Nonlinear expectations and nonlinear pricing

Transcription

1 Nonlinear expecaions and nonlinear pricing Zengjing Chen Kun He Deparmen of Mahemaics Shandong Universiy, Jinan China Reg Kulperger Deparmen of saisical and acuarial Science The Universiy of Wesern Onario London,Onario, Canada Absrac As he generalizaions of mahemaical expecaions,coheren and convex risk measures, Choque expecaion and Peng s g-expecaions all have been widely used o sudy he quesion of hedging coningen claims in incomplee markes. Obviously, he differen risk measures or expecaions will ypically yield differen pricing. In his paper we invesigae differences amongs hese risk measures and expecaions in he framework of he coninuous-ime asse pricing. We show ha he coheren pricing is always less han he corresponding Choque pricing. This propery and inequaliy fails in general when one uses pricing by convex risk measures. Finally, we show ha g-expecaions are he bes way for he pricing opions for some coninuous models. Keywords: risk measure, coheren risk, convex risk, Choque expecaion, g- expecaion, backward sochasic differenial equaion, converse comparison heorem, BSDE, Jensen s inequaliy. AMS 2000 subjec classificaions 60H10. The work was done while Zengjing Chen was visiing The Universiy of Wesern Onario in 2004, whose hospialiy he deeply appreciaed. Z. Chen received funding suppor from he ESSO Mahemaics Educaion fund and from he NSERC gran of RK. Research suppored parly by NSFC( ) and FANEDD(200118). This work has been suppored by grans from he Naural Sciences and Engineering Research Council of Canada. 1

2 Nonlinear pricing 2 1 Inroducion The celebraed papers of Black and Scholes (1973) and Meron (1973) paved he way for he pricing opions in a complee marke, hey showed ha for a complee marke in which he wealh process saisfies a linear sochasic differenial equaion (SDE)(see for example he one in Cvianic and Karazas (1993), El Karoui a al.(1997)), he fair price of every coningen claim is equal o he (linear) mahemaical expecaion of he discouned value of he claim under a new, so called risk-neural probabiliy measures. The forgoing argumen fails, however, if he marke is an incomplee marke in which he wealh process is a nonlinear sochasic differenial equaion(sde). To aack his more general problem, many auhors ry o use nonlinear expecaion (or risk measures). Thus several classes of financial risk measures or expecaions have been proposed in he lieraure. Among hese are coheren and convex risk measures, Choque expecaion and Peng s g-expecaion, which preserve many properies of he classical mahemaical expecaions excep lineariy (for convenience of he explosion, we someimes call all of hem nonlinear expecaion wihou confusion). Coheren risk measures were firs inroduced by Arzner, Delbaen, Eber and Heah [1] and Delbaen [6]. As an exension of coheren risk measures, convex risk measures in general probabiliy spaces were inroduced by Föllmer and Schied [8]. g-expecaions were inroduced by Peng [10] via a class of nonlinear backward sochasic differenial equaions (BSDEs), his class of nonlinear BSDEs being inroduced earlier by Pardoux and Peng [9]. Choque [4] exended probabiliy measures o nonaddiive probabiliy measures (capaciy), and inroduced he so called Choque expecaions. Obviously, he differen risk measures or expecaions will ypically yield differen pricing even for he same model. In his paper we invesigae differences amongs hese risk measures and expecaions in he framework of of El Karoui a al.(1997). We show ha (i) in he family of convex risk measures, only coheren risk measures saisfy Jensen s inequaliy; (ii) coheren risk measures are always bounded by he corresponding Choque expecaion, bu such an inequaliy in general fails for convex risk measures. In finance, coheren and convex risk measures and Choque expecaion are ofen used in he pricing of a coningen claim. Resul (ii) implies coheren pricing is always less han Chouqe pricing, bu he pricing by a convex risk measure no longer has his propery. Finally, we show ha g-expecaions are he bes way for he pricing opions in some coninuous models. In order o prove hese resuls, we esablish in Secion 3, Theorem 1, a new converse comparison heorem of g-expecaions. The paper is organized as follows. Secion 2 reviews and gives he various definiions needed here. Secion 3 gives he main resuls and proofs. Secion 4 gives a summary of he resuls, puing hem ino a Table form for convenience of he various relaions.

3 Nonlinear pricing 3 2 Expecaions and risk measures In his secion, we briefly recall he definiions of g-expecaion, Choque expecaion, coheren and convex risk measures. 2.1 g-expecaion Peng [10] inroduced g-expecaion via a class of backward sochasic differenial equaions(bsde). Some of he relevan definiion and noaion are given here. Fix T [0, ) and le (W ) 0 T be a d-dimensional sandard Brownian moion defined on a compleed probabiliy space (Ω, F, P ). Suppose {F } 0 T is he naural filraion generaed by (W ) 0 T, ha is We also assume F T = F. Denoe F = σ{w s ; s }. L 2 (Ω, F, P ) = {ξ : ξ is F -measurable random variables wih E ξ 2 < }, [0, T ]; L 2 (0, T, R d ) = {X : X is R d -valued, F -adaped processes wih E 0 X s 2 ds < }. Le g : Ω R R d [0, T ] R saisfy (H1) For any (y, z) R [ R d, {g(y, z, )} 0 is a coninuous progressively measurable process wih E g(y, z, 0 s) 2 ds < ] T. (H2) There exiss a consan K 0 such ha for any (y 1, z 1 ), (y 2, z 2 ) R R d g (y 1, z 1, ) g (y 2, z 2, ) K ( y 1 y 2 + z 1 z 2 ), [0, T ]. (H3) g(y, 0, ) = 0, (y, ) R [0, T ]. In Secion 3, Corollary 3 we will consider a special case of R d wih d = 1. Under he assumpions of (H1) and (H2), Pardoux and Peng [9] showed ha for any ξ L 2 (Ω, F, P ), he BSDE y = ξ + g(y s, z s, s)ds has a unique pair soluion (y, z ) 0 L 2 (0, T, R) L 2 (0, T, R d ). z s dw s, 0 T (1) Using he soluion y of BSDE (1), which depends on ξ, Peng [10] inroduced he noion of g-expecaions.

4 Nonlinear pricing 4 Definiion 1 Assume ha (H1),(H2) and (H3) hold on g and ξ L 2 (Ω, F, P ). Le (y s, z s ) be he soluion of BSDE (1). E g [ξ] defined by E g [ξ] := y 0 is called he g-expecaion of he random variable ξ. E g [ξ F ] defined by E g [ξ F ] := y is called he condiional g-expecaion of he random variable ξ. Peng [10] also showed ha g-expecaion E g [ ] and condiional g-expecaion E g [ F ] preserve mos of basic properies of mahemaical expecaion, excep for lineariy. The basic properies are summarized in he nex Lemma. Lemma 1 (Peng [10]) Suppose ha ξ, ξ 1, ξ 2 L 2 (Ω, F, P ). (i) Preservaion of consans: For any consan c, E g [c] = c. (ii) Monooniciy: If ξ 1 ξ 2, hen E g [ξ 1 ] E g [ξ 2 ]. (iii) Sric monooniciy: If ξ 1 ξ 2, and P (ξ 1 > ξ 2 ) > 0, hen E g [ξ 1 ] > E g [ξ 2 ]. (iv) Consisency: For any [0, T ], E g [E g [ξ F ]] = E g [ξ]. (v) If g does no depend on y, and η is F -measurable, hen In paricular, E g [ξ E g [ξ F ] F ] = 0. E g [ξ + η F ] = E g [ξ F ] + η. (vi) Coninuiy: If ξ n ξ as n in L 2 (Ω, F, P ), hen lim n E g [ξ n ] = E g [ξ]. The following lemma is from Briand e al. [2, Theorem 2.1]. We can rewrie i as follows. Lemma 2 (Briand e al. [2]) Suppose ha {X } is of he form X = x + 0 σ s dw s, 0 T, where {σ } is a coninuous bounded process. Then lim s E g [X s F ] E[X s F ] s where he limi is in he sense of L 2 (Ω, F, P ). = g(x, σ, ), 0,

5 Nonlinear pricing Choque Expecaion Choque [4] exended he noion of a probabiliy measure o nonaddiive probabiliy (called capaciy) and defined a kind of nonlinear expecaion, which is now called Choque expecaion. Definiion 2 (1) A real valued se funcion V : F [0, 1] is called a capaciy if (i) V ( ) = 0, V (Ω) = 1; (ii) V (A) V (B), whenever A, B F and A B. (2) Le V be a capaciy. For any ξ L 2 (Ω, F, P ), he Choque expecaion C V (ξ) is defined by C V (ξ) := 0 [V (ξ ) 1] d + 0 V (ξ )d Remark 1 A propery of Choque expecaion is posiive homogeneiy, i.e. for any consan a 0, C V (aξ) = ac V (ξ). 2.3 Risk Measures A risk measure is a map ρ : G R, where G is inerpreed as he habia of he financial posiions whose riskiness has o be quanified. In his paper, we shall consider G = L 2 (Ω, F, P ). The following modificaions of coheren risk measures (Arzner e al.[1]) is from Roorda e. al. [11]. Definiion 3 A risk measure ρ is said o be coheren if i saisfies (1) Subaddiiviy: ρ(x 1 + X 2 ) ρ(x 1 ) + ρ(x 2 ), X 1, X 2 G; (2) Posiive homogeneiy: ρ(λx) = λρ(x), for all real number λ 0; (3) Monooniciy: ρ(x) ρ(y ), whenever X Y ; (4) Translaion invariance: ρ(x + α) = ρ(x) + α for all real number α. As an exension of coheren risk measures, Föllmer and Schied [8] inroduced he axiomaic seing for convex risk measures. The following modificaions of convex risk measures of Föllmer and Schied [8] is from Frielli and Rosazza Gianin [7]. Definiion 4 A risk measure is said o be convex if i saisfies

6 Nonlinear pricing 6 (i) Convexiy: ρ(λx 1 +(1 λ)x 2 ) λρ(x 1 )+(1 λ)ρ(x 2 ), λ [0, 1], X 1, X 2 G; (ii) Normaliy: ρ(0) = 0; (iii) Properies (3) and (4) in Definiion 3. The funcional ρ( ) in Definiions 3 and 4 is usually called a saic risk measure. Obviously, a coheren risk measure is a convex risk measure. Insead of he funcional ρ( ), Arzner e al. [1], Frielli and Rosazza Gianin [7] inroduced he noion of dynamic risk measure ρ ( ), which is random and depends on a ime parameer. Definiion 5 A dynamic risk measure ρ ( ) : L 2 (Ω, F, P ) L 2 (Ω, F, P ) is a random funcional which depends on, such ha for each i is a risk measure. If ρ ( ) saisfies for each [0, T ] he condiions in Definiion 3, we say ρ ( ) is a dynamic coheren risk measure. Similarly if ρ ( ) saisfies for each [0, T ] he condiions in Definiion 4, we say ρ ( ) is a dynamic convex risk measure. 3 Main resuls In order o prove our main resuls, we esablish a general converse comparison heorem of g-expecaion. This heorem plays an imporan role in his paper. Theorem 1 Suppose ha g, g 1 and g 2 saisfy (H1), (H2) and (H3). Then he following conclusions are equivalen. (i) For any ξ, η L 2 (Ω, F, P ), E g [ξ + η] E g1 [ξ] + E g2 [η]. (2) (ii) For any (y 1, z 1, ), (y 2, z 2, ) R R d [0, T ], g(y 1 + y 2, z 1 + z 2, ) g 1 (y 1, z 1, ) + g 2 (y 2, z 2, ). (3) Proof: We firs show ha inequaliy (ii) implies inequaliy (i). Le (y 1, z 1 ), (y 2, z 2 ) and (Y, Z ) be he soluions of he following BSDE corresponding o he erminal value X = ξ, η and ξ + η, and he generaor g = g 1, g 2 and g, respecively y = X + g(y s, z s, s)ds Then E g1 [ξ] = y 1 0, E g2 [η] = y 2 0, E g [ξ + η] = Y 0. z s dw s. (4)

7 Nonlinear pricing 7 For fixed (y 1, z 1 ), consider he BSDE y = ξ + η + [ g2 (y s ys, 1 z s zs, 1 s) + g 1 (ys, 1 zs, 1 s) ] ds z s dw s. (5) I is easy o check ha (y 1 + y 2, z 1 + z 2 ) is he soluion of he BSDE (5). Comparing BSDEs (5) and (4) wih X = ξ + η and g = g, assumpion (ii) (3) yields g(y 1 + y 2, z 1 + z 2, ) g 1 (y 1, z 1, ) + g 2 (y 2, z 2, ), 0. Applying he comparison heorem of BSDE in Peng [10], we have Y y 1 + y 2, 0. Taking = 0, hus by he definiion of g-expecaion, he proof of his par is complee. We now prove ha inequaliy (i) implies (ii). We disinguish wo cases: he former where g does no depend on y, he laer where g may depend on y. Case 1, g does no depend on y. The proof of his case 1 is done in wo seps. Case 1, Sep 1: We now show ha for any [0, T ], we have E g [ξ + η F ] E g1 [ξ F ] + E g2 [η F ], ξ, η L 2 (Ω, F, P ). Indeed, for [0, T ], se A = {ω : E g [ξ + η F ] > E g1 [ξ F ] + E g2 [η F ] }. If for all [0, T ], we have P (A ) = 0, hen we obain our resul. If no, hen here exiss [0, T ] such ha P (A ) > 0. We will now obain a conradicion. For his, I A E g [ξ + η F ] > I A (E g1 [ξ F ] + E g2 [η F ]). Tha is I A (E g [ξ + η F ] E g1 [ξ F ] E g2 [η F ]) > 0. Taking g-expecaion on boh sides of he above inequaliy, and apply he sric monooniciy of g-expecaion in Lemma 1 (iii), i follows Bu by Lemma 1 (iv) and (v), E g [I A (E g [ξ + η F ] E g1 [ξ F ] E g2 [η F ])] > 0. E g [I A (E g [ξ+η F ] E g1 [ξ F ] E g2 [η F ])] = E g [I A (ξ + η) E g1 [I A ξ F ] E g2 [I A η F ])].

8 Nonlinear pricing 8 Noe ha by by Lemma 1(v) E gi [I A ξ E gi [I A ξ F ]] = 0, i = 1, 2. Thus 0 < E g [ I A (ξ + η) E g1 [I A ξ F ] E g2 [I A η F ] ] = E g [ I A ξ E g1 [I A ξ F ] + I A η E g2 [I A η F ] ] E g1 [ I A ξ E g1 [I A ξ F ] ] + E g2 [I A η E g2 [I A η F ]] = 0. This induces a conradicion, hus concluding he proof of his Sep 1. Case 1, Sep 2: For any τ, [0, T ] wih τ and z i Xτ i = z i (W τ W ), i = 1, 2. Obviously, Xτ i L 2 (Ω, F, P ). By Sep 1, Thus E g [X 1 τ + X 2 τ F ] E g1 [X 1 τ F ] + E g2 [X 2 τ F ], [0, T ]. R d, le us choose E g [X 1 τ + X 2 τ F ] E[X 1 τ + X 2 τ F ] τ E g 1 [X 1 τ F ] E[X 1 τ F ] τ + E g 2 [Xτ 2 F ] E[Xτ 2 F ]. τ Le τ, applying Lemma 2, since g does no depend on y, we rewrie g(y, z, ) simply as g(z, ), hus The proof of Case 1 is complee. g(z 1 + z 2, ) g 1 (z 1, ) + g 2 (z 2, ), 0. Case 2, g depends on y. The proof is similar o he proof of Theorem 2.1 in Coque e al. [5]. For each ɛ > 0 and (y 1, z 1 ), (y 2, z 2 ) R R d, define he sopping ime τ ɛ = τ ɛ (y 1, z 1 ; y 2, z 2 ) = inf{ 0; g 1 (y 1, z 1, )+g 2 (y 2, z 2, ) g(y 1 +y 2, z 1 +z 2, ) ɛ} T. Obviously, if for each (y 1, z 1 ), (y 2, z 2 ) R R d, P (τ ɛ (y 1, z 1 ; y 2, z 2 ) < T ) = 0, for all ɛ, hen he proof is done. If i is no he case, hen here exis ɛ > 0 and (y 1, z 1 ), (y 2, z 2 ) R R d such ha P (τ ɛ (y 1, z 1 ; y 2, z 2 ) < T ) > 0. Fix ɛ, y i, z i, (i = 1, 2), and consider he following (forward) SDEs defined on he inerval [τ ɛ, T ] { dy i () = g i (Y i (), z i, )d + z i dw, Y i (τ ɛ ) = y i, τ ɛ, i = 1, 2, and { dy 3 () = g(y 3 (), z 1 + z 2, )d + (z 1 + z 2 )dw, Y 3 (τ ɛ ) = y 1 + y 2, τ ɛ.

9 Nonlinear pricing 9 Obviously, he above equaions admi a unique soluion Y i which is progressively measurable wih E[sup 0 T Y i () 2 ] <. Define he following sopping ime τ δ := inf{ τ ɛ ; g 1 (Y 1 (), z 1, ) + g 2 (Y 2 (), z 2, ) g(y 3 (), z 1 + z 2, ) ɛ 2 } T. I is clear ha τ ɛ τ δ T and noe ha τ δ = T whenever τ ɛ = T, hus, Hence P (τ ɛ < τ δ ) > 0. Moreover, we can prove {τ ɛ < τ δ } = {τ ɛ < T }. Y 1 (τ δ ) + Y 2 (τ δ ) > Y 3 (τ δ ), on {τ ɛ < τ δ }. In fac, seing Ŷ () = Y 3 () Y 1 () Y 2 (), hen dŷ () = [ g(y 3 (), z 1 + z 2, ) + g 1 (Y 1 (), z 1, ) + g 2 (Y 2 (), z 2, )]d. Thus { I follows ha on [τ ɛ, τ δ ), This implies dŷ () d ɛ 2, [τ ɛ, τ δ ), Ŷ (τ ɛ ) = 0. Ŷ (τ δ ) ɛ 2 (τ δ τ ɛ ) < 0. P ( Y 3 (τ δ ) < Y 1 (τ δ ) + Y 2 (τ δ ) ) P (τ ɛ < τ δ ) > 0. (6) By he definiion of Y 1, Y 2 and Y 3, he pair processes (Y 1 (), z 1 ), (Y 2 (), z 2 ) and (Y 3 (), z 1 +z 2 ) are he soluions of he following BSDEs wih erminal values Y 1 (T ), Y 2 (T ) and Y 3 (T ), y = Y i (T ) + g i (y s, z i, s)ds z i dw s, i = 1, 2 and Hence, y = Y 3 (T ) + g(y s, z 1 + z 2, s)ds E g1 [Y 1 (τ δ ) F τɛ ] = E g1 [Y 1 (T ) F τɛ ] = y 1, (z 1 + z 2 )dw s. E g2 [Y 2 (τ δ ) F τɛ ] = E g2 [Y 2 (T ) F τɛ ] = y 2

10 Nonlinear pricing 10 and E g [Y 3 (τ δ ) F τɛ ] = E g [Y 3 (T ) F τɛ ] = y 1 + y 2. Applying he sric comparison heorem of BSDE and inequaliy (6), by he assumpions of his Theorem, we have y 1 + y 2 = E g [Y 3 (τ δ )] < E g [Y 1 (τ δ ) + Y 2 (τ δ )] E g1 [Y 1 (τ δ )] + E g2 [Y 2 (τ δ )] = y 1 + y 2. This induces a conradicion. The proof is complee. Lemma 3 Suppose ha g saisfies (H1), (H2) and (H3). For any consan c 0, le g(y, z, ) = cg( 1 c y, 1 c z, ). Then for any ξ L2 (Ω, F, P ), E g [cξ] = ce g [ξ]. Proof: Leing y = E g [cξ F ], hen y is he soluion of BSDE y = cξ + g(y s, z s, s)ds z s dw s. Since g(y, z, ) = cg( 1y, 1 z, ), he above BSDE can be rewrien as c c y = cξ + Le y = E g [ξ F ], hen cy saisfies cy = cξ + cg( 1 c y s, 1 c z s, s)ds cg(y s, z s, s)ds z s dw s. (7) cz s dw s. (8) Comparing wih BSDE (7) and BSDE (8), by he uniqueness of he soluion of BSDE, we have (cy, cz ) = (y, z ). Le = 0, hen cy 0 = y 0. The conclusion of he Lemma now follows by he definiion of g-expecaion. This concludes he proof. Applying Theorem 1 and Lemma 3, immediaely, we obain several relaions beween g-expecaion E g [ ] and g. These are given in he following Corollaries. Corollary 1 The g-expecaion E g [ ] is he classical mahemaical expecaion if and only if g does no depend on y and is linear in z. Proof : Applying Theorem 1, E g [ ] is linear if and only if g(y, z, ) is linear in (y, z). By assumpion (H3), ha is g(y, 0, ) = 0 for all (y, ). Thus g does no depend on y. The proof is complee.

11 Nonlinear pricing 11 Corollary 2 The g-expecaion E g [ ] is a convex risk measure if and only if g does no depend on y and is convex in z. Proof: Obviously, g-expecaion E g [ ] is convex risk measure if and only if for any λ (0, 1) E g [λξ + (1 λ)η] λe g [ξ] + (1 λ)e g [η], ξ, η L 2 (Ω, F, P ). (9) For a fixed λ (0, 1), le ( 1 g 1 (y, z, ) = λg λ y, 1 ) ( ) 1 λ z,, g 2 (y, z, ) = (1 λ)g 1 λ y, 1 1 λ z,. Applying Lemma 3, E g1 [λξ] = λe g [ξ], E g2 [(1 λ)ξ] = (1 λ)e g [ξ]. Inequaliy (9) becomes E g [λξ + (1 λ)η] E g1 [λξ] + E g2 [(1 λ)η], ξ, η L 2 (Ω, F, P ). (10) Applying Theorem 1, for any (y i, z i, ) R R d [0, T ], i = 1, 2, g(λy 1 + (1 λ)y 2, λz 1 + (1 λ)z 2, ) g 1 (λy 1, λz 1, ) + g 2 ((1 λ)y 2 + (1 λ)z 2, ) = λg(y 1, z 1, ) + (1 λ)g(y 2, z 2, ) which hen implies ha g is convex. By he explanaion of Remark for Lemma 4.5 in Briand e al. [2], he convexiy of g and he assumpion (H3) imply ha g does no depend on y. The proof is complee. The funcion g is posiively homogeneous in z if for any a 0, g(, az, ) = ag(, z, ). Corollary 3 The g-expecaion E g [ ] is a coheren risk measure if and only if g does no depend on y and i is convex and posiively homogenous in z. In paricular, if d = 1, g is of he form g(z, ) = a z + b z wih a 0. Proof : By Corollary 2, he g-expecaion E g [ ] is a convex risk measure if and only if g does no depend on y and is convex in z. Applying Theorem 1 and Lemma 3 again, i is easy o check ha g-expecaion E g [ ] is posiively homogeneous if and only if g is posiively homogeneous (ha is for all a > 0 and ξ, E g [aξ] = ae g [ξ] if and only if for any a 0, g(, az, ) = ag(, z, )). In paricular, if d = 1, noice he fac ha g is convex and posiively homogeneous on R, and ha g does no depend on y. We wrie i as g(z, ) hen g(z, ) = g(z, )I [z 0] + g(z, )I [z 0] = g(1, )zi [z 0] + g( 1, )( z)i [z 0]. (11)

12 Nonlinear pricing 12 Noe ha zi [z 0] = z +, ( z)i [z 0] = z, bu Thus from (11) z + = z + z 2, z = z z 2. g(z, ) = g(1, ) + g( 1, ) z + 2 g(1, ) g( 1, ) z. 2 Defining a := g(1,)+g( 1,), b 2 := g(1,) g( 1,). Obviously a 0, since he convexiy 2 of g yields g(1, ) + g( 1, ) g(0, ) = 0. 2 The proof is complee. Remark 2 Corollaries 2 and 3 give us an inuiive explanaion for he disincion beween coheren and convex risk measures. In he framework of g-expecaions, convex risk measures are generaed by convex funcions, while coheren measures only by convex and posiively homogenous funcions. In paricular, if d = 1, i is generaed only by he family g(z, ) = a z + b z wih a 0. Thus he family of coheren risk measures is much smaller han he family of convex risk measures. Jensen s inequaliy for mahemaical inequaliy is imporan in probabiliy heory. Chen e al. [3] sudied Jensen s inequaliy for g-expecaion. We say ha g-expecaion saisfies Jensen s inequaliy if for any convex funcion ϕ : R R, hen ϕ (E g [ξ]) E g [ϕ (ξ))], whenever ξ, ϕ (ξ) L 2 (Ω, F, P ). (12) Lemma 4 [Chen e al. [3] Theorem 3.1] (H1), (H2) and (H3). Then Le g be a convex funcion and saisfy (i) Jensen s inequaliy (12) holds for g-expecaions if and only if g does no depend on y and is posiively homogeneous in z; (ii) If d = 1, he necessary and sufficien condiion for Jensen s inequaliy (12) o hold is ha here exis wo adaped processes a 0 and b such ha g(z, ) = a z + b z. Now we can easily obain our main resuls. Theorem 2 below shows he relaion beween saic risk measures and dynamic risk measures.

13 Nonlinear pricing 13 Theorem 2 If g-expecaion E g [ ] is a saic convex (coheren) risk measure, hen he corresponding condiional g-expecaion E g [ F ] is dynamic convex (coheren) risk measure for each (0, T ). Proof: This follows direcly direc from Theorem 1. Theorem 3 below shows ha in he family of convex risk measure, only coheren risk measure saisfies Jensen s inequaliy. Theorem 3 Suppose ha E g [ ] is a convex risk measure. Then E g [ ] is a coheren risk measure if and only if E g [ ] saisfies Jensen s inequaliy. Proof: If E g [ ] is a convex risk measure, hen by Corollary 2, g is convex. Applying Lemma 4, E g [ ] saisfies Jensen s inequaliy if and only if g is posiively homogenous. By Corollary 2, he corresponding E g [ ] is coheren risk measure. The proof is complee. Theorem 4 and Counerexample 1 below give he relaion beween risk measures and Choque expecaion. Theorem 4 If E g [ ] is a coheren risk measure, hen E g [ ] is bounded by he corresponding Choque expecaion, ha is E g [ξ] C V (ξ), ξ L 2 (Ω, F, P ) where V (A) = E g [I A ]. If E g [ ] is a convex risk measure hen inequaliy above fails in general. By consrucion here exiss a convex risk measure and random variables ξ 1 and ξ 2 such ha E g [ξ 1 ] C V (ξ 1 ) and E g [ξ 2 ] > C V (ξ 2 ). The prove his heorem uses he following lemma. Lemma 5 Suppose ha g does no depend on y. Suppose ha he g-expecaion E g [ ] saisfies (i) E g [I A + I B ] E g [I A ] + E g [I B ], A, B F (ii) For any posiive consan a < 1, E g [aξ] ae g [ξ], ξ L 2 (Ω, F, P ). Then for any ξ L 2 (Ω, F, P ) he g-expecaion E g [ ] is bounded by he corresponding Choque expecaion, ha is E g [ξ] 0 [E g [I {ξ x} ] 1]dx + 0 E g [I {ξ x} ]dx. (13)

14 Nonlinear pricing 14 Proof : The proof is done in hree seps. Sep 1. We show ha if ξ 0 is bounded by N > 0, hen inequaliy (13) holds. In fac, for he fixed N, denoe ξ (n) by ξ (n) := 2 n 1 i=0 Then ξ (n) ξ, n in L 2 (Ω, F, P ). Moreover ξ (n) can be rewrien as ξ (n) = in 2 n I { in 2 n ξ< (i+1)n 2 n }. 2 n i=1 N 2 n I {ξ in 2 n }. Bu by he assumpions (i) and (ii) in his lemma, we have Noe ha and E g [ξ (n) ] = E g [ 2 n i=1 2 n i=1 N 2 E g[i n {ξ in 2 n } ] N 2n 2 I n {ξ in 2 n } ] N 2 E g[i n {ξ in 2 n }]. (14) 0 i=1 E g [I {ξ x} ]dx, E g [ξ (n) ] E g [ξ], n. n Thus, aking limis on boh sides of inequaliy (14), i follows ha The proof of Sep 1 is complee. E g [ξ] 0 E g [I {ξ x} ]dx. Sep 2. We show ha if ξ is bounded by N > 0, ha is ξ N, hen inequaliy (13) holds. Le ξ N = ξ + N, hen 0 ξ N 2N. Applying Sep 1, Bu by Lemma 1(v), E g [ξ + N] 0 E g [ξ + N] = E g [ξ] + N. E g [I {ξ+n x} ]dx. (15) On he oher hand, E 0 g [I {ξ+n x} ]dx = 2N E 0 g [I {ξ x N} ]dx = N E N g[i {ξ x} ]dx = 0 E N g[i {ξ x} ]dx + N E 0 g[i {ξ x} ]dx.

15 Nonlinear pricing 15 Thus by (15) Therefore E g [ξ] + N E g [ξ] 0 N 0 N E g [I {ξ x} ]dx + [E g [I {ξ x} ] 1]dx + N 0 N 0 E g [I {ξ x} ]dx. E g [I {ξ x} ]dx. Sep 3. For any ξ L 2 (Ω, F, P ), le ξ N = ξi [ ξ N], hen ξ N N. By Sep 2, E g [ξ N ] 0 N Leing N, i follows ha E g [ξ] The proof is complee. 0 [E g [I {ξ x} ] 1]dx + [E g [I {ξ x} ] 1]dx + N 0 0 E g [I {ξ x} ]dx. E g [I {ξ x} ]dx. Proof of Theorem 4: If he g-expecaion E g [ ] is a coheren risk measure, hen i is easy o check ha he g-expecaion E g [ ] saisfies he condiions of Lemma 5. Le V (A) = E g [I A ] A F. By Lemma 5 and he definiion of Choque expecaion, we have E g [ξ] C V [ξ]. The firs par of his heorem is complee. Counerexample 1 shows ha his propery of coheren risk measures fails in general for more general convex risk measures. This complees he proof of Theorem 4. Counerexample 1 Suppose ha {W } is 1-dimensional Brownian moion (i.e. d = 1). Le g(z) = (z 1) + where x + = max{x, 0}. Then E g [ ] is a convex risk measure. Le ξ 1 = 1 2 I {W T 1} and ξ 2 = 2I {WT 1}. Then However E g [ξ 1 ] C V (ξ 1 ). E g [ξ 2 ] > C V (ξ 2 ). Where capaciy V in he Choque expecaion C V ( ) is given by V (A) = E g [I A ].

16 Nonlinear pricing 16 Proof of he Inequaliy in Counerexample 1 : The convex funcion g(z) = (z 1) + saisfies (H1), (H2) and(h3). Thus, by Corollary 2, g-expecaion E g [ ] is a convex risk measure. This ogeher wih he propery of Choque expecaion in Remark 1 implies E g [ξ 1 ] = E g [ 1I 2 {W T 1}] 1E 2 g[i {WT 1}] = 1C 2 V (I {WT 1}) = C V ( 1I 2 {W T 1}) = C V (ξ 1 ). Moreover, since d = 1, by Corollary 3, E g [ ] is a convex risk measure raher han a coheren risk measure. We now prove ha E g [ξ 2 ] > C V (ξ 2 ). In fac, since C V (2I {WT 1}) = 2C V (I {WT 1}) = 2E g [I {WT 1}], we only need o show E g [2I {WT 1}] > 2E g [I {WT 1}]. Le (y, z) be he soluion of he BSDE Firs we prove ha y = 2I {WT 1} + (z s 1) + ds z s dw s. (16) (L P )((, ω) [0, T ) Ω : z (ω) > 1) > 0, (17) where L is Lebesgue measure on [0, T ]. If i is no rue, hen z 1 a.e. [0, T ] and BSDE (16) becomes Thus By he Markov propery, y = 2I {WT 1} y = 2E[I {WT 1} F ] = 2E[I {WT W 1 W } F ]. z s dw s. (18) y = 2P (W T W 1 W σ(w )). Recall ha W T W and W are independen and W T W N(0, T ). Thus y = 2 1 x ϕ(y)dy x=w,

17 Nonlinear pricing 17 where ϕ(x) is he densiy funcion of he normal disribuion N(0, T ). Thus z = D y = 2ϕ(1 W ), where D is he Malliavin derivaive. Thus z can be greaer han 1 whenever is near 0 and W is near 0. Thus (17) holds, which conradics he assumpion z 1 a.e. [0, T ]. Secondly we prove ha Le (Y, Z ) be he soluion of he BSDE Obviously, which means ( Y, Zs 2 2 Y = 2I {WT 1} + Y 2 = I {W T 1} + E g [2I {WT 1}] > 2E g [I {WT 1}]. ) is he soluion of BSDE y = I {WT 1} + 2( Z s 2 1)+ ds ( Z s 2 1)+ ds (z s 1) + ds Z s dw s. (19) Z s 2 dw s, z s dw s. Bu y = E g [I {WT 1} F ]. Thus by he uniqueness of he soluion of BSDE, Y 2 = E g[i {WT 1} F ]. On he oher hand, le (y, z ) be he soluion of he BSDE y = 2I {WT 1} + (z s 1) + ds Comparing BSDE(20) wih BSDE (19), noice (17) and he fac and (z 1) + 2( z 2 1)+ (z 1) + > 2( z 2 1)+, whenever z > 1. By he sric comparison heorem of BSDE, we have Seing = 0, hus The proof is complee. y > Y, [0, T ). E g [2I {WT 1}] > 2E g [I {WT 1}] = C V (2I {WT 1}). z s dw s. (20)

18 Nonlinear pricing 18 Table 1: Relaions Among Coheren and Convex Risk Measures, Choque Expecaion and Jensen s Inequaliy E g [ξ] Risk Measures Relaion o Jensen inequaliy Choque Expecaion g is linear mah. expecaion E g [ξ] = C V (ξ) rue g is CPH coheren E g [ξ] C V (ξ) rue (*) g is convex convex Neiher nor no rue excep (*) Remark 3 In mahemaical finance, coheren and convex risk measures and Choque expecaion are used in he pricing of coningen claim. Theorem 4 shows ha coheren pricing is always less han Choque pricing, while Counerexample 1 demonsraes ha pricing by a convex risk measure no longer has his propery. In fac he convex risk price may be greaer han or less han he Choque expecaion. 4 Summary Coheren risk measures are a generalizaion of mahemaical expecaions, while convex risk measures are a generalizaion of coheren risk measures. In he framework of g-expecaion, he summary of our resuls is given in Table 1. In ha Table, he Choque expecaion is V (A) := E g [I A ] and CPH is an abbreviaion for convex and posiively homogeneous. Counerexample 1 shows ha convex risk may be or Choque expecaion. Only in he case of coheren risk here is an inequaliy relaion wih Choque expecaion. 5 The Models Linear Pricing and Nonlinear Pricing In his secion, we shall employ our resul in he pricing of coningen claims. we begin wih he Black-Scholes model for coninuous-ime asse pricing. Consider a financial marke, he basic securiies consis of n + 1 asses. One of hem is a locally riskless asse wih price per uni P 0 governed by he linear equaion dp 0 i = P 0 i r d, where r is he shor rae. In addiion o he bond, n risk securiies(he socks)are coninuously raded. The price process P i for one share of ih sock is modeled by

19 Nonlinear pricing 19 he linear sochasic differenial equaion dp i = P i [b i d + n j=1 σ i,j dw j ], where W = (W 1,..., W n ) is a sandard Brownian moion on R n and defined on a probabiliy space. Making he following assumpions, he marke is complee, The shor rae r is a predicable and bounded process. I is generally nonnegaive. The column vecor of he sock appreciaion raes b = (b 1,..., b n ) is a predicable and bounded process. The volailiy marix σ = (σ i,j ) is a predicable and bounded process. σ has full rank a.s. for all [0, T ] and he inverse marix σ 1 is a bounded process. Consider a small invesor whose acions canno affec marke prices and who can decide a ime [0, T ] wha amoun π i of he wealh V o inves in he ih sock, i = 1,..., n. Following Harrison and Pliska(1981),Cvianic and Karazas (1993) and El Karoui e al. (1997), a sraegy is self-financing if he wealh process V = V V 0,π, which depend on he iniial wealh V 0 and he porfolio process π, saisfies he linear sochasic differenial equaion(sde) dv = r V d + π (b r 1)d + π σ dw Given a European coningen claim ξ, he quesion of hedging coningen claim ξ in fac is o seek an iniial endowmen x and porfolio process π such ha he wealh process V x,π T = ξ. The fair price of claim ξ is defined as he minimal endowmen x. In he above case, Black-Scholes[ formula shows ha here exiss a neural probabiliy measures such ha x = E Q ξe R ] T 0 rsds. I concludes ha he fair price of every coningen claim ξ is equal o he (linear) mahemaical expecaion of he discouned value of he claim ξ under a new, so called risk-neural probabiliy measures, he price is a linear expecaion due o he lineariy of he wealh process V in SDE(). Recenly, in sudying he pricing of coningen claim wih consrain on wealh or porfolio processes, many auhors have inroduced some nonlinear wealh processes for he fair price of claims. Among hem are so-called higher ineres rae for borrowing(bergman(1991), Korn(1992), and Cvianic and Karazas(1993)) and shor sales consrain(jouini and Kallal 1995, He and Pearson 1991). The marke value processes V 0,π are governed by he following nonlinear SDEs, respecely: n dv = r V d + π σ θ d + π σ dw (R r )(V π) i d. i=1

20 Nonlinear pricing 20 And dv = r V d + π σ θ 1 d + [π ] σ [θ 1 θ 2 ]d + π σ dw. Thus, he goal of pricing a coningen claim ξ is o find (x, π) such ha he wealh processes V x,π saisfies he above equaion and V x,π T = ξ. How o calculae x? The classical resul idenifies x as he linear expecaion, under a neural measure, of he claim s discouned value when he wealh process is linear. Can we idenify x as nonlinear expecaions when he wealh processes are SDE() or SDE()? Definiion 6 Given a wealh process V x,π, which depends corresponding o he iniial value x and porfolio π, if here is a nonlinear expecaion E[ ] such ha for any claim ξ L 2 (Ω, F, P ), le x = E[e R T 0 rsds ξ], here exiss a porfolio π such ha = ξ, we say ha he marke value can be priced by nonlinear expecaion E[ ]. The fair price of a claim corresponding o nonlinear expecaion E[ ] is sill defined as he minimal endowmen o finance a sraegy which guaranees ξ a ime T. V x,π T Applying our resuls, immediaely, we obain he following resuls. Theorem 5 Claims wih higher ineres rae for borrowing can be priced by g- expecaions, bu no by convex risk measures, while claims wih shor-sales consrains can be priced by boh g-expecaions and coheren risk measures. References [1] Arzner, Ph., F. Delbaen, J. M. Eber, and D. Heah (1999): Coheren measures of risk. Mahemaical Finance, 9, [2] Briand, P., F. Coque, Y. Hu, J. Mémin, and S. Peng (2000): A converse comparison heorem for BSDEs and relaed problems of g-expecaion. Elecronic Communicaions in Probabiliy, 5, [3] Chen, Z., R. Kulperger, and L. Jiang (2003): Jensen s inequaliy for g- expecaion: par 1. C. R. Acad. Sci. Paris, Ser.I, 337, [4] Choque, G. (1953): Theory of capaciies. Ann. Ins. Fourier (Grenoble), 5, [5] Coque, F., Hu, Y., Mémin, J., S., Peng (2001): A general converse comparison heorem for backward sochasic differenial equaions. C. R. Acad. Sci. Paris, Ser. I, 333,

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