INDIFFERENCE PRICING OF DEFAULTABLE CLAIMS

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INDIFFERENCE PRICING OF DEFAULTABLE CLAIMS

Tomasz R. Bielecki

¹

Department of Applied Mathematics Illinois Institute of Technology

Chicago, USA bielecki@iit.edu Monique Jeanblanc

Equipe d’Analyse et Probabilit´es Universit´e d’´ Evry-Val-d’Essonne

Evry, France ´

monique.jeanblanc@maths.univ-evry.fr

July 7, 2004

1The first author was supported in part by NSF Grant 0202851.

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INDIFFERENCE PRICING OF

DEFAULTABLE CLAIMS

The goal of this chapter is to give an application of the theory of indifference prices in the context of defaultable claims within the reduced-form approach.

In this approach the defaultable market is incomplete and there does not exist a (perfect) hedging strategy for claims which depend on the occurrence of the default. An important issue is the issue of choice of relevant information.

The chapter is organized as follows. Section 1.1 contains a brief description of the basic concepts of default risk that are used in the sequel. The second section is devoted to indifference pricing in the filtration of default-free assets.

The following section studies the case where the investor additionally uses the information on the default in the choice of the portfolio and is endowed with an exponential utility function. In a last section, we present the quadratic hedging problem.

For details on credit risk, the reader can refer to the books of Bielecki and Rutkowski (2002), Duffie and Singleton (2003), Sch¨onbucher (2003) and to the survey papers of Bielecki et al (2004a, 2004b) where many references are given.

1.1 Preliminaries

In this section, we introduce the basic notions that will be used in what follows.

First, we define a default-free market model. Then, we examine the concept of a default time and we present the associated hazard process. We make precise the choice of the filtration, which is an important aspect of our presentation.

5

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1.1.1 Default-Free Market

Consider an economy in continuous time, with the time parameter t∈IR+. A probability space (Ω,G,P) endowed with a one-dimensional standard Brow- nian motion (Wt, t ≥ 0) is given. We assume that the reference filtration F= (F_t, t≥0) is theP-augmented and right-continuous version of the natural filtration generated byW. We haveFt⊂ G, for anyt∈IR+, however we do not assume thatG=F∞.

In the first step, we introduce a Black and Scholes arbitrage-freedefault-free market. In this market, we have the following primary assets:

• A money market accountB satisfying

dBt=rBtdt, B0= 1,

or, equivalently,B_t= exp(rt), where the interest rateris assumed to be constant.

• A default-free asset whose price (St, t≥0) follows a geometric Brownian motion dynamics

dSt=St(νdt+σdWt), where ν andσare two constants, withσ6= 0.

It is not difficult to extend the study to the case wherer, ν andσareF-adapted process as soon as some regularity is assumed in order that the default-free market is arbitrage free. In the last part of the chapter, we shall turn to a more general model of the primary market.

As it is well known, the Black and Scholes default-free market is arbitrage- free and complete, and the (unique) risk-neutral probabilityQ is obtained via its Radon-Nikodym density, i.e.

dQ|Ft =ηtdP|Ft, where (ηt, t≥0) is the (P,F)-martingale given as

ηt= exp(−θWt−1 2θ²t),

whereθ= (ν−r)σ⁻¹is the risk premium. From Girsanov’s theorem, the process W_t^Q=Wt+θt, is a (Q,F)-Brownian motion.

1.1.2 Default Time

The default timeτ is defined as a non-negative random variable on the probability space (Ω,G,P). We introduce the default process Ht = 11_{τ≤t} and we denote by H = (Ht, t ≥ 0) the filtration generated by this process (this

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1.1. PRELIMINARIES 7 filtration is right-continuous, and, as usual, we take the completion of this filtration). Note that Ht=σ(t∧τ), hence, anyHt-measurable random variable if a deterministic function of the random variableτ∧t.

Hazard process. It is generally assumed that the investor knows when the default takes place, that is the observation of the investor includes the filtration H. At time t, the investor knows whether or not the default has occurred. If the default has not occurred in the past, the investor has no information on the date when the default will appear. Therefore, we consider the filtration of information which takes into account the information on the asset price and of the occurrence of default: G=F∨H so that Gt=Ft∨ Ht =σ(Ft∪ Ht) for everyt∈IR+. The filtrationGis referred to as to thefull filtration. It is clear thatτ is anH-stopping time, as well as aG-stopping time (but not necessarily anF-stopping time). The concept of the hazard process of a random time τ is closely related to the process (Ft, t≥0) which is defined as follows:

Ft=P{τ≤t| Ft}, ∀t∈IR+.

Let us denote Gt= 1−Ft=P{τ > t| Ft} and let us assume thatGt>0 for every t ∈IR+ (hence, we exclude the case where τ is an F-stopping time – a case that corresponds to the so-called structural approach). Then the process (Γt, t≥0), given by the formula

Γt=−ln(1−Ft) =−lnGt, ∀t≥0,

is well defined. It is termed the hazard process of the random timeτ with respect to the reference filtration F. We postulate thatF∞ = 1 (i.e. τ is finite with probability one).

We now formulate an important

Hypothesis: We assume in this chapter that the Brownian motion (Wt, t≥0) is a (P,G)-Brownian motion.

In other words, we assume that the so-called (H) hypothesis is satisfied and, as a consequence, the process F (hence Γ) is increasing. We do not comment here on that hypothesis, we simply mention that this hypothesis is necessary in order that there is no-arbitrage in the default-free market using G-adapted strategies. See Elliott et al. (2000) or Bielecki et al. (2004b) for comments.

Note that, due to (H) hypothesis, the process (ηt, t ≥ 0) is a (P,G)- martingale.This allows us to define the probability Q^∗ whose the restriction to Gt is

dQ^∗|Gt =ηtdP|Gt.

Obviously, the restriction ofQ^∗toFis equal toQ. We shall omit the superscript

∗ in what follows.

Moreover, for simplicity, we assume that the process (Ft, t≥0) is absolutely continuous, that is,

Ft= Z _t

0

fudu

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for some density processf :IR+ →IR+. Then we have Ft= 1−e^−Γ^t = 1−exp

µ

− Z _t

0

γudu

¶

, ∀t≥0 where

γt= ft

1−Ft, ∀t≥0.

The process γ is non-negative and satisfies R_∞

0 γudu = ∞. It is called the stochastic intensityofτ (or thehazard rate). It can be checked by direct calculations that the process

Mt=Ht− Z _t∧τ

0

γudu=Ht− Z _t

0

(1−Hu−)γudu (1.1) is a (purely discontinuous) (P,G) -martingale. This implies that the random time τ is totally inaccessible in the filtration G. We emphasize that, in our setting, the intensity process is uniquely defined up to infinity and isF-adapted.

Moreover, from the definition ofQ(relative to the full filtrationG) the process M is a (Q,G) -martingale. Indeed, the change of probability has an effect only on the Brownian motionW and no effect on the martingale M, which is orthogonal toW. In the particular case where the random timeτis independent of the filtrationF, the hazard process is deterministic.

Furthermore, note that, for any G-predictable process ψ such that ψs >

−1,∀s, a.s., andR_t

0(1 +ψs)(1−Hs)γsds <∞, the processM_t^ψ=Mt−R_t

0(1− Hs)ψsγsdsis aQ^ψ-martingale, where

dQ^ψ|Gt =ηtE(ψ•M)tdP|Gt.

Here, the processE(ψ•M)t – the Dol´eans-Dade exponential ofψ•M – is the unique processY, which is the solution of dYt=Yt−ψtdMt. The restriction of Q^ψ to theσ-algebraFtis equal toQ, andQequals toQ⁰onGt.

1.1.3 Defaultable Claims

Adefaultableclaim (X1, X2, τ) with maturity date T consists of:

• The default time τ specifying the random time of default and thus also the default events {τ ≤t} for everyt∈[0, T]. It is always assumed that τ is strictly positive with probability 1.

• Thepromised payoffX1, which represents the random payoff received by the owner of the claim at timeT,if there was no default prior to or at time T. The actual payoff at timeT associated withX1thus equalsX111_{{τ >T}_}. We assume thatX1 is anFT-measurable random variable.

• The recovery payoffX2, where X2 is an FT-measurable random variable which is received by the owner of the claim at maturity, provided that the default occurs prior to or at maturity date T.

In what follows, we shall denote by X =X111T <τ+X211τ≤T the value of the defaultable contingent claim at maturity.

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1.1. PRELIMINARIES 9

1.1.4 Hodges Indifference Price

In this section we discuss the concept of Hodges indifference price in our setup.

The difference between our approach and the approach of Barrieu and El Karoui (see the corresponding chapter in the present volume) is that we study two different problems, corresponding to the choice of two different filtrations (i.e.

the reference filtration and the full filtration). When considering Hodges indifference prices one starts with a given utility function, say u. Typically, u is assumed to be strictly increasing and strictly concave. We shall also apply a similar methodology in the case where u is assumed to be strictly convex (namely u(x) =x²) for quadratic hedging. In this case howevere one can not use the term indifference price and one solves a minimization problem.

Problem (P): Optimization in the default-free market.

The agent invests his initial wealthv >0 in the default-free financial market using a self-financing strategy. The associated optimization problem is,

(P) : V(v) := sup

φ∈Φ(F)

EP

©u¡

V_T^v(φ)¢ª , where the wealth process (Vt=V_t^v(φ), t≤T),is solution of

dVt=rVtdt+φt(dSt−rStdt), V0=v. (1.2) Here Φ(F) is the class of allF-adapted, self-financing trading strategies.

Problem (P_F^X): Optimization in the default-free market using F- adapted strategies and buying the defaultable claim.

The agent buys the defaultable claimX at pricep, and invests his remaining wealth v−pin the default-free financial market, using a trading strategy φ∈ Φ(F). The resultingglobal terminal wealthwill be

V_T^v−p,X(φ) =V_T^v−p(φ) +X.

The associated optimization problem is (P_F^X) : V_X^F(v−p) := sup

φ∈Φ(F)

EP

©u¡

V_T^v−p(φ) +X¢ª ,

where the processV^v−p(φ) is solution of (1.2) with the initial conditionV₀^v−p(φ) = v−p. We emphasize that the class Φ(F) of admissible strategies is the same as in the problem (P), that is, we restrict here our attention to trading strategies that are adapted to the reference filtrationF.

Problem (P_G^X): Optimization in the default-free market using G- adapted strategies and buying the defaultable claim.

The agent buys the defaultable contingent claimX at pricep, and invests the remaining wealthv−pin the financial market, using a strategy adapted to the enlarged filtrationG. The associated optimization problem is

(P_G^X) : V_X^G(v−p) := sup

φ∈Φ(G)

EP

©u¡

V_T^v−p(φ) +X¢ª ,

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where Φ(G) is the class of allG-admissible trading strategies.

Remark. It is easy to check that the solution of (PG) : sup

φ∈Φ(G)

EP

©u¡

V_T^v(φ)¢ª , is the same as the solution of (P).

Definition 1.1 For a given initial endowmentv, theF-Hodges buying priceof the defaultable claimX is the real numberp^∗_F(v) such that

V(v) =V_X^F¡

v−p^∗_F(v)¢ .

Similarly, theG-Hodges buying priceofX is the real numberp^∗_G(v) such that V(v) =V_X^G¡

v−p^∗_G(v)¢ .

Remark. We can define theF-Hodges selling pricep^F_∗(v) ofX by considering

−p, wherepis the buying price of−X, as specified in Definition 1.1.

If the contingent claimX isFT-measurable, then (See Rouge and ElKaroui (2000)) the F- and the G-Hodges selling and buying prices coincide with the hedging price ofX, i.e.,

p^∗_F(v) =p^∗_G(v) =EP(ζTX) =EQ(X) =p^G_∗(v) =p^F_∗(v), where we denote byζ the deflator processζt=ηte^−rt.

1.2 Hodges prices relative to the reference fil- tration

In this section, we study the problem (P_F^X) (i.e., we use strategies adapted to the reference filtration). First, we compute the value function, i.e.,V_X^F(v−p).

Next, we establish a quasi-explicit representation for the Hodges price ofX in the case of exponential utility. Finally, we compare the spread obtained via the risk-neutral valuation with the spread determined by the Hodges price of a defaultable zero-coupon bond.

1.2.1 Solution of Problem (P

_F^X

)

In view of the particular form of the defaultable claimX it follows that V_T^v−p,X(φ) = 11{τ >T}(V_T^v−p(φ) +X1) + 11{τ≤T}(V_T^v−p(φ) +X2).

Since the trading strategies areF-adapted, the terminal wealth V_T^v−p(φ) is an FT-measurable random variable. Consequently, it holds that

EP

h u¡

V_T^v−p,X(φ)¢i

=

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1.2. HODGES PRICES RELATIVE TO THE REFERENCE FILTRATION11

= EP

¡u¡

V_T^v−p(φ) +X1

¢11_{{τ >T}}+u¡

V_T^v−p(φ) +X2

¢11_{τ≤T_}¢

= EP

¡EP

£u¡

V_T^v−p(φ) +X1

¢11_{{τ >T}_}+u¡

V_T^v−p(φ) +X2

¢11_{τ≤T_}|FT

¤¢

= EP

£u¡

V_T^v−p(φ) +X1

¢(1−FT) +u¡

V_T^v−p(φ) +X2

¢FT

¤,

whereFT =P{τ≤T| FT}. Thus, problem (P_F^X) is equivalent to the following problem:

(P_F^X) : V_X^F(v−p) := sup

φ∈Φ(F)

EP

¡JX

¡V_T^v−p(φ),·¢¢

, where

JX(y, ω) =u(y+X1(ω))(1−FT(ω)) +u(y+X2(ω))FT(ω),

for every ω ∈Ω and y ∈IR. The real-valued mapping JX(·, ω) is strictly concave and increasing. Consequently, for any ω ∈Ω, we can define the mapping IX(z, ω) by setting IX(z, ω) = ¡

J_X⁰ (·, ω)¢₋₁

(z) for z ∈ IR, where (J_X⁰ (·, ω))⁻¹ denotes the inverse mapping of the derivative ofJXwith respect to the first variable. To simplify the notation, we shall usually suppress the second variable, and we shall write IX(·) in place of IX(·, ω).

The following lemma provides the form of the optimal solution for the problem (P_F^X),

Lemma 1.1 The optimal terminal wealth for the problem (P_F^X) is given by V_T^v−p,∗=IX(λ^∗ζT),P-a.s., for some λ^∗ such that

v−p=EP

¡ζTV_T^v−p,∗¢

. (1.3)

Thus the optimal global wealth equals V_T^v−p,X,∗=V_T^v−p,∗+X =IX(λ^∗ζT) +X and the value function of the objective criterion for the problem(P_F^X)is

V_X^F(v−p) =EP(u(V_T^v−p,X,∗)) =EP(u(IX(λ^∗ζT) +X)). (1.4) Proof. It is well known (see, e.g., Karatzas and Shreve (1998)) that, in order to find the optimal wealth it is enough to maximize u(∆) over the set of square-integrable andFT-measurable random variables ∆, subject to the budget constraint, given by

EP(ζT∆)≤v−p.

The mappingJX(·) is strictly concave (for allω). Hence, for every pair of FT-measurable random variables (∆,∆^∗) subject to the budget constraint, by tangent inequality, we have

EP

©JX(∆)−JX(∆^∗)ª

≤ EP

©(∆−∆^∗)J_X⁰ (∆^∗)ª . For ∆^∗=V_T^v−p,∗ given in the formulation of the Lemma we obtain

EP

©JX(∆)−JX(V_T^v−p,∗)ª

≤λ^∗EP

©ζT(∆−V_T^v−p,∗)ª

≤0,

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where the last inequality follows from the budget constraint and the choice of λ^∗. Hence, for anyφ∈Φ(F),

EP

©JX(V_T^v−p(φ))−JX(V_T^v−p,∗)ª

≤0.

To end the proof, it remains to observe that the first order conditions are also sufficient in the case of a concave criterion. Moreover, by virtue of strict concavity of the functionJX, the optimal strategy is unique. ¤

1.2.2 Exponential Utility: Explicit Computation of the Hodges Price

For the sake of simplicity, we assume here thatr= 0.

Proposition 1.1 Let u(x) = 1−exp(−%x) for some % >0. Assume that the random variablesζTe^−%Xⁱ, i= 1,2areP-integrable. Then theF-Hodges buying price is given by

p^∗_F(v) =−1

%EP

¡ζTln¡

(1−FT)e^−%X¹+FTe^−%X²¢¢

=EP(ζTΨ), where theFT-measurable random variableΨequals

Ψ =−1

%ln¡

(1−FT)e^−%X¹+FTe^−%X²¢

. (1.5)

Thus, the F-Hodges buying price p^∗_F(v) is the arbitrage price of the associated claimΨ. In addition, the claimΨenjoys the following meaningful property

EP

©u¡

X−Ψ¢ ¯¯FT

ª= 0. (1.6)

Proof. In view of the form of the solution to the problem (P), we obtain V_T^v,∗=−1

%ln µµ^∗ζT

%

¶ .

The budget constraintEP(ζTV_T^v,∗) =vimplies that the Lagrange multiplierµ^∗ satisfies

1

%ln µµ^∗

%

¶

=−1

%EP

¡ζTlnζT

¢−v. (1.7)

The solution to the problem (P_F^X) is obtained in a general setting in Lemma 1.1. In the case of an exponential utility, we have (recall that the variable ω is suppressed)

JX(y) = (1−e^−%(y+X¹⁾)(1−FT) + (1−e^−%(y+X²⁾)FT, so that

J_X⁰ (y) =% e^−%y(e^−%X¹(1−FT) +e^−%X²FT).

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1.2. HODGES PRICES RELATIVE TO THE REFERENCE FILTRATION13 Thus, setting

A=e^−%X¹(1−FT) +e^−%X²FT =e^−%Ψ, we obtain

IX(z) =−1

%ln µ z

A%

¶

=−1

%ln µz

%

¶

−Ψ.

It follows that the optimal terminal wealth for the initial endowment v−pis V_T^v−p,∗=−1

%ln µλ^∗ζT

A%

¶

=−1

%ln µλ^∗

%

¶

−1

%lnζT−Ψ,

where the Lagrange multiplier λ^∗ is chosen to satisfy the budget constraint EP(ζTV_T^v−p,∗) =v−p, that is,

1

%ln µλ^∗

%

¶

=−1

%EP

¡ζTlnζT

¢−EP

¡ζTΨ¢

−v+p. (1.8)

¿From definition, the F-Hodges buying price is a real numberp^∗ =p^∗_F(v) such that

EP

¡exp(−%V_T^v,∗)¢

=EP

¡exp(−%(V_T^v−p^∗^,∗+X))¢ ,

where µ^∗ andλ^∗ are given by (1.7) and (1.8), respectively. After substitution and simplifications, we arrive at the following equality

EP

n exp

³

−%¡

EP(ζTΨ)−p^∗+X−Ψ¢´o

= 1. (1.9)

It is easy to check that

EP

¡e^−%(X−Ψ)¯

¯FT

¢= 1 (1.10)

so that equality (1.6) holds, and EP

¡e^−%(X−Ψ)¢

= 1. Combining (1.9) and (1.10), we conclude thatp^∗_F(v) =EP(ζTΨ). 4 We briefly provide the analog of (1.5) for theF-Hodges selling price ofX . We have p^F_∗(v) =EP(ζTΨ), wheree

Ψ =e 1

%ln¡

(1−FT)e^%X¹+FTe^%X²¢

. (1.11)

Remark. It is important to notice that theF-Hodges prices p^∗_F(v) and p^F_∗(v) do not depend on the initial endowment v. This is an interesting property of the exponential utility function. In view of (1.6), the random variable Ψ will be called theindifference conditional hedge.

From concavity of the logarithm function we obtain

ln((1−FT)e^−%X¹+FTe^−%X²)≥(1−FT)(−%X1) +FT(−%X2).

Hence, using thatζT isFT-measurable,

p^∗_F(v)≤EP(ζT((1−FT)X1+FTX2)) =EQ(X).

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Comparison with the Davis price. Let us present the results derived from the marginal utility pricing approach. The Davis price (see Davis (1997)) is given by

d^∗(v) = EP

©u⁰¡ V_T^v,∗¢

Xª V⁰(v) . In our context, this yields

d^∗(v) =EP

©ζT

¡X1FT +X2(1−FT)¢ª .

In this case, the risk aversion%has no influence on the pricing of the contingent claim. In particular, when F is deterministic, the Davis price reduces to the arbitrage price of each (default-free) financial asset Xⁱ, i = 1,2, weighted by the corresponding probabilitiesFT and 1−FT.

1.2.3 Risk-Neutral Spread Versus Hodges Spreads

In our setting the price process of theT-maturity unit discount Treasury (default- free) bond isB(t, T) =e^−r(T^−t).Let us consider the case of a defaultable bond with zero recovery, i.e., X₁ = 1 and X₂ = 0. It follows from (1.11) that the F-Hodges buying and selling prices of the bond are (it will be convenient here to indicate the dependence of the Hodges price on maturityT)

D_F^∗(0, T) =−1

%E_P©

ζ_Tln(e^−%(1−F_T) +F_T)ª and

D^F_∗(0, T) = 1

%EP

©ζTln(e^%(1−FT) +FT)ª , respectively.

Let ˜Q be a risk-neutral probability for the filtration G, that is, for the enlarged market. The “market” price at timet= 0 of defaultable bond, denoted asD⁰(0, T), is thus equal to the expectation under ˜Qof its discounted pay-off, that is,

D⁰(0, T) =EQ˜

¡11{τ >T}RT

¢=EQ˜

¡(1−FeT)RT

¢,

where Fet= ˜Q{τ ≤t| Ft} for everyt∈[0, T]. Let us emphasize that the risk- neutral probability ˜Qis chosen by the market, via the price of the defaultable asset. The Hodges buying and selling spreads at timet= 0 are defined as

S^∗(0, T) =−1

T lnD^∗_F(0, T) B(0, T) and

S∗(0, T) =−1

T lnD^F_∗(0, T) B(0, T) ,

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1.2. HODGES PRICES RELATIVE TO THE REFERENCE FILTRATION15

respectively. Likewise, therisk-neutral spread at timet= 0 is given as S⁰(0, T) =−1

T lnD⁰(0, T) B(0, T) .

SinceD^∗_F(0,0) =D_∗^F(0,0) =D⁰(0,0) = 1, the respectivebackward short spreads at time t= 0 are given by the following limits (provided the limits exist)

s^∗(0) = lim

T↓0S^∗(0, T) =−d⁺lnD^∗_F(0, T) dT

¯¯

¯T=0−r and

s∗(0) = lim

T↓0S∗(0, T) =−d⁺lnD^F_∗(0, T) dT

¯¯

¯T=0−r, respectively. We also set

s⁰(0) = lim

T↓0S⁰(0, T) =−d⁺lnD⁰(0, T) dT

¯¯

¯T=0−r.

Assuming, as we do, that the processes FeT and FT are absolutely continuous with respect to the Lebesgue measure, and using the observation that the restriction of ˜Qto FT is equal toQ, we find out that

D^∗_F(0, T)

B(0, T) = −1

%EQ

©ln¡

e^−%(1−FT) +FT

¢ª

= −1

%EQ

n ln

³ e^−%

³ 1−

Z _T

0

ftdt

´ +

Z _T

0

ftdt

´o , and

D^F_∗(0, T)

B(0, T) = 1

%E_Q© ln¡

e^%(1−F_T) +F_T¢ª

= 1

%EQ

n ln³

e^%³ 1−

Z _T

0

ftdt´ +

Z _T

0

ftdt´o . Furthermore,

D⁰(0, T)

B(0, T) =EQ(1−FeT) =EQ

³ 1−

Z _T

0

fetdt

´ . Consequently,

s^∗(0) = 1

%

¡e^%−1¢

f0, s∗(0) = 1

%

¡1−e^−%¢ f0,

ands⁰(0) =fe0.Now, if we postulate, for instance, thats∗(0) =s⁰(0) (it would be the case if the market price is the selling Hodges price), then we must have

fe0=1

%

¡1−e^−%¢ f0= 1

%

¡1−e^−%¢ γ0

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so that eγ0 < γ0. Similar calculations can be made for anyt∈[0, T[. It can be noticed that, if the market price is the selling Hodges price,fe0 corresponds to the risk-neutral intensity at time 0 whereas γ0 is the historical intensity. The reader may refer to Bernis and Jeanblanc (2002) for other comments.

1.2.4 Recovery paid at time of default

Assume now that the recovery payment is made at time τ, if τ ≤ T. More precisely, let (X_t³, t≥0) be some F-adapted process. If τ < T, the payoffX_t³ is paid at time t=τ and re-invested in the riskless asset. The terminal global wealth is now

(V_T^v−p(π) +X1)11T <τ+ (V_T^v−p(π) +Zτ)11τ≤T

where Zt=X_t³e^r(T^−t), and we are still interested in optimization of wealth at timeT.

The corresponding optimization problem is (Pb_F^Z) : V(v−p) := sup

φ∈Φ(F)

EP

¡U(V_T^v−p(φ) +X1)11T <τ+U(V_T^v−p(φ) +Zτ)11τ≤T

¢.

The supremum part above can be written as sup

φ∈Φ(F)

EP

©Je¡

V_T^v−p(φ)¢ª , where, forP-a.e. ω∈Ω,

Je(y, ω) =U(y+X1(ω))(1−FT(ω)) + Z _T

0

U(y+Zt(ω))ftdt.

Let us introduce the conditional indifference hedge:

Φ :=−1

%ln

³Z T

0

exp(−%Zt)ftdt+ exp(−%X1)(1−FT)

´

. (1.12)

We have the following result,

Theorem 1.2.1 Assume thatsup_0≤t≤Texp(−%Zt)andexp(−%X¹)areQ-integrable.

The Hodges price of (X¹, X_·³) is the arbitrage price of the indifference condi- tional hedgeΦ, the pay-off of which is given by (1.12).

Proof. Observe first that problem (Pb_F^Z) can be written as V(x−p) = sup

φ∈Φ(F)

EP

©exp¡

−%[V_T^v−p(φ) + Φ]¢ª .

Thus, problem (Pb_F^Z) is the same as problem (P_F^X) withX= Φ, so that finding the Hodges price of (X¹, X_·³) amounts to finding the Hodges price of Φ. But now, the claim Φ is aFT-measurable random variable. Thus, its Hodges price must coincide with its arbitrage price.

¤ Observe that Φ is a pay-off at time T. However, at time of default selling the derivative Φ yields enough money to obtain the utility needed.

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1.3. OPTIMIZATION PROBLEMS AND BSDES 17

1.3 Optimization Problems and BSDEs

We now consider strategiesφthat are predictable with respect to the full filtra- tionG. The dynamics of the risky asset (St, t≥0) are

dSt=St(νdt+σdWt). (1.13) In order to simplify notation, we denote by (ξt, t ≥ 0) the G-predictable process such that dMt=dHt−ξtdt is aG-martingale, i.e.,ξt=γt(1−Ht−).

(See equation (1.1).)

We assume for simplicity thatr= 0, so that now θ=ν/σ, and we change the definition of admissible portfolios to one that will be more suitable for problems considered here: instead of using the number of shares φ as before, we set π = φS, so that π represents the value invested in the risky asset.

In addition, we adopt here the following relaxed definition of admissibility of trading strategies.

Definition 1.2 The class Π(F) (Π(G), respectively) ofF-admissible(G-admissi- ble, respectively) trading strategies is the set of allF-adapted (G-predictable, respectively) processesπsuch thatR_T

0 π²_tdt <∞,P-a.s.

The wealth process of a strategyπsatisfies dVt(π) =πt

¡νdt+σdWt

¢. (1.14)

LetX be a given contingent claim, represented by aGT-measurable random variable. We shall study the following problem:

sup

π∈Π(G)

EP

©u¡

V_T^v(π) +X¢ª .

in the case of the exponential utility. In a last step, for the determination of Hodges’ price, we shall changev intov−p.

1.3.1 Optimization Problem

Our first goal is to solve an optimization problem for an agent who sells a claim X. To this end, it suffices to find a strategyπ∈ Π(G) that maximizes EP(u(V_T^v(π)+X)), where the wealth process (Vt=V_t^v(π), t≥0) (for simplicity, we shall frequently skipv andπfrom the notation) satisfies

dVt=φtdSt=πt(νdt+σdWt), V0=v.

We consider the exponential utility function u(x) = 1−e^−%x, with % > 0.

Therefore, sup

π∈Π(G)

EP

= 1− inf

π∈Π(G)EP

¡e^−%V^T^v^(π)e^−%X¢ . We shall give three different methods to solve inf_π∈Π(G)EP

¡e^−%V^T^v^(π)e^−%X¢ .

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Direct method

We describe the idea of a solution; the idea follows the dynamic programming principle.

Suppose that we can find a G-adapted process (Zt, t≥0) withZT =e^−%X, which depends only on the claimX and parameters%, σ, ν, and such that the process (e^−%V^t^v^(π)Zt, t≥0) is a (P,G)-submartingale for any admissible strategy π, and is a martingale under Pfor some admissible strategy π^∗ ∈ Π(G).

Then, we would have

EP(e^−%V^T^v^(π)ZT)≥e^−%V⁰^v^(π)Z0=e^−%vZ0

for anyπ∈ Π(G), with equality for some strategy π^∗ ∈Π(G). Consequently, we would obtain

π∈Π(G)inf EP

¡e^−%V^T^v^(π)e^−%X¢

=EP

¡e^−%V^T^v^(π^∗⁾e^−%X¢

=e^−%vZ0, (1.15) and thus we would be in the position to conclude thatπ^∗is an optimal strategy.

In fact, it will turn out that in order to implement the above idea we shall need to restrict further the class ofG-admissible trading strategies to such strategies that the ”martingale part” in (1.17) determines a true martingale rather than a local-martingale.

In what follows, we shall use the BSDE framework. We refer the reader to the chapter by ElKaroui and Hamad´ene in this volume and to the papers of Barles (1997), Rong (1997) and the thesis of Royer (2002) for BSDE with jumps.

We shall search the process Z in the class of all processes satisfying the following BSDE

dZt=ztdt+bztdWt+zetdMt, t∈[0, T[, ZT =e^−%X, (1.16) where the processz = (zt, t≥0) will be determined later (see equation (1.19) below). By applying Itˆo’s formula, we obtain

d(e^−%V^t) =e^−%V^t¡¡₁

2%²π_t²σ²−%πtν¢

dt−%πtσ dWt

¢, so that

d(e^−%V^tZt) = e^−%V^t¡

zt+Zt(¹₂%²π²_tσ²−%πtν)−%πtσbzt

¢dt +e^−%V^t¡

(bz_t−%π_tσZ_t)dW_t+ez_tdM_t¢

. (1.17)

Let us chooseπ^∗= (π_t^∗, t≥0) such that it minimizes, for everyt, the following expression

Zt

¡₁

2%²π_t²σ²−%πtν¢

−%πtσbzt=−%πt(νZt+σbzt) +¹₂%²π_t²σ²Zt. It is easily seen that, assuming that the processZ is strictly positive, we have

π^∗_t = νZt+σbzt

%σ²Zt = 1

%σ

³ θ+ bzt

Zt

´

. (1.18)

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1.3. OPTIMIZATION PROBLEMS AND BSDES 19 Now, let us choose the processz as follows

z_t=Z_t¡

%π_t^∗ν−¹₂%²(π_t^∗)²σ²¢

+%π_t^∗σbz_t

= %π_t^∗(Ztν+σbzt)−¹₂%²(π^∗_t)²σ²Zt=(νZt+σbzt)² 2σ²Zt

= 1

2θ²Zt+θbzt+ 1

2Ztzb_t². (1.19)

Note that with the above choice of the process z the drift term in (1.17) is positive for any admissible strategyπ, and it is zero forπ=π^∗.

Given the above, it appears that we have reduced our problem to the problem of solving the BSDE (1.16) with the processzgiven by (1.19), i.e.,





dZt= (¹₂θ²Zt+θbzt+_2Z¹

tzb_t²)dt+bztdWt+zetdMt, t∈[0, T), ZT =e^−%X.

(1.20) In fact, assuming that (1.20) admits a solution (Z,z,bz), so that withe π =π^∗ the ”martingale part” in (1.17) is a true martingale part rather than a local- martingale part, then the process

π^∗_t = 1

%σ

³ θ+ bzt

Zt

´ , will be an optimal portfolio, i.e.,

π∈Π(G)inf EP

¡e^−%V^T^v^(π)e^−%X¢

=EP

¡e^−%V^T^v^(π^∗⁾e^−%X¢ .

However, this BSDE is not of standard. This is a BSDE with jumps, and existence theorems and comparison theorems are known only if the driver is Lipschitz. Hence, we shall establish the existence using another approach, an approach due to Mania and Tevzadze.

Mania and Tevzadze approach

In a very general setting, when the underlying asset is of the form dSt=dµt+λtdhµit

where µ is a continuous local martingale, Mania and Tevzadze (2003a) study the family of processes

Vt(v) = max

φ EP(U(v+ Z _T

t

φsdSs)|Gt)

wherevis a real-valued deterministic parameter. They establish that the process (V(t, v) = Vt(v), t ≥ 0) (which depends on the parameter v) is solution of a BSDE

dV(t, v) = 1 2

1

Vvv(t, v)(ϕv(t, v) +λtVv(t, v))²dhµit+ϕ(t, v)dµt+dNt(v),

V(T, v) = U(v), (1.21)

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whereN is a martingale orthogonal toµ, and the optimal portfolio is proved to be

φ^∗_t =−Stϕv(t, V_t^∗)−λtVv(t, V_t^∗) Vvv(t, V_t^∗) .

Analysis of the proof of the equation (1.4) in Mania and Tevzadze (2003a) reveals that their results carry to the case when

Vt(v) = max

φ E(U(v+ Z _T

t

φsdSs+X)|Gt)

for a claim X satisfying appropriate integrability conditions, in which case the process (V_t(v), t ≥ 0) satisfies the BSDE (1.21) with terminal condition V(T, v) = U(v+X). We note however that there are several technical conditions postulated in Mania and Tevzadze (2003a) that need to be verified before their results can be adopted.

In the particular case when the dynamics of the underlying asset follows dSt=St(νdt+σdWt)

we havedµt=StσdWtandλt=ν/(Stσ²), and the BSDE (1.21) reads dV(t, v) = S_t²σ²

2Vvv(t, v)(ϕ(t, v) + ν

σ²StVv(t, v))²dt+ϕ(t, v)StσdWt+dNt

= 1

2σ²Vvv(t, v)(ϕ(t, v)σ²St+νVv(t, v))²dt+ϕ(t, v)StσdWt+dNt

whereN is a martingale orthogonal toW (hence, in our setting a martingale of the formR_t

0ψsdMs). The terminal condition is V(T, v) =U(v+X). and the optimal portfolio is

φ^∗_t =−Stϕv+Vvν/(σ²St) Vvv .

Here,U is an exponential function. Thus, it is convenient to factorize processV asV(t, v) =e^−%vZt, and to factorize processϕasϕ(t, v) =ϕ(t)eb ^−%v. It follows thatZ satisfies

dZt=

(ϕ(t) +b ν σ²StZt)²

2Zt S_t²σ²dt+ϕ(t)Sb tσdWt+dNt, ZT =e^−%X. Settingzbt=ϕ(t)σSb t, we get

dZt= 1 2Zt

(bzt+ ν

σZt)²dt+zbtdWt+dNt, ZT =e^−%X,

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1.3. OPTIMIZATION PROBLEMS AND BSDES 21 which is exactly equation (1.19), where N is a stochastic integral w.r.t. the martingale M, orthogonal to W. Thus, it appears that a solution to equation (1.19) is given as

Zt=e^%vV(t, v), zbt=ϕ(t)σSb t, and zet= dNt

dMt. The optimal portfolio is

σbzt+Ztν

%σ²Zt

which is exactly (1.18).

Remark. Analogous results follow from by Mania and Tevzadze (2003b) where a more general case of utility function is studied.

Duality Approach

We present now the duality approach (See for example Delbaen et al. (2002), or Mania and Tevzadze (2003b)). In the case dSt = St(νdt+σdWt), the set of equivalent martingale measure (emm) is the set of probability measures Q^ψ defined as

dQ^ψ|Gt =LtdP|Gt

where

dLt=Lt−(−θdWt+ψtdMt)

where ψ is a G-predictable process, with ψ > −1 and θ is the risk premium θ=ν/σ. Indeed, using Kusuoka representation theorem (1999), we know that any strictly positive martingale can be written of the form

dLt=Lt−(`tdWt+ψtdMt).

The discounted price of the default-free asset is a martingale under the change of probability, hence, it is easy to check that`t=−θ. (We have already noticed that the restriction of any emm to the filtrationFis equal toQ.) Let us denote by W_t^Q =Wt+θt andMct =Mt−R_t

0ψsξsds. The processes W^Q and Mcare Q^ψ martingales. Then,

Lt = exp µ

−θWt−1 2θ²t+

Z _t

0

ln(1 +ψs)dHs− Z _t

0

ψsξsds

¶

= exp µ

−θW_t^Q+θ²t 2 +

Z _t

0

ln(1 +ψs)dMcs+ Z _t

0

[(1 +ψs) ln(1 +ψs)−ψs]ξsds

¶

Hence, the relative entropy ofQ^ψ with respect toPis H(Q^ψ|P) =E_Q^ψ(lnLT) =E_Q^ψ

Ã1 2θ²T+

Z _T

0

[(1 +ψs) ln(1 +ψs)−ψs]ξsds

! .

INDIFFERENCE PRICING OF DEFAULTABLE CLAIMS

INDIFFERENCE PRICING OF DEFAULTABLE CLAIMS

Tomasz R. Bielecki

Department of Applied Mathematics Illinois Institute of Technology

Chicago, USA bielecki@iit.edu Monique Jeanblanc

Equipe d’Analyse et Probabilit´es Universit´e d’´ Evry-Val-d’Essonne

Evry, France ´

monique.jeanblanc@maths.univ-evry.fr

July 7, 2004

Contents

Chapter 1

INDIFFERENCE PRICING OF

DEFAULTABLE CLAIMS

1.1 Preliminaries

1.1.1 Default-Free Market

1.1.2 Default Time

1.1.3 Defaultable Claims

1.1.4 Hodges Indifference Price

1.2 Hodges prices relative to the reference fil- tration

1.2.1 Solution of Problem (P

)

1.2.2 Exponential Utility: Explicit Computation of the Hodges Price

1.2.3 Risk-Neutral Spread Versus Hodges Spreads

1.2.4 Recovery paid at time of default

1.3 Optimization Problems and BSDEs

1.3.1 Optimization Problem