SALDOS Y OPERACIONES CON ENTIDADES VINCULADAS

ζ = 0 > −1,

so that the martingale measure Q¹ exists and is unique.

Observe that, since ζ = 0, the default intensity under Q¹coincides here with the default intensity under the real-life probability Q. It is interesting to note that, in a more general situation when all three assets are defaultable with non-zero recovery, the default intensity under Q¹ coincides with the default intensity under the real-life probability Q if and only if the process Y¹ is continuous.

For more details, the interested reader is referred to Bielecki at al. [14] where the general case is studied.

4.3.4 Two Defaultable Assets with Zero Recovery

We shall now assume that we have only two assets and both are defaultable assets with zero recovery.

This case was recently examined by Carr [44], who studied an imperfect hedging of digital options.

Note that here we present results for replication, that is, perfect hedging.

We shall briefly outline the analysis of hedging of a survival claim. Under the present assumptions, we have, for i = 1, 2,

dY_tⁱ= Y_t−ⁱ ¡

µi,tdt + σi,tdWt− dMt

¢, (4.39)

where W is a one-dimensional Brownian motion, so that

Y_t¹= 1{t<τ }Ye_t¹, Y_t²= 1{t<τ }Ye_t², with the pre-default prices governed by the SDEs

d eY_tⁱ= eY_tⁱ¡

(µi,t+ γt) dt + σi,tdWt

¢. (4.40)

The wealth process V associated with the self-financing trading strategy (φ¹, φ²) satisfies, for every t ∈ [0, T ],

Vt= Y_t¹ µ

V₀¹+ Z _t

0

φ²_ud eY_u^2,1

¶ ,

4.3. MARTINGALE APPROACH 155 where eY_t^2,1= eY_t²/ eY_t¹. Since both primary traded assets are subject to zero recovery, it is clear that the present model is incomplete, in the sense, that not all defaultable claims can be replicated.

We shall check in what follows that, under the assumption that the driving Brownian motion W is one-dimensional, all survival claims satisfying mild technical conditions are hedgeable, however.

In the more realistic case of a two-dimensional noise, we will still be able to hedge a large class of survival claims, including options on a defaultable asset and options to exchange defaultable assets.

Hedging a Survival Claim

For the sake of expositional simplicity, we assume in this subsection that the driving Brownian motion W is one-dimensional. Arguably, this is not the right choice, since we deal here with two risky assets, so that they will be perfectly correlated. However, this assumption is convenient for the expositional purposes, since it ensures the model completeness with respect to survival claims.

We will later relax this temporary assumption so it is fair to say that this assumption is not crucial.

We shall now argue that in a market model with two defaultable assets that are subject to zero recovery, the replication of a survival claim (X, 0, τ ) is in fact equivalent to replication of an associated promised payoff X using the pre-default price processes.

Lemma 4.3.6 If a trading strategy φⁱ, i = 1, 2, based on pre-default values eYⁱ, i = 1, 2, is a repli-cating strategy for an FT-measurable claim X, that is, if φ is such that the process

Vet(φ) = φ¹_tYe_t¹+ φ²_tYe_t² satisfies, for every t ∈ [0, T ],

d eVt(φ) = φ¹_td eY_t¹+ φ²_td eY_t², VeT(φ) = X,

then for the process Vt(φ) = φ¹_tY_t¹+ φ²_tY_t² we have, for every t ∈ [0, T ], dVt(φ) = φ¹_tdY_t¹+ φ²_tdY_t²,

VT(φ) = 1{T <τ }X.

This means that the strategy φ replicates the survival claim (X, 0, τ ).

Proof. It is clear that Vt(φ) = 1{t<τ }Vt(φ) = 1{t<τ }Vet(φ). From the equality

φ¹_tdY_t¹+ φ²_tdY_t²= −(φ¹_tYe_t¹+ φ²_tYe_t²) dHt+ (1 − Ht−)(φ¹_td eY_t¹+ φ²_td eY_t²), it follows that

φ¹_tdY_t¹+ φ²_tdY_t²= − eV_t(φ) dH_t+ (1 − H_t−)d eV_t(φ), that is,

φ¹_tdY_t¹+ φ²_tdY_t²= d(1{t<τ }Vet(φ)) = dVt(φ).

It is also easily seen that the equality VT(φ) = X1{T <τ }holds. ¤ Combining the last result with Lemma 4.2.1, we see that a strategy (φ¹, φ²) replicates a survival claim (X, 0, τ ) whenever we have

Ye_T¹³ x +

Z _T

0

φ²_td eY_t^2,1´

= X

for some constant x and some F-predictable process φ², where, in view of (4.40), d eY_t^2,1 = eY_t^2,1³¡

µ2,t− µ1,t+ σ1,t(σ1,t− σ2,t)¢

dt + (σ2,t− σ1,t) dWt

´ .

We introduce a probability measure eQ, equivalent to P on (Ω, GT), and such that eY^2,1 is an F-martingale under eQ. It is easily seen that the Radon-Nikod´ym density η satisfies, for t ∈ [0, T ],

d eQ |Gt = ηtdP |Gt= Et

µZ _·

0

θsdWs

¶ dP |Gt

with

θt=µ2,t− µ1,t+ σ1,t(σ1,t− σ2,t) σ1,t− σ2,t

,

provided, of course, that the process θ is well defined and satisfies suitable integrability conditions.

We shall show that a survival claim is attainable if the random variable X( eY_T¹)⁻¹ is eQ-integrable.

Indeed, the pre-default value eVtat time t of a survival claim equals Vet= eY_t¹EQe

¡X( eY_T¹)⁻¹| Ft

¢

and, from the predictable representation theorem, we deduce that there exists a process φ² such that

E_Qe¡

X( eY_T¹)⁻¹| Ft

¢= E_eQ¡

X( eY_T¹)⁻¹¢ +

Z _t

0

φ²_ud eY_u^2,1.

The component φ¹ of the self-financing trading strategy φ = (φ¹, φ²) is then chosen in such a way that, for every t ∈ [0, T ],

φ¹_tYe_t¹+ φ²_tYe_t²= eVt.

To conclude, by focusing on pre-default values, we have shown that the replication of survival claims can be reduced here to classic results on replication of (non-defaultable) contingent claims in a default-free market model.

Option on a Defaultable Asset

In order to get a complete model with respect to survival claims, we postulated in the preceding subsection that the driving Brownian motion in dynamics (4.39) is one-dimensional. This assumption is questionable, since it clearly implies the perfect correlation between risky assets. However, we may relax this restriction and work instead with the two correlated one-dimensional Brownian motions.

The model will no longer be complete, but options on a defaultable asset will still be attainable.

The payoff of a (non-vulnerable) call option written on the defaultable asset Y² equals CT = (Y_T²− K)⁺= 1_{{T <τ }}( eY_T²− K)⁺,

so that it is natural to interpret this contract as a survival claim with the promised payoff X = ( eY_T²− K)⁺.

To deal with this option in an efficient way, we consider a model in which dY_tⁱ= Y_t−ⁱ ¡

µi,tdt + σi,tdW_tⁱ− dMt

¢,

where W¹ and W² are two one-dimensional correlated Brownian motions with the instantaneous correlation coefficient ρt. More specifically, we assume that Y_t¹= D⁰(t, T ) = 1{t<τ }De⁰(t, T ) repre-sents a defaultable ZCB with zero recovery, and Y_t² = 1_{{t<τ }}Ye_t²is a generic defaultable asset with zero recovery. Within the present setup, the payoff can also be represented as follows

CT = (Y_T²− KY_T¹)⁺= g(Y_T¹, Y_T²),

where g(y1, y2) = (y2− Ky1)⁺, and thus it can also be seen as an option to exchange the second asset for K units of the first asset.