M. Jeanblanc Credit risk

April 2002 MUNICH ii

Contents

1 Structural approach 3 1.1 Merton’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 First passage times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Black and Cox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Term Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Elementary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.3 CreditMetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.4 KMV approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.5 CreditRisk+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.6 CreditPortfolioView . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Portfolio management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5.1 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5.2 Davis and Lo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.3 Portfolio management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Random barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6.1 Pan approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8.1 Defaultable swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8.2 First-to-default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Hazard process approach 13

2.1 A toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Payment at Maturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Payment at hit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.3 Risk neutral probability measures . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Toy model and martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.3 Intensity process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.4 Incompleteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.5 Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.6 Partial information: Duffie and Lando’s model . . . . . . . . . . . . . . . . 19 2.2.7 Representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Successive default times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Two times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 More than two times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Cox Processes and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Construction of Cox Processes with a given stochastic intensity . . . . . . . 23 2.4.2 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.3 Choice of filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.4 Key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.5 Conditional Expectation of F_{∞}-Measurable Random Variables . . . . . . . 25 2.4.6 Defaultable Zero-Coupon Bond . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.7 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.9 Duffee’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.10 Jarrow and Turnbull’s model . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.2 Key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.3 H Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.4 Stochastic Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.5 Representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.6 Hedging contingent claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5.7 Intensity approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Correlated defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.1 Copula and threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.2 Correlated intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.3 Jarrow and Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.4 Kusuoka exemple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.7.1 Explosive defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.7.2 Partial observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7.3 Noisy information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Appendix A : Hitting times 41

1.1 Hitting times of a level and law of the maximum for Brownian motion . . . . . . . 41 1.1.1 Law of the pair of the random variables (W_{t}, M_{t}) . . . . . . . . . . . . . . . 41 1.1.2 Law of the supremum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.1.3 Law of the hitting time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.1.4 Law of the infimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.1.5 Laplace transform of the hitting time . . . . . . . . . . . . . . . . . . . . . 43

1.2 Hitting times for a Drifted Brownian motion . . . . . . . . . . . . . . . . . . . . . 44 1.2.1 Laws of the pairs M, X and m, X at time t . . . . . . . . . . . . . . . . . . 44 1.2.2 Laws of maximum, minimum and hitting times . . . . . . . . . . . . . . . . 45 1.2.3 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.3 Hitting Times for Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . 46

1.3.1 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.4 Other processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.4.1 OU Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.5 Non-constant Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.5.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.5.2 Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2 Appendix B: Poisson processes 51 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2 Martingale properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Introduction

In an arbitrage-free market, a contingent claim H can be priced as soon as there is an hedging strategy, i.e., a self-financing portfolio whose T time value is exactly H. The market is said to be complete if any contingent claim can be hedged, and it is well known that this property is ”equivalent” to the uniqueness of the so-called equivalent martingale measure (e.m.m.). Hence, the t-time value of the hedging portfolio is the conditional expectation of the discounted terminal payoff.

Credit risk, or default risk, concerns the case where a promised payoff is not delivered if some event happens before the delivery date. The default occurs at time τ where τ is a random variable.

We shall present the two main approaches of this problem: the structural one and the hazard process one.

We quote some of the seminal papers; the reader can refer to the second edition of the book of Ammann [2] or to the recent book of Bielecki and Rutkowski [5] for more information. At the end of the bibliography, we also mention some web addresses.

I would like to thanks the organizers and the participants to Munich School. They will find here the answer of some exercieces, as well a a new (and short!) proof of the hedging strategy.

Begin at the beginning, and go on till you come to the end. Then, stop.

L. Carroll, Alice’s Adventures in Wonderland 2 Munich Spring school

Chapter 1

Structural approach

In this approach, default event is specified in terms of the evolution of the firm assets. Given the value of the assets of the firm, the aim is to deduce the value of corporate debt.

1.1 Merton’s Model

In the approach –pioneered by Merton [54] – the default occurs if the assets of the firm are insufficient to meet payments on debt at maturity. The firm is financed by the issue of bonds, and the face value L of the bonds must be paid at time T. At time T, the bondholders will receive min(V_{T} , L) where L is the debt value and V_{T} the value of the firm. Thus, writing

min(V_{T} , L) = L − (L − V_{T})^{+}

we essentially deal with an option pricing problem. Merton assumes that

dV_{t} = V_{t}(rdt + σdW_{t}), V_{0} = v > L ,

where r is the (constant) risk-free interest rate, and σ is the constant volatility. In that context, the contingent claim pricing methodology can be used : the market where (V_{t}, t ≥ 0) is a tradeable asset is complete and arbitrage free, the equivalent martingale measure is the historical one, hence the value of the corporate bonds at time t is

E(e^{−r(T −t)} min(V_{T} , L)|F_{t}) = Le^{−r(T −t)} − P (t, V_{t}, L)

where P (t, x, L) is the t-time value of a put option on the underlying V with strike L and maturity T. From Black and Scholes formula

P (t, x, L) = Le^{−r(T −t)}N (−d_{2}) − xN (−d_{1})

where N is the cumulative distribution of the normal distribution and

d_{1} = \frac{1}{σ\sqrt{T − t}} (ln(x/L) + (r + σ^{2}/2)(T − t))
d_{2} = d_{1} − σ\sqrt{T − t} .

We denote by B(t, T) = e^{−r(T −t)} the value of a default-free zero-coupon and by B_{d}(t, T) the value of the defaultable zero-coupon of maturity T, with payment L = 1. Then

B_{d}(t, T) = V_{t}N (−d_{1}(V_{t}, T − t)) + B(t, T)N (d_{2}(V_{t}, T − t)) .

We denote by

Y (t, T) = −\frac{ln B(t, T)}{T − t} ,

and

Y_{d}(t, T) = −\frac{ln B_{d}(t, T)}{T − t} ,

the yield to maturity. The spread on corporate debt, i.e.,

S(t, T) = Y_{d}(t, T) − Y (t, T)

is

S(t, T) = −\frac{1}{T − t} ln (N (d_{2}) + \frac{V_{t}}{B(t, T)}N (−d_{1})) .

We can specify the probability of default given the information at date t: if the dynamics of the firm are

dV_{t} = V_{t}(μdt + σdB_{t})

under the historical probability,

P (V_{T} ≤ L|F_{t}) = N (−d_{t})

where

d_{t} = \frac{1}{σ\sqrt{T − t}} (ln(V_{t}/L) + (μ − σ^{2}/2)(T − t))

is the so-called distance-to-default.

1.2 First passage times

1.2.1 Black and Cox

The Merton’s model does not allow for a premature default; Black and Cox [6] extend Merton’s model to the case where safety covenants provide the firm’s bondholders with the right to force the firm to bankruptcy. They postulate that as soon as the firm’s asset crosses a lower threshold, the bondholders take over the firm. The firm defaults when its value falls below a pre specified level, i.e.,

τ = inf{t : V_{t} ≤ L}.

In this case, the default time τ is a stopping time in the asset’s filtration. The valuation of a defaultable claim X reduces here to the problem of the pricing of the claim X11_{{T <τ}} which is measurable with respect to the filtration of the prices at time T.

The valuation of the defaultable claim within the structural approach is a standard (but difficult) problem, which needs the knowledge of the law of the pair (τ, X). Let us assume that

dV_{t} = V_{t}((r − δ)dt + σdW_{t}) ,

where δ stands for the dividend yield. The value of a defaultable T -maturity bond with face value 1 and L ≤ 1

B_{d}(t, T) = B(t, T) (N (d_{1}(V_{t}, T − t)) − [\frac{L}{V_{t}} ]^{2Lσ^{−2}} N (d_{2}(V_{t}, T − t)))

where d_{1}(x, T − t) = \frac{1}{σ\sqrt{T − t}} (ln(x/L) + (r − δ − σ^{2}/2)(T − t)) .

1.2.2 Extensions

Longstaff and Schwarz’s model

Longstaff and Schwarz [51] allow for stochastic interest rate.

Zhou’s Model

Zhou [63] studies the case where the dynamics of the firm is

dV_{t} = V_{t−} ((μ − λν)dt + σdW_{t} + (Φ − 1)dN_{t})

where W is a Brownian motion, N a Poisson process and Φ a random variable such that ln Φ =^{law} N (a, b^{2}) with ν = exp(a + b^{2}/2) − 1. This choice of parameters implies that V e^{μt} is a martingale. In a first part, Zhou studies Merton’s problem in that setting. In a second part, he gives an approximation for the first passage problem when the default time is τ = inf{t V_{t} ≤ L}

Deterministic boundary

One can study the problem

τ = inf{t : V_{t} ≤ L(t)}

where L(t) is a deterministic function. However, there exists few explicit results. See the appendix for some references.

1.3 Term Structure Models

A substantial literature proposes to model both the default free term structure and the term structure representing the relative prices of different maturities of default-risky debt, using an extension of the method developed by Heath-Jarrow-Morton.

Major papers in this area include Jarrow and Turnbull [41], Schönbucher [58, 59], Hubner [39, 38], and Bielecki and Rutkowski [5].

Other authors choice to model directly credit spreads Duffie [23], Douady [20].

1.4 Ratings

1.4.1 Elementary case

The default process can be interpreted as a continuous time Markov chain with two states 0 and 1. The state 1 is the state where the default has appeared, and the firm remains in default. If

P_{0,0}(t) = P (τ > t) = e^{−λt}, P_{0,1}(t) = 1 − e^{−λt}, P_{1,0}(t) = 0, P_{1,1}(t) = 1

The transition matrix is

P (t) = [ \begin{matrix}e^{−λt} & 1 − e^{−λt} \\ 0 & 1 \end{matrix} ]

and can be written in the form

P (t) = e^{Λt} = ∑_{n} \frac{(tΛ)^{n}}{n!}

with Λ = [ \begin{matrix}−λ & λ \\ 0 & 0 \end{matrix} ]. The matrix Λ is called the generator of the Markov chain. The probability

for going from state 0 to state 1 between the date t and t + dt is λdt.

1.4.2 Markov Chains

More generally, let us assume that X is a continuous time Markov chain on a finite state space E = {1, …, K, K + 1} with transition probability P (t). (See Karlin and Taylor [46]). In credit rating’s model, the states are the rating classes AAA, AA, … as used by Standard and Poors or Moody’s agencies. The state K + 1 corresponds to bankruptcy, and it is assumed that a firm which goes in bankruptcy disappears, that is, remains in that state. Let us call P_{i,j}(t) the probability that, at time t, the Markov chain starting from state i is in state j. Therefore, P_{K+1,i} = 0, ∀i ≠ K + 1, P_{K+1,K+1} = 1. This leads to P_{K+1,i} = 0 for all i. Let us remark that ∑_{j} P_{i,j}(t) = 1, P_{i,j}(t) ≥ 0. The generator matrix of X is the matrix Λ such that

P (t) = exp(tΛ) = ∑_{n} \frac{(tΛ)^{n}}{n!}

Λ = [ \begin{matrix}−λ_{1,1} & λ_{1,2} & … & λ_{1,K−1} & λ_{1,K} \\ λ_{2,1} & −λ_{2,2} & … & … & λ_{2,K} \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ λ_{K−1,1} & λ_{K−1,2} & ⋯ & −λ_{K−1,K−1} & λ_{K−1,K} \\ 0 & 0 & ⋯ & 0 & 0 \end{matrix} ]

Here λ_{i,j} ≥ 0, i ≠ j and ∑_{j} λ_{i,j} = 0. By definition λ_{i,,j} = lim t → 0\frac{P_{i,j}(t) − P_{i,j}(0)}{t}.

Here is an example (Lando’s thesis)

Initial  Rating at the end of the year
Rating AAA AA A BBB BB B CCC D
AAA 0,8910 0,0963 0,0078 0,0019 0,0030 0,000 0,0000 0,0000
AA  0,0086 0,9010 0,0747 0,0099 0,0029 0,0029 0,0000 0,0000
A  0,0009 0,0291 0,8894 0,0649 0,0101 0,0045 0,0000 0,0009
BBB  0,0006 0,0043 0,0656 0,8427 0,0644 0,0160 0,0018 0,0045
BB  0,0004 0,0022 0,0079 0,0719 0,7764 0,1043 0,0127 0,0241
B  0,000 0,0019 0,0031 0,0066 0,0517 0,8246 0,0435 0,0685
0,000 0,0000 0,0116 0,0116 0,0203 0,0754 0,6493 0,2319
D  0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 1,0000

and Λ is given by

Initial  Rating at the end of the year
Rating AAA AA A BBB BB B CCC D
AAA -0,1154 0,1019 0,0083 0,0020 0,0031 0,000  0,0000 0,0000
AA  0,0091 -0,1043 0,0787 0,0105 0,0030 0,0030 0,0000 0,0000
A  0,0010 0,0309 -0,1172 0,0688 0,0107 0,0048 0,0000 0,0010
BBB  0,0007 0,0047 0,0713 -0,1711 0,0701 0,0174 0,0020 0,0049
BB  0,0005 0,0025 0,0089 0,0813 -0,2530 0,1181 0,0144 0,0273
B  0,0000 0,0021 0,0034 0,0073 0,0568 -0,1929 0,0479 0,0753
CCC 0,0000 0,0000 0,0142 0,0142 0,0250 0,0818 -0,4318 0,2756
D  0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

In this model, the ”basic” filtration is the trivial one. Lando generalizes the study to the case where the coefficients of the matrix Λ depend on a random process X. It seems important to underline the fact that the data is a transition matrix conditioned to the information given by the factor at time t. Lando introduces nonnegative functions λ_{i}, λ_{i,j} such that λ_{i}(x) = ∑_{j,j≠i} λ_{i,j}(x) and the generator matrix Λ(X_{t}).

1.4.3 CreditMetrics

CreditMetrics (from J.P. Morgan) is based on credit migration analysis and the estimation of the distribution of the changes in value of credit quality at a given horizon. The first step is to provide a transition matrix (Moody’s or Standard & Poor’s). The spreads are evaluated for each of the 7 possible credit qualities. Then, assuming that the firm’s asset value (V_{t}, t ≥ 0) follows a standard geometric Brownian motion

dV_{t} = V_{t}(μdt + σdW_{t}),

one has to slice the distribution of asset returns into bands in such a way that one reproduces the migration frequencies given in the transition matrix. The terminal value V_{T} has a distribution given by

V_{T} = V_{0} exp ((μ − \frac{σ^{2}}{2} )T + σ\sqrt{T} G) ,

where G is a standard Gaussian variable. If p is the probability that a BB-rated obligor defaults, the barrier L is defined as

p = P (V_{T} ≤ L) = P (G ≤ −\frac{ln(V_{0}/L) + (μ − \frac{σ^{2}}{2} )T}{σ\sqrt{T}} ) = N (−d)

where

d = \frac{ln(V_{0}/L) + (μ − \frac{σ^{2}}{2} )T}{σ\sqrt{T}}

is the distance to default. The barrier L can be translated in a threshold g for the process Z_{t} = ln V_{t}.

1.4.4 KMV approach

KMV (Kealhofer-McQuown, Vacisek) implements an intermediate phase before computing the probabilities of default. From observation, they define Distance to Default (DD) as follows : the default point DPT is equal to STD+(1/2)LTD where STD is the short term debt and LTD the long term debt. The DD is

DD = \frac{ln(V_{0}/DP T) + (μ − σ^{2}/2)T}{σ\sqrt{T}}

and can be viewed as a measure for the distance between the expected return on assets and the DPT. KMV estimates the proportion of firms of a given rating which actually defaults after one year, this proportion is called the EDF (Expected Default Frequencies) and deduces the implied rating.

1.4.5 CreditRisk+

CreditRisk+ (Creduit Suisse Financial products) focuses on default. They assume that default for individual bonds or loans follow a Poisson process.

1.4.6 CreditPortfolioView

CreditPortfolioView measures default risk a a discrete time multiperiod model where default probabilities are functions of macro-economics variable.

See the paper of Crouhy et al. [13] for a comparative study of these approaches.

1.5 Portfolio management

In that setting, only the state of the firm et maturity is taken into account. A large portfolio, involving different counterparties is studied.

Let Y_{i} be a random variable, valued in {0, 1}, such that Y_{i} = 1 if the i-th counterparty has defaulted before time T. Two approaches are used in order to correlate these variables.

a. Latent variable model

This method is used by CreditRisk and KMV. The default occurs if a latent variable X_{i} is smaller than a fixed level D_{i}.

b. Mixture model

This model is used by CreditRisk+. Default probabilities are made stochastic by the introduction of a r.v. Q_{i} [0, 1]-valued. The conditional law of Y_{i}, given Q_{i} is a Bernoulli law of parameter Q_{i}

The correlation between Y_{i} is made using copulas function.

1.5.1 Copulas

A recent approach for modeling dependent credit risks is the use of copulas. The reader is refereed to various works, e.g. Durrleman et al [25], Nelsen [55] for definitions and properties of Copulas and to Frey and McNeil [30], Embrechts [28], for finance purpose.

We recall the main obvious lemma:

Lemma 1.5.1 If X is a r.v. with continuous distribution function F, the r.v. F (X) has a uniform distribution on [0, 1].

Definition 1.5.1 A copula C is a joint cumulative distribution of n random variables uniformly distributed on [0, 1].

A mapping C defined on [0, 1]^{d} is a copula if and only if it satisfies :

(i) C(u_{1}, …, u_{n}) is increasing with respect to each component u_{i}, (ii) C(1, …, u_{i}, …, 1) = u_{i}, for any i, for any u_{i} ∈ [0, 1], (iii) For any a, b ∈ [0, 1]^{d} with a ≤ b (i.e., a_{i} ≤ b_{i}, ∀i)

∑_{i_{1}=1}^{2} … ∑_{i_{n}=1}^{2}(−1)^{i_{1}+...+i_{n}}C(u_{1,i_{1}}, …, u_{n,i_{n}}) ≥ 0 ,

where u_{j,1} = a_{j}, u_{j,2} = b_{j}. The survival copula is Ĉ(u_{1}, …, u_{n}) = P (U_{1} > u_{1}, …, U_{n} > u_{n}).

Theorem 1.5.1 Sklar Theorem. Let F be an n-dimensional cumulative distribution with margin F_{i}. Then there exists a copula C such that

F (x) = C(F_{1}(x_{1}), ⋯, F_{n}(x_{n}))

A particular copula is the independent copula C(u_{1}, …, u_{n}) = ∏_{i=1}^{n} u_{i}.

1.5.2 Davis and Lo

Davis et Lo [15, 16] assume that a counterparty can default for two reasons:

- either directly, i.e. Y_{i} = 1

- or may be infected by default of company j

The random variable Z_{i} is equal to 1 if the firm i has defaulted, where

Z_{i} = Y_{i} + (1 − Y_{i})(1 − ∏_{j}(1 − Y_{j}C_{ji}))

Let N be the number of firms in default at date T. If (Y_{i}, C_{ji}) are independent and

P (Y_{i} = 1) = p, P (C_{ij} = 1) = q .

we get P (N = k) = C_{k}^{n}α_{p,q,n}(k) where

α_{p,q,n}(k) = p^{k}(1 − p)^{n−k}q̄^{k(n−k)} + ∑_{i=1}^{k−1} C_{i}^{k}p^{i}(1 − p)^{n−i}(1 − q̄^{i})^{k−i}q̄^{i(n−k)}

where q̄ = 1 − q.

1.5.3 Portfolio management

A portfolio is made of positions on different counterparties. Let

V = ∑_{i=1}^{n} Z_{i}

be the global position, where

Z_{i} = ρZ + ϵ_{i}

the r.v. ϵ_{i} are independent and independent of the common factor Z. The default for the i-th component occurs if Z_{i} < d. KMV assumes that Z and ϵ are Gaussian variables. The coefficient ρ stands for the correlation between the Z_{i}. They study

P (\frac{1}{N} ∑_{i=1}^{N} Z_{i} ≤ d)

This can be approximated, as follows

P (\frac{1}{N} ∑_{i=1}^{N} Z_{i} ≤ d|Z = z) = Φ_{N}(z)

where Φ_{N}(z) = P (\frac{1}{N} ∑_{i=1}^{N} ϵ_{i} ≤ d − ρz) is a non standard Gaussian cumulative function.

1.6 Random barrier

1.6.1 Pan approach

In [56] Pan studies the following model. Let W be a Brownian motion and Z a random variable, with law μ, i.e. P (Z ∈ dx) = μ(dx), independent of W. The default time is the first time τ where

X_{t} = x exp(at + σW_{t} − tσ^{2}/2)

hits the stochastic level L = D exp(λZ − λ^{2}/2). The sequence of inequalities

{τ ≤ t} = {min_{s≤t} X_{s} ≤ L} = {min_{s≤t}(as + σW_{s} − sσ^{2}/2) ≤ λZ − λ^{2}/2 + ln(D/x)}
= {min_{s≤t}(νs + W_{s}) ≤ ζ}

where ν = aσ − σ/2, ζ = (λZ − λ^{2}/2 + ln(D/x))/σ allows us to write

P (τ ≤ t) = ∫ μ(dz)P (min_{s≤t}(νs + W_{s}) ≤ αz + β)

with α = λ/σ, β = (−λ^{2}/2 + ln(D/x))/σ. We know that (for αz + β ≤ 0)

P (min_{s≤t}(νs + W_{s}) ≤ αz + β) = N (\frac{αz + β − νt}{\sqrt{t}}) + e^{2ν(αz+β)}N (\frac{αz + β + νt}{\sqrt{t}}) .

It remains to compute

P (τ ≤ t) = ∫_{αz+β≤0} dze^{−z^{2}/2} (N (\frac{αz + β − νt}{\sqrt{t}}) + e^{2ν(αz+β)}N (\frac{αz + β + νt}{\sqrt{t}}))
+ ∫_{αz+β≥0} dze^{−z^{2}/2}

1.7 Correlation

We consider n firms

dV_{i}(t) = V_{i}(t)(μ_{i}dt + σ_{i}dW_{i}(t)

In order to evaluate the correlation between default, we have to evaluate

P (τ_{i} ≤ t, τ_{j} ≤ t) .

This involves the joint law of the supremum of correlated Brownian motion. Two approaches are known. The He et al. and Zhou [62] ones, who gave the joint law using Fokker Plank equation and Bessel function, or the Bielecki and Rutkowski one (page 119). See also Giesecke [32]for an approach of first passage time with incomplete information.

1.8 Credit Derivatives

1.8.1 Defaultable swap

Counterparty B agrees to pay L to counterparty A if the credit event occurs. In return of this counterparty A pays a fee to B, at dates t_{i}, till τ ∧ T where τ is the default time. From the counterparty A perspective, the cash flows of payments are

L11_{{τ<T }} − ∑ κ11_{{t_{i}<T ∧τ}}

1.8.2 First-to-default

A first-to-default contract, we consider a portfolio of n reference credits. Counterparty A pays a regular fee, if the credit event has happened, B pays to A the default payment. After the first default, the basket defaul swap is terminated.

12 CHAPTER 1. STRUCTURAL APPROACH

Hazard process approach

Our goal is to furnish results which cover the reduced form methodology. We provide a detailed analysis of the relatively simple case when the flow of informations available to an agent reduces to the observations of the random time which models the default event. The focus is on the evaluation of conditional expectations with respect to the filtration generated by a default time with the use of the hazard function. Then, we study the case when an additional information flow - formally represented by some filtration F - is present, with the use of the hazard process.

2.1 A toy model

Let us begin with the case where a riskless asset, with deterministic interest rate (r(s); s ≥ 0) is the only asset available on the default-free market. We denote as usual by R(t) = exp (− ∫_{0}^{t} r(s)ds) the discount factor. The price of a zero-coupon with maturity T is B(0, T) = R(T), whereas the t-time price B(t, T) of a zero-coupon with maturity T is

B(t, T) = R_{T}^{t} =^{def} exp (− ∫_{t}^{T} r(s)ds) .

The default appears at time τ (where τ is a non-negative random variable with density f). We denote by F the cumulative function of τ defined as F (t) = P (τ ≤ t) and we assume that F (t) < 1 for any t < T, where T is the maturity date (Otherwise there exists t_{0} < T such that F (t_{0}) = 1, and the default occurs a.s. before t_{0}.) We emphasize that we do not use the term “risk-neutral” probability nor “equivalent martingale measure ” (e.m.m.) in what follows because the risk is not hedgeable for the moment.

Remark 2.1.1 It is not difficult to generalize this study to the case where τ does not admit a density, dealing with the right-continuous version of the cumulative function.

2.1.1 Payment at Maturity

A defaultable zero-coupon with maturity T, and rebate paid at maturity, consists of • the payment of one monetary unit at time T if the default has not occured before time T,

• A payment of δ, done at maturity, if τ < T, where 0 < δ < 1.

Value of the defaultable zero-coupon

The value of the defaultable zero-coupon is

B_{d}(0, T) = E(R(T) [11_{T <τ} + δ11_{τ≤T}])

= E(R(T) [1 − (1 − δ)11_{τ≤T}])
= B(0, T)[1 − (1 − δ)F (T)] .

The t-time value depends whether or not the default has happened before this time. If the default has occured before time t, the payment of δ will be done at time T, and the value of the defaultable zero coupon is δR_{T}^{t}. If the default has not yet occured, nobody knows when it will appear. The value B_{d}(t, T) of the defaultable zero-coupon is the expectation of the discounted payoff R_{T}^{t} [11_{T <τ} + δ11_{τ≤T}] knowing that t < τ, i.e.,

B_{d}(t, T) = E(R_{T}^{t} [11_{T <τ} + δ11_{τ≤T}] |t < τ)
= R_{T}^{t} [1 − (1 − δ)P (τ ≤ T |t < τ)]
= R_{T}^{t} (1 − (1 − δ)\frac{P (t < τ ≤ T)}{P (t < τ)})
= B(t, T) [1 − (1 − δ)\frac{F (T) − F (t)}{1 − F (t)}]  (2.1)
= B(t, T) [\frac{1 − F (T)}{1 − F (t)} + δ \frac{F (T) − F (t)}{1 − F (t)}] .  (2.2)

Let us remark that the value of the defaultable zero-coupon is discontinuous at time τ, except if F (T) = 1 (or δ = 1). In this case, the default appears with probability one before maturity.

In the case when the payment δ(τ) is a function of the default time, the value of this defaultable zero-coupon is

B_{d}(0, T) = E(R(T) 11_{T <τ} + R(T)δ(τ)11_{τ≤T})
= B(0, T) [P (T < τ) + ∫_{0}^{T} δ(s)f(s)ds] .

The t-time value B_{d}(t, T) satisfies, if the default has not occured before t

B_{d}(t, T) = B(t, T) [E( 11_{T <τ} + δ(τ)11_{τ≤T} |(t < τ))]
= B(t, T) [\frac{P (T < τ)}{P (t < τ)} + \frac{1}{P (t < τ)} ∫_{t}^{T} δ(s)f(s)ds)] .

Particular case

In the case where F is differentiable, let us introduce the hazard function Γ defined as Γ(t) =

− ln(1 − F (t)) and its derivative γ defined as γ(t) = \frac{f(t)}{1 − F (t)} where f(t) = F′(t), i.e.,

1 − F (t) = exp (− ∫_{0}^{t} γ(s)ds) = e^{−Γ(t)} .

Then, formula (2.2) reads

B_{d}(t, T) = exp (− ∫_{t}^{T} (r + γ)(s)ds) + δ exp (− ∫_{t}^{T} r(s)ds) (1 − exp (− ∫_{t}^{T} γ(s)ds)) .

In particular, for δ = 0

B_{d}(t, T) = exp (− ∫_{t}^{T} (r + γ)(s)ds) .

Hence, the spot rate has to be adjusted by means of a spread in order to evaluate defaultable zero-coupon.

If γ and δ are constant, the credit spread is

\frac{1}{T − t} ln \frac{B(t, T)}{B_{d}(t, T)} = γ − \frac{1}{T − t} ln (1 + δ(e^{γ(T −t)} − 1))

Remark 2.1.2 In the case where τ is the first jump of a Poisson process with deterministic intensity λ,

f(t) = P (τ ∈ dt) = λ(t) exp (− ∫_{0}^{t} λ(s)ds) = λ(t)e^{−Λ(t)}

and P (τ < t) = F (t) = 1 − e^{−Λ(t)}, hence Λ(t) = Γ(t). Reciprocally, if τ is a random time with a density f, setting Λ(t) = − ln(1−F (t)) allows us to interpret τ as the first jump time of a Poisson process with intensity Λ′.

2.1.2 Payment at hit

Here, a defaultable zero-coupon with maturity T consists of • The payment of one monetary unit at time T if the default has not yet occured,

• A payment of δ(τ), where δ is a deterministic function, done at time τ if τ < T.

Value of the defaultable zero-coupon

The value of this defaultable zero-coupon is

B_{d}(0, T) = E(R(T) 11_{T <τ} + R(τ)δ(τ)11_{τ≤T})
= P (T < τ)R(T) + ∫_{0}^{T} R(s)δ(s)dF (s) .

The t-time value B_{d}(t, T) satisfies, if the default has not occured before t

R(t)B_{d}(t, T) = E(R(T) 11_{T <τ} + R(τ)δ(τ)11_{τ≤T} |(t < τ))
= \frac{P (T < τ)}{P (t < τ)}R(T) + ∫_{t}^{T} R(s)δ(s)\frac{dF (s)}{1 − F (t)}

Hence,

R(t)G(t)B_{d}(t, T) = G(T)R(T) − ∫_{t}^{T} R(s)δ(s)dG(s)

with G = 1 − F. If τ < t, the payment of δ(τ) was done, and if this payment is not re-invested in the market, the value of B_{d}(t, T) is 0.

In terms of the hazard function,

B_{d}(0, T) = e^{−Γ(T)}R(T) + ∫_{0}^{T} R(s)e^{−Γ(s)}δ(s)dΓ(s) .  (2.3)

The t-time value B_{d}(t, T) satisfies

R(t)e^{−Γ(t)}B_{d}(t, T) = e^{−Γ(T)}R(T) + ∫_{t}^{T} R(s)e^{−Γ(s)}δ(s)dΓ(s) .

Particular case

if F is differentiable, the function γ = Γ′ satisfies f(t) = γ(t)e^{−Γ(t)}. Then,

B_{d}(0, T) = e^{−Γ(T)}R(T) + ∫_{0}^{T} R(s)γ(s)e^{−Γ(s)}δ(s)ds ,  (2.4)

and

R_{d}(t)B_{d}(t, T) = R_{d}(T) + ∫_{t}^{T} R_{d}(s)γ(s)δ(s)ds

with R_{d}(t) = exp (− ∫_{0}^{t}[r(s) + γ(s)]ds).

2.1.3 Risk neutral probability measures

It is usual to translate the absence of arbitrage opportunities as the existence of a risk-neutral probability, i.e., a probability such that the discounted assets are martingales. Here, so far, we do not have any martingale, since we do not have a reference filtration. The only source of noise is the default time. As we have noticed, the t-time value of the defaultable zero-coupon takes different form whether or not the default has occured before time t. If the default has not happened, the value of the zero-coupon is deterministic, if the default has happened, the value depends on this time.

If there exist tradeable defaultable zero-coupons (DZC in short), their prices are given by the market, and the equivalent martingale measure Q, chosen by the market, is such that, on the set t < τ,

B_{d}(t, T) = B(t, T)E_{Q} ( [11_{T <τ} + δ11_{t<τ≤T}] |(t < τ)) .

Therefore, we can characterize the cumulative function of τ under Q from the market prices of the DZC as follows.

Fixed Payment at maturity

If the prices of defaultable zero-coupon with different maturities are known, then

\frac{B(0, t) − B_{d}(0, t)}{B(0, t)(1 − δ)} = F_{Q}(t)

where F_{Q}(t) = Q(τ ≤ t), so that the law of τ is known under the e.m.m. However, as noticed in Hull and White [40] ,extracting default probabilities from bond prices [is] in practice, usually more complicated. First, the recovery rate is usually non-zero. Second, most corporate bonds are not zero-coupon bonds.

Payment at hit

In this case the cumulative function can be obtained using the derivative of the defaultable zero-

coupon price with respect to the maturity. Indeed, denoting by \frac{∂B_{d}}{∂t} the derivative of the value of the DZC at time 0 with respect to the maturity, we get from (2.3) in the case where G = 1 − F is differentiable

\frac{∂B_{d}}{∂t} (0, t) = e^{−Γ(t)}R(t) (−r(t) − γ(t) + δ(t)γ(t))
= −G(t)R(t)r(t) + (1 − δ(t))g(t)R(t) ,

where g(t) = G′(t). Therefore, solving this equation leads to

1−Q(τ ≤ t) = [1 + ∫_{0}^{t} \frac{∂B_{d}}{∂t} (0, s)\frac{1}{R(s)(1 − δ(s))} exp (− ∫_{0}^{s} \frac{r(u)}{1 − δ(u)}du) ds] exp (∫_{0}^{t} \frac{r(s)}{1 − δ(s)}ds) .

2.2 Toy model and martingales

We now present the result of the previous section in a different form. Let us start with some well known facts. Suppose that τ is an \overline{IR}_{+}^{∗}-valued random variable on some probability space (Ω, G, P). Let us denote by (D_{t}, t ≥ 0) the right-continuous increasing process D_{t} = 11_{t≥τ} and by (D_{t}) the natural filtration generated by the process D. The filtration D is the smallest filtration Δ = (Δ_{t}, t ≥ 0) (satisfying the usual hypotheses) on Ω such that τ is an Δ = (Δ_{t}, t ≥ 0)-stopping time. The σ-algebra D_{t} is generated by the sets {τ ≤ s} for s ≤ t (note that the set {τ > t} is an atom.) In this section, we deal with a right-continuous cumulative function F, and we denote by Γ the hazard function Γ(t) = − ln(1 − F (t)).

Any integrable D_{t}-measurable r.v. H is of the form h(τ ∧ t) = h(τ)11_{{τ≤t}} + h(t)11_{{t<τ}} where h is a Borel function. We now give some elementary tools to compute the conditional expectation w.r.t. D_{t}, presented in Brémaud [9], Dellacherie [17], Elliott [26].

2.2.1 Key lemma

Lemma 2.2.1 If X is any integrable, G-measurable r.v.

E(X|D_{t})11_{t<τ} = 11_{t<τ} \frac{E(X11_{t<τ})}{P (t < τ)} .

Proof: The r.v. E(X|D_{t}) is D_{t}-measurable, therefore it can be written of the form E(X|D_{t}) = h(τ ∧ t) = h(τ)11_{t≥τ} + h(t)11_{t<τ} for some function h. By multiplying both members by 11_{t<τ}, and taking the expectation, we obtain

E[11_{t<τ} E(X|D_{t})] = E[E(11_{t<τ} X|D_{t})] = E[11_{t<τ} X] = h(t)P (t < τ) .
□

Let F (t) = P (τ ≤ t) be the right-continuous cumulative function of τ. We denote by F (t−) the left limit of F (s) when s increases to t.

2.2.2 Martingales

Proposition 2.2.1 The process

M_{t} = D_{t} − ∫_{0}^{τ∧t} \frac{dF (s)}{1 − F (s−)} = D_{t} − ∫_{0}^{t}(1 − D_{s−})\frac{dF (s)}{1 − F (s−)}

is a D-martingale.

Proof: Let t be fixed and s < t. From the previous lemma

E(D_{t} − D_{s}|D_{s}) = E(11_{s<τ≤t}|D_{s}) = 11_{s<τ} A(s) + 11_{s≥τ} B(τ) .  (2.5)

In the one hand, from lemma 2.2.1 applied to X = 11_{T <τ} we get that A(s) = \frac{P (s < τ ≤ t)}{P (s < τ)} =

\frac{F (t) − F (s)}{1 − F (s)}. In the other hand, multiplying the two members of equality (2.5) by 11_{s≥τ}, we

obtain that the r.v. B(τ) is equal to 0. Hence,

E(D_{t} − D_{s}|F_{s}) = 11_{s<τ} \frac{F (t) − F (s)}{1 − F (s)} .  (2.6)

Let us study the term

C =^{def} E [∫_{0}^{τ∧t} \frac{dF (u)}{1 − F (u−)} − ∫_{0}^{τ∧s} \frac{dF (u)}{1 − F (u−)}|D_{s}] .

We get easily that

C = E [∫_{τ∧s}^{τ∧t} \frac{dF (u)}{1 − F (u−)}|D_{s}] = 11_{τ>s}g(s)

where from lemma 2.2.1

g(s) = \frac{1}{1 − F (s)}E [11_{τ>s} ∫_{s∧τ}^{t∧τ} \frac{dF (u)}{1 − F (u−)}]
= \frac{1 − F (t)}{1 − F (s)} ∫_{s}^{t} \frac{dF (u)}{1 − F (u−)} − \frac{1}{1 − F (s)} ∫_{s}^{t} dF (v) ∫_{s}^{v} \frac{dF (u)}{1 − F (u−)} .

Fubini’s theorem yields the equality g(s) = \frac{F (t) − F (s)}{1 − F (s)}. □

2.2.3 Intensity process

The function Λ(t) = ∫_{0}^{t} \frac{dF (s)}{1 − F (s−)} is continuous and increasing. When F is continuous,

Λ(t) = − ln(1 − F (t))

is the so-called hazard function, well known in fiability (See Cocozza [12].) From Prop. 2.2.1 we obtain the decomposition of the submartingale D_{t} as M_{t} + Λ(t ∧ τ). The process A_{t} = Λ_{t∧τ} is called the compensator of D. In particular, if F is differentiable, the law of τ admits a density f = F′, and the process

M_{t} = D_{t} − ∫_{0}^{τ∧t} λ(s)ds

is a martingale, where λ(s) = \frac{dF (s)/ds}{1 − F (s)} = \frac{f(s)}{1 − F (s)} is a deterministic positive function, called

the intensity of τ.

Proposition 2.2.2 The process L_{t} =^{def} 11_{τ>t} exp (∫_{0}^{t} λ(s)ds) is a D-martingale. In particular,

E(11_{τ>T} |D_{t}) = 11_{τ>t} exp (− ∫_{t}^{T} λ(s)ds)

Proof: Since the function λ is deterministic, for t > s

E(L_{t}|D_{s}) = exp(∫_{0}^{t} λ(u)du) E(11_{t<τ} |D_{s}) .

From (2.6)

E(11_{t<τ} |D_{s}) = 11s < τ \frac{1 − F (t)}{1 − F (s)} = 11_{τ>s} exp (−Λ(t) + Λ(s)) .

Hence,

E(L_{t}|D_{s}) = 11_{τ>s} exp (∫_{0}^{s} λ(u)du ) = L_{s}.

Another way is to apply Itô’s formula to the process L_{t} = (1 − D_{t}) exp (∫_{0}^{t} λ(s)ds)

dL_{t} = −dD_{t} exp (∫_{0}^{t} λ(s)ds) + λ(t) exp (∫_{0}^{t} λ(s)ds) (1 − D_{t})dt = − exp (∫_{0}^{t} λ(s)ds) dM_{t} .

A sophisticated way is to note that L is the exponential martingale solution of the SDE

dL_{t} = L_{t−}dM_{t} .
□

Example 2.2.1 Let us study the case where τ is the first time where a Poisson process N jumps.

Let λ(s) be the deterministic intensity and D_{t} = N_{t∧τ}. It is well known that N_{t} − ∫_{0}^{t} λ(s)ds is a

martingale. Therefore, the process stopped at time τ is also a martingale, i.e., D_{t} − ∫_{0}^{t∧τ} λ(s)ds

is a martingale. Furthermore, we have seen in Remark 2.1.2 that we can reduce our attention to that case.

2.2.4 Incompleteness

Now, the problem is to define the tradeable assets. If the primary market consists only of the risk-free asset, there exists infinitely many e.m.m. : if the support of the density of τ is IR^{+}, any probability such that τ admits a density with support IR^{+} is a change of law. The set Q of equivalent martingale measures is the set of probabilities, equivalent to the historical one, such that the discounted price of the tradeable assets are martingales. The range of price is defined as the set of prices which do not induce arbitrage opportunities, and is equal to

{E_{Q}(R_{T} (11_{{T <τ}} + δ11_{{τ<T }})), Q ∈ Q} .

This set is exactly the interval ]δR_{T} , R_{T} [, (obtained as limit cases : the default appears at time 0^{+}, or never.).

2.2.5 Spread

If a DZC of maturity T is traded at a price ρ(t, T), then, under any risk-neutral probability Q (for the moment, we do not know if the market is complete, so we can not claim that the e.m.m. is unique)

ρ(t, T)R(t) = E_{Q}(R(T)11_{T <τ} |D_{t}) = 11_{t<τ} exp (− ∫_{t}^{T} λ^{Q}(s)ds)

where λ^{Q}(s) = \frac{dF_{Q}(s)/ds}{1 − F_{Q}(s)} is the Q-intensity of τ. Therefore, the value of ∫_{0}^{t} λ^{Q}(s)ds is known

as soon as there are zero-coupon bonds for each maturity, and the risk-neutral intensity can be obtained from the market data.

2.2.6 Partial information: Duffie and Lando’s model

Duffie and Lando [22] study the case where τ = inf{t : V_{t} ≤ m} where V satisfies

dV_{t} = μ(t, V_{t})dt + σ(t, V_{t})dW_{t} .

Here the process W is a Brownian motion. If the information is the Brownian filtration, the time τ is a stopping time w.r.t. a Brownian filtration, therefore is predictable and admits no intensity. We will discuss this point latter on. If the agents do not know the behavior of V, but only the minimal information D_{t}, i.e. he knows when the default appears, the price of a zero-coupon is, in

the case where the default is not yet occured, exp (− ∫_{t}^{T} λ(s)ds) where λ(s) = \frac{f(s)}{G(s)} and G(s) =

P (τ > s), f = −G′, as soon as the cumulative function of τ is differentiable. Duffie and Lando

have obtained that the intensity is λ(t) = \frac{1}{2}σ^{2}(t, 0)\frac{∂f}{∂x} (t, 0) where f(t, x) is the conditional density

of V_{t} when T_{0} > t, i.e. the differential w.r.t. x of \frac{P (V_{t} ≤ x, T_{0} > t)}{P (T_{0} > t)}, where T_{0} = inf\{t; V_{t} = 0\}.

In the case where V is an homogenous diffusion, i.e. dV_{t} = μ(V_{t})dt + σ(V_{t})dW_{t}, the equality between Duffie-Lando and our result is not so obvious. See Elliott et al. [27] for comments.

2.2.7 Representation theorem

Proposition 2.2.3 Let h be a (bounded) Borel function. Then

E(h(τ)|D_{t}) = E(h(τ)) − ∫_{0}^{t∧τ} e^{ΔΓ(s)}(H_{s−} − h(s))dM_{s} ,

where M_{t} = D_{t} − Γ(t ∧ τ).

Proof: We give the proof in the case where F, hence Γ is continuous. Let H_{t} = E(h(τ)|D_{t}). From the measurability argument

H_{t} = h(τ)11_{τ≤t} + 11_{t<τ} \frac{E(h(τ)11_{t<τ})}{P (t < τ)} = h(τ)11_{\{τ≤t\}} + 11_{\{t<τ\}}e^{Γ(t)}E(h(τ)11_{\{t<τ\}}) .

Since F is the law of τ

E[h(τ)11_{{t<τ}}] = ∫_{t}^{∞} h(u)dF (u)) .

An integration by parts leads to

e^{Γ_{t}}E[h(τ)11_{{t<τ}}] = E(h(τ)) − ∫_{0}^{t} e^{Γ(s)}h(s)dF (s) + ∫_{0}^{t} E(h(τ)11_{{s<τ}})e^{Γ(s)}dΓ(s)

therefore, since H_{s} = e^{Γ(s)}E(h(τ)11_{{s<τ}}) on {s < τ},

11_{{t<τ}}e^{Γ_{t}}E[h(τ)11_{{t<τ}}] = 11_{{t<τ}} (E(h(τ)) − ∫_{0}^{t} e^{Γ(s)}h(s)dF (s) + ∫_{0}^{t} H_{s}dΓ(s)) .

Hence

11_{\{t<τ\}}E(h(τ)|D_{t}) = 11_{\{t<τ\}} (E(h(τ)) + ∫_{0}^{t∧τ} (H_{s} − h(s))\frac{dF (s)}{1 − F (s)} )
= 11_{{t<τ}} (E(h(τ)) − ∫_{0}^{t∧τ} (H_{s} − h(s))(dD_{s} − dΓ(s)) ) .

On the complementary set {s ≥ τ}, we have seen that E(h(τ)|D_{t}) = h(τ), whereas

E(h(τ)) − ∫_{0}^{t∧τ} (H_{s} − h(s))(dD_{s} − dΓ(s)) = H_{τ−} − (H_{τ} − h(τ)) .
□

2.3 Successive default times

The previous results can easily be generalized to the case of successive default times. We assume in this section that r = 0.

2.3.1 Two times

Let us first study the case with two random times τ_{1}, τ_{2}. We denote by T_{1} = inf(τ_{1}, τ_{2}) and T_{2} = sup(τ_{1}, τ_{2}), and we assume, for simplicity, that P (τ_{1} = τ_{2}) = 0. We denote by (D_{t}^{i}, t ≥ 0) the default process associated with T_{i}, (i = 1, 2), and by D_{t} = D_{t}^{1} + D_{t}^{2} the process associated with two defaults. As before, D is the filtration generated by the process D. The σ-algebra D_{t} is equal to σ(T_{1} ∧ t) ∨ σ(T_{2} ∧ t).

A D_{t}-measurable random variable is equal to - a constant on the set t < T_{1},

- a σ(T_{1})-measurable random variable on the set T_{1} ≤ t < T_{2},

- a σ(T_{2})-measurable random variable on the set T_{2} ≤ t.

Payment at maturity

Suppose that a payment of 1 monetary unit is done at maturity if no default has occured, δ_{1} if one and only one default has occured and δ_{2} in the remaining case, where 0 ≤ δ_{2} < δ_{1} < 1. Therefore, to obtain the value of this claim we have to compute

E(11_{T <T_{1}} + δ_{1}11_{T_{1}≤T <T_{2}} + δ_{2}11_{T_{2}≤T} |D_{t}) ;

It is then straightforward to prove, for t < T

P (T < T_{1}|D_{t}) = 11_{t<T_{1}} \frac{P (T < T_{1})}{P (t < T_{1})}
P (T_{1} ≤ T < T_{2}|D_{t}) = 11_{t<T_{1}} \frac{P (t < T_{1} ≤ T < T_{2})}{P (t < T_{1})} + 11_{T_{1}≤t<T_{2}} \frac{P (T < T_{2}|T_{1})}{P (t < T_{2}|T_{1})}
P (T_{2} ≤ T |D_{t}) = 11_{t<T_{1}} \frac{P (t ≤ T_{1} < T_{2} < T)}{P (t < T_{1})} + 11_{T_{1}≤t<T_{2}} \frac{P (t < T_{2} < T |T_{1})}{P (t < T_{2}|T_{1})} + 11_{T_{2}<t} .

In the very particular case where τ_{1} and τ_{2} are independent with densities f and g, the law of the pair T_{1}, T_{2} is easy to compute

P (T_{1} ∈ du, T_{2} ∈ dv) = 11_{u<v}[f(u)g(v) + g(u)f(v)]du dv

and the conditional law of T_{2} with respect to T_{1} follows

P (T_{2} ∈ dv|T_{1} = u) = 11_{u<v} \frac{f(u)g(v) + g(u)f(v)}{∫_{0}^{∞} dv[f(u)g(v) + g(u)f(v)]}dv .

In the general case,

P (t < T_{1} ≤ T < T_{2}) = P (t < τ_{1} ≤ T < τ_{2}) + P (t < τ_{2} ≤ T < τ_{1})

and

P (t < τ_{1} ≤ T < τ_{2}) = P (t < τ_{1}, T < τ_{2}) − P (T < τ_{1}, T < τ_{2})

can be computed via survival copula (See next section). In the same way,

P (T < T_{2}|T_{1}) = P (T < τ_{2}, τ_{1} < τ_{2}|τ_{1}) + P (T < τ_{1}, τ_{2} < τ_{1}|τ_{2})

and

P (T < τ_{2}, τ_{1} < τ_{2}|τ_{1} = t) = P (T < τ_{2}, τ_{1} < τ_{2}, τ_{1} ∈ dt)/P (τ_{1} ∈ dt)

can be computed from the copula of the pair (τ_{1}, τ_{2}) from P (T < τ_{2}, τ_{1} ≤ u) = P (T < τ_{2})−P (T < τ_{2}, u < τ_{1}).

Suppose that a payment of 1 monetary unit is done at maturity if no default has happened, δ_{1}(T_{1}) if the first default happens before T (and the second after T) and δ_{2}(T_{2}) in the remaining case, where 0 ≤ δ_{2} < δ_{1} < 1. Therefore, we have to compute

E(11_{T <T_{1}} + δ_{1}(T_{1})11_{T_{1}≤T <T_{2}} + δ_{2}(T_{2})11_{T_{2}≤T} |D_{t}) .

It is then straightforward to prove

E(δ_{1}(T_{1})11_{T_{1}≤T <T_{2}}|D_{t}) = 11_{t<T_{1}} \frac{E(δ_{1}(T_{1})11_{t<T_{1}≤T <T_{2}})}{P (t < T_{1})} + 11_{T_{1}≤T <T_{2}}δ(T_{1}) \frac{P (T < T_{2}|T_{1})}{P (t < T_{2}|T_{1})}
E(δ_{2}(T_{2})11_{T_{2}≤T} |D_{t}) = 11_{t<T_{1}} \frac{E(δ_{2}(T_{2})11_{t≤T_{1}<T_{2}<T})}{P (t < T_{1})} + 11_{T_{1}≤t<T_{2}} \frac{E(δ_{2}(T_{2})11_{t<T_{2}<T |T_{1}})}{P (t < T_{2}|τ_{1})}
+ 11_{T_{2}<t}δ_{2}(T_{2}) .

In order to compute the different terms, we need the joint law as well as the conditional joint law.

We reproduce now the result of Chou and Meyer [11], in order to obtain the martingales in the filtration G, in case of two default times. Let us denote by F_{1}(t) the distribution function of T_{1} and by F_{2}(t; u) the conditional distribution function of T_{2} with respect to T_{1}, i.e.,

F_{2}(t; u) = P (T_{2} ≤ t|T_{1} = u)

The process M_{t} =^{def} D_{t} − Λ_{t∧τ} is a D-martingale, where

Λ_{t} = Λ_{1}(t)11_{t<T_{1}} + [Λ_{1}(T_{1}) + Λ_{2}(t, T_{1})] 11_{T_{1}≤t<T_{2}} + [Λ_{1}(T_{1}) + Λ_{2}(T_{2}, T_{1})] 11_{T_{2}≤t}

with Λ_{1}(t) = ∫_{0}^{t} \frac{dF_{1}(s)}{1 − F_{1}(s)} and Λ(t; u) = ∫_{u}^{t} \frac{dF_{2}(s; u)}{1 − F_{2}(s; u)}. It is proved in Chou-Meyer [11] that

any D-martingale is a stochastic integral with respect to M. This result admits an immediate extension to the case of n successive defaults.

2.3.2 More than two times

Suppose that the default times are modeled via a Poisson process with intensity h. (See the appendix for definitions and main properties of Poisson processes) The terminal payoff is ∏_{T_{i}≤T} (1− δ(T_{i})), where δ is a deterministic function valued in [0, 1]. The value of this payoff is E(∏_{T_{i}≤T} (1− δ(T_{i}))). In the case of constant δ(s) = δ, we get

E ( ∏_{T_{i}≤T}(1 − δ(T_{i}))) = E ((1 − δ)^{N_{T}} ) = exp (δ ∫_{0}^{T} h(s)ds) .

In the general case,

E ( ∏_{T_{i}≤T}(1 − δ(T_{i}))) = E (exp (∑_{s≤T} ln(1 − δ(s))ΔN_{s}))) = E (exp (∫_{0}^{T} ln(1 − δ_{s})dN_{s})) .

Hence (See the appendix)

E ( ∏_{T_{i}≤T}(1 − δ(T_{i}))) = exp (∫_{0}^{T} δ(s)h(s)ds) .

2.4 Cox Processes and Extensions

We now present a case where the intensity is a stochastic process.

2.4.1 Construction of Cox Processes with a given stochastic intensity

Let (Ω, G, P) be a probability space endowed with a filtration F. A nonnegative F-adapted process λ is given. We assume that there exists a random variable Θ, independent of F_{∞}, with an exponential law: P (Θ ≥ t) = e^{−t}. We define the random time τ as the first time when the process Λ_{t} = ∫_{0}^{t} λ_{s} ds is above the random level Θ, i.e.,

τ = inf {t ≥ 0 : Λ_{t} ≥ Θ}.

In particular, {τ ≥ s} = {Λ_{s} ≤ Θ}.We assume that Λ_{t} < ∞, ∀t, Λ_{∞} = ∞.

Comments 2.4.1 Another example is to choose τ = inf {t ≥ 0 : Ñ_{Λ_{t}} = 1}, where Λ_{t} = ∫_{0}^{t} λ_{s} ds and Ñ is a Poisson process with intensity 1, independent of the filtration F. The second method is in fact equivalent to the first. Cox processes (or doubly stochastic Poisson processes) are used in a great number of studies (see, e.g., [48]).

2.4.2 Conditional Expectations

Lemma 2.4.1 The conditional distribution function of τ given the σ-field F_{t} is for t ≥ s

P (τ > s|F_{t}) = exp ( − Λ_{s}) .

Proof: The proof follows from the equality {τ > s} = {Λ_{s} < Θ}. From the independence assumption and the F_{t}-measurability of Λ_{s} for s ≤ t, we obtain

P (τ > s|F_{t}) = P (Λ_{s} < Θ | F_{t}) = exp ( − Λ_{s}).

In particular, we have

P (τ ≤ t|F_{t}) = P (τ ≤ t|F_{∞}),  (2.7)

and, for t ≥ s, P (τ > s|F_{t}) = P (τ > s|F_{s}). Let us notice that the process F_{t} = P (τ ≤ t|F_{t}) is here an increasing process. □

Remark 2.4.1 If the process λ is not non-negative, we get, for s < t

P (τ > s|F_{t}) = exp(− sup_{u≤s} Λ_{u}) .

2.4.3 Choice of filtration

We write as before D_{t} = 11_{{τ≤t}} and D_{t} = σ(D_{s} : s ≤ t). We introduce the filtration G_{t} = F_{t} ∨D_{t}, that is, the enlarged filtration generated by the underlying filtration F and the process D. (We denote by F the original Filtration and by G the enlarGed one.) We shall frequently write G = F ∨ D.

It is easy to describe the events which belong to the σ-field G_{t} on the set {τ > t}. Indeed, if G_{t} ∈ G_{t}, then G_{t} ∩ {τ > t} = B_{t} ∩ {τ > t} for some event B_{t} ∈ F_{t}.

Therefore any G_{t}-measurable random variable Y_{t} satisfies 11_{{τ>t}}Y_{t} = 11_{{τ>t}}y_{t}, where y_{t} is an F_{t}-measurable random variable.

2.4.4 Key lemma

Proposition 2.4.1 Let Y be an integrable r.v. Then,

11_{\{τ>t\}}E(Y |G_{t}) = 11_{\{τ>t\}} \frac{E(Y 11_{\{τ>t\}}|F_{t})}{E(11_{\{τ>t\}}|F_{t})} = 11_{\{τ>t\}}e^{Λ_{t}}E(Y 11_{\{τ>t\}}|F_{t}).

Proof: From the remarks on the G_{t}-measurability, if Y_{t} = E(Y |G_{t}), then there exists an F_{t}-measurable r.v. y_{t} such that

11_{{τ>t}}E(Y |G_{t}) = 11_{{τ>t}}y_{t}

and taking conditional expectation w.r.t. F_{t} of both members, we deduce y_{t} = \frac{E(Y 11_{\{τ>t\}}|F_{t})}{E(11_{\{τ>t\}}|F_{t})}.

□

Corollary 2.4.1 If X is an integrable F_{T} -measurable random variable

E(X11_{{T <τ}}|G_{t}) = 11_{{τ>t}}e^{Λ_{t}}E(Xe^{−Λ_{T}} |F_{t}) .  (2.8)

Proof: Let X be an F_{T} -measurable r.v. From prop. 2.4.1, E(X11_{{τ>T }}|G_{t}) is equal to 0 on the G_{t}-measurable set τ < t, whereas

E(X11_{{τ>T }}|F_{t}) = E(X11_{{τ>T }}|F_{T} |F_{t}) = E(Xe^{Λ_{T}} |F_{t}).
□

Comments 2.4.2 This corollary admits an interesting interpretation. If X11_{{T <τ}} is some defaultable payoff, its value is the value of the default free payoff X when the interest rate is higher that the spot rate and the difference, i.e., λ can be interpreted as a spread. However, we emphasize that we are not dealing with a risk neutral probability. If the market is assumed to be complete, that means in particular that a defaultable zero-coupon is traded. Then, the intensity has to be evaluated under the risk-neutral probability given by the market.

Definition 2.4.1 The process λ is called the intensity of τ.

We now compute the expectation of a τ-time value of a predictable process.

Lemma 2.4.2 (i) If h is an F-predictable (bounded) process then

E(h_{τ} |F_{t}) = E( ∫_{0}^{∞} h_{u}λ_{u} exp ( − Λ_{u}) du | F_{t})

and

E(h_{τ} |G_{t}) = E( ∫_{t}^{∞} h_{u}λ_{u} exp (Λ_{t} − Λ_{u}) du | F_{t})11_{{τ>t}} + h_{τ} 11_{{τ≤t}}.  (2.9)

In particular

E(h_{τ}) = E( ∫_{0}^{∞} h_{u}λ_{u} exp ( − Λ_{u}) du)

(ii) The process (D_{t} − ∫_{0}^{t∧τ} λ_{s}ds, t ≥ 0) is a G-martingale.

Proof: Let h_{t} = 11_{]v,w]}(t)B_{v} where B_{v} ∈ F_{v} be an elementary predictable process. Then, from Cor. 2.4.1

E(h_{τ} |F_{t}) = E(11_{]v,w]}(τ)B_{v}F_{t}) = E(E(11_{]v,w]}(τ)B_{v}|F_{∞}) | F_{t})
= E(B_{v}P (v < τ ≤ w|F_{∞}) | F_{t}) = E(B_{v}(e^{−Λ_{v}} − e^{−Λ_{w}})|F_{t})

It follows that

E(h_{τ} |F_{t}) = E(B_{v} ∫_{v}^{w} λ_{u}e^{−Λ_{u}} du | F_{t}) = E( ∫_{0}^{∞} h_{u}λ_{u}e^{−Λ_{u}}du | F_{t})

and the result is derived from the monotone class theorem.

The martingale property (ii) follows from integration by parts formula. Indeed, let t < s. Then, on the one hand from Cor 2.4.1

E(D_{s} − D_{t}|G_{t}) = P (t < τ ≤ s|G_{t}) = 11_{\{t<τ\}} \frac{P (t < τ ≤ s|F_{t})}{P (t < τ|F_{t})}
= 11_{{t<τ}}E(1 − exp(Λ_{s} − Λ_{t})|F_{t})

On the other hand, from part (i)

E( ∫_{t∧τ}^{s∧τ} λ_{u}du | G_{t}) = E(Λ_{s∧τ} − Λ_{t∧τ} |G_{t}) = 11_{{t<τ}}E( ∫_{t}^{∞} h_{u}λ_{u}e^{−(Λ_{u}−Λ_{t})}du | F_{t})

where h_{u} = Λ(s ∧ u) − Λ(t ∧ u). Consequently,

∫_{t}^{∞} h_{u}λ_{u}e^{−(Λ_{u}−Λ_{t})}du = ∫_{t}^{s}(Λ_{u} − Λ_{s})λ_{u}e^{−(Λ_{u}−Λ_{t})}du + (Λ_{t} − Λ_{s}) ∫_{s}^{∞} λ_{u}e^{−(Λ_{u}−Λ_{t})}du
= −(Λ_{s} − Λ_{t})e^{−(Λ_{s}−Λ_{t})} + ∫_{t}^{s} λ(u)e^{−(Λ_{u}−Λ_{t})}du + (Λ_{s} − Λ_{t})e^{−(Λ_{s}−Λ_{t})}
= 1 − e^{−(Λ_{s}−Λ_{t})}.

This ends the proof. □

2.4.5 Conditional Expectation of F_{∞}-Measurable Random Variables

Lemma 2.4.3 Let X be an F_{∞}-measurable r.v.. Then

E(X|G_{t}) = E(X|F_{t}) .  (2.10)

Proof: Let X be an F_{∞}-measurable r.v. To prove that E(X|G_{t}) = E(X|F_{t}), it suffices to check that

E(B_{t}h(τ ∧ t)X) = E(B_{t}h(τ ∧ t)E(X|F_{t}))

for any B_{t} ∈ F_{t} and any h = 11_{[0,a]}. For t ≤ a, the equality is obvious. For t > a, we have from (2.7)

E(B_{t}11_{{τ≤a}}E(X|F_{t})) = E(B_{t}E(X|F_{t})E(11_{{τ≤a}}|F_{∞})) = E(E(B_{t}X|F_{t})E(11_{{τ≤a}}|F_{t}))
= E(XB_{t}E(11_{{τ≤a}}|F_{t})) = E(B_{t}X11_{{τ≤a}})

as expected. □

Remark 2.4.2 Let us remark that (2.10) implies that every F-square integrable martingale is a G-martingale. However, equality (2.10) does not apply to any G-measurable random variable; in particular P (τ ≤ t|G_{t}) = 11_{{τ≤t}} is not equal to F_{t} = P (τ ≤ t|F_{t}).

2.4.6 Defaultable Zero-Coupon Bond

From Cor. 2.4.1, for t < T

E(11_{{T <τ}}|G_{t}) = 11_{{τ>t}}E( exp ( − ∫_{t}^{T} λ_{s} ds) | F_{t}).

Let B(t, T) be the price at time t of a default-free bond paying 1 at maturity t satisfies

B(t, T) = E_{Q}( exp ( − ∫_{t}^{T} r_{s} ds) | F_{t}).

The market price B_{d}(t, T) of a defaultable zero-coupon bond with maturity T is

B_{d}(t, T) = E_{Q}(11_{{T <τ}} exp ( − ∫_{t}^{T} r_{s} ds) | G_{t})
= 11_{{τ>t}}E_{Q}( exp ( − ∫_{t}^{T} [r_{s} + λ_{s}^{Q}] ds) | F_{t}).

If a recovery δ(τ) is paid at time T, then

B_{d}(t, T) = B(t, T) − E (e− ∫_{t}^{T} r_{s}ds ∫_{t}^{T} ds(1 − δ_{s})λ_{s}^{Q}e− ∫_{t}^{s} λ_{u}^{Q}du)

Therefore, given the price of a DZC, we can deduce the risk neutral intensity.

The t-time value of a corporate bond, which pays δ at time T in case of default and 1 otherwise, is given by

E_{Q}(e^{− ∫_{t}^{T} r_{s} ds} (δ11_{{τ≤T }} + 11_{{τ>T }}) | F_{t}).

The last quantity is equal to

δB(t, T) + 11_{{τ>t}}(1 − δ)E( exp ( − ∫_{t}^{T} [r_{s} + λ_{s}^{Q}] ds) | F_{t}).

In the case where the compensation is time dependent,

ρ(t, T) = B(t, T) − E (e− ∫_{t}^{T} r_{s}ds ∫_{t}^{T} ds(1 − δ_{s})λ_{s}e− ∫_{t}^{s} λ_{u}du)

2.4.7 Extension

In Wong [61], the time of default is given as

τ = inf{t : Λ_{t} ≥ Σ}

where Σ a non-negative r.v. independent of F_{∞}. This model reduces to the previous one: if Φ is the cumulative function of Σ, the r.v. Φ(Σ) has a uniform distribution and

τ = inf{t : Φ(Λ_{t}) ≥ Φ(Σ)} = inf{t : Ψ^{−1}[Φ(Λ_{t})] ≥ Θ}

where Ψ is the cumulative function of the exponential law. Then,

F_{t} = P (τ ≤ t|F_{t}) = P (Λ_{t} ≥ Σ|F_{t}) = 1 − exp (−Ψ^{−1}(Φ(Λ_{t}))) .

2.4.8 Examples

Some authors choose to model the intensity.

2.4.9 Duffee’s model

Duffee [21] assumes that the value of a default free bond is

E_{Q} (exp (− ∫_{t}^{T} r_{s}ds) |F_{t})

and that the defaultable bond is priced as

E_{Q} (exp (− ∫_{t}^{T} (r_{s} + γ_{s})ds) |F_{t})

where

r_{t} = s_{1,t} + s_{2,t}, h_{t} = β + β_{1}s_{1,t} + β_{2}s_{2,t} + s_{3,t}

and

ds_{i,t} = κ_{i}(θ_{i} − s_{i,t})dt + σ_{i}\sqrt{s_{i,t}}dW_{i}(t)

2.4.10 Jarrow and Turnbull’s model

Jarrow and Turnbull consider a situation where the interest rate follows a Vasicek’s dynamics and where the intensity is a linear function of the interest rate and a factor Z, modeled as a Brownian motion.

dr_{t} = κ(r_{∞} − r_{t})dt + σdW_{t}

and γ_{t} = a_{0}(t)+a_{1}(t)r_{t} +a_{2}(t)Z_{t}. The problem is that γ is not a non-negative value. nevertheless, the corporate bond follows

B_{d}(t, T) = δB(t, T) + (1 − δ) exp(−μ + \frac{1}{2}v)

where μ and v are the mean and variance of R_{T} + Γ_{T}.

2.5 General case

2.5.1 The model

In reduced form approach, we shall deal with two kinds of information : the information from the asset’s prices, denoted as (F_{t}, t ≥ 0) and the information from the default time, i.e. the knowledge of the time were the default occured in the past, it the default has appeared. More precisely, this information is modeled by the filtration D_{t} generated by the default process D.

At the intuitive level, F is generated by prices of some assets, or by other economic factors (e.g., interest rates). This filtration can also be a subfiltration of the prices. The case where F is the trivial filtration is exactly what we have studied in the toy example. Though in typical examples F is chosen to be the Brownian filtration, most theoretical results do not rely on such a specification of the filtration F. We denote by G_{t} = F_{t} ∨ D_{t}.

Special attention is paid here to the hypothesis (H), which postulates the invariance of the martingale property with respect to the enlargement of F by the observations of a default time. We establish a representation theorem, in order to understand the meaning of complete market in a defaultable world and we deduce the hedging strategies for credit derivatives. The main part of this section can be found in the surveys of Jeanblanc and Rutkowski [42, 43]. For a complete study of credit risk, see the book of Bielecki and Rutkowski [5]. The reader can find other interesting information on the web sites quoted at the end of the bibliography of this document.

2.5.2 Key lemma

It is straightforward to establish that any G_{t}-random variable is equal, on the set {τ > t}, to an F_{t}-measurable random variable.

We denote by F_{t} = P (τ ≤ t|F_{t}) the conditional law of τ given the information F_{t}.

Lemma 2.5.1 Let X be an F_{T} -measurable integrable r.v. Then,

E(X11_{T <τ} |G_{t}) = 11_{\{τ>t\}} \frac{E(X11_{\{τ>T \}}|F_{t})}{E(11_{\{τ>t\}}|F_{t})} = 11_{\{τ>t\}}e^{Γ_{t}}E(Xe^{−Γ_{T}} |F_{t}).  (2.11)

where Γ_{t} = − ln(1 − F_{t})

Proof: The proof is exactly the same as Cor. 2.4.1. Indeed,

11_{{τ>t}}E(X|G_{t}) = 11_{{τ>t}}x_{t}

where x_{t} is F_{t}-measurable, and taking conditional expectation w.r.t. F_{t} of both members, we deduce

x_{t} = \frac{E(X11_{\{τ>t\}}|F_{t})}{E(11_{\{τ>t\}}|F_{t})} = 11_{\{τ>t\}}e^{Γ_{t}}E(Xe^{−Γ_{T}} |F_{t}) .

□

The main point is that here, the process Γ is not necessarily increasing.

Lemma 2.5.2 Let h be an F-predictable process. Then,

E(h_{τ} 11_{τ<T} |G_{t}) = h_{τ} 11_{{τ<t}} + 11_{{τ>t}}E(∫_{t}^{T} h_{u}dF_{u}|F_{t})  (2.12)

We are not interested with G predictable processes, mainly because any G predictable process is equal, o, {t ≤ τ} to an F-predictable process.

Lemma 2.5.3 The submartingale F_{t} admits a decomposition as Z + A where Z is a martingale and A a predictable increasing process. In terms of A,

E(h_{τ} 11_{τ<T} |G_{t}) = h_{τ} 11_{{τ<t}} + 11_{{τ>t}}E(∫_{t}^{T} h_{u}dA_{u}|F_{t})

As we shall see, this elementary result will allow us to compute the value of credit derivatives, as soon as some elementary defaultable asset is priced by the market.

Comments 2.5.1 It can be useful to understand the meaning of the lemma in the case where, as in the structural model, the default time is an (F_{t}) stopping time.

Remark 2.5.1 We emphasize that, in the Cox process approach, the enlarged filtration G = F ∨ D is here the filtration which should be taken into account; the filtration generated by F_{t} and σ(Θ) is too large. In the latter filtration, in the case where F is a Brownian filtration, τ would be a predictable stopping time.

2.5.3 H Hypothesis

We discuss now the hypothesis on the modeling of default time that we require in order to avoid arbitrages in the defaultable market.

Proposition 2.5.1 Let S be a semi-martingale on (Ω, G, P) such that there exists a unique probability Q, equivalent to P on F_{T}, where F_{t} = F_{t}^{S} = σ(S_{s}, s ≤ t) such that (S̃_{t} = S_{t}R_{t}, 0 ≤ t ≤ T) is an F^{S}-martingale under the probability Q. We assume that there exists a probability Q̃, equivalent to P on G_{T} such that (S̃_{t}, 0 ≤ t ≤ T) is a G-martingale under the probability Q̃. Then, (H) holds under Q and the restriction of Q̃ to F_{T} is equal Q.

Proof: We give a ”financial proof”. Under the hypothesis, any square integrable F−Q martingale can be thought as the discounted value of a contingent claim ξ ∈ F_{T}. Since the same claim exists in the larger market, which is assumed to be arbitrage free, the claim process is also a G − Q̃ martingale. From the uniqueness of price for hedgeable claims, for any contingent claim X ∈ F_{T} and any G-e.m.m. Q̃,

E_{Q}(XR_{T} |F_{t}) = EQ̃(XR_{T} |G_{t}) .

In particular, E_{Q}(Z) = EQ̃(Z) for any Z ∈ F_{T} ( take t = 0 and X = ZR_{T}^{−1}), hence the restriction of any e.m.m. Q̃ to the σ-algebra F_{T} equals Q. Moreover, since any square integrable F-Q-martingale can be written as E_{Q}(X|F_{t}) = EQ̃(X|G_{t}), we get that any square integrable F-Q̃-martingale is a G-Q̃-martingale. □

Comments 2.5.2 In the literature, it is generally assumed that the defaultable market is complete and arbitrage free. If this assumption means that the set of contingent claims is the set of G_{T} -measurable random variable, then, in particular, any F_{T} -measurable random variable is a tradeable contingent claim, and S_{T} is a tradeable asset.

We shall now examine the hypothesis (H) which reads:

(H) Every F square-integrable martingale is a G square-integrable martingale.

This hypothesis implies that the F-Brownian motion remains a Brownian motion in the enlarged filtration. It was studied by Brémaud and Yor [10] and Mazziotto and Szpirglas [53], and for financial purpose by Kusuoka [47]. This hypothesis is equivalent to:

(H*) For any t, the σ-fields F_{∞} and G_{t} are conditionally independent given F_{t}.

This can be written in any of the equivalent forms (see, e.g. Dellacherie and Meyer [18]) :

(H1) ∀ F ∈ F_{∞}, ∀ G_{t} ∈ G_{t}, E(F G_{t}|F_{t}) = E(F |F_{t}) E(G_{t}|F_{t})

(H2) ∀ t ≥ 0, ∀ G_{t} ∈ G_{t}, E(G_{t}|F_{∞}) = E(G_{t}|F_{t})

(H3) ∀ t ≥ 0, ∀ F ∈ F_{∞}, E(F |G_{t}) = E(F |F_{t}).

Exercise 2.5.1 Prove the equivalence between (H_{i}).

Lemma 2.5.4 In our setting, (H) is equivalent to one of the equivalent following conditions (H′)

∀ s ≤ t, P (τ ≤ s|F_{∞}) = P (τ ≤ s|F_{t}).  (2.13)
∀ t, P (τ ≤ t∥F_{∞}) = P (τ ≤ t∥F_{t}).  (2.14)

Proof: If (H2) holds, then (2.13) holds too. If (2.13) holds, the fact that D_{t} is generated by the sets {τ ≤ s}, s ≤ t proves that F_{∞} and D_{t} are conditionally independent given F_{t}. The property follows. This result can be also found in [19]. The equivalence between (2.13) and (2.14) is left to the reader. □

Remark 2.5.2 (i) Equality (2.14) appears in several papers on default risk, typically without any reference to the (H) hypothesis. For example, in the Madan-Unal paper [52], the main theorem follows from the fact that (2.14) holds (See the proof of B9 in the appendix of their paper). This is also the case for Wong’s model [61]. (ii) If τ is F_{∞}-measurable, and if (2.14) holds, then τ is an F-stopping time. If τ is a F-stopping time, equality (2.13) holds. If F is the Brownian filtration, τ is predictable and Λ = D. (iii) Though condition (H) does not necessarily hold true, in general, it is satisfied when τ is constructed through a standard approach (See Cox processes). This hypothesis is quite natural under the historical probability, and is stable under some change of measure. However, Kusuoka provides an example where (H) holds under the historical probability and does not hold after a change of probability. This counter example is linked with dependency between default of different firms. (iv) Hypothesis (H) holds in particular if τ is independent from F_{∞}. See Greenfield thesis. [34].

We reduce our attention to the case where

∀t, P (τ ≤ t|F_{t}) = P (τ ≤ t|F_{∞}) .

In that case F is an increasing process.

Comments 2.5.3 See Elliott et al. [27] for more comments. It is worthwhile to note that this hypothesis does not hold in the example 2.7.2

2.5.4 Stochastic Barrier

Suppose that

P (τ ≤ t|F_{∞}) = 1 − e^{−Γ_{t}}

where Γ is an arbitrary continuous strictly increasing F-adapted process. Our goal is to show that there exists a random variable Θ, independent of F_{∞}, with exponential law of parameter 1, such that τ =^{law} inf {t ≥ 0 : Γ_{t} > Θ}. Let us set Θ =^{def} Γ_{τ} . Then

{t < Θ} = {t < Γ_{τ} } = {C_{t} < τ},

where C is the right inverse of Γ, so that Γ_{C_{t}} = t. Therefore

P (Θ > u|F_{∞}) = e^{−Γ_{Cu}} = e^{−u}.

We have thus established the required properties, namely, the probability law of Θ and its independence of the σ-field F_{∞}. Furthermore, τ = inf{t : Γ_{t} > Γ_{τ} } = inf{t : Γ_{t} > Θ}.

2.5.5 Representation theorem

Kusuoka [47] establishes the following representation theorem.

Theorem 2.5.1 Under (H), any G-square integrable martingale admits a representation as the sum of a stochastic integral with respect to the Brownian motion and a stochastic integral with respect to the discontinuous martingale M.

We assume for simplicity that F is continuous and F_{t} < 1, ∀t ∈ IR^{+}. Since (H) hypothesis holds, F is an increasing process. Then,

dF_{t} = e^{−Γ_{t}}dΓ_{t}

and

d(e^{Γ_{t}}) = e^{Γ_{t}}dΓ_{t} = e^{Γ_{t}} \frac{dF_{t}}{1 − F_{t}} .  (2.15)

Proposition 2.5.2 Suppose that hypothesis (H) holds under P and that any F-martingale is continuous. Then, the martingale H_{t} = E_{P} (h_{τ} | G_{t}) , where h is an F-predictable process such that E(h_{τ}) < ∞, admits the following decomposition as the sum of a continuous martingale and a discontinuous martingale

H_{t} = m_{0}^{h} + ∫_{0}^{t∧τ} e^{Γ_{u}}dm_{u}^{h} + ∫_{]0,t∧τ]}(h_{u} − J_{u}) dM_{u},  (2.16)

where m^{h} is the continuous F-martingale

m_{t}^{h} = E_{P} ( ∫_{0}^{∞} h_{u}dF_{u} | F_{t}) ,

J_{t} = e^{Γ_{t}}(m_{t}^{h} − ∫_{0}^{t} h_{u}dF_{u}) and M is the discontinuous G-martingale M_{t} = D_{t} − Λ_{t∧τ} where

dΛ_{u} = \frac{dF_{u}}{1 − F_{u}}.

Proof: From (2.9) we know that

H_{t} = E(h_{τ} | G_{t}) = 11_{{τ≤t}}h_{τ} + 11_{{τ>t}}e^{Γ_{t}}E( ∫_{t}^{∞} h_{u}dF_{u} | F_{t}) = 11_{{τ≤t}}h_{τ} + 11_{{τ>t}}J_{t} . (2.17)

From the facts that Γ is an increasing process and m^{h} a continuous martingale, and using the integration by parts formula, we deduce that

dJ_{t} = e^{Γ_{t}}dm_{t}^{h} + J_{t}e^{−Γ_{t}} d(e^{Γ_{t}}) − h_{t}(e^{Γ_{t}}dF_{t}) .

Therefore, from (2.15)

dJ_{t} = e^{Γ_{t}}dm_{t}^{h} + (J_{t} − h_{t})\frac{dF_{t}}{1 − F_{t}}

or, in an integrated form,

J_{t} = m_{0} + ∫_{0}^{t} e^{Γ_{u}}dm_{u}^{h} + ∫_{0}^{t}(J_{u} − h_{u})dΛ_{u}.

Note that J_{u} = H_{u} for u < τ. Therefore, on {t < τ}

H_{t} = m_{0}^{h} + ∫_{0}^{t∧τ} e^{Γ_{u}}dm_{u}^{h} + ∫_{0}^{t∧τ} (J_{u} − h_{u})dΛ_{u}

From (2.17), the jump of H at time τ is h_{τ} − J_{τ} = h_{τ} − H_{τ−}. Then, (2.16) follows. □

Remark 2.5.3 Since hypothesis (H) holds, the processes (m_{t}, t ≥ 0) and (∫_{0}^{t∧τ} e^{Γ_{u}}dm_{u}, t ≥ 0) are also G-martingales.

2.5.6 Hedging contingent claims

We suppose that the default-free market including the default-free zero-coupon and the risky asset S is complete and arbitrage free, we denote by Q the unique F_{T} -e.m.m. For any process V, we denote by Ṽ the discounted value of V, i.e., Ṽ_{t} = R_{t}V_{t}.

We assume that a defaultable zero-coupon, with maturity T and price ρ(t, T) is available on the market and that (H) holds under Q. We also assume that the process F is continuous.

We denote by Q̃ a G_{T} -e.m.m. We recall that Q̃ and Q are equal on the σ-algebra F_{T}. We now make precise the hedging of a defaultable claim and check that the G_{T} -market where the risky asset S and a DZC with maturity T and price ρ(t, T) is complete.

The market price of the DZC and the e.m.m. Q̃ are related by

ρ(t, T)R_{t} = EQ̃(R_{T} 11_{T <τ} |G_{t}) = 11_{t<τ} e^{Γ_{t}}EQ̃(R_{T} e^{−Γ_{T}} |G_{t}) = L_{t}m_{t}

where L_{t} = 11_{t<τ} e^{Γ_{t}}. We recall that a triple (a, b, c) of G-predictable processes is an hedging strategy for the contingent claim Z_{T} ∈ G_{T} if, denoting by Z_{t} = a_{t}S_{t}^{0} + b_{t}S_{t} + c_{t}ρ(t, T) the value of this strategy, the self-financing relation dZ_{t} = a_{t}dS_{t}^{0} + b_{t}dS_{t} + c_{t}dρ(t, T) holds and Z_{T} = a_{T} S_{T}^{0} + b_{T} S_{T} + c_{T} ρ(T, T). From no-arbitrage hypothesis,

Z_{t}R_{t} = EQ̃(Z_{T} R_{T} |G_{t}) .

Terminal payoff

We study in a first step the case where there is no rebate, i.e., we consider terminal payoff of the form Z_{T} = X11_{T <τ}. We compute EQ̃(X11_{T <τ} R_{T} |G_{t}) by mean of our representation theorem, and we give the hedging strategy for X11_{T <τ} based on riskless asset, risky asset and defaultable zero-coupon.

Theorem 2.5.2 The hedging strategy (a, b, c) for the defaultable contingent claim X11_{T <τ}, based on the riskless bond, the asset and the defaultable zero-coupon satisfies

c_{t}ρ(t, T) = R_{t}^{−1}EQ̃(XR_{T} 11_{T <τ} |G_{t}) .

Hence, a_{t}S_{t}^{0} + b_{t}S_{t} = 0. More precisely, let (Ṽ_{t}^{X} − v_{t}^{X} S̃_{t}, v_{t}^{X}) be the hedging strategy for the default-free contingent claim XG_{T}, and (Ṽ_{t} − v_{t}S̃_{t}, v_{t}) the hedging strategy for the default-free contingent claim G_{T}, i.e.

Ṽ_{t}^{X} = R_{t}V_{t}^{X} = E_{Q}(XG_{T} R_{T} |F_{t}) = x + ∫_{0}^{t} v_{s}^{X}dS̃_{s}
Ṽ_{t} = R_{t}V_{t} = E_{Q}(G_{T} R_{T} |F_{t}) = x + ∫_{0}^{t} v_{s}^{X}dS̃_{s}

Then, on t < τ

(i) c_{t} = \frac{V_{t}^{X}}{V_{t}}

(ii) b_{t} = e^{Γ_{t}}(v_{t}^{X} − \frac{V_{t}^{X}}{V_{t}} v_{t})

(iii) a_{t} = −e^{Γ_{t}}(v_{t}^{X} − \frac{V_{t}^{X}}{V_{t}} v_{t})S̃_{t} .

Proof: Let us give two different proofs. First proof Let us give a first proof in the simple case r = 0. The price of the defaultable claim X11_{T <τ} is Z_{t} defined by

Z_{t} = EQ̃(X11_{T <τ} |G_{t}) = 11_{t<τ} e^{Γ_{t}^{Q}}E_{Q}(Xe^{−Γ_{T}^{Q}}|F_{t})
= 11_{t<τ} e^{Γ_{t}^{Q}}m_{t}^{X} = L_{t}m_{t}^{X}

and the price of the DZC satisfies

ρ(t, T) = 11_{t<τ} e^{Γ_{t}^{Q}}E_{Q}(e^{−Γ_{T}^{Q}}|F_{t}) = 11_{t<τ} e^{Γ_{t}^{Q}}m_{t}

Hence

Z_{t} = EQ̃(X11_{T <τ} |G_{t}) = \frac{m_{t}^{X}}{m_{t}} ρ(t, T)

We now check that there exists a triple (a, b, c) determining a self financing portfolio such that a_{t}S_{t}^{0} + b_{t}S_{t} = 0. The self financing condition reads

b_{t}S_{t}σdW̃_{t} + c_{t}(L_{t−}dm_{t} + m_{t}dL_{t} = dZ_{t} = Ldm_{t}^{X} + m_{t}^{X}dL_{t}

From the choice of c, this equality reduces to

b_{t}S_{t}σdW̃_{t} + c_{t}L_{t−}dm_{t} = Ldm_{t}^{X}

Hence the form of b given in the theorem. Second proof

We now use the representation theorem as follows. Let G_{t} = Q̃(t < τ|F_{t}) where Q̃ is the G e.m.m.. Any contingent claim in F_{T} is hedgeable in the default-free market, in particular Xe^{Γ_{T}}; therefore there exists a predictable process (v_{t}^{X}, t ≥ 0) and a constant x such that

XG_{T} R_{T} = x + ∫_{0}^{T} v_{s}^{X}dS̃_{s} .

Hence, the t-time price V_{t}^{X} of XG_{T} is given via

Ṽ_{t}^{X} = V_{t}^{X}R_{t} = EQ̃(XG_{T} R_{T} |F_{t}) = x + ∫_{0}^{t} v_{s}^{X}dS̃_{s} .

The strategy (Ṽ_{t}^{X} −v_{t}^{X} S̃_{t}, v_{t}^{X}) is now a self-financing strategy hedging the contingent claim XG_{T}. From the representation theorem 2.5.2 applied to

H̃_{t} = EQ̃(XR_{T} 11_{T <τ} |G_{t}) = (1 − D_{t})e^{Γ_{t}}E_{Q}(XR_{T} G_{T} |F_{t}) = L_{t}Ṽ_{t}^{X} ,

we obtain

XR_{T} 11_{T <τ} = V_{0}^{X} + ∫_{0}^{T ∧τ} e^{Γ_{u}}dṼ_{u}^{X} − ∫_{]0,T ∧τ]} Ṽ_{u}^{X}e^{Γ_{u}}dM_{u} .

Let us denote Ṽ_{t} = E_{Q}(R_{T} G_{T} |F_{t}) = Ṽ_{t}^{1} the discounted price of G_{T} and v_{t} = v_{t}^{1} its hedging portfolio. Then, using that dρ̃_{t} − L_{t−}dṼ_{t} = −L_{t}Ṽ_{t}dM_{t},

XR_{T} 11_{T <τ} = V_{0}^{X} + ∫_{0}^{T ∧τ} e^{Γ_{s}}(v_{s}^{X} − \frac{Ṽ_{s}^{X}}{Ṽ_{s}} v_{s})dS̃_{s} + ∫_{]0,T ∧τ]} \frac{Ṽ_{s}^{X}}{Ṽ_{s}} dρ̃_{s}

= V_{0}^{X} + ∫_{0}^{T ∧τ} e^{Γ_{s}}(v_{s}^{X} − \frac{V_{s}^{X}}{V_{s}} v_{s})dS̃_{s} + ∫_{]0,T ∧τ]} \frac{V_{s}^{X}}{V_{s}} dρ̃_{s} .  (2.18)

Now, using that ρ_{t} = L_{t}V_{t}, the position on the riskless asset is the value H_{t} of the portfolio minus the wealth invested in the risky securities, i.e.

L_{t}V_{t}^{X} − \frac{V_{t}^{X}}{V_{t}} ρ_{t} − e^{Γ_{t}}(v_{t}^{X} − \frac{V_{t}^{X}}{V_{t}} v_{t})S_{t} = −e^{Γ_{t}}(v_{t}^{X} − \frac{V_{t}^{X}}{V_{t}} v_{t})S_{t} .

From the construction, this strategy is self-financing. The amount of money invested in the DZC is

\frac{V_{t}^{X}}{V_{t}} ρ(t, T) = L_{t}V_{t}^{X} = H_{t}

that is exactly the opposite of the loss that can occurs for immediate default. □

Remark 2.5.4 It is easy to check that the given pair (a, b) is the unique pair (a, b) such that

(a, b, (ρ(t, T)R_{t})^{−1}E_{Q}(XR_{T} 11_{T <τ} |G_{t})

is self-financing.

Remark 2.5.5 In the particular case where Γ is deterministic, we get immediately

XR_{T} 11_{T <τ} = h + ∫_{0}^{T ∧τ} e^{Γ_{s}}v_{s}^{X}dS̃_{s} + ∫_{]0,T ∧τ]} E_{Q}(XR_{T} |F_{s})dρ̃_{s} .

Rebate part

The representation theorem provides also an hedging strategy for a rebate h paid at hit. We denote by C_{t}^{h} the price of the contingent claim which consists of a dividend hf paid between time t and T, i.e.

C_{t}^{h}R_{t} = C̃_{t}^{h} = E_{Q}(∫_{t}^{T} R_{u}f_{u}h_{u}du|F_{t}) ,

and by μ^{h} the associated hedging strategy in the default-free world, i.e.,

m̃_{t}^{h} = C̃_{t}^{h} + ∫_{0}^{t} R_{u}f_{u}h_{u}du = C_{0}^{h} + ∫_{0}^{t} μ_{s}^{h}dS̃_{s} .

Proposition 2.5.3 Let (a, b, c) be the hedging strategy for the rebate part, i.e., the self-financing strategy such that

R_{t}(a_{t}S_{t}^{0} + b_{t}S_{t} + c_{t}ρ(t, T)) = EQ̃(h_{τ} 11_{τ≤T} R_{τ} |G_{t}) .

Then, c_{t}ρ(t, T) = R_{t}^{−1}EQ̃(h_{τ} 11_{τ≤T} R_{τ} |G_{t}) − h_{t}. More precisely, the hedging strategy before default time of the rebate part, paid at hit, consists of

(i) c_{t} = \frac{1}{V_{t}} (C_{t}^{h} − e^{−Γ_{t}}h_{t})

(ii) b_{t} = e^{Γ_{t}}[v_{t}^{h} − \frac{1}{V_{t}} v_{t}C_{t}^{h}] + \frac{1}{V_{t}} v_{t}h_{t}

(iii) a_{t}S_{t}^{0} = S_{t}[\frac{1}{V_{t}} v_{t}h_{t} − e^{Γ_{t}}[v_{t}^{h} + \frac{1}{V_{t}} v_{t}C_{t}^{h}]] + h_{t}.

We compute the quantity EQ̃(h_{τ} 11_{τ≤T} R_{τ} |G_{t}), which corresponds to the price of the rebate, when the compensation is payed at hit.

We denote by C_{t}^{h} the price of the contingent claim which consists of a dividend hdF paid between time t and T, i.e.

C̃_{t}^{h} = E_{Q}(∫_{t}^{T} R_{u}f_{u}h_{u}du|F_{t}) ,

and by v^{h} the associated hedging strategy in the default-free world, i.e.,

Ṽ_{t}^{h} = C̃_{t}^{h} + ∫_{0}^{t} R_{u}h_{u}dF_{u} = C_{0}^{h} + ∫_{0}^{t} v_{s}^{h}dS̃_{s} .

The representation theorem states that

EQ̃(h_{τ} R_{τ} 11_{τ≤T} |G_{t}) = C_{0}^{h} + ∫_{0}^{t∧τ} e^{Γ_{u}}V_{u}^{h}dS̃_{u} + ∫_{[0,t∧τ[}(h_{u}R_{u} − J_{u−})dM_{u} ,

where, on the set t < τ,

J_{t} = EQ̃(h_{τ} R_{τ} 11_{τ≤T} |G_{t}) = e^{Γ_{t}}E_{Q}(∫_{t}^{T} h_{u}R_{u}dF_{u}|F_{t}) = e^{Γ_{t}} C̃_{t}^{h} .

Hence, introducing Ṽ_{t}, the discounted price of G_{T}

EQ̃ (h_{τ} R_{τ} 11_{τ≤T} |G_{t}) = C_{0}^{h} + ∫_{0}^{t∧τ} e^{Γ_{u}}v_{u}^{h}dS̃_{u} − ∫_{[0,t∧τ[}(h_{u}R_{u} − e^{Γ_{u}} C̃_{u}^{h})\frac{1}{L_{u}Ṽ_{u}} [dρ̃_{u} − L_{u}dṼ_{u}]

which leads to

EQ̃ (h_{τ} R_{τ} 11_{τ≤T} |G_{t}) = V_{0}^{h} + ∫_{0}^{t∧τ} [e^{Γ_{u}} (v_{u}^{h} − C_{u}^{h} \frac{v_{u}^{h}}{V_{u}} ) + \frac{v_{u}h_{u}}{V_{u}} ] dS̃_{u}

− ∫_{[0,t∧τ[}(h_{u} − e^{Γ_{u}}C_{u}^{h})\frac{1}{L_{u}V_{u}} dρ̃_{u}

□

Corollary 2.5.1 Under (H) hypothesis, the defaultable-market is complete as soon as a defaultable zero-coupon is traded.

Comments 2.5.4 See also Bélanger et al. [4] for an extensive study of hedging strategies.

2.5.7 Intensity approach

In the so-called intensity approach, the starting point is the knowledge of default time τ and some filtration G such that τ is a G-stopping time. The intensity is defined as any non-negative process λ, such that

M_{t} =^{def} D_{t} − ∫_{0}^{t∧τ} λ_{s}ds

is a G-martingale. The existence of the intensity relies on the fact that D is an increasing process, therefore a sub-martingale and can be written as a martingale M plus a predictable increasing process A. The increasing process A is such that A_{t}11_{t≥τ} = A_{τ} 11_{t≥τ}. In the case where τ is a predictable stopping time, obviously A = D. The intensity exists only if τ is a totally inaccessible stopping time. Under some additional properties, Duffie and Singleton [23] establishes formulae similar to (2.8). We emphasize that, in that setting the intensity is not well defined after time τ, i.e., if λ is an intensity, for any non-negative predictable process g the process λ̃_{t} = λ_{t}11_{t≤τ} + g_{t}11_{{t>τ}} is also an intensity.

2.6 Correlated defaults

2.6.1 Copula and threshold

Let τ_{i} = inf{t, Λ_{i}(t) ≥ Θ_{i}} where Λ_{i} are F-adapted, increasing processes and Θ_{i} are random variables, independent of F_{∞}, with a survival copula

Ĉ(θ_{1}, …, θ_{n}) = P (F_{i}(Θ_{i}) > θ_{i}, ∀i ≤ n)

Then, the copula of default time is

P (τ_{i} > t_{i}, ∀i ≤ n) = E (Ĉ(G_{1}(t_{1}), …, G_{n}(t_{n})))

where G_{i}(t) = exp (−Λ_{i}(t)). In particular, if τ = inf_{i}(τ_{i}),

E(Y 11_{τ_{i}>T} |G_{t}) = 11_{τ>t}E (Y \frac{Ĉ(G_{1}(t), …, G_{i}(T), … G_{n}(t))}{Ĉ(G_{1}(t), …, G_{i}(t), … G_{n}(t))}|F_{t})

2.6.2 Correlated intensity

Let τ_{i} = inf{t, Λ_{i}(t) ≥ Θ} where Λ_{i} are F_{t}-adapted, increasing processes and Θ is a random variable, independent of F_{∞}. Let τ = τ_{1} ∧ τ_{2}. Then we can compute

P (τ > t|F_{t}) = P (Λ_{1}(t) ∧ Λ_{2}(t) ≥ Θ|F_{t}) = exp (−Λ_{1}(t) ∧ Λ_{2}(t)) .

It is more difficult to compute the law of τ.

2.6.3 Jarrow and Yu

Let τ_{i} = inf{t : Λ_{i}(t) ≥ Θ_{i}} where Λ_{i}(t) = ∫_{0}^{t} λ_{i}(s)ds and Θ are r.v. with exponential law,

independent and independent of F. Jarrow and Yu study the case where λ_{1} is a constant and λ_{2}(t) = λ_{2} + (α_{2} − λ_{2})11_{{τ_{1}≤t}}. Assume for simplicity that r = 0 and compute the value of

B_{i}(t, T) = P (τ_{i} > T + δ_{i}11_{τ_{i}<T} |G_{t}), for G_{t} = D_{t}^{1} ∨ D_{t}^{2}

From the key lemma, it is easy to establish that

B_{1}(t, T) = δ_{1} + 11_{{τ_{1}>t}}(1 − δ_{1})e^{−λ_{1}(T −t)}
B_{2}(t, T) = δ_{2} + (1 − δ_{2}) (11_{{τ_{1}≤t}}e^{−α_{2}(T −t)}

+11_{\{τ_{1}>t\}} \frac{1}{λ_{1} + λ_{2} − α_{2}} [(λ_{2} − α_{2})e^{−(λ_{1}+λ_{2})(T −t)} + λ_{1}e^{−α_{2}(T −t)}])

Indeed, the computation of B_{1} reduces (thanks to the key lemma) to

P (τ_{1} > T |G_{t}) = P (τ_{1} > T |F_{t} ∨ D_{t}^{1})

where F_{t} = D_{t}^{2}. Hence

P (τ_{1} > T |F_{t} ∨ D_{t}^{1}) = 11_{t<τ_{1}} \frac{P (τ_{1} > T |F_{t})}{P (τ_{1} > t|F_{t})}

The set τ_{1} > t is independent of τ_{2}, and the result follows using that τ_{1} has an exponential law. For the computation of B_{2}, we note that

P (τ_{2} > T |G_{t}) = P (τ_{2} > T |F_{t} ∨ D_{t}^{2})

where now F_{t} = D_{t}^{1}. Hence, from the key lemma,

P (τ_{2} > T |F_{t} ∨ D_{t}^{2}) = 11_{t<τ_{2}} \frac{P (τ_{2} > T |F_{t})}{P (τ_{2} > t|F_{t})}

Now,

P (τ_{2} > T |D_{t}^{1}) = 11_{t<τ_{1}} \frac{P (τ_{2} > T, t < τ_{1})}{P (τ_{1} > t)} + 11_{τ_{1}<t}P (τ_{2} > T |τ_{1})

The computation of P (τ_{2} > T, t < τ_{1}) follows from

P (τ_{2} > T, t < τ_{1}) = P (τ_{2} > T) − P (τ_{2} > T > t > τ_{1})
P (τ_{2} > T > t > τ_{1}) = ∫_{0}^{t} P (τ_{2} > T, τ_{1} ∈ du)
= ∫_{0}^{t} P (∫_{0}^{T} λ_{2}(s)ds < Θ_{2}, τ_{1} ∈ du)
= ∫_{0}^{t} P (∫_{0}^{u} λ_{2}(s)ds + ∫_{u}^{T} λ_{2}(s)ds < Θ_{2}, τ_{1} ∈ du)
= ∫_{0}^{t} P (λ_{2}u + α_{2}(T − u) < Θ_{2})P (τ_{1} ∈ du)

and the computation of P (τ_{2} > T |τ_{1}) follows from

P (τ_{2} > T |τ_{1} = u) = P (∫_{0}^{T} λ_{2}(s)ds < Θ_{2}|τ_{1} = u)
= P (λ_{2}u + α_{2}(T − u) < Θ_{2})

2.6.4 Kusuoka exemple

Kusuoka assumes that the default times τ_{i}, i = 1, 2 have joint density

f(x, y) = λ_{1}λ_{2}e^{−(λ_{1}x+λ_{2}y)}11_{{x≥0,y≥0}}

under P. He defines Q such that

dQ|_{F_{T}} = η_{T} dP |_{F_{T}}

where

dη_{t} = η_{t−}(κ_{t}^{1}dM_{t}^{1} + κ_{t}^{2}dM_{t}^{2})

with

M_{t}^{i} = D_{t}^{i} − ∫_{0}^{t∧τ_{i}} λ_{i}ds

and

κ_{t}^{i} = 11_{τ_{j}≤t}(\frac{α_{i}}{λ_{i}} − 1), fori ≠ j, i, j = 1, 2

The intensity of τ_{i} under P are λ_{i}, whereas they are

λ_{t}^{1} = λ_{1}11_{{τ_{2}>t}} + α_{1}11_{{τ_{2}≤t}}
λ_{t}^{2} = λ_{2}11_{{τ_{1}>t}} + α_{2}11_{{τ_{1}≤t}}

under Q. The two default times are no more independent under Q. The main problem is that the (H) hypothesis does not holds between D^{2} and D^{1} ∨ D^{2}, and therefore

Q(τ_{1} > T |D_{t}^{1} ∨ D_{t}^{2}) ≠ 11_{{τ_{1}>t}}E_{Q}(exp(− ∫_{t}^{T} λ_{u}^{1}du|D_{t}^{2}).

2.7 Extensions

2.7.1 Explosive defaults

Andreasen states that the intensity process is λ_{t} = 1/X_{t} where

dX_{t} = (θ − k(t)X_{t})dt + σ\sqrt{X_{t}}dW_{t}

before the first hitting time of 0, and assumes that X remains at 0 after hitting this level. The filtration of observation is F_{t} = F_{t}^{X}. The survival probability follows

P (τ > T |G_{t}) = 11_{t<τ} E (exp (− ∫_{t}^{T} λ_{s}ds) |F_{t})

Under some choice of parameter, the process X reaches 0 for a finite time, therefore, the intensity exploses at a predictable stopping time.

2.7.2 Partial observation

The previous problem admits an obvious extension to the case where the observation is F_{t} = σ(V_{t_{i}}, t_{i} ≤ [t]) where [t] is the greatest integer smaller than t, which corresponds to observation at discrete times t_{1}, …, t_{i}.

Let us study ([31]) the case where

dV_{t} = V_{t}(μdt + σdW_{t})

i.e., V_{t} = ve^{σ(W_{t}+νt)}, with ν = \frac{1}{σ} (μ − \frac{σ^{2}}{2} ). The default time is the hitting time of α with α < v, i.e.,

τ = inf{t : V_{t} ≤ α} = inf{t : X_{t} ≤ a}

where a = \frac{1}{σ} ln(α/v). We study the case where F_{t} is the filtration of the observations of V at discrete times t_{1}, ⋯ t_{n} and we compute F_{t} = P (τ ≤ t|F_{t}).

Case t < t_{1}

In that case, F_{t} is the cumulative function of τ. Since a < 0, we apply the results on hitting times of a drifted Brownian motion and obtain

F_{t} = P (τ ≤ t) = P (inf_{s≤t} X_{s} ≤ a)
= N (\frac{a − νt}{\sqrt{t}}) + e^{2νa}N (\frac{a + νt}{\sqrt{t}})

Case t_{1} < t < t_{2}

We denote by H_{t} = σ(W_{s}, s ≤ t) the natural filtration of the Brownian motion (this is also the natural filtration of X)

F_{t} = P (τ ≤ t|X_{t_{1}}) = 1 − P (τ < t|X_{t_{1}})
= P ( inf_{s<t_{1}} X_{s} > a P ( inf_{t_{1}≤s<t} X_{s} > a|H_{t_{1}})|X_{t_{1}})

Markov’s property implies that

P ( inf_{t_{1}≤s<t} X_{s} > a|H_{t_{1}}) = Φ(X_{t_{1}}) = P ( inf_{t_{1}≤s<t}(X_{s} − X_{t_{1}} > a − X_{t_{1}}|X_{t_{1}})

Independence and stationarity of the increments of X yield to

Φ(x) = P ( inf_{s<t−t_{1}} X_{s} > a − x)

Hence

F_{t} = Φ(X_{t_{1}})P ( inf_{s<t_{1}} X_{s} > a|X_{t_{1}})

The second term in the LHS is the cumulative function of the infimum of a drifted Brownian motion, and is independent of the drift. Let us check that:

Lemma 2.7.1 Let X_{t} = W_{t} + νt et m_{t}^{X} = inf_{s≤t} X_{s}. Then for y < 0, y < x

P (m_{t}^{X} ≤ y|X_{t} = x) = exp (−\frac{2}{t} (y(y − x))) .

2.7. EXTENSIONS 39 Proof: Differentiating with respect to x the equality

P (X_{t} ≤ x, m_{t}^{X} ≤ y) = e^{2νy}N (\frac{−x + 2y + νt}{\sqrt{t}}) ,

(valid for y ≤ 0, y ≤ x ) we get

P (X_{t} ∈ dx, m_{t}^{X} ≤ y) = \frac{1}{\sqrt{2πt}}e^{2νy} exp (−\frac{1}{2} (\frac{−x + 2y + νt}{\sqrt{t}})^{2}) .

From definition of X,

P (X_{t} ∈ dx) = \frac{1}{\sqrt{2πt}} exp (−\frac{1}{2} (\frac{x − νt}{\sqrt{t}})^{2})

Using that

2νy − \frac{1}{2} (\frac{−x + 2y + νt}{\sqrt{t}})^{2} + \frac{1}{2} (\frac{x − νt}{\sqrt{t}})^{2} = 2νy − \frac{1}{2t} (4y^{2} + 4y(−x + νt)) = −\frac{2}{t} (y^{2} − yx)

we get the result.

□

Therefore, for X_{t_{1}} > a

F_{t} = Φ(X_{t_{1}}) (1 − exp((−\frac{2}{t_{1}} (a(a − X_{t_{1}})))  (2.19)

et for X_{t_{1}} < a; F_{t} = 0.

When t goes to t_{1}, we obtain, for X_{t_{1}} > a, F_{t_{1}^{+}} = exp [−\frac{2}{t_{1}} a(a − X_{t_{1}})] The jump of F at time t_{1} is, for X_{t_{1}} > a

ΔF_{t_{1}} = exp [−\frac{2}{t_{1}} a(a − X_{t_{1}})] − N (\frac{a − νt_{1}}{\sqrt{t_{1}}} ) − e^{2νa}N (\frac{a + ν_{1}t}{\sqrt{t_{1}}})

and; for X_{t_{1}} < a

ΔF_{t_{1}} = −N (\frac{a − νt_{1}}{\sqrt{t_{1}}} ) − e^{2νa}N (\frac{a + νt_{1}}{\sqrt{t_{1}}} )

Let us denote

G(t, a) = P (m_{t}^{X} > a) = N (\frac{−a + νt}{\sqrt{t}}) − e^{2νa}N (\frac{a + νt}{\sqrt{t}})

Then, for Z ∈ F_{t_{2}}

E(Z11_{T <τ} |G_{t}) = 11_{t<τ} \frac{G(T, a)}{G(t, a)}E(Z) for T < t_{1}
= 11_{t<τ} E [\frac{G(T − t_{1}, a − X_{t_{1}})}{G(t, a)} (1 − exp (−\frac{2}{t_{1}} (a^{2} − aX_{t_{1}}))) Z] for t < t_{1} < T < t_{2}
= 11_{t<τ} \frac{G(T − t_{1}, a − X_{t_{1}})}{G(t − t_{1}, a − X_{t_{1}})}E(Z|X_{t_{1}}) for t_{1} < t < T < t_{2}

It follows that, in the case r = 0

B_{d}(t, T) = Q(T < τ|G_{t}) = 11_{t<τ} \frac{G(T, a)}{G(t, a)} for T < t_{1}
= 11_{t<τ} E_{Q} [\frac{G(T − t_{1}, a − X_{t_{1}})}{G(t, a)} (1 − exp (−\frac{2}{t_{1}} (a^{2} − aX_{t_{1}})))] for t < t_{1} < T < t_{2}
= 11_{t<τ} \frac{G(T − t_{1}, a − X_{t_{1}})}{G(t − t_{1}, a − X_{t_{1}})} for t_{1} < t < T < t_{2}

40 CHAPTER 2. HAZARD PROCESS APPROACH

2.7.3 Noisy information

It is also possible, as in Duffie and Lando, to assume that the observation at time [t] is only V_{[t]} +ϵ where ϵ is a noise, modeled as a random variable independent of V.

Appendix A : Hitting times

In this chapter, a Brownian motion (W_{t}, t ≥ 0) starting from 0 is given on a probability space (Ω, F, P), and F = (F_{t}, t ≥ 0) is its natural filtration. The function N (x) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{x} e^{−u^{2}/2}du is the cumulative function of the standard Gaussian law.

1.1 Hitting times of a level and law of the maximum for

Brownian motion

Let us study the law of the pair of random variables (W_{t}, M_{t}) where M is the maximum process of the Brownian motion, i.e., M_{t} =^{def} sup_{s≤t} W_{s}. The law of hitting times of a given level by the Brownian motion will be obtained.

1.1.1 Law of the pair of the random variables (W_{t}, M_{t})

Let us remark that the process M is an increasing process, with non negative values. Theorem 1.1.1 Let W be a Brownian motion starting from 0 and M_{t} = sup (W_{s}, 0 ≤ s ≤ t).

\{ \begin{matrix}for y ≥ 0, x ≤ y & P (W_{t} ≤ x, M_{t} ≤ y) & = & N (\frac{x}{\sqrt{t}}) − N (\frac{x − 2y}{\sqrt{t}}), \\ for y ≥ 0, x ≥ y & P (W_{t} ≤ x, M_{t} ≤ y) & = & P (M_{t} ≤ y) = N (\frac{y}{\sqrt{t}}) − N (\frac{−y}{\sqrt{t}}), \\ for y ≤ 0 & P (W_{t} ≤ x, M_{t} ≤ y) & = & 0 . \end{matrix} (1.1)

P (W_{t} ∈ dx, M_{t} ∈ dy) = 11_{y≥0}11_{x≤y}\sqrt{\frac{2}{πt^{3}}} (2y − x) exp (−\frac{(x − 2y)^{2}}{2t} ) .  (1.2)

Proof: • \underline{Reflexion Principle} Let us show that for 0 ≤ y, x ≤ y :

P (W_{t} ≤ x , M_{t} ≥ y) = P (W_{t} ≥ 2y − x) .  (1.3)

It will follow that

P (W_{t} ≤ x , M_{t} ≤ y) = P (W_{t} ≤ x) − P (W_{t} ≤ x , M_{t} ≥ y) = P (W_{t} ≤ x) − P (W_{t} ≥ 2y − x) , (1.4)

hence the first equality.

Let T_{y} = inf{t : W_{t} ≥ y} be the first time where the Brownian motion is greater than y. This is a stopping time in the filtration F and (T_{y} ≤ t) = (M_{t} ≥ y) for y ≥ 0. Furthermore, from y ≥ 0 and by relying on the continuity of Brownian motion paths, the stopping time T_{y} is also the hitting time of y, i.e., T_{y} = inf{t : W_{t} = y} and W_{T_{y}} = y. Therefore

P (W_{t} ≤ x , M_{t} ≥ y) = P (W_{t} ≤ x , T_{y} ≤ t) = P (W_{t} − W_{T_{y}} ≤ x − y , T_{y} ≤ t) .

For the sake of simplicity, we denote E_{P} (11_{A}|T_{y}) = P (A|T_{y}). By relying on the strong Markov property, we obtain

P (W_{t} − W_{T_{y}} ≤ x − y , T_{y} ≤ t) = E(11_{T_{y}≤t} P (W_{t} − W_{T_{y}} ≤ x − y |T_{y})) = E(11_{T_{y}≤t} Φ(T_{y}))

with Φ(u) = P (W̃_{t−u} ≤ x − y) where (W̃_{u} =^{def} W_{T_{y}+u} − W_{T_{y}}, u ≥ 0) is a Brownian motion independent of (W_{t}, t ≤ T_{y}), and with the same law as −W̃. Therefore Φ(u) = P (W̃_{t−u} ≥ y − x) and by proceeding backward:

E(11_{T_{y}≤t} Φ(T_{y})) = E[11_{T_{y}≤t}P (W_{t} − W_{T_{y}} ≥ y − x |T_{y})] = P (W_{t} ≥ 2y − x , T_{y} ≤ t).

Hence

P (W_{t} ≤ x, M_{t} ≥ y) = P (W_{t} ≥ 2y − x, M_{t} ≥ y)  (1.5)

The right-hand side of (1.5) is equal to P (W_{t} ≥ 2y − x) since 2y − x ≥ y which implies that, on the set {W_{t} ≥ 2y − x}, the hitting time T_{y} is smaller than t (or M_{t} ≥ y.)

• For {0 ≤ y ≤ x},

P (W_{t} ≤ x, M_{t} ≤ y) = P (W_{t} ≤ y, M_{t} ≤ y)

since M_{t} ≥ W_{t}. Furthermore, by setting x = y in (1.4)

P (M_{t} ≤ y) = P (W_{t} ≤ y, M_{t} ≤ y) = N (\frac{y}{\sqrt{t}}) − N (\frac{−y}{\sqrt{t}} ) .  (1.6)

• Finally, for y ≤ 0, P (W_{t} ≤ x, M_{t} ≤ y) = 0 since M_{t} ≥ M_{0} = 0. □

1.1.2 Law of the supremum

Proposition 1.1.1 The random variable M_{t} has the same law as |W_{t}|.

Proof: This follows from (1.6). In a different way, for x ≥ 0 :

P (M_{t} ≥ x) = P (M_{t} ≥ x, W_{t} ≥ x) + P (M_{t} ≥ x, W_{t} ≤ x) ,

therefore, from (1.3) (with x = y ) we obtain, using again that W_{t} =^{law} −W_{t},

P (M_{t} ≥ x) = P (W_{t} ≥ x) + P (W_{t} ≥ x) = P (W_{t} ≥ x) + P (W_{t} ≤ −x) = P (|W_{t}| ≥ x) .
□

1.1.3 Law of the hitting time

For x > 0, the law of T_{x} = inf{s : W_{s} ≥ x} can be now easily deduced from

P (T_{x} ≤ t) = P (x ≤ M_{t}) = P (x ≤ |W_{t}|) = P (x ≤ |G| \sqrt{t}) = P (\frac{x^{2}}{G^{2}} ≤ t) ,  (1.7)

1.1. HITTING TIMES OF A LEVEL AND LAW OF THE MAXIMUM FOR BROWNIAN MOTION43

where, as usual G stands for a Gaussian random variable, with zero expectation and unit variance.

Hence, T_{x} =^{law} \frac{x^{2}}{G^{2}} and the density of T_{x} follows:

P (T_{x} ∈ dt) = \frac{x}{\sqrt{2πt^{3}}} exp(−\frac{x^{2}}{2t})11_{t≥0} dt .

For x < 0, we have, using the symmetry of the BM

T_{x} = inf{t : W_{t} ≤ x} = inf{t : W_{t} = x} =^{law} T_{−x}

and

P (T_{x} ∈ dt) = \frac{|x|}{\sqrt{2πt^{3}}} exp(−\frac{x^{2}}{2t})11_{t≥0} dt .  (1.8)

Comments 1.1.1 The process M does not have the same law as the process |W |. Indeed, the process M is an increasing process, this is not the case for |W |. Nevertheless, there are some further equalities in law, as M − W =^{law} |W | between these processes.

1.1.4 Law of the infimum

The law of the infimum of a Brownian motion is obtained by relying on the same procedure. It can also be deduced by observing that

m_{t} =^{def} inf_{s≤t} W_{s} = − sup_{s≤t}(−W_{s}) = − sup_{s≤t}(B_{s})

where B = −W is a Brownian motion. Hence

for y ≤ 0, x ≥ y  P (W_{t} ≥ x, m_{t} ≥ y) = N (\frac{−x}{\sqrt{t}}) − N (\frac{2y − x}{\sqrt{t}}),
for y ≤ 0, x ≤ y  P (W_{t} ≥ x, m_{t} ≥ y) = N (\frac{−y}{\sqrt{t}}) − N (\frac{y}{\sqrt{t}}) ,
for y ≥ 0  P (W_{t} ≥ x, m_{t} ≥ y) = 0 .

(1.9)

In particular, P (m_{t} ≥ y) = N (\frac{−y}{\sqrt{t}}) − N (\frac{y}{\sqrt{t}}). As an immediate consequence, we obtain that, for

x > 0 and y > 0,

P_{x}(W_{t} ∈ dy, T_{0} > t) = P_{0}(W_{t} + x ∈ dy, T_{−x} > t) = P_{0}(W_{t} + x ∈ dy, m_{t} > −x)
= \frac{1}{\sqrt{2πt}} [exp (−\frac{(x − y)^{2}}{2t} ) − exp (−\frac{(x + y)^{2}}{2t} )] dx dy . (1.10)

1.1.5 Laplace transform of the hitting time

We have recalled that, for any λ > 0 the process (exp(λW_{t} − \frac{λ^{2}}{2} t), t ≥ 0) is a martingale. Let y ≥ 0, λ ≥ 0 and T_{y} be the hitting time of y. The martingale

(exp(λW_{t∧T_{y}} − \frac{λ^{2}}{2} (t ∧ T_{y})), t ≥ 0)

is bounded by e^{λy}. Doob’s optional sampling theorem yields

E[exp(λW_{T_{y}} − \frac{λ^{2}}{2} T_{y})] = 1 .

The case where y < 0 is obtained by studying the Brownian motion −W.

Warning 1 In order to apply Doob’s optional sampling theorem, we have to check carefully that

the martingale exp(λW_{t∧T_{y}} − \frac{λ^{2}}{2} (t ∧ T_{y})) is uniformely integrable. In the case λ > 0 and y < 0, a wrong use of this theorem would lead to the equality between 1 and

E[exp(λW_{T_{y}} − \frac{λ^{2}}{2} T_{y})] = e^{λy}E[exp(−\frac{λ^{2}}{2} T_{y})]

that is between E[exp(−\frac{λ^{2}}{2} T_{y})] and exp(−yλ) This is obvioulsly wrong since the quantity E[exp(−\frac{λ^{2}}{2} T_{y})] is smaller than 1 whereas exp(−yλ) is strictly greater than 1.

Proposition 1.1.2 Let T_{y} be the hitting time of y ∈ IR for a standard Brownian motion. Then, for λ > 0

E[exp(−\frac{λ^{2}}{2} T_{y})] = exp(−|y|λ) .

1.2 Hitting times for a Drifted Brownian motion

We study now the case where X_{t} = νt + W_{t}, where W is a Brownian motion. Let M_{t}^{X} = sup (X_{s}, s ≤ t), m_{t}^{X} = inf (X_{s}, s ≤ t) and T_{y}(X) = inf{t ≥ 0 | X_{t} = y}.

1.2.1 Laws of the pairs M, X and m, X at time t

Proposition 1.2.1 For y ≥ 0, y ≥ x

P (X_{t} ≤ x, M_{t}^{X} ≤ y) = N (\frac{x − νt}{\sqrt{t}}) − e^{2νy}N (\frac{x − 2y − νt}{\sqrt{t}})

and for y ≤ 0, y ≤ x

P (X_{t} ≥ x, m_{t}^{X} ≥ y) = N (\frac{−x + νt}{\sqrt{t}}) − e^{2νy}N (\frac{−x + 2y + νt}{\sqrt{t}}) .

Proof: From Cameron-Martin’s theorem

P^{(ν)}(X_{t} ≤ x, M_{t}^{X} ≥ y) = E [exp[νW_{t} − \frac{ν^{2}}{2} t] 11\{W_{t} ≤ x, M_{t}^{W} ≥ y\}] .

From the reflection principle (1.5) for y ≥ 0, x ≤ y,

P (W_{t} ≤ x, M_{t} ≥ y) = P (x ≥ 2y − W_{t}, M_{t} ≥ y)

hence

E [exp[νW_{t} − \frac{ν^{2}}{2} t] 11\{W_{t} ≤ x, M_{t}^{W} ≥ y\}] = E [exp[ν(2y − W_{t}) − \frac{ν^{2}}{2} t] 11\{2y − W_{t} ≤ x, M_{t}^{W} ≥ y\}]
= e^{2νy}E [exp[−νW_{t} − \frac{ν^{2}}{2} t] 11\{W_{t} ≥ 2y − x\}] .

We apply again Cameron-Martin’s theorem

E [exp[−νW_{t} − \frac{ν^{2}}{2} t] 11\{W_{t} ≥ 2y − x\}] = P (W_{t} − νt ≥ 2y − x) = P (X_{t} ≥ 2y − x + 2νt)
= P (W_{t} ≤ −2y + x − 2νt) .

1.2. HITTING TIMES FOR A DRIFTED BROWNIAN MOTION 45 It follows that for y ≥ 0, y ≥ x

P (X_{t} ≤ x, M_{t}^{X} ≥ y) = e^{2νy}P (X_{t} ≥ 2y − x + 2νt)
= e^{2νy}N (\frac{−2y + x − νt}{\sqrt{t}})
P (X_{t} ≤ x, M_{t}^{X} ≤ y) = P (X_{t} ≤ x) − P (X_{t} ≤ x, M_{t}^{X} ≥ y)
= P (X_{t} ≤ x) − e^{2νy}P (X_{t} ≥ 2y − x + 2νt)
= N (\frac{x − νt}{\sqrt{t}}) − e^{2νy}N (\frac{−2y + x − νt}{\sqrt{t}}) .
□

1.2.2 Laws of maximum, minimum and hitting times

In particular, the laws of the maximum and of the minimum are deduced :

P (M_{t}^{X} ≤ y) = N (\frac{y − νt}{\sqrt{t}}) − e^{2νy}N (\frac{−y − νt}{\sqrt{t}}), y ≥ 0
P (M_{t}^{X} ≥ y) = N (\frac{−y + νt}{\sqrt{t}}) + e^{2νy}N (\frac{−y − νt}{\sqrt{t}}), y ≥ 0

and for y > 0

P (T_{y}(X) ≥ t) = P (M_{t}^{X} ≤ y) .

The law of the variable T_{y}(X) has density

P (T_{y}(X) ∈ dt) = \frac{dt}{\sqrt{2πt^{3}}} ye^{νy} exp (−\frac{1}{2} (\frac{y^{2}}{t} + ν^{2}t)) = \frac{dt}{\sqrt{2πt^{3}}} y exp (−\frac{1}{2t} (y − νt)^{2}) ,

named inverse Gaussian law with parameter (y, ν). In particular, when t → ∞ in

P (T_{y} ≥ t) = N (\frac{y − νt}{\sqrt{t}}) − e^{2νy}N (\frac{−y − νt}{\sqrt{t}}) ,

we obtain P (T_{y} = ∞) = 1 − e^{2νy}, for ν ≤ 0 and y > 0.

P (m_{t}^{X} ≥ y) = N (\frac{−y + νt}{\sqrt{t}}) − e^{2νy}N (\frac{y + νt}{\sqrt{t}}), y ≤ 0  (1.11)
P (m_{t}^{X} ≤ y) = N (\frac{y − νt}{\sqrt{t}}) + e^{2νy}N (\frac{y + νt}{\sqrt{t}}), y ≤ 0 .  (1.12)

1.2.3 Laplace transforms

From Cameron-Martin’s theorem

E(exp −\frac{λ^{2}}{2} T_{y}(X)) = E (exp(νW_{T_{y}} − \frac{ν^{2} + λ^{2}}{2} T_{y}(W))) .

From Proposition 1.1.2, the right hand side equals

e^{νy}E[exp(−\frac{1}{2}(ν^{2} + λ^{2})T_{y}(W))] = e^{νy} exp[−|y|\sqrt{ν^{2} + λ^{2}}] .

Therefore

E(exp −\frac{λ^{2}}{2} T_{y}(X)) = e^{νy} exp[−|y|\sqrt{ν^{2} + λ^{2}}] .  (1.13)

Therefore, setting λ = 0 in (1.13), in the case ν > 0, y < 0, or ν < 0, y > 0

P (T_{y} < ∞) = e^{2νy} ,

which proves again that, in that case the probability that a Brownian motion with positive drift hits a negative level is not equal to one. In the case νy > 0, obviously P (T_{y} < ∞) = 1 . This is explained by the fact that (W_{t} + νt)/t goes to ν when t goes to infinity, hence the drift drives the process to infinity. In the case νy > 0, taking the derivative of (1.13) for λ = 0, we obtain E(T_{y}(X)) = y/ν. When νy < 0, the differentiation does not work, and obvioussly the expectation of the stopping time is equal to infinity.

1.3 Hitting Times for Geometric Brownian Motion

Let us assume that the dynamics of the risky asset are, under the risk neutral probability Q, given by

dS_{t} = S_{t}(μdt + σdW_{t}) , S_{0} = x  (1.14)

with σ > 0, i.e.,

S_{t} = x exp ((μ − σ^{2}/2)t + σW_{t}) = xe^{σX_{t}} ,

where μ = r − δ, X_{t} = νt + W_{t}, ν = \frac{μ}{σ} − \frac{σ}{2} . We denote the first hitting time of a by

T_{a}(S) = inf\{t ≥ 0 : S_{t} = a\} = inf\{t ≥ 0 : X_{t} = \frac{1}{σ} ln(a/x)\}

Then T_{a}(S) = T_{α}(X) where α = \frac{1}{σ} ln(a/x). When a level b is used for the geometric Brownian

motion S, we shall denote β = \frac{1}{σ} ln(b/x).

Law of the pair (maximum, minimum)

We deduce from Proposition 1.2.1 that for b > a, b > x

P (S_{t} ≤ a, M_{t}^{S} ≤ b) = P (X_{t} ≤ α, M_{t}^{X} ≤ β) = N (\frac{α − νt}{\sqrt{t}}) − e^{2νβ}N (\frac{α − 2β − νt}{\sqrt{t}})

whereas, for a > b, b < x

P (S_{t} ≥ a, m_{t}^{S} ≥ b) = P (X_{t} ≥ α, m_{t}^{X} ≥ β) = N (\frac{−α + νt}{\sqrt{t}}) − e^{2νβ}N (\frac{−α + 2β + νt}{\sqrt{t}})

It follows that, for a > x (or α > 0)

P (T_{a}(S) < t) = P (T_{α}(X) < t) = 1 − P (M_{t}^{X} ≤ α)
= 1 − P (X_{t} ≤ α, M_{t}^{X} ≤ α)  (1.15)
= 1 − N (\frac{α − νt}{\sqrt{t}}) + e^{2να}N (\frac{−νt − α}{\sqrt{t}})  (1.16)
= N (\frac{−α + νt}{\sqrt{t}}) + e^{2να}N (\frac{−νt − α}{\sqrt{t}})  (1.17)

and, for a < x (or α < 0)

P (T_{a}(S) < t) = P (T_{α}(X) < t) = 1 − P (m_{t}^{X} ≥ α)
= N (\frac{α − νt}{\sqrt{t}}) + e^{2να}N (\frac{νt + α}{\sqrt{t}}) .

1.3.1 Laplace transforms

From the previous remarks

E(exp −\frac{λ^{2}}{2} T_{a}(S)) = E^{(ν)}(exp −\frac{λ^{2}}{2} T_{α}(X)) .

Therefore, from (1.13)

E(exp −\frac{λ^{2}}{2} T_{a}(S)) = exp (να − |α|\sqrt{ν^{2} + λ^{2}}) .  (1.18)

1.4 Other processes

1.4.1 OU Process

Let (r_{t}, t ≥ 0) be defined as

dr_{t} = k(θ − r_{t}) dt + σ dW_{t}, r_{0} = 0,

and τ_{ρ} = inf {t ≥ 0 : r_{t} ≥ ρ}. For any ρ > r_{0} = 0, the density function of τ_{ρ} equals

f(t) = \frac{ρ}{σ\sqrt{2π}} (\frac{k}{sinh kt})^{3/2}e^{kt/2} exp [ − \frac{k}{2σ2}((ρ − θ)^{2} − θ^{2} + ρ^{2} coth kt)].

For the derivation of the last formula, the reader is referred to Göing and Yor [33]. The formula in Leblanc and Scaillet [49] is only valid for r_{0} = 0. The Laplace transform of the stopping time τ_{ρ} is known (see Borodin and Salminen [7]):

E_{r} (exp (−δτ_{ρ})) = \frac{Υ(r)}{Υ(ρ)}

where

Υ(r) = exp (\frac{(r − θ)^{2}}{4σ^{2}} ) D_{−k} (−\frac{r − k}{σ} )

where D is the parabolic cylinder function :

D_{−ν}(z) = exp (−\frac{z^{2}}{4} ) 2^{−ν/2}\sqrt{π}

\{\frac{1}{Γ ((ν + 1) /2)} (1 + ∑_{k=1}^{+∞} \frac{ν (ν + 2) … (ν + 2k − 2)}{3.5 … (2k − 1) k!} (\frac{z^{2}}{2} )^{k})

−\frac{z\sqrt{2}}{Γ (ν/2)} (1 + ∑_{k=1}^{+∞} \frac{(ν + 1) (ν + 3) … (ν + 2k − 1)}{3.5 … (2k + 1) k!} (\frac{z^{2}}{2} )^{k})\} .

1.5 Non-constant Barrier

The case of non-constant barrier would be of great interest. For example, the process X is a geometric Brownian motion with deterministic volatility

dS_{t} = S_{t}(rdt + σ(t)dB_{t}), S_{0} = x

and

T_{a}(S) = inf\{t : S_{t} = a\} = inf\{t : rt − \frac{1}{2}[∫_{0}^{t} σ(s)]^{2}ds + ∫_{0}^{t} σ_{s}dB_{s} = α\} ,

where α = ln(a/x) As we shall see below, the process U_{t} = ∫_{0}^{t} σ_{s}dB_{s} is a changed time Brownian

motion and can be written as Z_{A(t)} where Z is a brownian motion and A(t) = ∫_{0}^{t}[σ(s)]^{2}ds. Hence,

introducing the inverse C of the function A

T_{a}(S) = inf\{t : rt − \frac{1}{2}A(t) + Z_{A(t)} = α\} = inf\{C(u) : rC(u) − \frac{1}{2}u + Z_{u} = α\} ,

and we are reduced to the study of the hitting time of the non-constant boundary C(u) by the drifted Brownian motion Z_{t} − \frac{1}{2}t.

1.5.1 Bibliography

More generally, let τ_{f}(V) = inf {t ≥ 0 : V_{t} = f(t)}, where f is a deterministic function and V a diffusion process. There are only few cases for which the law of τ_{f}(V) is explicitly known; for instance, the previous case when V is a Brownian motion and f is an affine function.

This problem is studied in a general framework in Alili’s thesis [1], Barndorff-Nielsen et al. [3], Daniels [14], Durbin [24], Ferebee [29], Hobson et al. [37], Jennen and Lerche [44][45], Lerche [50], Salminen [57] and Siegmund and Yuh [60]. Breiman [8] studies the case of a square root boundary, i.e. T = inf\{t : x + B_{t} = α\sqrt{t}\} .

Groeneboom [35] studies the case T = inf{t : x + B_{t} = αt^{2}}. For any x > 0 and α < 0,

P_{x}(T ∈ dt) = 2(αc)^{2}Σ_{n=0}^{∞} exp(−μ_{n} − \frac{2}{3}α^{2}t^{3})\frac{Ai(λ_{n} − 2αcx)}{Ai′(λ_{n})}

here λ_{n} are the zeros on the negative half-line of the Airy function Ai, the unique bounded solution of u′′ − xu = 0, u(0) = 1, and μ_{n} = −λ_{n}/c. This last expression was obtained by Salminen [57]. The Airy function is defined as

(Ai)(x) =^{def} \frac{1}{π} (\frac{x}{3})^{1/2} K_{1/3} (\frac{2}{3}x^{3/2}) .

1.5.2 Fokker-Planck equation

Let

dX_{t} = b(t, x_{t})dt + σ(t, X − t)dW_{t}

be a diffusion.

Proposition 1.5.1 Let h be a deterministic function, τ = inf{t ≥ 0 : X_{t} ≤ h(t)} and

g(t, x)dx = P (X_{t} ∈ dx, τ > t) .

The measure g(t, x)dx satisfies

\frac{d}{dt}g(t, x) = −\frac{∂}{∂x}(b(t, x)g(t, x)) + \frac{1}{2} \frac{∂^{2}}{∂x^{2}} σ^{2}(t, x)g(t, x) ; x > h(t)

and the boundary conditions

g(t, x)dx|_{t=0} = δ(x − X_{0})
g(t, x)|_{x=h(t)} = 0

Proof: The proof is done as for the backward Kolmogorov equation. If φ is a C^{2} function with compact support, for s < t

φ(X_{t∧τ}) = φ(X_{s∧τ}) + ∫_{s∧τ}^{t∧τ} φ′(X_{u})dX_{u} + \frac{1}{2} ∫_{s∧τ}^{t∧τ} φ′′(X_{u})σ^{2}(u, X_{u})du .  (1.19)

Hence, on the one hand

E(φ(X_{t∧τ})) = E(φ(X_{t})11_{{t≤τ}}) + E(φ(X_{τ})11_{{τ<t}})
= ∫ φ(x)g(t, x)dx + E(φ(h(τ))11_{{τ<t}})
= ∫ φ(x)g(t, x)dx + ∫_{0}^{t} φ(h(v))μ(dv)

where μ is tha law of τ, and on the other hand

E(φ(X_{t∧τ})) = E(φ(X_{s∧τ})) + E (∫_{s∧τ}^{t∧τ} +φ′(X_{u})b(u, X_{u})du)

\frac{1}{2}E (∫_{s∧τ}^{t∧τ} φ′′(X_{u})σ^{2}(u, X_{u})du)

= ∫ φ(x)g(s, x)dx + ∫_{0}^{s} φ(h(v))μ(dv)
+ ∫_{s}^{t} ∫ (φ′(x)b(u, x) + \frac{1}{2}φ′′(x)σ^{2}(u, x)) g(x, u)du .

This identity stands for any function φ, therefore, using integration by parts, and differentiation with respect to t, we get the result. □

Using Fokker Planck equation, He et al. [36] established the following result

Proposition 1.5.2 Let X_{i}(t) = α_{i}t+σ_{i}W_{i}(t) where W_{1}, W_{2} are two correlated Brownian motion, with correlation ρ, and M_{i}, m_{i} the running maximum and minimum. The probability density

\begin{matrix}P (X_{1}(t) ∈ dx_{1}, X_{2}(t) ∈ dx_{2}, m_{1}(t) ∈ dm_{1}, m_{2}(t) ∈ dm_{2}) \\ = p(x_{1}, x_{2}, t; m_{1}, m_{2}, α_{1}, α_{2}, σ_{1}, σ_{2}, ρ) \end{matrix}

is

\frac{e^{a_{1}x_{1}+a_{2}x_{2}+bt}}{σ_{1}σ_{2}\sqrt{1 − ρ^{2}}} h(x_{1}, x_{2}, t; m_{1}, m_{2}, α_{1}, α_{2}, σ_{1}, σ_{2}, ρ)

with

h(x_{1}, x_{2}, t; m_{1}, m_{2}, α_{1}, α_{2}, σ_{1}, σ_{2}, ρ) = \frac{2}{βt} ∑_{n=1}^{∞} e^{−(r^{2}+r_{0}^{2})/(2t)} sin (\frac{nπθ_{0}}{β} ) sin (\frac{nπθ}{β} ) I_{(nπ)/β} (\frac{rr_{0}}{t} )

and

a_{1} = \frac{α_{1}σ_{2} − ρα_{2}σ_{1}}{(1 − ρ^{2})σ_{1}^{2}σ_{2}} a_{2} = \frac{α_{2}σ_{1} − ρα_{1}σ_{2}}{(1 − ρ^{2})σ_{1}σ_{2}^{2}}

b = −α_{1}a_{1} − α_{2}a_{2} + \frac{1}{2} (σ_{1}^{2}a_{1}^{2} + σ_{2}^{2}a_{2}^{2}) + ρσ_{1}σ_{2}a_{1}a_{2}
tan β = −\frac{\sqrt{1 − ρ^{2}}}{ρ} ,
z_{1} = \frac{1}{\sqrt{1 − ρ^{2}}} [(\frac{x_{1} − m_{1}}{σ_{1}} ) − ρ (\frac{x_{2} − m_{2}}{σ_{2}} )] , z_{2} = \frac{x_{2} − m_{2}}{σ_{2}}

z_{10} = \frac{1}{\sqrt{1 − ρ^{2}}} [−\frac{m_{1}}{σ_{1}} ) + ρ\frac{m_{2}}{σ_{2}} , z_{20} = −\frac{m_{2}}{σ_{2}}
r = \sqrt{z_{1}^{2} + z_{2}^{2}}, tan θ = \frac{z_{2}}{z_{1}} , θ ∈ [0, β]
r_{0} = \sqrt{z_{10}^{2} + z_{20}^{2}}, tan θ_{0} = \frac{z_{20}}{z_{10}} , θ_{0} ∈ [0, β]

\begin{matrix}P (X_{1}(t) ∈ dx_{1}, X_{2}(t) ∈ dx_{2}, m_{1}(t) ≥ m_{1}, M_{2}(t) ≤ M_{2}) \\ = p(x_{1}, −x_{2}, t; m_{1}, −M_{2}, α_{1}, −α_{2}, σ_{1}, σ_{2}, −ρ)dx_{1}dx_{2} \end{matrix}

See Bielecki and Rutkowsky for a direct proof.

Chapter 2

Appendix B: Poisson processes

We give some results on Poisson Processes with deterministic intensity.

2.1 Definition

The Poisson process N with deterministic intensity (λ(t), t ≥ 0) where λ is a IR^{+} valued function satisfying ∫_{0}^{t} λ(u)du < ∞, forallu, is a counting process with independent increments which satisfy

P (N_{t} − N_{s} = n) = e^{−Λ(s,t)} \frac{(Λ(s, t))^{n}}{n!}  (2.1)

where Λ(t) = E(N_{t}) = ∫_{0}^{t} λ(u) du and Λ(s, t) = Λ(t) − Λ(s). If (T_{n}) is the stopping time sequence associated with N, the following equality is satisfied :

P (T_{n} ≤ t) = \frac{1}{n!} ∫_{0}^{t} exp(−Λ(s)) (Λ(s))^{n−1} dΛ(s) .

It can easily be shown that a Poisson process with deterministic intensity is a Markov process.

2.2 Martingale properties

Proposition 2.2.1 Let N be a PP with deterministic intensity λ. The process

(M_{t} = N_{t} − ∫_{0}^{t} λ(s)ds, t ≥ 0)

is a F-martingale, and ∫_{0}^{t} λ(s)ds is called the compensator. Let ϕ be a F- predictable process such that for every t, E(∫_{0}^{t} |ϕ_{s}|λ(s)ds) < ∞, then the process (∫_{0}^{t} ϕ_{s}dM_{s}, t ≥ 0) is an F-martingale. In particular,

E(∫_{0}^{t} ϕ_{s} dN_{s}) = E(∫_{0}^{t} ϕ_{s}λ(s)ds)  (2.2)

The increasing function Λ(t) = ∫_{0}^{t} λ(s)ds is the compensator of N. In what follows, we denote by ϕ·M the process ((ϕ·M)_{t} =^{def} ∫_{0}^{t} ϕ_{s}dM_{s}, t ≥ 0). As in the constant 52 CHAPTER 2. APPENDIX B: POISSON PROCESSES intensity case, the following processes are martingales for any bounded predictable process H

M_{t}^{H} =^{def} ∫_{0}^{t} H_{s}dN_{s} − ∫_{0}^{t} λ(s)H_{s}ds = (H · M)_{t}  (2.3)
Y_{t}^{H} =^{def} (M_{t}^{H})^{2} − ∫_{0}^{t} λ(s)H_{s}^{2}ds  (2.4)
Z_{t}^{H} =^{def} exp (∫_{0}^{t} H_{s}dN_{s} + ∫_{0}^{t} λ(s)(1 − e^{H_{s}})ds)  (2.5)

Bibliography

[1] L. Alili. Fonctionnelles exponentielles et certaines valeurs principales des temps locaux browniens. Thèse, Paris VI, 1995.

[2] M Ammann. Credit risk valuation: Methods, model and application. Springer Verlag, Berlin, 2001.

[3] O.E. Barndorff-Nielsen, P. Blaesild, and C. Halgreen. First hitting times models for the generalized inverse Gaussian distribution. Stochastic processes and Appl., 7:49–54, 1978.

[4] A. Bélanger, S.E. Shreve, and D. Wong. A unified model for credit derivatives. Preprint, 2001.

[5] T. Bielecki and M. Rutkowski. Credit risk : Modelling valuation and Hedging. Springer Verlag, Berlin, 2001.

[6] F. Black and J.C. Cox. Valuing corporate securities. some effects of bond indenture provisions. J. Finance, 31:351–367, 1976.

[7] A. Borodin and P. Salminen. Handbook of Brownian motion – facts and formulae. Birkhäuser, 1996.

[8] L. Breiman. First exit times from a square root boundary. In L. Le Cam and J. Neyman, editors, Proc. 5th Berkeley Symp. Math. Statist. prob., 2, pages 9–16. U.C. Press, Berkeley, 1966.

[9] P. Brémaud. Point Processes and Queues. Martingale dynamics. Springer Verlag, Berlin, 1981.

[10] P. Brémaud and M. Yor. Changes of filtration and of probability measures. Z. f. W., 45:269– 295, 1978.

[11] C.S. Chou and P.A. Meyer. Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels. In J. Azéma and M. Yor, editors, Séminaire de Probabilités IX, volume 1557 of Lecture Notes in Maths., pages 226–236. Springer-Verlag, 1975.

[12] C. Cocozza-Thivent. Processus stochastiques et fiabilité des systèmes. Springer-Smai, Paris, 1997.

[13] M. Crouhy, D. Galai, and R Mark. A comparative analysis of current credit risk models. Journal of banking and finance, 24:49–117, 2000.

[14] H.E. Daniels. The first crossing time density for Brownian motion with a perturbed linear boundary. Bernoulli, 6:571–580, 2000.

[15] M. Davis and V. Lo. Modelling default correlation in bond portfolio. preprint, 1999.

54 BIBLIOGRAPHY

[16] M. Davis and V. Lo. Infectious default. Quantitative finance, 1:382–386, 2001.

[17] C. Dellacherie. Un exemple de la théorie générale des processus. In P.A. Meyer, editor, Séminaire de Probabilités IV, volume 124 of Lecture Notes in Maths., pages 60–70. Springer-Verlag, 1970.

[18] C. Dellacherie and P.A. Meyer. Probabilités et potentiel, chapitres I-IV. Hermann, Paris, 1975.

[19] C. Dellacherie and P.A. Meyer. A propos du travail de Yor sur les grossissements des tribus. In P.A. Meyer, editor, Séminaire de Probabilités XII, volume 649 of Lecture Notes in Maths., pages 69–78. Springer-Verlag, 1978.

[20] R. Douady and M. Jeanblanc. A rating-based model for credit derivatives. Preprint, 2001.

[21] G. Duffee. Estimating the price of default risk. The Review of financial studies, 12:197–226, 1999.

[22] D. Duffie and D. Lando. Term structure of credit spreads with incomplete accounting information. Econometrica, 69:633–664, 2000.

[23] D. Duffie and K. Singleton. Modeling term structure of defaultable bonds. Review of financial studies, 12:687–720, 1998.

[24] J. Durbin. The first passage density of the Brownian motion process to a curved boundary. J. Appl. Prob., 29:291–304, 1992.

[25] V. Durrleman, A. Nikeghbali, G. Riboulet, and T. Roncalli. Copulas for finance: a reading guide and some applications. Working paper, groupe de recherche opérationnelle, Crédit lyonnais, 2000.

[26] R. Elliott. Stochastic calculus and applications. Springer, Berlin, 1982.

[27] R.J. Elliott, M. Jeanblanc, and M. Yor. On models of default risk. Math. Finance, 10:179–196, 2000.

[28] P. Embrecht, A. McNeil, and D. Strautman. Correlation and dependance in risk management. preprint, Risklab, ETH, 1999.

[29] B. Ferebee. The tangent approximation to one sided Brownian exit densities. Z. Wahr. Verw. Gebiete, 61:309–326, 1982.

[30] R. Frey and A.J. McNeil. Modelling dependent defaults. Preprint, 2001.

[31] G. Gauthier and M. Jeanblanc. Partial information and hazard process. preprint, 2001.

[32] K. Giesecke. Correlated default with incomplete information. Preprint, 2001.

[33] A. Göing and M. Yor. A survey and some generalizations of Bessel processes. preprint, 1999.

[34] Y.M. Greenfeld. Hedging of the credit risk embedded in derivative transactions. Ph. d., Carnegie Mellon, May 2000.

[35] P Groeneboom. Brownian motion with a parabolic drift and Airy functions. Probab. Th. Rel. Fields, 81:79–109, 1989.

[36] H. He, W.P. Keirstead, and J. Rebholz. Double lookbacks. Math. Finance, 8:201–228, 1998.

[37] D. Hobson, D. Williams, and A. Wood. Taylor expensions of curve-crossing probabilities. Bernoulli, 5:779–795, 1999.

BIBLIOGRAPHY 55

[38] G. Hubner. The analytic pricing of asymmetric defaultable swaps. Preprint, 1997.

[39] G. Hubner. A two factor gaussian model of default risk. Preprint, 1997.

[40] J. Hull and A. White. Valuing credit default swaps i: no counterparty default risk. Preprint, Rotman School Toronto, 2000.

[41] R.A. Jarrow and S.M. Turnbull. Pricing derivatives on financial securities subject to credit risk. The journal of finance, 50:53–85, 1995.

[42] M. Jeanblanc and M. Rutkowski. Modeling default risk: an overview. Mathematical Finance: theory and practice, Fudan University. Modern Mathematics Series, High Education press. Beijin, pages 171–269, 2000.

[43] M. Jeanblanc and M. Rutkowski. Modeling default risk: Mathematical tools. Fixed Income and Credit risk modeling and Management, New York University, Stern school of business, Statistics and operations research department, Workshop, 2000.

[44] C. Jennen and H.R. Lerche. First exit densities of Brownian motion through one- sided moving boundaries. Z. Wahr. Verw. Gebiete, 55:133–148, 1981.

[45] C. Jennen and H.R. Lerche. Asymptotic densities of stopping times associated with test of power one. Z. Wahr. Verw. Gebiete, 61:501–511, 1982.

[46] S. Karlin and H. Taylor. A first course in stochastic processes. Academic Press, San Diego, 1975.

[47] S. Kusuoka. A remark on default risk models. Adv. Math. Econ., 1:69–82, 1999.

[48] D. Lando. On Cox processes and credit risky securities. Review of Derivatives Research, 2:99–120, 1998.

[49] B. Leblanc and O. Scaillet. A correction note on the first passage time of an Ornstein-Uhlenbeck process to a boundary. Finance and Sto., 4:109–111, 2000.

[50] H.R. Lerche. Boundary crossing of Brownian motion, volume 40 of Lecture notes in statistics. Springer-Verlag, 1986.

[51] F.A; Longstaff and E.S. Schwartz. Valuing credit derivatives. The journal of fixed income, june, pages 6–12, 1995.

[52] D. Madan and H. Unal. Pricing the risks of default. Review of Derivatives Research, 2:121– 160, 1998.

[53] G. Mazziotto and J. Szpirglas. Modèle général de filtrage non linéaire et équations différentielles stochastiques associées. Ann. Inst. Henri Poincaré, 15:147–173, 1979.

[54] R. Merton. On the pricing of corporate debt : the risk structure of interest rates. J. of Finance, 3:449–470, 1974.

[55] R.B. Nelsen. An introduction to copulas, volume 139 of Lecture Notes in Stats. Springer-Verlag, Berlin, 1999.

[56] G. Pan. Equity to credit pricing. Risk magazine, November, pages 99–102, 2001.

[57] P. Salminen. On the first hitting time and the last exit for a Brownian motion to/from a moving boundary. Adv. Appl. prob., 20:411–426, 1988.

[58] P.J. Schönbucher. The pricing of credit risk and credit risk derivatives. Preprint, 1997.

[59] P.J. Schönbucher. Term structure modelling of defaultable bonds. Review of Derivatives Research, 2:161–192, 1998.

[60] D. Siegmund and Y-S. Yuh. Brownian approximations to first passage probabilities. Z. Wahr. Verw. Gebiete, 59:239–248, 1982.

[61] D. Wong. A unifying credit model. Technical report, Capital Markets Group, 1998.

[62] C. Zhou. Default correlation: an analytical result. preprint, 1997.

[63] C. Zhou. A jump-diffusion apprach to modeling credit risk and valuing defaultable securities. preprint, Federal Reserve Board, 1997.

Websites: www.defaulrisk.com www.risklab.com www.kmv.com www.csfb.com/creditrisk (CreditRisk+) www.risksmetrics.com/research (JP Morgan) www.creditlyonnais.com (Copulas)

Index

Airy function, 48

Defaultable swap, 11 Defaultable zero-coupon, 15 Distance-to-default, 4

Explosive default, 37

First-to-default, 11 Fokker-Planck equation, 49

Hazard function, 14, 18

intensity, 18 Intensity approach, 34 Intensity process, 18

Kusuoka example, 36

Laplace transform hitting time for a GBM, 47 hitting time for OU, 47 of hitting time for a BM, 43 of hitting time for a drifted BM, 45

Law Inverse Gaussian -, 45

Noisy information, 39

Partial information, 19, 37 Predictable representation theorem, 30

Reflection principle, 41 Risk neutral probability measure, 16

Spread, 19