Credit Default Swaps

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Credit Default Swaps

R. Jansen

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Masterthesis Econometrics-Actuarial Studies Supervision:

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Credit Default Swaps

R. Jansen

Abstract

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CHAPTER

1

Introduction

Bond holders cope with different types of risk. The most important risk types are: market risk, credit risk, liquidity risk, operational risk and systematic risk. Credit derivatives allow market participants to exchange credit risk. The first agreements of credit derivatives arose in the early 1990’s. The main difference with more traditional credit risk protections is that the former could not be traded separately from the underlying obligation, whereas this is allowed for credit derivatives. Several types of credit derivatives exist, among which Credit Default Swaps (CDSs) form the largest group. The CDS is invented in the early 1990s by the mathematican Blythe Masters who worked at JP Morgan at that time and was in charge of the group that invented CDSs. The last years, before the credit crisis, the number of CDSs sold has increased enormously. In 2007 the size of the CDS market was about two times the size of the New York stock market, where the size of the CDS market is measured as the notional amounts outstanding in dollars. CDSs played a large role in the credit crisis allowing banks to reduce their risk on Collateralized Debt Obligations (CDOs).

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CHAPTER

2

Basic Theory

This chapter discusses some basic theory and concepts useful in the discus-sion about CDSs. The first section introduces some definitions and gives an example of a CDS. In the second section the cash flows of a CDS are considered. Section 2.3 and 2.4 explain how to price a single claim using a constant and a stochastic interest rate respectively. Section 2.5 discusses some standard interest rate models. The last sections introduce some basic probability theory, including the Radon-Nikodym derivative and the concepts of predictable and totally inaccessible stopping times.

2.1 Introduction to Credit Default Swaps

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for his loss if the reference entity defaults. The exact amount of compensa-tion is stated in the contract. We will consider an example of a CDS. Assume that the protection buyer, bank A, possesses bonds of a single company with a total face value of 100 million dollar. Bank A buys a five year CDS contract from the protection seller, bank B. Each year bank A pays a premium. If no default occurs bank B simply earned the accumulated amount of premium payments. On the other hand if the bond defaults bank A delivers all his bonds to bank B and receives the total face value of 100 million dollar in return. The exact amount of premium payments bank A should pay depends on the probability of default of the underlying bonds. In the next section we will consider the cash flows of a CDS.

2.2 Cash flows of a credit default swap

We examine the cash flows of a CDS. First note that a corporate bond bears the risk of default, hence a corporate bond is an example of a defaultable bond or also called risky bond. Consider the following framework. Set the notional amount (also called face value) of the corporate bond equal to L and take a fixed maturity T . Let D(t, T ) be the arbitrage-free price of the bond at time t. Note that D(T, T ) = L if no default has occurred before or at maturity. Furthermore, let B(t, T ) denote the arbitrage free price at time t of a default-free bond with maturity T and face value 1. In the sequel sometimes a default-free bond is given with only one argument, B(T ). If this is the case then this single argument denotes the maturity of the bond. Since we consider a default-free bond the value at maturity equals the face value, hence B(T, T ) = 1. We want to model the cash flows of a Credit Default Swap. Let τ be the time of default. The premium amount κ is paid m times at Ui, with i = 1, 2, ..., m where Ui ∈ [0, U ] with U the maturity of the CDS.

From the buyers perspective the cash flows on the CDS can be represented as (L − D(t, T ))1τ(t)1[0,U ](t) − m X i=1 e−r(Ui−t)_κ1 [Ui,∞](τ )1[Ui,∞](t), (2.1)

where T is the maturity of the corporate bond such that T ≥ U . The first term of Eq. (2.1) represents the recovery payment in case default occurs, in mathematical terms 1τ(t) and if it occurs, it should occur before or at

maturity, hence the term 1[0,U ](t). The second term of Eq. (2.1) represents

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payment is given as the difference between the face value of the bond L and the value of the bond D(t, T ) at the time of default τ (or actually the value immediately after default). Other possibilities for the recovery payment are

(LB(τ, T ) − D(τ, T )) (2.2) and

(D(τ −, T ) − D(τ, T )) , (2.3) where τ − is the time immediately before default. Or more precisely, τ − is defined by D(τ −, T ) := limt↑TD(t, T ) where t 6= τ . The recovery payment as

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2.3 A short note on option pricing

This section considers the pricing of a contingent claim, also called a deriva-tive instrument. Most of what follows can be found in Bj¨ork (2004). A contingent claim is a contract of which the price is defined in terms of an underlying asset, S(t). A contingent claim with maturity time T is called a T -claim. In this section we only consider simple claims since under that restriction we can obtain an explicit formula for the hedging portfolio. A simple claim is a claim of the form

χ = φ(S(T )), (2.4) where φ is called the contract function. If we want to price a simple claim we thus know that at time T we receive an amount φ(S(T )). Therefore, the price at time T of this claim, Π(T, χ), is

Π(T, χ) = φ(S(T )). (2.5) We want to know Π(t, χ), the price of the claim at each time instant t, where t < T . Since the price of the claim is derived from the price of the underlying asset, a reasonable assumption is that Π(t, χ) = F (t, St) where F is a smooth

function. This assumption is justified in case St is a Markov process since

then the claim is a function of the last value St only and not of the whole

process S0, ..., St.

The claim exists in a certain market. We consider the framework of the Black-Scholes model, but we could also have chosen a different market. In the Black-Scholes model we have two assets: a bond and a stock defined by the following equations

dB(t) = rB(t)dt (2.6)

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In the above equation we left out the arguments for sake of notational sim-plicity, but of course F means F (t, S(t). We form a self-financing portfolio (us, uπ) based on the underlying stock and the derivative asset. Constructing

this self-financing portfolio based on the derivative asset is allowed only if the derivative asset is tradable in the market. This is an important, and yet very strong assumption made in this derivation. We will come back to this later. Since we have a self-financing portfolio, the value V of the portfolio will have the dynamics

dV = V (us[αdt + σdW ] + uπ[απdt + σπdW ]) (2.8)

and must satisfy that us+ uπ = 1. Rearranging Eq. (2.8) gives

dV = V (us[αdt + σdW ] + uπ[απdt = σπdW ]) . (2.9)

We require the market to be free from arbitrage and therefore, we must have dV = rV dt. (2.10) Hence, if we set usσ + uπσπ = 0 then we know that

usα + uπ = r. (2.11)

The next steps are simple algebra and therefore, we only mention which steps should be taken. Solving the system of Eq’s

us+ uπ = 1

usσ + uπσπ = 0

gives us expressions for us and uπ. These expressions can be substituted

in the no-arbitrage condition, Eq. (2.11) and rewriting gives the following equation ∂F ∂t + 1 2σ 2_S∂ 2_F ∂S2 + S ∂F ∂S = rF. (2.12) Also we must have Π(T ) = φ(S(T )) since we consider a simple claim of this form. Under very weak assumptions the distribution of S(t), ∀t has support on the entire positive line, so S(t) ∈R+ _{∀t. Since this holds for all S, we get}

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The question now rises: has this PDE a solution? And is the assumption we made in the beginning that F is a smooth function satisfied? We must verify this, since only then it was correct to apply Ito’s formula. This PDE indeed has a solution. It can be solved using a stochastic representation `a la Feynman-Ka˜c. The next section considers stochastic interest rates. We will see that one of the differences has to do with the fact that the interest rate is not a tradable asset.

2.4 Term structure equation

In this section we consider stochastic interest rates and the so called term structure equation is derived. The dynamics of the stochastic short rate is, under the probability measure P, given as

dr(t) = µ(t, r(t))dt + σ(t, r(t))dW. (2.15) The risk-free asset follows the dynamics

dB(t) = r(t)B(t)dt, (2.16) where r(t) is now a stochastic process given by Eq. (2.15). We turn to the problem of pricing a simple claim of the form

χ = φ(r(T )). (2.17) Our first guess could be to consider the value of the claim Π(t, χ) as a function

Π(t, χ) = F (t, r(t))

and to proceed analogue to the procedure used in the previous section to derive the Black-Scholes PDE. However, a major difference is that the interest rate is not a tradable asset, whereas the stock price S(t) in the Black-Scholes model is. Therefore, it is not possible to form a self-financing portfolio based on the interest rate. There are no possibilities of forming interesting portfolios and it is not possible to derive a unique price based on the no-arbitrage condition only. A different way to see this is to look at the completeness of the model. The model is incomplete and thus there is no unique martingale measure and no unique market price of risk.

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Inclusion of this benchmark bond makes the model complete and makes it possible to find an unique price for a simple claim of the form χ = φ(r(T )). We assume that the price of a bond, p(t, T ), has the form

p(t, T ) = F (t, r(t); T ) := FT(t, r) ∀T, (2.18) where F is a smooth function. We form a self-financing portfolio consisting of bonds with maturity times S and T , (us, uT). If we apply Ito’s lemma on

FT we obtain dFT = ∂F T ∂t dt + ∂FT ∂r dr + 1 2σ 2∂ 2_FT ∂r2 dt := αTdt + σTdW

and anologue for FS _{we find}

dFS = ∂F S ∂t + ∂FS ∂r dr + 1 2σ 2∂ 2_FS ∂r2 dt := αSdt + σSdW.

The value of the relative portfolio (uS, uT) has the form

dV = V uT dFT FT + us dFS FS = V ((uTαT + uSαS) dt + (uTσT + uSσS) dW ) ,

with uT + uS = 1. In the second line we substituted the expressions for dFT

and dFS. If we set uTσT + uSσS = 0 then only the dt-parts remain. Since in

a market that is free from arbitrage we must have that dV = V rdt we obtain

uTαT + uSαS = r. (2.19)

Solving the system of Eq.’s

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Substituting Eq.’s (2.20) and (2.21) into Eq. (2.11) gives αS− r(t) σT = αT − r(t) σS . (2.22)

Rewriting the above equation gives the following result αs− r(t)

σS

= αT − r(t) σT

. (2.23)

Notice that if we define

λ(t) = αS− r(t) σT

, (2.24)

where λ(t), called the market price of risk, then λ(t) has the property that it has the same value for every value of the maturity T . Substitution of the expressions for αS and σT into Eq. (2.24) gives after some rewriting the term

structure equation of interest rates ∂FT ∂t + (µ − λσ) ∂FT ∂r + 1 2 ∂2_FT ∂r2 = rF T _(2.25)

with boundary condition

FT(T, r) = φ(r). (2.26)

Term structure models of interest rates: the martingale

approach

In the previous section a derivation of the term structure is given under the probability measure P. It is also possible to give a derivation of the term structure equation using a different pricing measure Q. We will first note something about risk-neutral pricing. Consider a continuous time model with a filtered probability space defined on this interval. The probability space {(Ω, F , P), (Ft : t ∈ [0, T ])} satisfies the usual conditions, see Protter

(1990) page 3, where P is a probability measure. In bond pricing one mostly uses the measure Q, sometimes called risk neutral measure or martingale-measure, which is a measure such that under Q all discounted bond prices are martingales with respect to the information set {Ft : t ∈ [0, T ])}. This

risk-neutral measure is a tool used to simplify calculations. Under measure Q the process dWQ(t) = dWP(t) + λdt follows a Brownian motion, where λ is the market price of risk and WP_{(t) is a Brownian motion under measure}

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economy. It is the measure obtained by choosing the riskless asset as the numeraire, hence it’s name. The discount factor is e−R0tr(s)ds, where r(s) is

the interest rate. The measure Q is unique only if markets are complete. The general form of the short-term interest rate in a one-factor interest rate model under the measure Q is

dr(t) = ˜µ(t, r(t))dt + ˜σ(t, r(t))dWQ_(t), _(2.27)

where ˜µ(t, r(t)) and ˜σ(t, r(t)) are the drift and the diffusion of the interest rate process under measure Q respectively and WQ_{(t) is a Wiener process}

under Q.

We derive the term structure equation using a slightly different approach compared to the method used in the previous section. Assume that the interest rate follows the process according to Eq. (2.27). Let the bank account B(t) = e−R0tr(s)ds be the discount factor, where dB(t)t = B(t)r(t)dt.

As in the previous section we want to price a simple claim X = g(rT). Then

the value of the claim at time t denoted by V (t) is V (t) = EQhXe−RtTr(s)ds

i

(2.28)

and can be written as a function V (t) = F (t, r(t)). If F (t, r(t)) is a smooth function then we can apply Ito. This gives:

dV (t) = ∂F ∂tdt + ∂F ∂rdr + 1 2σ 2∂2F ∂r2dt. We now compute d V (t) B(t) = V (t)d 1 B(t) + 1 B(t)dV (t) = −V (t) B(t)2 dB(t) + 1 B(t)dV (t). Substitution of the equations for dV (t), dB(t) and dr(t) gives

d V (t)t B(t) = 1 B(t) ∂F ∂tdt + ∂F ∂rµ(t, r(t))dt + σ(t, r(t))dW˜ Q_{(t) +} 1 2σ 2∂2F ∂r2 dt − F (t)r(t) B(t) dt.

Since V (t)_B(t)_{t is a martingale under Q the finite variation parts should sum to} zero. Therefore, we obtain the following PDE

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with boundary condition F (T, r) = g(r) for all r ∈ R since X = g(rT).

For a default-free bond B(t, T ) the boundary condition is F (T, r) = 1 since then the payment at time T corresponds to X = 1.

Under the probability measure P the models will be slightly adapted by adding a term involving the market price of risk λ(t, r(t)) since dWQ_{(t) =}

dWP_{(t) + λ(t, r(t))dt, where W}P_{(t) is a Wiener process under the measure P.}

Under probability measure P Eq. (2.27) becomes

dr(t) = (˜µ(t, r(t)) + ˜σ(t, r(t))λ(t, r(t)))dt + ˜σ(t, r(t))dWP_(t)

= µ(t, r(t))dt + σ(t, r(t))dW P(t) (2.30)

where µ(t, r(t)) and σ(t, r(t)) are the drift and volatility under measure P respectively. We conclude that

µ(t, r(t)) = ˜µ(t, r(t)) + ˜σ(t, r(t))λ(t, r(t)) (2.31)

and

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2.5 Some standard interest rate models

This section discusses two of the most frequently used stochastic interest rate models: The Vasicek model introduced by Vasicek (1977) and the Cox-Ingersoll-Ross (CIR) model introduced by Cox, Ingersoll, and Ross (1985). The Vasicek model is the first model that incorporates a stochastic interest rate and is still frequently used. In the Vasicek model the short rate follows, under measure Q, the SDE

dr(t) = a(b − r(t))dt + σdWQ_(t), _(2.33)

where a, b and σ are positive constants and WQ_{(t) is a Wiener process under}

measure Q. The Vasicek model is a mean reverting process, which makes the model attractive since this is consistent with observations of the interest rate in practice. The Vasicek model allows for negative interest rates, but the probability of such an outcome is very small. We simulated the Vasicek model to verify this. For a description of the simulation method and a summary of the results we refer to Appendix A.

In the Cox-Ingersoll-Ross (CIR) model the short rate follows, under the risk-neutral measure Q, the SDE

dr(t) = κ(θ − r(t))dt + σpr(t)dWQ_(t), _(2.34)

where κ, θ, σ and r0 are positive constants and WQ(t) is a Wiener process

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2.6 Relation between the Radon-Nikodym

deriva-tive and the market price of risk

In this section the relationship between the Radon-Nikodym derivative and the market price of risk is investigated. Most of what follows can be found in Shreve (2004), Jacod and Protter (2004) Lamberton and Lapeyre (2008) and Bj¨ork (2004). First we discuss the concept of equivalence of two proba-bility measures. Let (Ω, F , P) be a probaproba-bility space. Then two probaproba-bility measures P and Q are equivalent if and only if

P(F ) = 0 ⇔ Q(F ) = 0, ∀F ∈ F . If it only holds that

P(F ) ⇒ Q(F ) = 0, ∀F ∈ F

then we say that Q is absolutely continuous with respect to P. This last definition is used in the Radon-Nikodym theorem, but of course the theo-rem then holds if P and Q are two equivalent probability measures as well. The Radon-Nikodym theorem states that if Q is absolutely continuous with respect to P then there exist a positive random variable Z such that

Q(F ) = EP(1FZ), (2.35)

where Z is unique almost surely. It is convention to denote Z = dQ_dP. Z is called the Radon-Nikodym derivative. An alternative notation of Eq. (2.35) is Z F dQ(ω) = Z F Z(ω)dP(ω), ∀F ∈ F . (2.36) We discuss two properties of the Radon-Nikodym derivative. One property of the Radon-Nikodym derivative is that EP_{(Z) = 1:}

Q(Ω) = EP(1ΩZ) = EP(Z) (2.37)

and we know that Q(Ω) = 1 since Q is a probability measure. Then it follows immediately from the equation above that EP_{(Z) = 1. Another property of}

the Radon-Nikodym derivative is that Z > 0. This can be shown as follows:

Q(Z = 0) = EP(1Z=0Z) = 0 (2.38)

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Suppose now that we have a filtration Ft then we can define the

Radon-Nikodym process as

Z(t) = E(Z | Ft), 0 ≤ t ≤ T, (2.39)

where Z = Z(T ). The Radon-Nikodym process is a Ft-measurable function

Zt : Ω → R+, ∀ 0 ≤ t ≤ T . We can show that the Radon-Nikodym process

is a P-martingale:

E [Z(t) | Fs] = E [E [Z | Ft] | Fs]

= E [Z | Fs]

= Z(s).

The first step applies the definition of Z(t). The second step is justified by the law of iterated expectations and in the third step the definition of Z(t) is used again. The property that the Radon-Nikodym process is a P-martingale allows us to represent Ztin terms of a stochastic integral. Namely,

according to the Martingale representation theorem every square-integrable martingale with respect to the natural filtration can be represented in terms of a stochastic integral. To be somewhat more precise, the theorem states that if (Mt), 0 ≤ t ≤ T , is a square-integrable martingale with respect to the

filtration Ft then there exist an adapted process Ht such that E(

RT 0 H 2 s) < +∞ and Mt= M0+ Z t 0 HsdWsP. (2.40) An alternative representation is ( dMt = HtdWsP M0 = M0 (2.41)

The martingale representation theorem leads us to the Girsanov theorem. Choose for example Ht = φtZt and note that this process is adapted to Ft

since Zt is a Ft-measurable function. Using the martingale representation

theorem we can denote the Radon-Nikodym process as (

dZt = φtZtdWtP

Z0 = 1

(2.42)

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lemma to Y to obtain a solution for Y : dY = 1 ZdZ + 1 2 −1 Z2[Z, Z]t = 1 Z φtZtdW P t − 1 2 1 Z2φ 2 Z2dt = φtdWtP− 1 2φ 2 t Integration yields Yt= Z t 0 φsdWsP− 1 2φ 2 tdt. As a result we find Zt= exp Z t 0 φsdWsP− 1 2φ 2 sds . (2.43)

The Girsanov theorem states the following: let WP _{be a d-dimensional}

stan-dard Wiener process under measure P and φ is a d-dimensional adapted vector process, called the Girsanov kernel. Let Zt be as given in Eq. (2.43)

then

dWP_{(t) = φ(t)dt + dW}Q_(t),

where WQ_{(t) is a Wiener process under measure Q. The following}

relation-ship exists between the market price of risk λ(t) and the Girsanov kernel φ(t):

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2.7 Stopping times

In this section a short introduction to stopping times is given. The concepts of a predictable stopping time and a totally inaccessible stopping time are discussed briefly. Recall that τ is a stopping time if and only if

{τ ≤ t} ∈ Ft ∀t ≥ 0. (2.45)

The characterization of a stopping time is that at any time t we can decide whether τ has occurred or not, based upon the information available at that time instant. The concepts of a predictable and a totally inaccessible stopping time will turn out to be useful in the modeling of credit risk, hence we discuss them here. A stopping time τ is said to be predictable if there exist an increasing sequence of stopping times τ1 ≤ τ2 ≤ ... ≤ τn such that

τn< τ and lim

n→∞τn= τ ∀ω ∈ Ω with {τ (ω) > 0}. (2.46)

Before the event {τ ≤ t} actually occurs, it is already announced by this sequence of signals τn.

The contrary of a predictable stopping time is a totally inaccessible stopping time which is not announced on forehand. A stopping time τ is totally inaccessible if for every predictable time ˜τ

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CHAPTER

3

Modeling credit risk

So far we discussed some basic theory that turns out to be useful when dis-cussing some methods to price a CDS. In this chapter we turn to the pricing of a CDS and in that respect the modeling of credit risk is discussed. Sev-eral classes of models exist to model credit risk. We discuss three models that can be used for pricing purposes: structural models, reduced form mod-els and hybrid modmod-els, or incomplete information modmod-els. The structural approach, sometimes also called value-of-the-firm approach, is based on the Black-Scholes Merton model invented by Black and Scholes (1973) and Mer-ton (1974). Jarrow and Turnbull (1995) introduced reduced form models in a discrete framework. Duffie and Singleton (1999) among others work in a continuous framework. Some models contain elements of both models, the so called hybrid models. Examples of hybrid models are Zhou (1997) and Madan and Unal (1998). Zhou (1997) uses a jump-diffusion process to model the firm-value process which allows default to occur as a surprise. The jumps are modeled as a Poisson process. Madan and Unal (1998) present a reduced form model in which the probability of default depends on the firms equity.

3.1 Structural form models

3.1.1 Merton’s model

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Merton’s model it is assumed that the total value of the assets Vt of a firm

follows a process that satisfies under the pricing measure Q the stochastic differential equation (SDE)

dVt = Vt((r − κ)dt + σVdWt), with V0 > 0 (3.1)

where κ is the constant payout ratio (dividend yield), σV the volatility and W

is a Wiener process. The interest rate r is assumed to be constant. Assume that the firm has a single liability: a single zero-coupon bond with face value L and maturity T . Default occurs only at time T if {VT < L}. Therefore,

the default time τM equals

τM = T1{VT<L}+ ∞1{VT≥L}. (3.2)

In Merton’s model the default time τ is a Ft-predictable stopping time, where

Ft = σ(Vs, s ≤ t), since the sequence of stopping times that can be chosen

is τn = inf{t ≥ T − 1_n : Vt ≤ L}. Merton’s model requires knowledge of the

value of the assets Vt at each time instant. However, in reality it is hard

to obtain the value of the assets directly, since the assets usually are not a tradable security. Another way to obtain the value of the assets is to add the values of the debt and the equity. The equity is traded, but also the debt is usually not traded. This makes Merton’s model very hard to use in practice. We consider the payoff of a corporate bond at maturity date T , D(T, T ). The payoff of the bond equals D(T, T ) = min(VT, L). Note that the payoff of

the bond is the debt of the firm. We can rewrite the expression for D(T, T ) as

D(T, T ) = min(VT, L)

= L − max(L − VT, 0)

= L − (L − VT)+. (3.3)

In Merton’s model the asset value Vt can be seen as the stock price of an

equity option and the notional value of the debt L can be seen as the strike price of an equity option. Therefore, the solution for the price of the second term of Eq. (3.3) equals that of an European put option written on the firm’s value V with strike price equal to the bond’s face value L and expiration date T . Denote the price of this option with strike price L and maturity T by Pt.

We want to price the bond at time t < T . We have to discount the first term with B(t, T ) = e−r(T −t) to get the present value and we obtain

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Recall that the equity of the firm is the difference between it’s assets and it’s debt. Using Eq. (3.3) we find for the equity:

E(Vt) = Vt− D(t, T )

= Vt− LB(t, T ) + Pt

= Ct, (3.5)

where Ctis the price of a call option at time t with strike price L and maturity

T . This can be seen as follows:

E(VT) = VT − D(VT)

= VT − min(VT, L)

= (VT − L)+. (3.6)

This could also be seen differently by recalling the put-call parity for Euro-pean options

Ct− Pt = Vt− LB(t, T ). (3.7)

Hence the firm’s equity can be seen as a call option on the firm’s assets. This call option on the firm’s assets is sold by the debt holders. In case the firm does well, the equity holders pay off the debt holders using the assets and keep the remainder of the firm. If the firm defaults the firm’s equity holders sell the firm to the debt holders for the face value L and return this amount to the debt holders. This is the equity interpretation. In the debt interpretation the debt holders sell a put option to the equity holders. If the firm defaults the equity holders sell the firm to the bond holders and receive the strike price L. The bond holders are now left with the remaining assets. The price of the firm’s debt, D(t, T ), can be solved in close form from Eq (3.4) using the Scholes framework for option pricing. The Black-Scholes formula for the price of a European put option with strike price L is given by

Pt= LB(t, T )N (−d1(Vt, T − t)) − Vte−κ(T −t)N (−d1(Vt, T − t)). (3.8)

Substitution into Eq. (3.4) gives for the price of the firm’s debt, D(t, T ) D(t, T ) = LB(t, T ) − Pt

= LB(t, T ) − LB(t, T )N (−d2(Vt, T − t)) + Vte−κ(T −t)N (−d1(Vt, T − t))

= (1 − N (−d2(Vt, T − t))LB(t, T ) + Vte−κ(T −t)N (−d1(Vt, T − t))

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where d1,2(Vt, T − t) = ln(Vt L) + (r − κ ± 1 2σ 2 V)(T − t) σV √ T − t (3.10) and where N (x) denotes the standard Gaussian cumulative distribution func-tion N (x) = √1 2π Z x −∞ e−u22 du ∀x ∈ R. (3.11)

In the above derivation we have used that N (−x) = 1 − N (x). Consider Eq. (3.10) and note that since B(t, T ) = e−r(T −t) we can write r = _{T −t}ln B. If we now substitute this expression for the interest rate r into Eq. (3.10) we obtain d1,2(Vt, T − t) = ln(Vt L) − ln B + (−κ ± 1 2σ 2_{)(T − t)} σV √ T − t = ln Vt LB + (−κ ± 1 2σ 2_{)(T − t)} σV √ T − t (3.12)

In terms of Γt:= _LBVt we can rewrite d1,2(Vt, T − t) as

d1,2(Vt, T − t) =

ln(Γt) + (−κ ± 1₂σ2)(T − t)

σV

√

T − t . (3.13) The expression for D in terms of Γt will be

D(t, T ) = LB(T, t)ΓtN (−d)e−κ(T −t)+ N (d − σV p (T − t)), (3.14) where d = d1 = ln Γt+ (1₂σ2V − κ)(T − t) σV √ T − t . (3.15) Set κ = 0 for notational simplicity which means that we assume that no dividend is paid out. We are interested in the credit spread. The credit spread S(t, T ) is defined as the difference between the yield on a defaultable bond and the yield on a default-free bond. In other words

S(t, T ) = rd(t, T ) − r(t, T ), (3.16)

where rd ₌ − ln D(t,T )+ln L

T −t by definition and r(t, T ) = − ln B

T −t . For the credit

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If we now substitute the expression for D(t, T ) as given in Eq. (3.14) we obtain S(t, T ) = −1 T − tln ΓtN (−d) + N (d − σV p (T − t))> 0. (3.18)

To understand why it follows from the expression above that S(t, T ) > 0, first conclude from Eq. (3.4) that D(t, T ) < LB(t, T ). If we now look at Eq. (3.14) we conclude that ΓtN (−d)e−κ(T −t)+ N (d − σVp(T − t))

< 1 and hence S(t, T ) > 0. This is what we would expect since in reality the expected return of corporate bonds exceeds that of Treasury bonds. We consider the short-term credit spreads, in other words we take limt→TS(t, T ). Observe

that limt→TΓt= V_LT. Also note that

limt→T D(t, T ) LB(t, T ) = D(T, T ) L = min(VT, L) L (3.19)

Then according to Eq. (3.14) the following must hold: ( N (−d) = 1 if VT < L N (−d) = 0 if VT > L (3.20) and ( N (d − σVp(T − t)) = 0 if VT < L N (d − σVp(T − t)) = 1 if VT > L. (3.21) It follows that limt→TS(t, T ) = ( +∞ VT < L 0 VT > L. (3.22)

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3.1.2 Extension of Merton’s model

In Merton’s model default can occur only at maturity time T . Several exten-sions of Merton’s model are possible. An extension of the model is to consider stochastic interest rates. This can be easily solved by assuming for instance that the interest rate r follows Vasicek’s interest rate model. Jahmshidian (1989) already derived a closed-form solution for the price of a European put option on stocks for interest rates that follow Vasicek’s interest rate model. Another extension of Merton’s approach is introduced by Zhou (1997) who models the firm’s value process as a geometric jump-diffusion process. The first-passage time models allow default to occur at any time before maturity T . As an example of a first-passage-time model the Black and Cox model is discussed in more detail in the next section.

3.1.3 First-passage-time models: Black and Cox model

Black and Cox (1976) extend the Black-Scholes Merton model by defining the default time as a first passage time. The value process of the assets Vt follows the same process as in Merton’s model, Eq. (3.1). The interest

rate in the Black and Cox model is assumed to be constant. Black and Cox introduce safety convenants that provide the bondholders with the right to take over the firm if the value of the firm becomes lower than some barrier ¯v. The barrier ¯v is set to be deterministic and time-dependent: ¯vt= Ke−γ(T −t),

where t ∈ [0, T ] and K some constant. The constant γ effects the value of the barrier. If γ equals zero then the barrier has constant value K. If γ goes to infinity the value of the barrier approaches zero and we get back Merton’s model. Let vt be vt= ( ¯ vt, t < T, L, t = T. (3.23)

Default occurs the first time that Vt < vt, so default can take place prior to

maturity as well. The default time τ is

τ = inf{t ∈ [0, T ] : Vt < vt} (3.24)

The default time is a predictable stopping time with respect to the filtration Ft, where Ft = σ(Vs, s ≤ t), as is the case in the Black-Scholes Merton

model also. This can be understood by choosing the following sequence of stopping times: τn= inf{t ∈ [0, T ] : Vt≤ vt+_n1}. Furthermore, observe that

τ = τM ∧ ˆτ , where τM is the stopping time of Merton’s model and ˆτ is the

early default time given by ˆ

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We assume ¯vt≤ LB(t, T ) ∀t ∈ [0, T ] since then the payoff to the bondholder

never exceeds the face value of the discounted debt. We know that ¯v = Ke−γ(T −t)and B(t, T ) = e−r(T −t), hence Ke−γ(T −t)≤ Le−r(T −t) _{, ∀t ∈ [0, T ].}

Since this holds for all t ∈ [0, T ] we must have K ≤ L. We now derive a PDE for the price of a defaultable bond with constant interest rate. This derivation is analogous to the derivation of the price of a simple claim as given in section 2.3. It is reasonable to assume that the price of the defaultable bond with constant interest rate, D(t, T ), is a function of the firm’s assets, such that D(t, T ) = F (Vt, t). This notation might be misleading in the sense

that the bond is not traded at [0, T ) but at [0, τ ). The second variable T just notes the maturity of the bond. Notice that we assume that the price of the bond depends on the most recent value of the assets only, rather than on the whole process. If we assume that that the value of the assets is a Markov process this indeed holds. We also assume that F (Vt, t) is a smooth function,

so that we can apply Ito to find

dD(t, T ) = ∂F ∂tdt + ∂F ∂V dVt+ ∂2_F ∂V2d[V, V ]t.

We apply Ito on the discounted bond price. Since the discounted bond price is a martingale under Q we know that the finite variation parts will vanish. Applying Ito gives

d D(t, T ) Bt = D(t, T )d 1 Bt + 1 Bt dD(t, T ) = 1 Bt ∂F ∂tdt + ∂F ∂V Vt((r − κ)dt + σVdWt) + ∂2F ∂V2σ 2 Vdt − F r Bt dt.

Since the discounted bond price is a martingale under measure Q the finite variation parts sum to zero and we obtain the following equation

−F r +∂F ∂t + ∂F ∂V Vt((r − κ) + ∂2F ∂V2σ 2 Vdt = 0. (3.26)

Under weak assumptions Vt has support on the entire positive line and this

leads to the following PDE

−F r + ∂F ∂t + ∂F ∂vv((r − κ) + ∂2F ∂v2σ 2 vdt = 0, (3.27)

with boundary conditions: (

F (Ke−γ(T −t), t) = Ke−γ(T −t) F (VT, T ) = min(VT, L).

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Since default can occur before maturity we obtain an additional boundary condition. The first boundary condition holds in case default occurs before maturity. The second boundary condition is used when no default has oc-curred before maturity. An alternative approach is to denote the price in terms of expectations. This leads us to the following equation

D(t, T ) = EQLe−r(T −t) 1(T,∞](τ (ω)) | Ft

+ EQVTe−r(T −t) 1T(τ (ω)) | Ft

+ EQ¯vte−r(τ −t) 1[0,T )(τ (ω)) | Ft . (3.29)

The first term of the above expression represents the case when there is no default at all. In the second term default occurs at maturity. The third term describes the occurrence of default before maturity. For a solution of the price in the Black and Cox model we refer to Bielecki and Rutkowski (2002). To understand the relation between the partial differential equation approach and the probabilistic notation it is neccessary to explain something about the Feyman-Kac theorem, see for example Shreve (2004) and Bj¨ork (2004). Consider a stochastic differential equation of the form

dX(s) = µ(s, X(s))ds + σ(s, X(s))dW (s) (3.30)

with inititial condition X(t) = x. Define

F (t, x) = Et,x[φ(XT)] , (3.31)

where X satisfies the SDE as given in Eq. (3.30). The notation Et,x _means

that the expectation is taken given the initial condition X(t) = x. The theorem of Feyman-Kac states that F is a solution to the PDE

∂F ∂t(t, x) + µ(t, x) + 1 2σ 2∂2F ∂x2(t, x) = 0 (3.32)

with boundary condition

F (T, x) = φ(x). (3.33) One could also state it the other way around and start with a PDE and then show that it can be representated in terms of an expectation. For our purposes we need the discounted Feynman-Kac formula. Assume that F is a solution to the PDE of Eq. (3.27) with boundary condition then according to Feynman-Kac F has the representation (see Bj¨ork (2004))

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where V satisfies the SDE from Eq. (3.1) with initial condition V (t) = v. Note that in Eq. (3.29) the price is indeed a function of VT. Another thing

to realise is that the expectations in Eq. (3.29) are conditioned on Ft which

is in fact also the case in the equation above, except that here the notation Et,v is used to indicate that the initial condition V (t) = v is assumed to be given.

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3.2 Reduced form models

In this section we discuss reduced form models. We saw that in structural form models default is modeled by specifying the value of the firm’s assets and it’s capital structure. In reduced form models these values are not modeled at all. Two types of reduced form models exist: intensity based models and credit migration models. The intensity based models model the default time using an intensity process. The credit migration models model migrations between credit rating classes. In this thesis we focus on intensity based models only.

In intensity models the probability of default is modeled by a doubly stochastic Poisson process, also called Cox process. Before we discuss Cox processes we first recall the definitions of a Poisson process and a inhomo-geneous Poisson process. Most of what follows can be found in Sch¨onbucher (2003). Consider an increasing sequence of stopping times τn. A counting

process N (t) is a stochastic process defined by N (t) =X

n

1{τn≤t}. (3.35)

In other words, N (t) counts the number of times that τn ≤ t. If τ is the

default time then N (t) is a measure for the number of times default has occurred at time t. Sometimes N (t) is defined without a summation sign, since most of the time we are only interested in the first time default occurs. A Poisson process is a process with independent increments that satisfies N (0) = 0 and

P (N (t) − N (s) = k) = 1

k!(t − s)

k_λk_{exp [−(t − s)λ] ,}

where λ is the intensity of the Poisson process. Since N (0) = 0 the survival probability is

P (N (t) = 0) = P (τ > t) = exp (λt) ,

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We can further extend the Poisson process by allowing the intensity not only to be time-dependent but to be random as well. A random intensity can be obtained in two ways. One way is to let the intensity itself be random such that it follows for example the SDE

dλ(t) = µ(t, λ(t))dt + σ(t, λ(t))dWt. (3.36)

Another possiblity is to let the intensity be a function of a state (observable) variable X(t) where X(t) is the solution of the SDE

dX(t) = µ(t, X(t))dt + σ(t, X(t))dWt. (3.37)

The state variable X contains the information that is relevant to predict default. This information might include for example time, stock prices and credit ratings. If we let λ = λ(X(t)) then this way we introduce randomness in the intensity as well. In fact, in both cases we assume that there exist a background process containing the relevant information to predict default. In the first case the background process is X(t) = (W (t)).

We thus consider a process with λ = λ(t, ω). If the realization ω is known then we obtain the inhomogeneous Poisson again. That is exactly what de-fines a Cox process. In a Cox process the intensity does not only change in time, but it changes in time according to a probabilistic law. Loosely speaking, a Cox process is a Poisson process whose intensity follows a second stochastic process. An example of the use of a Cox process is in the mod-eling of communication systems, see for example Manton, Krishnamurthy, and Elliott (1996). Photons strike a photondector according to a Poisson distribution. The Poisson rate corresponds to the intensity of light. If the intensity of light varies stochastically we have an example of a Cox process. Before we compute the survival probability in a Cox process, we first need to define the filtration of the background process as Gt= σ{Xs: 0 ≤ s ≤ t}.

Both the interest rate and the intensity process are adapted to (Gt)t≥0. In

other words, the filtration (Gt)t≥0 contains the information needed to

deter-mine the realization of λ(t) and r(t). In the sequal we always write λ(Xt)

to denote the intensity of a Cox process and we write r(Xt) for the interest

rate. Notice this is shorthand for λ(t, Xt) and r(t, Xt). In a Cox process the

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In the third line of the above equation the law of iterated expectations is used.

3.2.1 The default time

We are interested in the default time. Assume that the counting process N (t) is a Cox process. Define Λ(t) = R₀tλ(Xs)ds. The expected number of

default occurrences in the interval (0,t] is given by

E [N (t)] = E Z t 0 dN (s) = E Z t 0 λ(Xs)ds = E [Λ(t)] .

We justify the second step later in this section. Now obviously default occurs if N (t) ≥ N (τ ), or in other words if

Λ(t) ≥ Λ(τ ).

It can be shown that Λ(τ ) is an exponential random variable with parameter 1, independent of G∞, where G∞ = σ (Gt| t ∈ R+). We refer to Filipovi´c

(2009) page 235 for a proof of this result. The discussion above leads to the following definition of the default time

τ = inf t : Z t 0 λ(Xs)ds ≥ E1 , (3.38)

where E1is an exponential random variable with mean 1, independent of G∞.

Sometimes an alternative definition of the default time is given. If we define the random variable η as a random variable that is uniform with density equal to 1, then from Eq.(3.38) we can conclude that the default time is

τ = min{t > 0 | Z t 0 λ(Xs)ds ≥ − ln(η)} = min{t > 0 | Z t 0 λ(Xs)ds ≤ ln(η)} = min{t > 0 | exp − Z t 0 λ(Xsds ≤ η}. (3.39)

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It remains to verify that E]hRt 0 dN (s) i = EhRt 0 λ(Xs)ds i . According to the Doob-Meyer decomposition theorem it holds that if a process X(t) is a submartingale then there exist a local martingale M (t) and a unique increasing predictable process A(t) that is locally integrable such that

X(t) = X(0) + M (t) + A(t). (3.40) A proof of the Doob-Meyer decomposition theorem can be found for example in Rogers and Williams (1987). Since the process N (t) is a submartingale it follows that the process

M (t) = N (t) − Z t

0

λ(Xs)ds (3.41)

is a martingale. As M (t) is a martingale it follows that E [dN(t) | Ft] =

E [λ(Xt)dt | Ft] from which it follows that

E [N (t)] = E Z t 0 dN (z) = E Z t 0 λ(Xz)dz . (3.42)

3.2.2 The short term credit spread

In section 3.1.1 we discussed the short term credit spread in structural form models and saw that it converges to zero. This is a consequence of the fact that the default time is a predictable stopping time. In reduced form models, however, the default time is a totally inaccessible stopping time. This can be understood by noting that the time of default is modeled by a Cox process, so that conditional on the background process the default time follows and inhomogeneous Poisson process. An important property of Poisson processes is that such processes are memoryless. In other words, the probability of default at each time instant is independent of the probability of default at other time instants. Since the counting process can suddenly jump to one it is not possible to predict the default time. This implies that the default time is totally inaccessible in these models, and therefore it follows that the credit spreads at a time instant close to maturity are nonzero. This makes reduced form models more appropriate to use in practice since in empirical credit spreads the short term credit spreads are nonzero as well.

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(3.17) we have that the credit spread is

S(t, T ) = −1

T − tln P [τ > T | Gt] . (3.43) The short term credit spread is the excess yield over the risk-free yield over the infinitesimal time period (t, t + dt] and is defined by, see Giesecke (2001)

lim

T ↓t S(t, T ) =

∂

∂TP [τ ≤ T | Gt] |T =t (3.44) where t < τ . We can write

Therefore we can conclude that in reduced form models the short term credit spread converges to the intensity at time t and does not converge to zero as is the case in structural form models.

3.2.3 Filtrations

In the framework of reduced form modeling we work in the probability space (Ω, F , P) with filtration (Ft)t≥0. The filtration (Gt)t≥0 is introduced in the

previous subsection. This filtration contains the information about the back-ground process. Since both the interest rate and the intensity process are functions of the background process (Xt) it follows that both the interest rate

and the intensity are adapted to (Gt)t≥0. The filtration (Gt)t≥0 is assumed

not to contain any information about the default occurrence. However, for pricing purposes one information is required about whether default has oc-curred yet or not. This information is contained in the following filtration

Ht = σ{1{τ ≤t} : 0 ≤ s ≤ t}. The filtration carrying all this information is Ft,

where Ft= Gt∨ Ht and Gt∨ Ht denotes the smallest sigma-algebra

contain-ing Gt∪ Ht. As we mentioned before, the filtration (Gt)t≥0 does not contain

information about the default event, so the inclusion (Gt)t≥0 ⊂ (Ft)t≥0 is

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3.2.4 Pricing the building blocks

In the previous section we discussed the properties of reduced form models. In this section we explore the basic ingredients to price a Credit Default Swap and a corporate bond in the framework of reduced form modeling. Recall from section 2.2 that a CDS contract generally consist of two types of payments: the premium leg and the protection leg. These two types of payments can be described as follows:

1. the premium leg consists of a stream of payments X that must be paid as long as no default has occured yet;

2. the protection leg consists of a recovery payment Z paid in case of default.

To find the premium, the premium leg and the protection leg must be equal at the start of the contract (t = 0) to avoid arbitrage possibilities.

We follow the approach of Lando (1998) to derive expressions for the premium leg and the protection leg. For simplicity we assume both X and Z to be constant. To find the present value at t=0 of the premium leg we must evaluate the following expression:

EQ X Z T 0 exp − Z t 0 r(Xs)ds 1{τ >t} | Ft (3.45)

which is the expected present value of the premium payments. The present value at t=0 of the protection leg can be obtained by evaluating the following expression: EQ Z exp − Z τ 0 r(Xs)ds 1{τ <T }| Ft . (3.46)

In order to to be able to price a corporate bond in this frame it is necessary to consider a final payment in case no default has occurred before expiry of the contract. If we assume that the final payment is one, then the present value at t = 0 is EQ exp − Z T 0 r(Xs)ds 1{τ >T }| Ft . (3.47)

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that substituting t = 0 indeed gives the expressions we need. To obtain such expressions we first need to show that

EQ 1{τ >T } | GT ∨ Ht =1{τ >t}exp − Z T t λ(Xs)ds . (3.48) We condition on GT ∨ Ht since

• all information about the relevant economic variables, like stock prices for example, is needed and this information is contained in GT.

Con-ditional on this information both the interest rate and the intensity follow an inhomogeneous Poisson process;

• one should know whether default has occurred at time t or not, this information is contained in Ht.

Notice that if we set t = 0 in Eq. (3.48) this simplifies things a lot, since then we condition on H0. We assume that at t = 0 no default has occurred yet and

therefore H0 is the trivial σ-algebra. However, we derive the more general

expression, Eq. (3.48). The first thing to notice is that this expectation is zero for 1{τ ≤t} since if this is the case then it is impossible that {τ ≥ T }

because t ≤ T . Therefore, we can rewrite Eq. (3.48) as follows EQ 1{τ >T }| GT ∨ Ht = EQ 1 {τ >T }| GT ∨ Ht 1{τ ≤t}+1{τ >t} = 1{τ >t}EQ 1{τ >T }| GT ∨ Ht .

A different way to obtain this result is by noting that since t ≤ T it follows that 1{τ >T } = 1{τ >t}1{τ >T } and since 1{τ >t} is adapted to Ht we find the

same result. Recall that Ht = σ{1{τ ≤t} : 0 ≤ s ≤ t}. The atoms of Ht are

{τ ≤ t} and {τ > t}.

Before we continue the calculation we first have a short intermezzo about conditional expectations. Define a probability space (Ω, F , P) and let E be a σ−field generated by the atoms {D1, ..., Dk}. Then the expectation

condi-tional on E of the random variable X that takes values xi is, see for example

Klebaner (2005)

E (X | E ) = X

i

xiP (Ai| E) , (3.49)

where the event Ai = {X = xi}. Furthermore, we know that the conditional

probability of an event A with respect to the filtration E is

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We are now able to continue our computation. Eq.’s (3.49) and (3.50) applied to Eq. (3.48) gives EQ 1{τ >T } | GT ∨ Ht = 1{τ >t}EQ 1{τ >T } 1{τ ≤t} P(τ ≤ t) | GT 1{τ ≤t} + 1{τ >t}EQ 1{τ >T } 1{τ >t} P(τ > t) | GT 1{τ >t}.

The first term of the equation above is obviously zero, therefore we find

EQ 1{τ >T }| GT ∨ Ht = 1{τ >t}E Q 1 {τ >T }1{τ >t} | GT P(τ > t | GT) = 1{τ >t}P (τ > T ) ∩ (τ > t) | G T) P(τ > t | GT) = 1{τ >t}P (τ > T | GT ) P(τ > t | GT) .

Since the counting process is a Cox process we know that P (τ > t | GT) =

exp

−Rt

0 λ(Xs)ds

and therefore, it follows that

EQ 1{τ >T }| GT ∨ Ht = 1{τ >t} exp−RT 0 λ(Xs)ds exp−Rt 0 λ(Xs)ds = 1{τ >t}exp − Z T t λ(Xs)ds

which gives indeed the correct result. It remains to obtain expressions for Eq.’s (3.45), (3.46) and (3.47) using Eq. (3.48). Start with Eq. (3.45). Using the tower property we find

EQ exp − Z T t r(Xs)ds X1{τ >T }| Ft = EQ EQ exp − Z T t r(Xs)ds X1{τ >T }| GT ∨ Ht | Ft = EQ exp − Z T t r(Xs) + λ(Xs)ds X1{τ >t} | Ft = 1{τ >t}EQ exp − Z T t r(Xs) + λ(Xs)ds X | Ft .

In the last line it is used that 1{τ >t} is adapted to Ft. The next step is

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random variable E1 is independent of GT. Moreover, E1 is independent of

σexp−RT

t (r(Xs) + λ(Xs))ds

X∨ Gt. According to Williams (1991),

property 9.7 (k), it then follows that we can replace conditioning on Gt∨σ(E1)

by Gt. But we also have

Gt⊂ Gt∨ Ht⊂ Gt∨ σ(E1). (3.51)

That the equation above holds can be understood by noting that Htcontains

the information whether default has occurred at time t or not. However, σ(E1) contains more information, since if E1 is known together with Gt then

it follows from the definition of the default time, Eq. (3.38) that the default time is known. As a final result we obtain

EQ exp − Z T t r(Xs)ds X1{τ >T }| Ft = 1{τ >t}EQ exp − Z T t r(Xs) + λ(Xs)ds X | Gt . (3.52)

The derivation of Eq. (3.47) is analogue to the derivation of Eq. (3.45) and will not be repeated here. We only state the final result:

EQ exp − Z T t r(Xs)ds 1{τ >T } | Ft 1{τ >t}EQ exp − Z T t (r(Xs) + λ(Xs))ds | Gt . (3.53)

To derive Eq. (3.46) first note that the probability density of the default time is ∂ ∂sP (τ ≤ s | τ > t, GT) = λ(Xs) exp − Z s t λ(Xu)du . (3.54)

Using the tower property we find

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Computation of the inner expectation yields EQ Z exp − Z τ 0 r(Xs)ds 1{τ <T }| GT ∨ Ht =1{τ >t}Z Z T t exp − Z s t r(Xs)ds ∂ ∂sP (τ ≤ s | τ > t, GT) ds =1{τ >t}Z Z T t exp − Z s t r(Xs)ds λ(Xs) exp − Z s t λ(Xu)du ds.

Again it is allowed to replace conditioning on Ftby conditioning on Gt using

the same arguments as before. We obtain

EQ Z exp − Z τ t r(Xs)ds 1{τ <T }| Ft = 1{τ >t}ZE Z T t exp − Z s t r(Xu) + λ(Xu)du λ(Xs) | Gt (3.55)

Substituting t = 0 in the obtained expressions gives the following three ex-pressions. Notice that F0 is the trivial σ−algebra, so conditioning on F0

can be removed since F0 gives no additional information. The value of the

premium leg (or coupon payments) at t = 0 is

EQ X Z T 0 exp − Z t 0 r(Xs)ds = EQ exp − Z T 0 r(Xs) + λ(Xs)ds X (3.56)

The value of the protection leg (or a recovery payment) at t = 0 is

EQ Z exp − Z τ 0 r(Xs)ds 1{τ <T } = ZE Z T 0 exp − Z s 0 r(Xu) + λ(Xu)du λ(Xs) (3.57)

The value of a final payment in case no recovery has occurred is

EQ exp − Z T 0 r(Xs)ds 1{τ >T } = EQ exp − Z T 0 (r(Xs) + λ(Xs))ds . (3.58)

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3.3 Connection between both models:

infor-mation based perspective

In the previous sections two approaches to model default risk are discussed: structural form models and reduced form models. We saw that in structural form models the default time is a predictable stopping time. Since the default time is a predictable stopping time the credit spread at a time instant close to maturity converges to zero. In reduced form models the default time is a totally inaccessible stopping time and the short term credit spread does not converge to zero. We are now inclined to conclude that the main difference between reduced form and structural form models is whether the stopping time is predictable or not.

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3.3.1 Complete information models

Let (Ω, H, P) be a probability space to model the uncertainty of investors. The filtration (Ht)t≥0 contains the information available to investors.

Each default model consists of two elements:

• the default time τ , where τ is a Ht− stopping time;

• a model filtration (Ft)t≥0 ⊆ (Ht)t≥0.

We assume that investors can observe the default time, therefore τ must be a Ht− stopping time. In structural form models the default time is defined

as

τ = inf{t > 0 : Vt≤ Lt}, (3.59)

where Vtrepresents the value of the firm’s assets and Ltdenotes the value of

the liabilities of the firm, or in other words Lt is the default barrier. See also

section 3.1 for a discussion of structural form models.

The model filtration (Ft)t≥0 describes the information available to

in-vestors relative to Eq. 3.59. Suppose that inin-vestors can observe both the asset value Vt and the value of the default barrier Lt at each time instant.

Then the model filtration (Ft)t≥0 is generated as follows

Ft= σ(Vs : s ≤ t) ∨ σ(Lt : s ≤ t). (3.60)

In this case we have a complete information model. The model filtration (Ft)t≥0 contains all the information needed to determine the default time.

At each time instant one knows whether default has occurred or not, hence τ is a Ft−stopping time. The investor information then can be defined as

(Ht)t≥0 = (Ft)t≥0. In all structural form models the value of the assets at

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3.3.2 Incomplete information models

Again we work in the probability space (Ω, H, P), where the filtration (Ht)t≥0

contains the information available to investors. In incomplete models this filtration contains less information compared to the information set in the complete information models. Our approach is to first consider several possi-bilities for the model filtration (Ft)t≥0 and then construct (Ht)t≥0 such that

is contains (Ft)t≥0 and makes τ a Ht−stopping time. In other words, we

assume investors to know at each time instant whether default has occurred or not.

In the models with incomplete information the assumption about the available information is more realistic. Either the value of the assets is not known continuously but for example only available at discrete time instants or is not available at all. Also the default barrier might be unknown. Combined this gives several incomplete information models. We will consider some of them. We first assume that investors do not observe the value of the default barrier, but only some information about the values of the assets. We give four examples, see also Giesecke (2001):

• Investors have complete information about the value of the assets, then the model filtration is Ft = σ(Vs : 0 ≤ s ≤ t);

• The totally opposite possibility is that investors have no information about the value of the assets at all, then Ft= {φ, Ω};

• Another possibility is that investors receive information about the value of the assets, but only at discrete time instants t1 < t2 <.. < tn, then

Ft= σ(Vs: s ≤ t, s ∈ {t1, ..., tn});

• Again investors observe the value of the assets at discrete time instants t1 < t2 < ... < tn, but with some random noise. Let Yti = Vti + ti

be the observed asset value for i = 1, ..., n, where Vti is the true assets

value and ti is some random noise variable independent of Vti. In this

case we have Ft= σ(Ys, s ≤ t, s ∈ {t1, ..., tn}.

Of course it is also possible that investors observe both some information about the value of the firm’s assets and some information about the default barrier, thus about the liabilities of the firm, as well. Assume for example that investors receive information about the assets at discrete time instants t1 < t2 <.. < tn and that the barrier is known at all time instants. Then the

model filtration is

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The investors filtration is such that it contains the model filtration and such that τ is a stopping time with respect to this filtration. Therefore, we could define

Ht= Ft∨ σ(Ns, s ≤ t) (3.62)

or equivalently

Ht= {B ∈ G : ∃Bt∈ Ft, B ∩ {τ > t} = Bt∩ {τ > t}}. (3.63)

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3.3.3 A generalized reduced form model framework

In reduced form models we do not have a model definition such as in struc-tural form models. Therefore, a model of the default time should be in-cluded in the model filtration here as well. Further information includes the background process. Hence the model filtration in reduced form models is Ft = σ(τ, Xs: s ≤ t) or we could also say that

Ft = Gt∨ σ(τ ). (3.64)

where Gt = σ(Xs : s ≤ t) is the filtration as defined before in the section

about reduced form models. In reduced form models the default time is not an Ft− predictable stopping time as is the case in incomplete information

models also. However, not all incomplete information models allow for the existence of an intensity.

Giesecke (2001) describes a general reduced form model framework. Let (Ht)t≥0 contain the information available to investors again. The exact

in-formation contained in this filtration is not specified. According to the Doob-Meyer decomposition, see Eq. (3.40) the process N (t) − A(t) is a Ht−martingale, where A(t) is called the trend or compensator and N (t) is

the counting process.

If A(t) = R₀t∧τλ(s)ds then we say that τ admits the intensity λ. If the intensity exist then τ is a totally inaccessible stopping time. On the other hand is τ is a totally inaccessible stopping time this does not necessarily imply the existence of an intensity. In the general framework the conditional default probability for t ≤ T is

P (τ ≤ t | Ht) = 1 − E [exp (At− AT) | Ht] . (3.65)

According to Proposition 3.2 of Giesecke (2001) the value of a default-contingent claim with face value X and maturity T at t < τ is

E exp − Z T t r(s)ds X1{τ >T } = E X exp − Z T t r(s)ds + At− AT | Ht , t ≤ T.

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CHAPTER

4

Derivation of a Credit Default Swap premium

The model of Longstaff et al. (2005) is a reduced form model. A Cox process is used to model the stochastic default intensity λ(Xt). For more information

about reduced form models and Cox processes we refer to section 3.2. The riskless interest rate r(Xt) is assumed to be stochastic as well. The authors

Longstaff et al. (2005) use a third stochastic process, the liquidity process γ(t) which we introduce in this section. We assume that the liquidity process is adapted to the filtration Gt= σ(Xs : s ≤ t) as well so in the sequal we use

the notation γ(Xt) which is shorthand for γ(t, Xt).

4.1 Liquidity process

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more realistic, in this thesis we restrict to the first method, since this is in correspondence with the approach of Longstaff et al. (2005). We first give a short introduction to convenience yields. Convenience yields were already used before in the pricing of commodities, e.g. oil, where an arbitrage-free but incomplete debt market is assumed. Convenient yields represent the benefit of holding the underlying product rather than the derivative. A convenience yield can arise for example when the cost of short-term changes in output are high, or when there is a time delay. Holding inventory of an input can sometimes lower the unit output costs, also in this case an convenience yield can arise, see Fama and French (1998). Let S be the spot price of a commodity at time t and let F (S, T ) be the price of a futures contract written on the same commodity with a maturity time T . Define δ as a constant instantaneous convenience yield and assume that the interest rate r is constant as well. Then according to Gibson and Schwartz (1990) the relationship between the spot price and the futures price of the commodity is

F (S, T ) = S exp ((r − δ)(T − t)) . (4.1) See also Brennan and Schwartz (1985) for pricing of commodities with a con-venience yield. After this introduction to the origin of the use of concon-venience yields, we now use a convenience yield to model liquidity risk.

Jarrow (2001) argues that the relation between the price of a risky bond in an illiquid market compared to the price of a comparable bond in a liquid market can be described using a convenience yield. Let D(t, T ) denote the price of a corporate bond at time t with maturity T . Define Dil(t, T ) to be

the price of the price of a corporate bond with the same maturity time T but with the only difference that this bond is traded in an illiquid market. Jar-row (2001) argues that when there are shortages the corporate bond cannot always be shorted and thus the following no-arbitrage relationship holds:

Dil(t, T ) ≥ D(t, T ).

Since the bond traded in an illiquid market is harder to get when there are shortages, this will increase the price. The reverse holds when there is an oversupply in the illiquid market, so then it holds that

Dil(t, T ) ≤ D(t, T ).

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where γ(t, T ) is a forward convenience yield.

At this stage it might be convenient to recall the difference between for-ward rates and short rates, see also Bj¨ork (2004). Denote the price of a zero coupon bond with maturity date T by B(t, T ) as usual. The instantaneous forward rate with maturity T , contracted at t, is defined as

f (t, T ) = −∂ log B(t, T )

∂T . (4.3)

The instantaneous short rate at time t is

r(t) = f (t, t). (4.4) The bank account process is given by

B(t) = exp Z t 0 r(s)ds . (4.5)

Analogue to the short rate, the instantaneous convenience yield γ(s), also called liquidity is

γ(t) = γ(t, t). (4.6) Taking liquidity risk into account, the discount factor becomes

B_tliq = exp Z t 0 r(Xs) + γ(Xs)ds , (4.7)

where we adopted the notation r(Xs) and γ(Xs) again. The interpretation

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4.2 Bond pricing with liquidity risk

In this section we apply the basic building blocks obtained in section 3.2.4 to derive an expression for the premium of a CDS in the framework of intensity based models. We apply these building blocks again to derive an expression for a corporate bond, but in this case liquidity risk is included.

Denote the value of the CDS premium by p. Then according to Eq. (3.56) the present value of the premium leg is given by

P = E p Z T 0 exp − Z t 0 r(Xs) + λ(Xs) dt , (4.8)

The protection leg is the payout in case of default and can be expressed as, see Eq. (3.57) P R = E w Z T 0 λ(Xt) exp − Z t 0 r(Xs) + λ(Xs) dt . (4.9)

Therefore, by setting the premium and the protection leg equal to each other we obtain for the premium p

p = E h wRT 0 λ(Xt) exp −Rt 0r(Xs) + λ(Xs)ds dti E h RT 0 exp −Rt 0 r(Xs) + λ(Xs)ds dt i , (4.10)

where 1−w is the recovery rate of the bond. Note that a recovery rate of 1−w means that a fraction 1 − w of the bond is recovered. The remaining fraction w of the bond is paid by the protection seller to the protection buyer of the CDS. According to Eq.’s (3.56), (3.57) and (3.58) the price of a defaultable bond with coupon payments c and recovery rate 1 − w is

CB = B(0, T ) + R(0, T ) + C(0, T )¯ = EQ exp − Z T 0 r(Xs) + γ(Xs) + λ(Xs)ds + EQ (1 − w) Z T 0 λ(Xt) exp − Z t 0 r(Xs) + γ(Xs) + λ(Xs)ds + EQ c Z T 0 exp − Z T 0 r(Xs) + γ(Xs) + λ(Xs)ds dt . (4.11)

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4.3 Closed-form solutions

In order to derive closed-form solutions, we must specify the dynamics of the three stochastic processes: the interest rate, the intensity process and the liquidity process. Longstaff et al. (2005) assumes these processes to be mutually independent. For the interest rate the only requirement for the dynamics is that the price of a riskless zero-coupon bond B(T ) with maturity T is given by the following expression

B(T ) = E exp − Z T 0 r(Xt)dt . (4.12)

The dynamics of the intensity process under measure Q is assumed to be

dλ = (α − βλ)dt + σ√λdW1, (4.13)

where W1 is a Wiener process and α, β are positive constants.

Other choices for the intensity process could have been made as long as the choice of the drift and the volatility ensures that the intensity process is always non-negative. For the liquidity process it is assumed that under measure Q

dγ = ηdW2, (4.14)

where W2 is again a Wiener process, independent of W1 and η a positive

constant.

In Appendix B it is shown that the price of a corporate bond can be expressed as

CB = c Z T

0

A(t) exp(D(t)λ(0))C(t)B(t) exp(−γ(0)t)dt

+ (1 − w) Z T

0

exp(D(t)λ(0))C(t)B(t)(G(t) + H(t)λ(0)) exp(−γt)dt + A(T ) exp(D(T )λ(0))C(T )B(T ) exp(−γ(0)T ), (4.15)

where λ(0) and γ(0) are the initial values of the processes of Eq.’s 4.13 and 4.14 respectively. Appendix B shows that the CDS premium can be expressed as p = w RT 0 exp(D(t)λ(0))B(t)(G(t) + H(t)λ(0))dt RT 0 A(t) exp(D(t)λ(0))B(t)dt . (4.16)

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A(t) = exp α(β + φ) σ2 t 1 − κ 1 − κ exp(φt) 2α σ2 B(t) = E exp − Z t 0 r(Xs)ds C(t) = exp η 2_t3 6 D(t) = 1 σ2 (β − φ) + 2φ 1 − κ exp(φt) G(t) = α φ(exp(φt − 1)) exp α(β + φ) σ2 t 1 − κ 1 − κ exp(φt) 2α σ2+2 H(t) = exp α(β + φ) + φσ 2 σ2 t 1 − κ 1 − κ exp(φt) 2α_σ2+2 .

In the above equations φ and κ are defined as

φ = p2σ2_{+ β}2

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CHAPTER

5

Conclusion

This thesis describes two approaches to model the probability of default: structural form models and reduced form models. The first structural form model is introduced by Merton (1974). In this model the firm is assumed to have issued a single obligation. Default occurs at maturity if the value of the firm’s assets at that time instant is less than the face value of this single obligation. In Merton’s model the unrealistic assumption is made that the value of the firm’s assets is known at each time instant. Another disadvantage of the model is that for a firm that is not in financial distress short before maturity the short term credit spreads converge to zero. This is in contradiction with empirical evidence. Other structural models extend Merton’s model. The model of Black and Cox (1976) introduces a safety convenant that allows default to occur prior to maturity in case the value of the assets crosses a certain barrier. Although this model is somewhat more realistic, the problem of short term credit spreads converging to zero remains in all structural form models. This can be understood by noting that the default time in structural form models is a predictable stopping time.

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CHAPTER

6

Acknowledgement

The subject of this thesis is put forward by my supervisor at Dutch Central Bank, Drs. D.W.G.A. Broeders. I am gratefull that he gave me the opportu-nity to work on such an interesting subject as the pricing of a Credit Default Swap. Writing my thesis at Dutch Central Bank was a great experience from which I a learned a lot.

It took me some time to finish my thesis and all this time Dr. J.W. Nieuwenhuis was patient and willing to help me by answering my questions. He gave me suitable literature that enabled me to find most of the answers myself. He gave me a few private lectures on some of the important, but rather complicated basics. He inspired me to reach a better understanding of the material than I thought was possible. I would like to thank him for this.

Credit Default Swaps

Credit Default Swaps

R. Jansen

Credit Default Swaps

R. Jansen

Contents

CHAPTER

1

Introduction

CHAPTER

2

Basic Theory

2.1

Introduction to Credit Default Swaps

2.2

Cash flows of a credit default swap

2.3

A short note on option pricing

2.4

Term structure equation

Term structure models of interest rates: the martingale

approach

2.5

Some standard interest rate models

2.6

Relation between the Radon-Nikodym

deriva-tive and the market price of risk

2.7

Stopping times

CHAPTER

3

Modeling credit risk

3.1

Structural form models

3.1.1

Merton’s model

3.1.2

Extension of Merton’s model

3.1.3

First-passage-time models: Black and Cox model

3.2

Reduced form models

3.2.1

The default time

3.2.2

The short term credit spread

3.2.3

Filtrations

3.2.4

Pricing the building blocks

3.3

Connection between both models:

infor-mation based perspective

3.3.1

Complete information models

3.3.2

Incomplete information models

3.3.3

A generalized reduced form model framework

CHAPTER

4

Derivation of a Credit Default Swap premium

4.1

Liquidity process

4.2

Bond pricing with liquidity risk

4.3

Closed-form solutions

CHAPTER

5

Conclusion

CHAPTER

6

Acknowledgement