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) =VXF¡
v−p∗F(v)¢ .
Similarly, theG-Hodges buying priceofX is the real numberp∗G(v) such that V(v) =VXG¡
v−p∗G(v)¢ .
Remark. We can define theF-Hodges selling pricepF∗(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) =pG∗(v) =pF∗(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 (PFX) (i.e., we use strategies adapted to the reference filtration). First, we compute the value function, i.e.,VXF(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
FX)
In view of the particular form of the defaultable claimX it follows that VTv−p,X(φ) = 11{τ >T}(VTv−p(φ) +X1) + 11{τ≤T}(VTv−p(φ) +X2).
Since the trading strategies areF-adapted, the terminal wealth VTv−p(φ) is an FT-measurable random variable. Consequently, it holds that
EP
h u¡
VTv−p,X(φ)¢i
=
1.2. HODGES PRICES RELATIVE TO THE REFERENCE FILTRATION11 con-cave and increasing. Consequently, for any ω ∈Ω, we can define the mapping IX(z, ω) by setting IX(z, ω) = ¡
JX0 (·, ω)¢−1
(z) for z ∈ IR, where (JX0 (·, ω))−1 denotes the inverse mapping of the derivative ofJXwith respect to the first vari-able. 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 prob-lem (PFX),
Lemma 1.1 The optimal terminal wealth for the problem (PFX) is given by VTv−p,∗=IX(λ∗ζT),P-a.s., for some λ∗ such that
v−p=EP
¡ζTVTv−p,∗¢
. (1.3)
Thus the optimal global wealth equals VTv−p,X,∗=VTv−p,∗+X =IX(λ∗ζT) +X and the value function of the objective criterion for the problem(PFX)is
VXF(v−p) =EP(u(VTv−p,X,∗)) =EP(u(IX(λ∗ζT) +X)). (1.4) Proof. It is well known (see, e.g., Karatzas and Shreve (1998)) that, in or-der 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
where the last inequality follows from the budget constraint and the choice of λ∗. Hence, for anyφ∈Φ(F),
EP
©JX(VTv−p(φ))−JX(VTv−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 con-cavity 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 VTv,∗=−1
%ln µµ∗ζT
%
¶ .
The budget constraintEP(ζTVTv,∗) =vimplies that the Lagrange multiplierµ∗ satisfies
1
%ln µµ∗
%
¶
=−1
%EP
¡ζTlnζT
¢−v. (1.7)
The solution to the problem (PFX) 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
JX0 (y) =% e−%y(e−%X1(1−FT) +e−%X2FT).
1.2. HODGES PRICES RELATIVE TO THE REFERENCE FILTRATION13
It follows that the optimal terminal wealth for the initial endowment v−pis VTv−p,∗=−1
where the Lagrange multiplier λ∗ is chosen to satisfy the budget constraint EP(ζTVTv−p,∗) =v−p, that is,
¿From definition, the F-Hodges buying price is a real numberp∗ =p∗F(v) such that
where µ∗ andλ∗ are given by (1.7) and (1.8), respectively. After substitution and simplifications, we arrive at the following equality
EP
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 pF∗(v) =EP(ζTΨ), wheree 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).
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¡ VTv,∗¢
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)
DF∗(0, T) =−1
%EP©
ζTln(e−%(1−FT) +FT)ª and
DF∗(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 lnDF∗(0, T) B(0, T) ,
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
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 re-striction of ˜Qto FT is equal toQ, we find out that be the case if the market price is the selling Hodges price), then we must have
fe0=1
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 (Xt3, t≥0) be some F-adapted process. If τ < T, the payoffXt3 is paid at time t=τ and re-invested in the riskless asset. The terminal global wealth is now
(VTv−p(π) +X1)11T <τ+ (VTv−p(π) +Zτ)11τ≤T
where Zt=Xt3er(T−t), and we are still interested in optimization of wealth at timeT.
The corresponding optimization problem is (PbFZ) : V(v−p) := sup
φ∈Φ(F)
EP
¡U(VTv−p(φ) +X1)11T <τ+U(VTv−p(φ) +Zτ)11τ≤T
¢.
The supremum part above can be written as sup
φ∈Φ(F)
EP
©Je¡
VTv−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 thatsup0≤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 (PbFZ) can be written as V(x−p) = sup
φ∈Φ(F)
EP
©exp¡
−%[VTv−p(φ) + Φ]¢ª .
Thus, problem (PbFZ) is the same as problem (PFX) 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.