Encuesta a Mipymes

where the process V^v−p(φ) is solution of (6.1) with the initial condition V₀^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 filtration F.

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 claim X at price p, and invests the remaining wealth v − p in the financial market, using a strategy adapted to the enlarged filtration G. 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 , where Φ(G) is the class of all G-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 6.1.1 For a given initial endowment v, the F-Hodges buying price of the defaultable claim X is the real number p^∗_F(v) such that

V(v) = V_X^F¡

v − p^∗_F(v)¢ .

Similarly, the G-Hodges buying price of X is the real number p^∗_G(v) such that V(v) = V_X^G¡

v−p^∗_G(v)¢ . Remark 6.1.1 We can define the F-Hodges selling price p^F_∗(v) of X by considering −p, where p is the buying price of −X, as specified in Definition 6.1.1.

If the contingent claim X is FT-measurable, then (See Rouge and ElKaroui[79]) the F- and the G-Hodges selling and buying prices coincide with the hedging price of X, i.e.,

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

6.2 Hodges prices relative to the reference filtration

In this section, we study the problem (P_F^X) (i.e., we use strategies adapted to the reference filtra-tion). First, we compute the value function, i.e., V_X^F(v − p). Next, we establish a quasi-explicit representation for the Hodges price of X 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.

6.2.1 Solution of Problem (P

_F^X

)

In view of the particular form of the defaultable claim X 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 are F-adapted, the terminal wealth V_T^v−p(φ) is an FT-measurable random variable. Consequently, it holds that

for every ω ∈ Ω and y ∈ IR. The real-valued mapping JX(·, ω) is strictly concave and increasing. Con-sequently, 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 of JX with 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 6.2.1 The optimal terminal wealth for the problem (P_F^X) is given by V_T^v−p,∗= IX(λ^∗ζT), 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)). (6.3) Proof: It is well known (see, e.g., Karatzas and Shreve [130]) that, in order to find the optimal wealth it is enough to maximize u(∆) over the set of square-integrable and FT-measurable random variables ∆, subject to the budget constraint, given by

EP(ζT∆) ≤ v − p.

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

EP

where the last inequality follows from the budget constraint and the choice of λ^∗. Hence, for any φ ∈ 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 function JX, the optimal strategy

is unique. ¤

6.2.2 Exponential Utility: Explicit Computation of the Hodges Price

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

Proposition 6.2.1 Let u(x) = 1 − exp(−%x) for some % > 0. Assume that the random variables ζTe^−%Xi, i = 1, 2 are P-integrable. Then the F-Hodges buying price is given by where the FT-measurable random variable Ψ equals

Ψ = −1

%ln¡

(1 − FT)e^−%X1+ FTe^−%X2¢

. (6.4)

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

EP

©u¡

X − Ψ¢ ¯

¯ F_Tª

= 0. (6.5)

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

The budget constraint EP(ζTV_T^v,∗) = v implies that the Lagrange multiplier µ^∗satisfies 1

The solution to the problem (P_F^X) is obtained in a general setting in Lemma 6.2.1. 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,

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

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

¡exp(−%V_T^v,∗)¢

= EP

¡exp(−%(V_T^v−p∗,∗+ X))¢ ,

where µ^∗ and λ^∗ are given by (6.6) and (6.7), respectively. After substitution and simplifications, we arrive at the following equality

E_Pn exp³

− %¡

E_P(ζ_TΨ) − p^∗+ X − Ψ¢´o

= 1. (6.8)

It is easy to check that

EP

¡e^−%(X−Ψ)¯

¯ FT

¢= 1 (6.9)

so that equality (6.5) holds, and EP

¡e^−%(X−Ψ)¢

= 1. Combining (6.8) and (6.9), we conclude that

p^∗_F(v) = EP(ζTΨ). ¤

We briefly provide the analog of (6.4) for the F-Hodges selling price of X . We have p^F_∗(v) = EP(ζTΨ),e where

Ψ =e 1

%ln¡

(1 − FT)e^%X1+ FTe^%X2¢

. (6.10)

Remark 6.2.1 It is important to notice that the F-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 (6.5), the random variable Ψ will be called the indifference conditional hedge.

From concavity of the logarithm function we obtain

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

Hence, using that ζT is FT-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 [58]) 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 probabilities FT and 1 − FT.

6.2.3 Risk-Neutral Spread Versus Hodges Spreads

In our setting the price process of the T -maturity unit discount Treasury (default-free) bond is B(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 (6.10) 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 maturity T )

D_F^∗(0, T ) = −1

%EP

©ζTln(e^−%(1 − FT) + FT)ª

and

D_∗^F(0, T ) = 1

%EP

©ζTln(e^%(1 − FT) + FT)ª ,

respectively.

Let eQ be a risk-neutral probability for the filtration G, that is, for the enlarged market. The

“market” price at time t = 0 of defaultable bond, denoted as D⁰(0, T ), is thus equal to the expec-tation under eQ of its discounted pay-off, that is,

D⁰(0, T ) = EeQ Q is chosen by the market, via the price of the defaultable asset. The Hodges buying and sellinge spreads at time t = 0 are defined as

S^∗(0, T ) = −1 respectively. Likewise, the risk-neutral spread at time t = 0 is given as

S⁰(0, T ) = −1

T lnD⁰(0, T ) B(0, T ) .

Since D^∗_F(0, 0) = D_∗^F(0, 0) = D⁰(0, 0) = 1, the respective backward 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 eFT and FT are absolutely continuous with respect to the Lebesgue measure, and using the observation that the restriction of eQ to FT is equal to Q, we find out that

Consequently,

s^∗(0) = 1

%

¡e^%− 1¢

f0, s∗(0) =1

%

¡1 − e^−%¢ f0,

and s⁰(0) = ef0. Now, if we postulate, for instance, that s∗(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

so that eγ0 < γ0. Similar calculations can be made for any t ∈ [0, T [. It can be noticed that, if the market price is the selling Hodges price, ef0 corresponds to the risk-neutral intensity at time 0 whereas γ0 is the historical intensity. The reader may refer to Bernis and Jeanblanc [11] for other comments.

6.2.4 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 payoff X_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 Z_t= X_t³e^{r(T −t)}, and we are still interested in optimization of wealth at time T . The corresponding optimization problem is

( bP_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, for P-a.e. ω ∈ Ω,

J(y, ω) = U (y + Xe 1(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)

´

. (6.11)

We have the following result,

Th´eor`eme 6.1 Assume that sup_0≤t≤Texp(−%Zt) and exp(−%X¹) are Q-integrable. The Hodges price of (X¹, X_·³) is the arbitrage price of the indifference conditional hedgeΦ, the pay-off of which is given by (6.11).

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

φ∈Φ(F )

EP

©exp¡

−% [V_T^v−p(φ) + Φ]¢ª .

Thus, problem ( bP_F^Z) is the same as problem (P_F^X) with X = 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 a FT-measurable random variable. Thus, its Hodges price must coincide with its arbitrage price.

In document Follow this and additional works at: (página 34-39)