves to be able to capture default risk in short maturity contracts as well.

6 Data.

6.1 Data description.

Every business day, CDS spread quotes for a wide variety of financial reference entities are published and traded in the market. Usually, on each trading day and for each reference entity corresponds a series of CDS contracts with a varying maturity date.

Our data set was taken from Bloomberg’s Data-stream and consist of daily quotes of Sovereign CDS contract spreads of Greece and Germany. The German market proved much more stable than the Greek Credit Default swap market that presented hight volatility during the crisis. A three year period of Sovereign Credit Default Swap spreads is covered, for both countries, from 19-02-2008 to 18-02-2011 and for contracts with maturities 1 year, 3 years, 5 years, 7 years and 10 years, resulting to 784 observations for each maturity. This data set provides us with ample data curves to use in order to test for the replicating capabilities of our models.

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6.2 Data visualization.

Figure 6.1, presents two CDS spread term structures of Greece and two of Germany. Each circle represents a CDS contract spread with a corresponding maturity of 1 year, 3 years, 5 years, 7 years and 10 years. We can see that the CDS term structures present a variety of shapes. The term structure of Germany on 30/04/2009 is upward sloping up to the 5th_{year and then presents}

a flat behavior for contracts with maturities higher than 5 years.

The most common shape CDS term structures form is upward sloping curves, but the term structure of Greece on 26/08/2010 presents a different behavior. Intuitively, CDS contracts with higher maturities are more risky, in consequence of the higher uncertainty of the farther future time-point, therefore would nor-mally expect that they present higher spreads resulting in upward sloping curves. Though, we can see that this is not the case for the CDS curve of Greece on 26-08-2010. We see that the term structure presents a negative 10yr/1year slope of approximately 200bps. Merton (1974) explained this phenomenon, in times of financial distress short-term risk of default can be very high, but if the reference entity is able to survive the current difficult situation, it is more likely that obligations in the long term future will be honored. It is interesting to see if our models can capture downward sloping curves as well.

Figure 6.1. Sovereign CDS spreads of Germany and Greece on selected dates. Maturities of 1, 3, 5, 7 and 10 years.

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7 Calibration.

The data of the market term structures as presented in the previous section will be used in our calibration. We saw that each of our models depend on a set of parameters to estimate probability of default and consequently CDS spreads. In section 5.3 for example, we chose randomly those parameters and we received some results that served our comparison purposes. In this section we will invert the procedure and, instead of choosing values for the parameters, we will use our data and find the parameters that are implied by them. Our models are forward looking and their implied parameters reflect future estimations. Therefore, these implied parameters can be seen as the instant market estimation of the future price movement of the underlying asset. Historical estimates are not appropriate for our context, because they incorporate past behavior.

To extract these implied parameters we are going to use the Root Mean Square Error statistic (RMSE). RMSE can be seen as a type of distance be-tween our model resulting values and the real market data. Optimal implied parameters will be considered the set of parameters, for each of our models, that when they are plugged in our model the result will present a minimum RMSE.

The Root Mean Square Error statistic is defined as:

rmse = v u u t X numberof maturities

(M arket spread − M odel spread)2

number of CDS maturities (20)

Statistical packages offer search algorithms who look for optimal parameters by the trial and error technique. These search algorithms change the required initial values (user specified) of the parameters and return those who minimize the input function, which is RMSE function in our case. There are plenty of opti-mization methods in the literature such as the “quasi - Newton Method”, “Nelder - Mead simplex method” and “Conjugate Gradients Method”, these methods dif-fer mainly on the tradeoff between time consumption and accuracy depending on the type of the function under calibration of course.

8 Results.

We test the performance of our models on a randomly selected CDS curves from our dataset. The recovery rate is taken to be R = 40%, the risk free rate is constant at r = 2%, the asset value V0= 100 and a barrier level L of 40. The

CDS spreads are the result of the formulas (1) , (19) and (12). Our calibration method of choice is Nelder - Mead simplex method because of robustness.

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Market spreads vs Model Spreads

Results of the fits of our models to the CDS term structure.

Country Date Model 1 year 3 year 5 year 7 year 10 year

Greece 23/07/2008 Market 21 36 49 53 60 Merton 0 12 33 55 74 Cox 7 26 47 56 63 VG 20 35 46 54 62 Greece 20/01/2010 Market 374 355 345 334 317 Merton 60 199 312 409 466 Cox 95 350 392 386 387 VG 313 369 372 361 346 Germany 30/04/2009 Market 15 26 39 39 39 Merton 0 7 21 38 53 Cox 7 18 33 40 43 VG 14 27 36 38 39 Germany 27/10/2010 Market 4 11 21 25 32 Merton 0 3 13 25 37 Cox 2 10 21 27 33 VG 3 12 19 25 31

Table 2. Comparison of resulting CDS spreads in bps from different models.

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Figure 7.2. Calibration results. Market spreads are depicted with “_{#” signs.}

On Table 2 and graphically on Figures 7.1 and 7.2, we can see the results of our calibration exercise. Merton model completely fails to capture any features of the CDS curves. The Black-Cox model presents low performance for the low maturity contracts but presents seemingly better fits for contracts with higher maturity. The Variance Gamma model follows more closely the shapes of the CDS curves but it seems to miss the downward sloping behavior in the case of Greece on 20/01/2010, it underestimates the CDS spread of the contract with maturity 1 and over estimates the spreads of the remaining maturities.

RMSE and implied optimal parameters

Country Model σ ν θ rmse

Greece 23/07/2008 Merton(σ) 0.1746 – – 17.2895 Cox(σ) 0.1823 – – 8.0397 VG(σ, ν, θ) 0.1151 2.2490 -0.0534 1.8853 Greece 20/01/2010 Merton(σ) 0.3066 – – 174.1717 Cox(σ) 0.4215 – – 132.2041 VG(σ, ν, θ) 0.1813 1.7325 0.2085 35.2012 Germany 30/04/2009 Merton(σ) 0.1622 – – 14.7877 Cox(σ) 0.184 – – 7.3096 VG(σ, ν, θ) 0.1550 3.2601 0.0021 3.4641 Germany 27/10/2010 Merton(σ) 0.1515 – – 5.6418 Cox(σ) 0.1684 – – 3.4963 VG(σ, ν, θ) 0.0350 3.1998 -0.051 1.3178 Table 3. Calibration output of model parameters and rmse.

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Probability of default given the implied parameters.

Country Pdef ault(T ) 1 y 3 y 5y 7y 10y

Greece 23/07/2008 Merton 0 0.6% 2.75% 6.41% 12.33% Cox 0.12% 1.3% 3.91% 6.53% 10.5% VG 0.3% 1.75% 3.83% 6.3% 10.32% Greece 20/01/2010 Merton 0.9% 9.95% 25.99% 47.72% 70% Cox 1.57% 17.5% 32% 45.03% 64.49% VG 5.21% 18.44% 31.12% 42.11% 57%.66 Germany 30/04/2009 Merton 0% 0.35% 1.75% 4.33% 8.07% Cox 0.16% 0.92% 2.75% 4.66% 7.14% VG 2.56% 1.05% 2.83% 4.2% 6.5% Germany 27/10/2010 Merton 0% 0.15% 1.07% 2.91% 6.15% Cox 0.03% 0.49% 1.72% 3.15% 5.5% VG 0.05% 0.6% 1.57% 2.9% 5.1%

According to our results, the VG model presents a better fit to the market features of CDS compared to the rest of the models in term of RMSE. Though, all three models fail to capture the downward sloping term structure of the Greek curve on 20/01/2010.

9 Conclusion.

The purpose of our paper was to test for the capability of three selected struc-tural credit models to price CDS contracts and replicate their market spread term structures. Among the credit risk estimation methods found in the lit-erature, the two most popular approaches are Structural models and Intensity Based models. We focused on the first approach and analyzed the conception of Merton, which is the prototype of the structural models approach. Merton linked the equity of the reference entity with the stock market. That allowed the equity value to be seen as an option on the asset value with strike price a fixed barrier that can be priced through the Black-Scholes formula. Default happens when the asset value falls below the barrier. The normality assumptions of the Black-Scholes framework allow for efficient calculations of the option prices and probabilities of survival. In the Merton model, default can only happen at maturity and this is the reason that our second model, which is a Cox type structural model, was chosen. The Black- Cox model allows for early default, which is a more realistic assumption, resulting in lower probabilities of survival but continues to take advantage of the efficiency of the Black-Scholes framework. Though, the normality assumptions of the previous two models are not consid-ered realistic, due to the fact that default events can be shock driven, and this is the reason for our third model of choice, the Lévy Variance Gamma Default Model. This structural model assumes that the asset value follows a Variance Gamma process which is entirely driven by jumps, overcoming the flaws of the normality assumptions.

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We tested the models on: Firstly, estimating probabilities of default, sec-ondly, on pricing CDS contracts and finally, we explored the capability of the chosen structural credit risk models in replicating the features of Credit Default Swap term structures presented in the market.

Our findings suggest that the Merton model fails completely to capture the market CDS term structure trends. The Black-Cox model underestimates the probability of default for contracts with short maturity but presents better fits for contracts with longer maturity. The Lévy Variance Gamma model presents encouraging results in its ability to capture the behavior of the CDS term struc-tures and it is the only one that is able to capture CDS term strucstruc-tures for con-tracts with low maturities. This provides us with evidence that incorporating jumps in the modeling process has a positive impact in estimating probabilities of default even for short maturities and indicates an added value to Structural credit risk modeling and Credit derivative pricing. Though, all three of our selected models fail to capture the downward sloping term structure presented in our data. One possible reason for this can be in our assumptions, where we took interest rates and recovery rates as constant.

We would recommend to further investigate by modifying the models under stochastic interest rates and a variable recovery rate. As far as the Merton model is concerned, it would be interesting to investigate if its performance can be improved by incorporating the concept of the volatility smile. The results of our exercise indicate that a fixed volatility parameter for all maturities is not appropriate to describe the market Sovereign CDS term structures. That would be possible by calibrating the model on each CDS contract separately extracting a maturity specific implied volatility each time.

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APPENDIX

A.1. Probability of survival in Black-Cox model

with constant Barrier.

Following Gokgoz (2012) and Harrison M. J. (1990) , we are going to derive the probability of survival Psurv. To do so, we should initially determine the

joint distribution of the brownian motion and itself. Suppose the maximum of a Brownian motion with zero drift and arbitrary σ , Mt= sup{Xs, 0 ≤ s ≤ t}

and the maximum of a standard brownian motion, mt = sup[{X, 0 ≤ s ≤ t},

where Xt = σWt and Wt ∼ N (0, t) a standard Brownian motion. The joint

distribution function is:

Ft(x, y) = P{Xt≤ x, Mt≤ y}

Since Xt≤ Mt, for all t ∈ [0, T ] we can to compute Ft(x, y) only for x ≤ y

Ft(x, y) = P{Xt≤ x, Mt≤ y} = P{Xt≤ x} − P{Xt≤ x, Mt> y}. as a result of P{Xt≤ x} = P{Xt≤ x, Mt≤ y} + P{Xt≤ x, Mt> y} = P{σWt≤ x, σmt≤ y} + P{σWt≤ x, σmt> y} = P{Wt≤ x/σ, mt≤ y/σ} + P{σWt≤ x/σ, σm > y/σ} Thus, Ft(x, y) = Φ _x σ√t − P{Wt≤ x σ, mt> y σ}.

At this point, to solve P{Wt≤ x_σ, mt> y_σ} we need to use the reflection property

of the Brownian motion. Firstly we substitute x

σ = υ and y

σ = u for simplicity,

so we have P{Wt≤ υ, mt> u}. The reflection property is based on the fact that

the normal distribution is symmetrical around its mean,then, for every sample path of W that touches u earlier that t but ends under υ at time t, there exists a symmetric path with the same probability that touches u earlier than tand then goes upwards to at least u − υ and to end at u + (u − υ) = 2u − υ at time t.

Therefore, we have :

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= P{Wt< υ − 2u} = Φ υ − 2u√ t = Φ x − 2y σ√t .

Suppose that T is the first time that Wt= u and WtQ= Wt+T− WT. From

the Strong Markov property we have that :

P{Wt≤ υ, mt> u} = P{T ≤ t, Wt−TQ ≤ υ − u} = P{T ≤ t, WQ t−T ≥ υ − u}. By definition, W_t∗− T = Wt− u therefore, P{Wt≤ υ, mt> u} = Φ υ − 2u √ t P{Wt≤ x σ, mt> y σ} = Φ x − 2y σ√t hence, Ft(x, y) = Φ _x σ√t − Φ x − 2y σ√t (21) Proposition A1. 1.

Suppose a standard brownian motion W, Xt= σWtand Mt= sup{Xs, 0 ≤

s ≤ t}. Then, P{Xt≤ x, Mt≤ y} = Φ _x σ√t − Φ x − 2y σ√t (22) Corollary A1.2. P{Xt∈ dx, Mt≤ y} = gt(x, y)dx with gt(x, y) = Φ _x σ√t − Φ x − 2y σ√t ₁ σ√t, (23) Φ(z) = √1 2πexp −z 2 2 .

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dFt(x, y) dx = Φ _x σ√t − Φ x − 2y σ√t ₁ σ√t. Proposition A1.3.

Suppose Xt ∼ N (µt, σ2t) and ft(x, y) = PQ{Xt ∈ dx, Mt ≤ y}, then

ft(x, y) = ξtgt(x, y)dx, where ξ = dP Q dP = exp{ µ σWt− 1 2 µ2 σ2}.

By the Girsanov Theorem, WQ

t = Wt+

´t 0 µ

σdt is a standard Brownian

Motion under the risk neutral measure Q, θ = µ−rσ is market price of risk,

for r = 0 θ = µ_σ. Since Xt = µt + σWt under P, Wt = Xt_σ−µt, and hence,

WQ t = Xt−µt σ + µt σ = Xt σ . Also, ξ = exp{ µ σWt− 1 2 µ2 σ2} = exp{ µ σ2Xt−1₂µ 2 σ2},we have: PQ[Xt≤ x, Mt≤ y] = EQ[1{Xt≤x,Mt≤y}] = E[ξt1{Xt≤x,Mt≤y}] = E exp{ µ σ2Xt− 1 2 µ2 σ2}1{Xt≤x,Mt≤y} = ˆ x −∝ exp{µ σ2Xt− 1 2 µ2 σ2}P{Xt∈ dz, Mt≤ y} = x ˆ −∝ exp{ µ σ2Xt− 1 2 µ2 σ2}gt(z, y)dz. Corollary A1.4.

Let Ft(x, y) = P{Xt ≤ x, Mt ≤ y} where Xt ∼ N (µt, σ2t) and Mt =

sup{Xs, 0 ≤ s ≤ t}, we have that :

Ft(x, y) = Φ( x − µt σ√t ) − exp( 2µy σ2 )Φ( x − 2y − µt σ√t ) (24) Proof. Ft(x, y) = x ˆ −∝ ft(z, y)dz = x ˆ −∝ exp µz σ2 − µ2t 2σ2 ₁ σ√t Φ _z σ√t − Φ z − 2y σ√t dz = exp −µ 2_t 2σ2 ˆx exp{µz σ2} 1 σ√t Φ z σ√t − Φ z − 2y σ√t dz

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with z0 = z − x we have, Ft(x, y) = exp −µ 2_t 2σ2 ˆ0 −∝ exp{µ(z + x) σ2 } 1 σ√t Φ z + x σ√t − Φ z + x − 2y σ√t dz = exp −µ 2_t 2σ2+ µx σ2 ˆ0 −∝ exp{µz σ2} 1 σ√t Φ z + x σ√t − Φ z + x − 2y σ√t dz Hence, Ft(x, y) = exp −µ 2_t 2σ2 + µx σ2 {Ψ(x) − Ψ(x − 2y)} (25) with Ψ(x) = 0 ˆ −∝ expnµz σ2 o 1 σ√tΦ z + x σ√t dz Let h(x, t) = x−µt

σ√t and by writing out Φ() we have,

Ψ(x) = 0 ˆ −∝ 1 σ√texp nµz σ2 o 1 √ 2πexp −(z + x) 2 2σ2_t dz = 0 ˆ −∝ 1 σ√2πexp µz σ2 − (z + x)2 2σ2_t dz = 0 ˆ −∝ 1 σ√2πexp 1 σ2 µz −z 2_{+ 2xz + x}2 2t dz = 0 ˆ −∝ 1 σ√2πexp 1 σ2 −z 2_{+ 2(x − µt)z + (x − µt)}2 2t + µ 2_t 2 − µx dz = exp µ 2_t 2σ2 − µx σ2 ˆ0 −∝ 1 σ√2πtexp ( −1 2 ₁ σ√t(z + x − µt) 2) dz = exp µ 2_t 2σ2− µx σ2 ˆ 0 −∝ 1 σ√tΦ (z + x − µt) dz

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= exp µ 2_t 2σ2 − µx σ2 h(x,t)ˆ −∝ Φ(u)dz with u = z+x−µt σ√t , by taking z = 0 we have u = x−µt σ√t. Thus, Ψ(x) = exp µ 2_t 2σ2− µx σ2 Φ x − µt σ√t . (26)

By plugging equation (24) into (23) we get,

Ft(x, y) = exp −µ 2_t 2σ2+ µx σ2 exp µ 2_t 2σ2 − µx σ2 Φ (h(t, x)) = exp −µ 2_t 2σ2 + µx σ2 exp µ 2_t 2σ2 − µx σ2 Φ x − µt σ√t − exp µ 2_t 2σ2 − µ(x − 2y) σ2_t Φ x − 2y − µt σ√t = Φ x − µt σ√t − exp 2µy σ2 Φ x − 2y − µt σ√t .

Suppose that Ty is the first time that the process Xt is equal to y, Ty =

inf{t : Xt= y}, Ty> y =⇒ Mt< y and by letting x % y :

P(Ty > t) = P(Mt< y) = Ft(y, y) =⇒ Ft(y, y) = Φ y − µt σ√t − exp 2µy σ2 Φ −y − µt σ√t (27)

To calculate the probability of the asset value crossing the barrier L for the first time we need to estimate the probability P {infs≤tXs> y}. Because of the

symmetric property of the brownian motion, Wt∼ −Wt, instead of calculating

infs≤t(Xs) we can use the equal expression − sups≤t(−Xs), therefore we have,

P inf s≤tXs> y = P − sup s≤t (−Xs) > y = P sup s≤t (−Xs) < −y = P − sup s≤t (−µs − σWs) < −y = P − sup s≤t (−µs + σWs) < −y .

To calculate this probability suppose Yt = −µ + σWt and Nt = sups≤tYs

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Gt(x, y) = P {Yt≤ −x, Nt≤ −y}

for y < x < 0 or for 0 < −x < −y and by plugging −x, −y, −µt instead of x, y, and µt we have the following proposition.

Proposition 1.4.

The joint distribution of Ytand it supremum Ntis :

Gt(x, y) = P {Yt≤ −x, Nt≤ −y} = Φ −x + µt σ√t −exp{2µσ−2y}Φ −x + 2y + µt σ√t (28) and by letting x % y Gt(y, y) = P inf s≤t(Xs) > y = Φ −y + µt σ√t − exp2µσ−2_{y Φ} y + µτ σ√t . (29) Corollary 1.5

As a result of proposition (1.4) for every t ≥ u , u ∈ (0, T ], while t < τ , we have: P(τ > u|Ft) = Φ Zt+ µ(u − t) σ√u − t − exp(−2µσ−2Zt)Φ −Zt+ µ(u − t) σ√u − t . (30) Since the asset value follows a log-normal diffusion process we have

P(τ > u|Ft) = P(τ > u|τ > t) P(τ > u|τ > t) = Φ(d3) − L Vt _σ22r−1 Φ(d4) = PsurvQ (T |Ft) (31)

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Credit default swap term structures and structural models