BSDEs with default jump arXiv:1612.05681v1 [math.PR] 16 Dec 2016
Roxana Dumitrescu∗
Marie-Claire Quenez
†
Agn`es Sulem
‡
December 20, 2016
Abstract In this paper, we study the properties of nonlinear BSDEs driven by a Brownian motion and a martingale measure associated with a default jump with intensity process (λt ). We give a priori estimates for these equations and prove comparison and strict comparison theorems. These results are generalized to drivers involving a singular process. The special case of a λ-linear driver is studied, leading to a representation of the solution of the associated BSDE in terms of a conditional expectation of an adjoint exponential semi-martingale. We then apply these results to nonlinear pricing of European contingent claims in an imperfect financial market with a defaultable risky asset. The case of claims paying dividends is also included via the singular process.
Key-words: backward stochastic differential equations, default jump, nonlinear pricing, dividends
1
Introduction
The aim of the present paper is to study BSDEs driven by a Brownian motion and a martingale measure associated with a default jump process with intensity process (λt ). The applications we have in mind are the pricing and hedging issues for contingent claims in an imperfect financial market with default. The theory on BSDEs driven by a Brownian motion and a Poisson random measure has been studied extensively by several authors (we refer e.g. to Barles, Buckdahn and Pardoux [3], Royer [21], Quenez and Sulem [20]), and the present study relies on many arguments which are used in this literature. Nevertheless, the treatment of the default jump requires some specific adjustments which are not straightforward and we present here a rigorous analysis of these BSDEs with default jump. We moreover allow the driver of these equations to have some singular component, in the sense that the driver may be of the generalized form g(t, ω, y, z, k)dt + dDt(ω), where D is a finite variation c`al`ag process with square integrability conditions. This will allow us to treat the case of dividends ∗ Department of Mathematics, King’s College London, United Kingdom, email:
[email protected] † LPMA, Universit´e Paris 7 Denis Diderot, Boite courrier 7012, 75251 Paris cedex 05, France, email:
[email protected] ‡ INRIA Paris, France, and Universit´e Paris-Est, email:
[email protected]
1
in our financial application. Moreover, g is not necessarily Lipschitz with respect to y, which allows to consider the case of unbounded interest rate process. The paper is organized as follows: in Section 2, we provide the theory of BSDEs with default jumps, including a priori estimates, existence and uniqueness results, comparison and strict comparison theorems. Some interesting counterexamples of comparison theorems are given when the necessary assumptions for these theorems to hold are violated. In the special case of a λ-linear driver, where λ refers to the intensity of the jump process, we provide an explicit solution in terms of a conditional expectation of an adjoint exponential semi-martingale. We then turn to the financial application in section 3. We consider a financial market with a defaultable risky asset and we study pricing and hedging issues for a European option paying a payoff ξ at the maturity T and intermediate dividends modelized via a singular process D. The case of a perfect market model is first studied via the theory of λ-linear BSDEs with default jump, while the case of imperfections, expressed via the nonlinearity of the wealth dynamics, is then treated by the theory of nonlinear BSDEs with generalized driver developped in Section 2. In this setting, the pricing system is expressed g,D as a nonlinear expectation/evaluation E g,· : (ξ, D) 7→ E (ξ), induced by a nonlinear BSDE with default jump (solved under the primitive probability measure P ) with generalized driver g(t, ·)dt + dDt . Properties of consistency, monotonicity, convexity, non-arbitrage rely on the properties of the associated BSDE. As an illustrative example of market imperfections, we consider the case when the seller of the option is a large investor whose trading strategy may affect the market asset prices and the default intensity.
2
BSDEs with a default jump
2.1
Mathematical setup
Let (Ω, G, P) be a complete probability space equipped with two stochastic processes: a unidimensional standard Brownian motion W and a jump process N defined by Nt = 1ϑ≤t for any t ∈ [0, T ], where ϑ is a random variable which modelizes a default time. We assume that this default can appear at any time that is P (ϑ ≥ t) > 0 for any t ≥ 0. We denote by G = {Gt , t ≥ 0} the complete natural filtration of W and N. We suppose that W is a G-Brownian motion. Let (Λt ) be the predictable compensator of the non decreasing process (Nt ). Note that (Λt∧ϑ ) is then the predictable compensator of (Nt∧ϑ ) = (Nt ). By uniqueness of the predictable compensator, Λt∧ϑ = Λt , t ≥ 0 a.s. We assume that Λ is absolutely continuous w.r.t. Lebesgue’s measure,Rso that there t exists a nonnegative process λ, called the intensity process, such that Λt = 0 λs ds, t ≥ 0. Since Λt∧ϑ = Λt , λ vanishes after ϑ. We denote by M the compensated martingale which satisfies Z t Mt = Nt − λs ds . 0
Let T > 0 be the finite horizon. We introduce the following sets: 2
• S 2 is the set of G-adapted RCLL processes ϕ such that E[sup0≤t≤T |ϕt |2 ] < +∞. • A2 is the set of real-valued non decreasing RCLL adapted processes A with A0 = 0 and E(A2T ) < ∞. hR i T 2 2 2 • H is the set of G-predictable processes such that kZk := E 0 |Zt | dt < ∞ . •
H2λ
is the set of G-predictable processes such that
kUk2λ
hR i T 2 := E 0 |Ut | λt dt < ∞ .
In the following, P denotes the G-predictablehσ-algebra on Ωi × [0, T ]. R T ∧ϑ Note that for each U ∈ H2λ , we have kUk2λ = E 0 |Ut |2 λt dt because the G-intensity λ
vanishes after ϑ. Moreover, we can suppose that for each U in H2λ (= L2 (Ω×[0, T ], P, dP λt dt)), its representant, still denoted by U, vanishes after ϑ. Moreover, T is the set of stopping times τ such that τ ∈ [0, T ] a.s. and for each S in T , TS is the set of stopping times τ such that S ≤ τ ≤ T a.s. We recall the martingale representation theorem (see e.g. [15]): Lemma 2.1. Any G-local martingale m = (mt )0≤t≤T has the representation Z t Z t mt = m0 + zs dWs + ls dMs , ∀ t ∈ [0, T ] a.s. , 0
(2.1)
0
where z = (zt )0≤t≤T and l = (lt )0≤t≤T are predictable such that the two above stochastic integrals are well defined. If m is a square integrable martingale, then z ∈ H2 and l ∈ H2λ . We now introduce the following definitions. Definition 2.2 (Driver, λ-admissible driver). A function g is said to be a driver if g : [0, T ] × Ω × R3 → R; (ω, t, y, z, k) 7→ g(ω, t, y, z, k) which is P ⊗ B(R3 )− measurable, and such that g(., 0, 0, 0) ∈ H2 . A driver g is called a λ-admissible driver if moreover there exists a constant C ≥ 0 such that dP ⊗ dt-a.s. , for each (y, z, k), (y1, z1 , k1 ), (y2, z2 , k2 ), √ (2.2) |g(ω, t, y, z1, k1 ) − g(ω, t, y, z2, k2 )| ≤ C(|z1 − z2 | + λt |k1 − k2 |), and (g(ω, t, y1, z, k) − g(ω, t, y2, z, k))(y1 − y2 ) ≤ C|y1 − y2 |2 .
(2.3)
The positive real C is called the λ-constant associated with driver g.
1 √ Note that condition (2.2) holds if for example g is C in z, k with |∂z g| ≤ C and |∂k g| ≤ λt C. Moreover, condition (2.2) implies that for each t > ϑ, since λt = 0, g does depend on k. In other terms, for each (y, z, k), we have:
g(t, y, z, k) = g(t, y, z, 0),
t > ϑ dP ⊗ dt − a.s.
Moreover, condition (2.3) is satisfied if e.g. g is Lipschitz with respect to y uniformly with respect to ω, t, z, k (which corresponds to the usual case for BSDEs). It is also satisfied if g is non increasing with respect to y, or if g is C 1 in y with ∂y g ≤ C. 3
Definition 2.3 (BSDE with default jump). Let g be a λ-admissible driver, let ξ ∈ L2 (GT ). A process (Y, Z, K) in S 2 × H2 × H2λ is said to be a solution of the BSDE with default jump associated with terminal time T , driver g and terminal condition ξ if it satisfies: − dYt = g(t, Yt , Zt , Kt )dt − Zt dWt − Kt dMt ;
2.2
YT = ξ.
(2.4)
First properties of BSDEs with a default jump
RT For β > 0, φ ∈ IH 2 , and l ∈ IHλ2, we introduce the norms kφk2β := E[ 0 eβs φ2s ds], and RT kkk2λ,β := E[ 0 eβs ks2 λs ds]. In this section, we first show some a priori estimates for BSDEs with a default jump, from which we derive the existence and uniqueness of the solution. 2.2.1
A priori estimates for BSDEs with a default jump
Proposition 2.4. Let ξ 1 , ξ 2 ∈ L2 (FT ). Let g 1 and g 2 be two drivers. Suppose that g 1 is λ-admissible, that is satisfies conditions (2.2) and (2.3). For i = 1, 2, let (Y i , Z i , K i ) be a solution of the BSDE associated with terminal time T , driver g i and terminal conditions ξ 1 and ξ 2 . Let η, β > 0 be such that β ≥ η3 + 2C and η ≤ C12 . Let g¯(s) := g 1(s, Ys2 , Zs2 , Ks2 ) − g 2(s, Ys2 , Zs2 , Ks2 ). For each t ∈ [0, T ], we then have Z T βt 1 2 2 βT ¯ 2 e (Yt − Yt ) ≤ E[e ξ | Ft ] + η E[ eβs g¯(s)2 ds | Gt ] a .s. (2.5) t
Moreover, If η
0, we have √ for all non √ λ, y, z, k, 2 2 2y(Cz + Ck λ + g) ≤ yε2 + ε2 (Cz + Ck λ + g)2 ≤ yε2 + 3ε2 (C 2 y 2 + C 2 k 2 λ + g 2 ). Hence, Z T Z T βt ¯ 2 βs ¯ 2 βs ¯ 2 2 ¯ e Yt + E β e Ys ds + e (Zs + Ks λs )ds | Gt ≤ E eβT Y¯T2 | Gt t t Z T Z T Z T 1 βs ¯ 2 2 2 βs ¯ 2 2 2 βs 2 ¯ λs )ds + 3ε e Ys ds + 3C ε e (Z s + K e g¯s ds | Gt . (2.10) +E (2C + 2 ) s ε t t t Let us make the change of variable η = 3ǫ2 . Then, for each β, η > 0 chosen as in the proposition, these inequalities lead to (2.5). By integrating (2.5), we obtain (2.6). Using inequality (2.10), we derive (2.7). Remark 2.5. By classical results on the norms of semimartingales, one similarly shows g kIH 2 , where K is a positive constant only depending on T and that kY¯ kS 2 ≤ K E[ξ¯ 2 ] + k¯ C. 2.2.2
Existence and uniqueness result for BSDEs with a default jump
Using the above a priori estimates, we derive the following existence and uniqueness result. Proposition 2.6. Let g be a λ-admissible driver, let ξ ∈ L2 (GT ). There exists an unique solution (Y, Z, K) in S 2 × H2 × H2λ of the BSDE (2.4). Proof. The arguments are classical and a short proof is given for completeness. Let us first consider the case when the driver g(t) does not depend on the solution. By using the representation property of G-martingales (Lemma 2.1) together with classical computations, one can show that there exists a unique solution of the BSDE (2.4) associated with terminal condition ξ ∈ L2 (FT ) and driver process g(t) ∈ IH 2. Let us consider the case with a general λ-admissible driver g(t, y, z, k). Denote by IHβ2 the space IH 2 × IH 2 × IHλ2 equipped with the norm kY, Z, Kk2β := kY k2β + kZk2β + kKk2λ,β . We define a mapping Φ from IHβ2 into itself as follows. Given (U, V, L) ∈ IHβ2 , let (Y, Z, K) = Φ(U, V, L) be the the solution of the BSDE associated with driver g 1 (s) := g(s, Us , Vs , Ls ). Let us prove that the mapping Φ is a contraction from IH 2β into IH 2β . Let (U ′ , V ′ , L′ ) be another element of IH 2β and let (Y ′ , Z ′, k ′ ) := Φ(U ′ , V ′ , L′ ), that is, the solution of the 5
RBSDE associated with driver process g(s, Us′ , Vs′ , L′s ). ¯ = L − L′ , Y¯ = Y − Y ′ , Z¯ = Z − Z ′ , K ¯ = K − K ′. Set U¯ = U − U ′ , V¯ = V − V ′ , L ′ ′ ′ Let ∆g· := g(·, U, V, L) − g(·, U , V , L ). Using the estimates given in the last assertion of the above property, with η ≤ 2C1 2 and λ-constant equal to 0 (since the driver g 1 does not depend on the solution), we get ¯ 2 + kKk ¯ 2 ≤ η(T + 2)k∆gk2 ≤ η(T + 2)2C 2kU¯ k2 + kV¯ k2 + kLk ¯ 2 ), kY¯ k2β + kZk β λ,β β β β λ,β where the second inequality follows from the λ-admissible property of g with λ-constant C. 1 1 2 2 Choosing η = (T +2)4C 2 , we deduce k(Y , Z, K)kβ ≤ 2 k(U , V , K)kβ . Hence, Φ is a contraction and admits a unique fixed point (Y, Z, K) in IH 2β , which is the solution of BSDE (2.4). By similar arguments, we have the following generalized result. Proposition 2.7. [BSDEs with default jump and “generalized driver”] Let g be a λ-admissible driver, let ξ ∈ L2 (GT ) and let D be a finite variational RCLL adapted process with square integrable total variation process. There exists an unique solution (Y, Z, K) (also denoted by (Y D (T, ξ), Z D (T, ξ), K D (T, ξ))) in S 2 × H2 × H2λ of the BSDE associated with “generalized driver” g(t, ·)dt + dDt and terminal condition ξ, that is − dYt = g(t, Yt, Zt , Kt )dt + dDt − Zt dWt − Kt dMt ;
YT = ξ.
(2.11)
Remark 2.8. Let D be a finite variational RCLL adapted process. Its associated total variation process is square integrable if and only if D can be decomposed as follows: D = A − A′ , with A, A′ ∈ A2 .
2.3
λ-linear BSDEs with default jump
We introduce the notion of λ-linear BSDEs in our framework with default jump. Definition 2.9 (λ-linear driver). A driver g is called λ-linear if it is of the form: g(t, y, z, k) = ϕt + δt y + βt z + γt k λt ,
(2.12)
where (ϕt ) ∈ H2 , and where (βt ), (γt ) (respectively (δt )) are bounded (resp. upper bounded) R-valued predictable processes. Remark 2.10. Note first that a λ-linear driver is λ-admissible. We also stress on the fact that the λ-linear driver g does not depend on k after the default time, due to the presence of λ in the last term of (2.12). We will now prove that the solution of a λ-linear BSDE, that is the solution of BSDE (2.4) associated with a λ-linear driver, can be written as a conditional expectation via an exponential semimartingale. We first show a preliminary result. Proposition 2.11. Let (βs ) and (γs ) be two bounded real-valued G-predictable processes, and let (ζs ) be the process satisfying the forward SDE: dζs = ζs− (βs dWs + γs dMs ) with ζ0 = 1. The process (ζs ) satisfies the so-called Dol´eans-Dade formula, that is Z s Z Z s 1 s 2 ζs = exp{ βr dWr − β dr} exp{− γr λr dr}(1 + γϑ 1{s≥ϑ} ). 2 0 r 0 0 6
For each T > 0, the process (ζs )0≤s≤T is a martingale and satisfies E[sup0≤s≤T ζsp ] < +∞, for all p ≥ 2. Moreover, if γϑ ≥ −1 (resp. > −1) a.s. , then ζs ≥ 0 (resp. > 0) a.s. for each s ∈ [0, T ]. Proof. By definition, the process (ζs ) is a local martingale. Let T > 0. Let us show that E[sup0≤s≤T ζs2 ] < +∞. By Itˆo’s formula applied to ζs2 , we get dζs2 = 2ζs− dζs + d[ζ, ζ]s. We have d[ζ, ζ]s = ζs2− βs2 ds + ζs2− γs2 dNs . Now, ζs2− γs2 dNs = ζs2− γs2 dMs + ζs2− γs2 λs ds. We thus derive that dζs2 = ζs2− [2βs dWs + (2γs + γs2 )dMs + (βs2 + γs2 λs )ds]. It follows that ζ 2 is an exponential semimartingale which can be written: Z s 2 ζs = ηs exp{ (βr2 + γr2 λr ) dr}, (2.13) 0
where η is the exponential local martingale satisfying dηs = ηs− [2βs dWs + (2γs + γs2 )dMs ] with η0 = 1. By equality (2.13), the local martingale η is non negative. Hence, it is a supermartingale, which yields that E[ηT ] ≤ 1. Now, by assumption, β, γ and λ are bounded. By (2.13), it follows that E[ζT2 ] ≤ E[ηT ] K ≤ K, where K is a positive constant. By martingale inequalities, we derive that E[sup0≤s≤T ζs2] < +∞. Hence, the process (ζs )0≤s≤T is a martingale. By an induction argument as in the proof of Proposition A.1 in [20], one can prove that the integrability property of ζ holds for all integer p ≥ 2. The last assertion follows from the Dol´eans-Dade formula. Remark 2.12. The inequality γϑ ≥ −1 a.s. inequality γt ≥ −1, R +∞ is equivalent to Rthe +∞ dt ⊗ dP a.s.. Indeed, we have E[1 ]= E[ 1 dN ] = E[ 1γr −1. If Yt0 = 0, for some t0 ∈ [0, T ], then the inequalities in (2.22) are equalities. Proof.
¯ s = Ks1 − Ks2 , we have Setting Y¯s = Ys1 − Ys2 ; Z¯s = Zs1 − Zs2 ; K ¯ s dMs ; −dY¯s = hs ds − Z¯s dWs − K
Y¯s = ξ1 − ξ2 ,
where hs := f1 (s, Ys1 , Zs1 , Ks1 ) − f2 (s, Ys2 , Zs2 , Ks2 ). Set δs := (f1 (s, Ys1− , Zs1 , Ks1 ) − f1 (s, Ys2− , Zs1 , Ks1 ))/Y¯s if Y¯s 6= 0, and 0 otherwise. Set βs := (f1 (s, Ys2− , Zs1 , Ks1 ) − f1 (s, Ys2− , Zs2 , Ks1 ))/Z¯s if Z¯s 6= 0, and 0 otherwise. By classical linearization techniques, we obtain hs = δs Y¯s + βs Z¯s + f1 (s, Ys2 , Zs2 , Ks1 ) − f1 (s, Ys2 , Zs2 , Ks2 ) + ϕs , where ϕs := f1 (s, Ys2− , Zs2 , Ks2 ) − f2 (s, Ys2− , Zs2 , Ks2 ). Using the assumption (2.20), we get ¯ s λs + ϕs hs ≥ δs Y¯s + βs Z¯s + γs K
ds ⊗ dP − a.s.
(2.23)
Since f1 satisfies condition (2.3), the process (δs ) is upper bounded. Also, since f1 satisfies condition (2.2), the predictable processes (βs ) and (γs ) are bounded. Fix t ∈ [0, T ]. Let Γt,. be the process defined by (2.15). Since δ is upper bounded, and β, γ are bounded, it follows from Remark 2.14 that Γt,. ∈ S 2 . Also, since γs ≥ −1, we have Γt,. ≥ 0 a.s. By Itˆo’s formula and similar computations as in the proof of Theorem 2.13, we derive that ¯ s λs ) ds − dms , −d(Y¯s Γt,s ) = Γt,s (hs − δs Y¯s − βs Z¯s − γs K ¯ ∈ IH 2 and β, γ are where m is a martingale (because Γt,. ∈ S 2 , Y¯ ∈ S 2 , Z¯ ∈ IH 2 , K λ bounded).Using the inequality (2.23) together with the non negativity of Γ, we thus get −d(Y¯s Γt,s ) ≥ Γt,s ϕs ds − dms . By integrating between t and T and by taking the conditional expectation, we obtain Z T ¯ Yt ≥ E [ Γt,T (ξ1 − ξ2 ) + Γt,s ϕs ds | Gt ], 0 ≤ t ≤ T, a.s. (2.24) t
By assumption (2.22), ϕs ≥ 0 and ξ1 − ξ2 ≥ 0, which, together with the non negativity of Γt,T , implies that Y¯ = Y 1 − Y 2 ≥ 0. Suppose now that γt > −1. By Remark 2.14, Γt,T > 0 a.s. The last assertion of the theorem thus follows from (2.24). We give here some counter-examples related to the comparison theorems for BSDEs with a default jump.
9
Remark 2.17. Let us give an example which shows that in the case where assumption eqrefrobisis violated, that is γ takes values < −1 with positive measure, then, even if the terminal condition is nonnegative, the solution Y of the linear BSDE with default jump may take strictly negative values. Hence, in this case, the comparison theorem does not hold. Let f be a λ-linear driver which takes the particular form f (t, ω, k) = γkλt (ω),
(2.25)
where γ is a real constant (this corresponds to the driver of the λ-linear BSDE (2.14) with δs = βs = ϕs = 0 and γs = γ). At terminal time T , the associated adjoint process Γt,s satisfies (see (2.15) and Remark 2.14) : Z T Γt,T = exp{− γr λr dr}(1 + γ1{T ≥ϑ} ). t
Using now the definition of the default jump process N, we get: Z T Γt,T = exp{− γr λr dr}(1 + γNT ).
(2.26)
t
Let Y be the solution of the BSDE associated with driver f and terminal condition ξ := NT . The representation property of linear BSDEs with default jump (see (2.16)) gives Yt = E[Γt,T ξ|Gt ] = E[Γt,T NT |Gt ]. Hence, by (2.26), we get Yt = E[Γt,T NT |Gt ] = E[e−γ
RT t
λs ds
(1 + γNT )NT ) |Gt ] = (1 + γ)E[e−γ
RT t
λs ds
NT |Gt ],
(2.27)
where for the last equality we have used the fact that NT = NT2 . Equation (2.27) at time t = 0 shows that when γ < −1, we have Y0 < 0 although ξ ≥ 0 a.s. This example gives also a counter-example for the strict comparison theorem by taking γ = −1. Indeed, in this case, the relation (2.27) at time 0 yields that Y0 = 0. On the other hand, we have E[NT ] = 1 − P (ϑ > T ) > 0.
(2.28)
Under the additional assumption P (ϑ > T ) < 1, since ξ = NT , we get that P (ξ > 0) = P (NT > 0) > 0 even though Y0 = 0. Proposition 2.18 (Comparison theorem for BSDEs with “generalized driver”). Let ξ1 and ξ2 ∈ L2 (GT ). Let f1 be a λ-admissible driver. Let f2 be a driver. Let D 1 and D 2 be finite variational RCLL adapted processes with square integrable total variation. Let (Y i , Z i , K i ) be a solution in S 2 × IH 2 × IHλ2 of the BSDE −dYti = fi (t, Yti , Zti , Kti )dt + dDti − Zti dWt − Kti dMt ;
YTi = ξi .
Assume that there exists a bounded predictable process (γt ) satisfying (2.20) with (2.21) and ¯ := D 1 − D 2 is non decreasing. We that (2.22) holds. Moreover, suppose that the process D 1 2 then have Yt ≥ Yt for all t ∈ [0, T ]. (Strict comparison theorem) Suppose moreover that the inequality (2.21) is strict, that is γt > −1. If Yt0 = 0, for some t0 ∈ [0, T ], then the inequalities in (2.22) are equalities and D 1 − D 2 is constant on the time interval [t0 , T ]. 10
Proof.
Using the same arguments and notation as above, we obtain: Z T ¯ ¯ s ) | Gt ], 0 ≤ t ≤ T, Yt ≥ E [ Γt,T (ξ1 − ξ2 ) + Γt,s (ϕs ds + dD
a.s.
t
Hence, Y¯t ≥ 0 a.s. Suppose moreover thatR Yt0 = 0 a.s. We thus have ξ1 = ξ2 a.s. and ϕt = 0, t ∈ [t0 , T ] ˜ t := ¯ s , for each t ∈ [t0 , T ]. By assumption, D ˜ T ≥ 0 a.s. dt ⊗ dP -a.s. Set D Γ dD t0 ,t t0 ,s ˜ T | Gt0 ] = 0 a.s. Hence D ˜ T = 0 a.s. Now, since Γt0 ,s > 0, we have D ¯T − D ¯ t0 = D Rand E[−1 ˜ s . Hence D ¯T = D ¯ t0 a.s. Γ dD t0 ,T t0 ,s
3 3.1
Nonlinear pricing in a financial market with default Financial market with defaultable risky asset
We consider a financial market which consists of one risk-free asset, whose price process S 0 satisfies dSt0 = St0 rt dt, and two risky assets with price processes S 1 , S 2 evolving according to the equations: dSt1 = St1 [µ1t dt + σt1 dWt ] dSt2 = St2− [µ2t dt + σt2 dWt − dMt ]. Note that the second risky asset is defaultable with total default. We have St2 = 0, t ≥ ϑ a.s. All the processes σ 1 , σ 2 , r, µ1 , µ2 are predictable (that is P-measurable). We set σ = −1 −1 (σ 1 , σ 2 )′ . We suppose that σ 1 , σ 2 > 0, and σ 1 , σ 2 , µ1 , µ2 , λ, λ−1 ,(σ 1 ) , (σ 2 ) are bounded. Moreover, the interest rate r is lower bounded (not necessarily upper bounded). Let us consider an investor who can invest in the three tradable assets. At time 0, he invests the amount x ≥ 0 in the three assets. For i = 1, 2, we denote by ϕit the amount invested in the ith risky asset. Since after time ϑ, the investor cannot invest his wealth in the defaultable asset (since its price is equal to 0), we have ϕ2t = 0 for each t ≥ ϑ. A process ϕ = (ϕ1 , ϕ2 )′ belonging to H2 × H2λ is called a risky assets stategy. Let Ct be the cumulated cash amount which has been withdrawn from the market portfolio between time 0 and time t. The process C belongs to A2 , that is, C is an RCLL adapted non decreasing process satisfying C0 = 0 and E[CT2 ] < +∞. The value of the associated portfolio (or wealth) at time t is denoted by Vtx,ϕ,C . The amount invested in the non risky asset at time t is then given by Vtx,ϕ,C − (ϕ1t + ϕ2t ).
11
3.2
Pricing of European options with dividends in a perfect market model
In this section, we suppose that the market model is perfect. In this case, by the self financing condition, the wealth process V x,ϕ,C (simply denoted by V ) follows the dynamics: dVt = (rt Vt + ϕ1t (µ1t − rt ) + ϕ2t (µ2t − rt ))dt − dCt + (ϕ1t σt1 + ϕ2t σt2 )dWt − ϕ2t dMt = (rt Vt + ϕ1t θt1 σt1 − ϕ2t θt2 λt )dt − dCt + ϕ′t σt dWt − ϕ2t dMt , µ2t − σt2 θt1 − rt µ1t − rt 2 1{t≤ϑ} . 1 , θt := − where := 1 σt λt Let T > 0. Let ξ be a GT -measurable random variable belonging to L2 , and let D be a non decreasing process belonging to A2 . We consider a European option with maturity T , payoff ξ and cumulative dividend process D. For each t ∈ [0, T ], dDt represents the dividend amount paid to the owner of the option between time t and time t + dt. The aim is to price this contingent claim. Let us consider a seller who wants to sell the option at time 0. With the amount he receives at time 0 from the buyer, he wants to be able to construct a portfolio which allows him to pay to the buyer the amount ξ at time T and the intermediate dividends. By Proposition 2.7, we derive the existence of an unique process (X, Z, K) ∈ S 2 ×H2 ×H2λ solution of the following λ-linear BSDE: θt1
− dXt = −(rt Xt + (Zt + σt2 Kt )θt1 + Kt θt2 λt )dt + dDt − Zt dWt − Kt dMt ;
XT = ξ. (3.1)
The solution (X, Z, K) corresponds to the replicating portfolio. More precisely, the hedging risky assets stategy ϕ is such that ϕt ′ σt = Zt ; −ϕ2t = Kt ,
(3.2)
where ϕt ′ σt = ϕ1t σt1 + ϕ2t σt2 . Note that this defines a change of variables Φ defined by: Φ : H2 × H2λ → H2 × H2λ ; (Z, K) 7→ Φ(Z, K) := ϕ, where ϕ = (ϕ1 , ϕ2 ) is given by (3.2), which is equivalent to ϕ2t = −Kt ; ϕ1t =
Zt − ϕ2t σt2 Zt + σt2 Kt = . σt1 σt1
(3.3)
The process D corresponds to the cumulated cash withdrawal. The process X coincides with V X0 ,ϕ,D , the value of the portfolio associated with initial wealth x = X0 , portfolio strategy ϕ and cumulated (dividend) cash withdrawal D. From the seller’s point of view, this portfolio is a hedging portfolio since, by investing the initial amount X0 in the reference assets along the strategy ϕ, it allows him to pay the amount ξ to the buyer at time T and the intermediate dividends. We derive that X0 is the initial price of the option, called hedging price, and denoted by X0D (ξ). Now, there exists an unique process (X, Z, K) ∈ S 2 × H2 × H2λ solution of the λ-linear BSDE associated with terminal condition ξ and “generalized driver” g(t, y, z, k)dt + dDt , where g(t, y, z, k) = −(rt y + (z + σt2 k)θt1 + θt2 λt k), (3.4) 1
Loosely speaking, dCt represents the amount withdrawn from the portfolio during the time period [t, t + dt].
12
that is satisfies: − dXt = −(rt Xt + (Zt + σt2 Kt )θt1 + θt2 λt θt2 Kt )dt + dDt − Zt dWt − Kt dMt ;
XT = ξ. (3.5)
The solution (X, Z, K) provides the replicating portfolio. Again, the hedging risky assets stategy ϕ is given by (3.3). Moreover, the process X (also denoted by X D (ξ)) is the value of the replicating portfolio, that is X = V X0 ,ϕ,D . Hence, the amount X0 allows the seller to pay the intermediate dividends and the terminal payoff to the buyer, by investing it along the strategy ϕ in the market. Since the driver g given by (3.4) is λ-linear the representation property (see Theorem 2.13) yields XtD (ξ)
−
= E[e
RT t
rs ds
ζt,T ξ +
Z
T
e−
Rs t
t
ru du
ζt,s dDs | Gt ],
(3.6)
where ζ satisfies dζt,s = ζt,s− [−θs1 dWs − θs2 dMs ];
ζt,t = 1.
(3.7)
This defines a linear price system X: (ξ, D) 7→ X D (ξ). Suppose now θt2 < 1, 0 ≤ t ≤ ϑ dt ⊗ dP -a.s. Then, by Proposition 2.11, ζ0,. is a square integrable positive martingale. By classical results, the probability measure with density ζ0,T on GT is the unique martingale probability measure. In this case, the price system X is increasing and corresponds to the classical free-arbitrage price system (see e.g. Proposition 7.9.11 in [15]).
3.3
Nonlinear pricing of European options with dividends in an imperfect market with default
From now on, we assume that there are imperfections in the market which are taken into account via the nonlinearity of the dynamics of the wealth. More precisely, we suppose that the wealth process Vtx,ϕ,C (or simply Vt ) associated with an initial wealth x, a strategy ϕ = (ϕ1 , ϕ2 ) in H2 × H2λ and a cumulated withdrawal process C satisfies the following dynamics: − dVt = g(t, Vt , ϕt ′ σt , −ϕ2t )dt − ϕt ′ σt dWt + dCt + ϕ2t dMt ; V0 = x,
(3.8)
where g is a nonlinear λ-admissible driver (see Definition 2.2). Equivalently, setting Zt = ϕt ′ σt and Kt = −ϕ2t , − dVt = g(t, Vt , Zt , Kt )dt − Zt dWt + dCt − Kt dMt ; V0 = x.
(3.9)
Note that in the special case of a perfect market (see (3.5)), g is λ-linear and of the form: g(t, y, z, k) = −(rt y + (z + σt2 k)θt1 + θt2 λt k). Let us consider a European option with maturity T , terminal payoff ξ ∈ L2 (GT ) and dividend process D ∈ A2 in this market model. Let (X D (T, ξ), Z D (T, ξ), K D (T, ξ)), also denoted by (X, Z, K), be the solution of BSDE associated with terminal time T , “generalized driver” g(·)dt + dDt and terminal condition ξ, that is satisfying 13
−dXt = g(t, Xt, Zt , Kt )dt + dDt − Zt dWt − Kt dMt ;
XT = ξ.
The process X = X D (T, ξ) is equal to the wealth process associated with initial value x = X0 , strategy ϕ = Φ(Z, K) (see (3.3)) and cumulated amount D of cash withdrawals that is X = V X0 ,ϕ,D . Its initial value X0 = X0D (T, ξ) is thus a sensible price (at time 0) of the option for the seller since this amount allows him/her to construct a trading strategy ϕ, called hedging strategy, such that the value of the associated portfolio is equal to ξ at time T . Note that the cash withdrawals perfectly replicate the dividends of the option. Similarly, Xt = XtD (T, ξ) is a sensible price for the seller at time t. For each maturity S ∈ [0, T ] and for each pair “payoff-dividend” (ξ, D) ∈ L2 (GS ) × A2 , g,D g,D we define the g-value process by Et,S (ξ) := XtD (S, ξ), t ∈ [0, S]. Note that Et,S (ξ) can g S ,D S
g,D
be defined on the whole interval [0, T ] by setting Et,S (ξ) := Et,T (ξ) for t ≥ S, where g S (t, .) := g(t, .)1t≤S and DtS := Dt∧S . Note that when D = 0, it corresponds to the g g,0 g so-called g-evaluation, denoted in [19] by E , that is E (ξ) = E (ξ). This leads to a nonlinear pricing system E
g,·
g,D
: (S, ξ, D) 7→ E·,S (ξ). g,0
When there are no dividends, it reduces to the nonlinear pricing system E (usually g denoted by E ), first introduced by El Karoui-Quenez ([14]) in a Brownian framework. g,· We now give some properties on this nonlinear pricing system E which complete those given in [14] in a Brownian framework to the case with default and dividends. g,·
• Consistency. By the flow property for BSDEs, E is consistent. More precisely, let S ′ ∈ [0, T ], ξ ∈ L2 (GT ), D ∈ A2 , and let S be a stopping time smaller than S ′ . Then for each time t smaller than S, the g-value of the option associated with payoff ξ, (cumulated) dividend process D and maturity S ′ coincides with the g-value of the g,D option associated with maturity S, payoff ES,T (ξ) and dividend process D, that is g,D
g,D
g,D
Et,S ′ (ξ) = Et,S (ES,S ′ (ξ)) a.s. g,·
Because of the presence of the default jump, the nonlinear pricing system E is not necessarily monotone with respect to (ξ, D). We introduce the following Assumption. Assumption 3.1. Assume that there exists a bounded map γ : [0, T ] × Ω × R4 → R ; (ω, t, y, z, k1, k2 ) 7→ γty,z,k1,k2 (ω) P ⊗ B(R4 )-measurable and satisfying dP ⊗ dt-a.s. , for each (y, z, k1 , k2 ) ∈ R4 , g(t, y, z, k1) − g(t, y, z, k2) ≥ γty,z,k1 ,k2 (k1 − k2 )λt ,
(3.10)
and P -a.s. , for each (y, z, k1, k2 ) ∈ R4 , γty,z,k1 ,k2 ≥ −1. Recall that λ vanishes after ϑ and g(t, ·) does not depend on k on {t > ϑ}. Hence, the inequality (3.10) is always satisfied on {t > ϑ}. 14
Note also that the above assumption is satisfied e.g. if g(t, ·) is non decreasing with respect to k, or if g is C 1 in k with ∂k g(t, ·) ≥ −λt on {t ≤ ϑ}. This assertion can be shown by classical analysis arguments, similar to those used in the proof of Proposition A.2 in [10]. Before giving some additional properties (which hold under this Assumption), we introduce the following partial order relation, defined for each fixed time S ∈ [0, T ], on the set of pairs ”payoff-dividends” by: for each (ξ 1 , D 1 ), (ξ 2 , D 2) ∈ L2 (GS ) × A2 by (ξ 1, D 1 ) ≻ (ξ 2 , D 2 )
if
ξ 1 ≥ ξ 2 a.s. and D 1 − D 2 is non decreasing.2 g,·
• Monotonicity. Under Assumption 3.1, the nonlinear pricing system E is non decreasing with respect to the payoff and the dividend. More precisely, for all maturity S ∈ [0, T ], for all payoffs ξ1 , ξ2 ∈ L2 (GS ), and cumulated dividend processes D 1 , D 2 ∈ A2 , the following property holds: 1
2
g,D g,D (MO) If (ξ 1, D 1 ) ≻ (ξ 2 , D 2 ), then we have Et,S (ξ1) ≥ Et,S (ξ2 ), t ∈ [0, S] a.s.
This property follows from the comparison theorem for BSDEs with “generalized drivers” (Proposition 2.18) applied to g 1 = g 2 = g and ξ 1, ξ 2 , D 1 , D 2 . Using this comparison theorem, we also derive the following property:
• Convexity. Under Assumption 3.1, if g is convex with respect to (y, z, k), then g,D the nonlinear pricing system E is convex, that is, for any λ ∈ [0, 1], S ∈ [0, T ], 2 1 2 2 ξ1 , ξ2 ∈ L (GS ), D , D ∈ A g,λD 1 +(1−λ)D 2
Et,S
g,D 1
g,D 2
(λξ1 + (1 − λ)ξ2 ) ≤ λEt,S (ξ1 ) + (1 − λ)Et,S (ξ2 ),
for all t ∈ [0, S].
Moreover, under the additional assumption γ > −1 in Assumption 3.1, using the strict comparison theorem (Proposition 2.18), we derive the following no arbitrage property: g,·
• No arbitrage. Under Assumption 3.1 with γ > −1, the nonlinear pricing system E satisfies the no arbitrage property: for all maturity S ∈ [0, T ], and for all payoffs ξ 1 , ξ 2 ∈ L2 (GS ), and cumulated dividend processes D 1 , D 2 ∈ A2 , t0 ∈ [0, S], and A ∈ Gt0 , g,D 1
g,D 2
(NA) Suppose that (ξ 1 , D 1) ≻ (ξ 2 , D 2 ), and Et0 ,S (ξ1 ) = Et0 ,S (ξ2 ) a.s. on A ∈ Gt0 . Then, ξ1 = ξ2 a.s. on A and (Dt1 − Dt2 )t0 ≤t≤S is a.s. constant on A, that is DS1 − Dt10 = DS2 − Dt20 a.s. on A. In other words, the payoffs and the dividends paid between t0 and S are equal a.s. on A. The NA property also ensures that when γ > −1, the nonlinear pricing system E g is strictly monotone. Remark 3.2. Several authors have studied dynamic risk measures defined as the solutions of BSDEs (see e.g. [19, 4, 20]). In our framework with a default jump, given a λ-admissible 2
In other words, the dividends corresponding to D1 are greater or equal to the ones corresponding to
2
D .
15
driver, one can define a dynamic measure of risk ρg as follows: for each S ∈ [0, T ] and ξ ∈ L2 (GS ), we set g ρg· (ξ, S) = −E·,S (ξ),
g where E·,S (ξ) denotes the solution of the BSDE associated with terminal condition ξ, terminal time T and driver g. Then, by the results of this section, the dynamic risk-measure g g,0 ρg satisfies analogous properties to the ones of the nonlinear pricing system E·,S = E·,S (corresponding to the case with no dividends).
We now introduce the definition of an E notion of E g -supermartingale.
g,D
-supermartingale which generalizes the classical g,D
Definition 3.3. Let D ∈ A2 and Y ∈ S 2 . The process Y is said to be a strong E g,D supermartingale (resp. martingale) if Eσ,τ (Yτ ) ≤ Yσ (resp. = Yσ ) a.s. on σ ≤ τ , for all σ, τ ∈ T0 . Proposition 3.4. For all S ∈ [0, T ], payoff ξ ∈ L2 (GS ) and dividend process D ∈ A2 , the g,D g,D associated g-value process E·,S (ξ) is an E -martingale. Moreover, for all x ∈ R, portfolio strategy ϕ ∈ H2 × H2λ and cash withdrawal process g,D D ∈ A2 , the associated wealth process V x,ϕ,D is an E -martingale. g,D
Proof. The first assertion follows from the consistency property of E . The second one is obtained by noting that V x,ϕ,D is the solution of the BSDE with “generalized driver” g(·)dt + dDt , terminal time T and terminal condition VTx,ϕ,D . Example (Large investor seller) When the seller is a large trader, his hedging portfolio may affect the prices of the risky assets and the default probability. He can take into account these feedback effect in his market model as follows. In order to simplify the presentation, we only consider the case when the seller’s strategy affects only the default intensity. We are given a family of probability measures parametrized by V and ϕ. More precisely, for each V ∈ S 2 and ϕ ∈ H2 , let QV,ϕ be the probability measure equivalent to P , which admits LV,ϕ as density with respect to P , where (LV,ϕ ) is the solution of the following SDE: dLV,ϕ = Lt− γ(t, Vt− , ϕt )dMt ; t
LV,ϕ = 1, 0
where γ is a measurable bounded map with γ(t, ·) > −1, where γ : (t, ω, y, ϕ1, ϕ2 ) 7→ γ(t, ω, y, ϕ1, ϕ2 ), is a P ⊗ B(R3 )-measurable function defined on R+ × Ω × A, Lipschitz with respect to (y, ϕ1, ϕ2 ), uniformly with respect to (t, ω). By Girsanov’s theorem (see [15] Chapter 9.4 Corollary 9.4.4.5), the process W is a QV,ϕ Brownian motion and the process M V,ϕ defined as Z t V,ϕ Mt := Nt − λs (1 + γ(s, Vs , ϕs ))ds (3.11) 0
is a QV,ϕ -martingale. Hence, under QV,ϕ , the default intensity is equal to λt (1 + γ(t, Vt , ϕt )). The process γ(t, Vt , ϕt ) represents the impact of the seller’s strategy on the default intensity. 16
The dynamics of the wealth process associated with an initial wealth x and a risky assets stategy ϕ satisfy dVt = (rt Vt + ϕ1t θt1 σt1 − ϕ2t θt2 λt )dt − dCt + ϕ′t σt dWt − ϕ2t dMtV,ϕ ,
(3.12)
Let us show that this model can be seen as a particular case of the model described above associated with an appropriate map λ-admissible driver g. First, note that the dynamics of the wealth (3.12) can be written dVt = (λt γ(t, Vt , ϕt ) + rt Vt + ϕ1t θt1 σt1 − ϕ2t θt2 λt )dt − dCt + ϕ′t σt dWt − ϕ2t dMt , Equivalently, setting Zt = ϕt ′ σt and Kt = −ϕ2t , − dVt = g(t, Vt , Zt , Kt )dt − Zt dWt + dCt − Kt dMt ,
(3.13)
where g(t, y, z, k) = −λt γ t, y, (σt1)−1 (z + σt2 k), −k − rt y − (z + σt2 k)θt1 − θt2 λt k.
We are thus led to the general model described above associated with this λ-admissible driver. This model can be easily generalized to the case when the coefficients µ1 , σ 1 , µ2 , σ 2 also depend on the hedging cost V (equal to the price of the option) and on the hedging strategy ϕ2 . 3
A
Appendix
For each p ≥ 2, we introduce the spaces S p , Hp and Hpλ defined as follows. Let S p be the set of G-adapted RCLL processes ϕ such that E[sup |ϕt |p ] < +∞. i h 0≤t≤T RT Let Hp be the set of G-predictable processes such that kZkpp := E ( 0 |Zt |2 dt)p/2 < ∞ . hR i T p p 2 p/2 Let Hλ be the set of G-predictable processes such that kUkp,λ := E ( 0 |Ut | λt dt) < ∞. BSDEs with a default jump in Lp Proposition A.1. Let p ≥ 2 and let T > 0. Let g be a λ-admissible driver such that g(t, 0, 0, 0) ∈ IH p. Let ξ ∈ Lp (GT ). There exists a unique solution (Y, Z, K) in S p × Hp × Hpλ of the BSDE with default (2.4). Remark A.2. The above result still holds in the case when there is a G-martingale representation theorem with respect to W and M, even if G is not generated by W and M. Proof. The proof relies on the same arguments as in the proof of Proposition A.2 in [20] together with the arguments used in the proof of Proposition 2.6. 3
The coefficients may also depend on ϕ = (ϕ1 , ϕ2 ), but in this case, we have to assume that at fixed (ω, t), the map Ψ : (ω, t, ϕ) 7→ (z, k) with z = ϕ′ σt (ω) and k = −ϕ2 is one to one with respect to ϕ, and such that its 2 inverse Ψ−1 ϕ (ω, t, ·) is P ⊗ B(R )-measurable.
17
BSDEs with a default jump and change of probability measure Let (βs ) and (γs ) be two bounded real-valued G-predictable processes, and let (ζs ) be the process satisfying the forward SDE: dζs = ζs− (βs dWs + γs dMs ) with ζ0 = 1. By Proposition 2.11, ζ is a p-integrable martingale, that is ζT ∈ Lp for all p ≥ 1. We suppose that γ > −1, which implies that ζs > 0, s ∈ [0, T ] a.s. Let Q be the probability measure equivalent to P which admits ζT as density with respect to P on GT . 9.4.4.5), the process Wtβ := Wt − R t By Girsanov’s theorem (see [15] Chapter 9.4 Corollary γ β ds is a Q-Brownian motion and the process M defined as 0 s Z t Z t γ Mt := Mt − λs γs ds = Nt − λs (1 + γs )ds (A.1) 0
0
is a Q-martingale. We now show a representation theorem for (Q, G)-local martingales with respect to W β and M γ . Proposition A.3. Let m = (mt )0≤t≤T be a (Q, G)-local martingale. There exists a unique pair of predictable processes (zt , kt ) such that Z t Z t β mt = m0 + zs dWs + ks dMsγ 0 ≤ s ≤ T a.s. (A.2) 0
0
Proof. Since m is a Q-local martingale, the process m ¯ t := ζt mt is a P -local martingale. By the martingale representation theorem (Lemma 2.1), there exists a unique pair of predictable processes (Z, K) such that Z t Z t m ¯t = m ¯0 + Zs dWs + Ks dMs 0 ≤ t ≤ T a.s. 0
0
Then, by applying Itˆo’s formula to mt = m ¯ t (ζt )−1 and by classical computations, one can derive the existence of (z, k) satisfying (A.2). From this result together with Proposition A.1 and Remark A.2, we derive the following corollary. Corollary A.4. Let p ≥ 2 and let T > 0. Let g be a λ-admissible driver such that g(t, 0, 0, 0) ∈ IHQp . Let ξ ∈ LpQ (GT ). There exists a unique solution (Y, Z, K) in SQp × HpQ × HpQ,λ of the BSDE with default: −dYt = g(t, Yt , Zt , Kt )dt − Zt Wtβ − Kt dMtγ ;
YT = ξ.
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