- Email: [email protected]
Hedging American contingent claims with arbitrage costs
Chaos, Solitons and Fractals 32 (2007) 598–603 www.elsevier.com/locate/chaos
Hedging American contingent claims with arbitrage costs Wang Bo b
a,*
, Meng Qingxin
b
a Wenzhou Medical College, Wenzhou 325035, China Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Accepted 2 November 2005
Abstract In a continuous-time market model, the wealth process have an arbitrage costs. we give a representation for the upper hedging prices hup of American contingent claims. Furthermore, we give some example of the arbitrage costs. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The valuation for American contingent claims, due to Bensoussan [1] and Karatzas [2], is extended to deal with constraints on portfolio choice, included incomplete markets and borrowing/short-selling constraints, or with different interest rats for borrowing and lending. Karatzas and Kou [3] obtained the hedging price of American contingent claims with constrained portfolios in 1998. Furthermore, Wang Bo and Meng Qingxin [7] obtained the hedging price of American contingent claims with constrained portfolios under proportional transaction costs. In this paper, we consider the wealth process have an arbitrage cost. The arbitrage cost is a lipschitz function, which can be transaction costs or the costs due to the different interest rats for borrowing and lending, and so on. In this model the upper hedging price hup is obtained. In the last we give some example of the arbitrage cost.
2. The model We consider a financial market with d + 1 assets. One of these asset, called the bond, has price S0(t) given by dS 0 ðtÞ ¼ S 0 ðtÞrðtÞ dt; 0 6 t 6 T ; S 0 ð0Þ ¼ 1. The remaining d assets are subject to systematic risk; we refer to them as stock, and assume that the price per share Si(t) of the ith satisfies the equation:
*
Corresponding author. E-mail addresses: [email protected] (W. Bo), [email protected] (M. Qingxin).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.007
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
599
" # 8 d > < dS ðtÞ ¼ S ðtÞ b ðtÞ dt þ P r ðtÞ dW ðtÞ ; i i i ij j j¼1 > : S i ð0Þ ¼ si 2 ð0; 1Þ for every i = 1, . . . , d. Here W = {(W1(t), . . . , Wd(t))*, 0 6 t 6 T} is a d-dimensional standard Brownian motion on a complete probability space ðX; F; P Þ, endowed with a filtration F ¼ fFðtÞ; 0 6 t 6 T g which is the P-augmentation of the natural filtration generated by W. The scalar processes r(Æ), b(Æ) = (b1(Æ), . . . , bd(Æ))* and r(Æ) = (rij(Æ))16i,j6d are bounded uniformly in (t, w) 2 [0, T] · X and progressively measurable with respect to F. Furthermore, r(t) is assumed to be invertible, with r1(t, x) bounded uniformly in (t, w) 2 [0, T] · X. We define the relative risk process of the market hðtÞ , r1 ðtÞðbðtÞ rðtÞ~ 1Þ; then h(t) is bounded and F-progressively measurable; thus Z t Z 1 t h ðsÞ dW ðsÞ khðsÞk2 ds ; 0 6 t 6 T Z 0 ðtÞ , exp 2 0 0 is a martingale, and W 0 ðtÞ , W ðtÞ þ
Z
t
hðsÞ ds;
06t6T
0
is a Brownian motion under the probability measure P 0 ðAÞ , E½Z 0 ðT Þ1A ;
A 2 FðT Þ
by the Girsanov theorem. Definition 1 RT (i) An F-progressively measurable process p : [0, T] · X ! Rd with 0 kpðtÞk2 dt < 1 a.s. is called portfolio process. (ii) An F-adapted process C : [0, T] · X ! [0, 1) with increasing, right continuous paths and C(0) = 0, C(T) < 1 a.s., is called cumulative consumption process. In this Market, the wealth process corresponding to a given portfolio/consumption (p, C) and initial capital X(0) = x satisfies ! " # d d d X X X pi ðtÞ rðtÞ dt þ pi ðtÞ bi ðtÞ dt þ rij ðtÞ dW j ðtÞ F ðt; pÞ dt dCðtÞ; ð1Þ dX ðtÞ ¼ X ðtÞ i¼1
i¼1
j¼1
here F(t, p) may expressed as an additional cost, F : [0, T] · X · Rd ! R is a P BðRd Þ progressively measurable proR T be 2 cess with E 0 F ðt; x; 0Þ dt < 1 and jF ðt; x; yÞ F ðt; x; y 0 Þj 6 kjy y 0 j. We rewrite (1) dbðtÞX ðtÞ ¼ bðtÞ dCðtÞ þ bðtÞp ðtÞrðtÞ dW 0 ðtÞ bðtÞF ðt; pÞ dt; where
Z t bðtÞ , exp rðsÞ ds ;
ð2Þ
0 6 t 6 T.
0
Finally, let us notice that the solution of (4) is given by Z t Z t Z t bðsÞ dCðsÞ þ bðsÞF ðp; sÞ ds ¼ x þ bðsÞp ðsÞrðsÞ dW 0 ðsÞ. bðtÞX ðtÞ þ 0
0
ð3Þ
0
Definition 2. We say that a portfolio/consumption process pair (p, C) is admissible for the initial wealth x, if the pair p(Æ), C(Æ) obeys the conditions of Definition 1.
600
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
The set of admissible pair (p, C) will be denoted by A0(x). Here and in the sequel, we denote Cs,t the class of F-stopping times s : X ! [s, t] for 0 6 s 6 t 6 T and let C , C0,T.
3. American contingent claims and constraint on portfolio choice Definition 3. An American contingent claim (ACC) is an F-adapted process B : [0, T] · X ! (0, 1) with continuous sample paths and E sup ðbðtÞBðtÞÞ2 < þ1. 06t6T
Let us consider now the following situation: at time t = 0, the buyer and the seller enter into an agreement. The seller agrees to provide the buyer with the random payment B(s(x), x) at time t = s(x) where s 2 f and at the disposal of the buyer. What is the amount of money x P 0 that the buyer should pay at time t = 0, in return for this commitment? Define ^
b 2 A0 ðxÞ s.t. X x;^p;C ðsÞ P BðsÞ 8s 2 Cg. U , fx P 0j9ð^ p; CÞ We introduce the upper hedging price hup , inffx P 0jx 2 Ug. In the next two sections, we shall consider the upper hedging price hup of ACC.
4. The drift F(t, x, Æ) In this section we study the case of a drift random field F. We make the following assumptions. Assumption 1. The function y ! F(t, x, y) is convex on Rd, for every (t, x) 2 [0, T] · X. Following Karoui et al. [5], we introduce for every fixed (t, x) 2 [0, T] · X the dual f ðt; x; lÞ , sup½lT y F ðt; x; yÞ;
l 2 Rd
y2Rd
of the convex function F(t, x, Æ), as well as its effective domain O , fðt; x; lÞ 2 ½0; T X Rd jf ðt; x; lÞ < 1g. As in Karoui et al. [5], one can show that each (t, x)-section of O is include in a bounded set Q in Rd, independent of (t,Rx). Let us also introduce the class D of F-progressively measurable processes l(Æ) : [0, T] · X ! Q which satisfies T E 0 f 2 ðt; lðtÞÞ dt < 1. Lemma 1. For any p(Æ) 2 A0(x), there exists a process l(Æ) 2 D such that f ðt; lðtÞÞ ¼ l ðtÞpðtÞ F ðt; pðtÞÞ;
0 6 t 6 T;
hold almost surely. This result is proved in Karoui et al. [5]. 5. The upper hedging For any given pair of processes l(Æ) 2 D, let us introduce the exponential martingale Z t Z 1 t 1 Z l ðtÞ , exp ðr1 ðsÞlðsÞÞ dW 0 ðsÞ kr ðsÞlðsÞk2 ds 2 0 0 as well as the probability measure P l ðAÞ , E½Z l ðT Þ1A ;
A 2 FT
ð4Þ
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
601
under which process Z
W l ðtÞ , W 0 ðtÞ
t
r1 ðsÞlðsÞ ds
0
is Brownian motion. we also denote by El the expectation with respect to the probability measure of Pl. Then, we can rewrite (3) for X(Æ) = Xx,p,C(Æ) equivalently as Z t Z t Z t bðsÞ dCðsÞ þ bðsÞp ðsÞrðsÞ dW l ðsÞ bðsÞ½F ðs; pÞ p ðsÞlðsÞ ds; 0 6 t 6 T bðtÞX ðtÞ ¼ x 0
0
ð5Þ
0
for every l 2 D. Define
Z s U l ð0Þ , sup El bðsÞBðsÞ bðsÞf ðs; lÞ ds ; s2C 0 Z T 1 b l ðtÞ , bðsÞf ðs; lÞ dsjFðtÞ ; X ess sup El bðsÞBðsÞ bðtÞ s2Ct;T s V , sup U l ð0Þ. l2D
b l ðT Þ ¼ BðT Þ, a.s. Finally, we introduce the process b l ð0Þ ¼ U l ð0Þ; X Clearly X b l ðtÞ. X ðtÞ , ess sup X l2D
Clearly that X ð0Þ ¼ V , X ðT Þ ¼ B. Then we have the following lemma. Lemma 2. The F-adapted process X ðÞ can be considered in its RCLL modification and bðÞX ðÞ is a Pl—supermartingale for every l 2 D. The proof is similar to A.2 lemma of Karatzas [3]. Theorem 1. The upper hedging price hup is given by Z T l bðsÞf ðs; lÞ ds . hup ¼ V ¼ sup sup E bðsÞBðsÞ l2D s2C
s
b 2 A0 ðV Þ such that X V ;^p;C ðsÞ ¼ X ðsÞ P BðsÞ holds almost surely for Furthermore, if V < 1, there exists a pair ð^ p; CÞ "s 2 C. ^
Proof. First we shall prove the inequality hup P V. If hup = 1, the inequality is obvious; if not, the set U is nonempty. b 2 A0 ðxÞ, s.t. X x;^p;C^ ðsÞ P BðsÞ 8s 2 C. With x P 0 an arbitrary element of U, there exists ð^ p; CÞ From (5), we have Z t Z t Z t ^ b ^Þ p ^ ðsÞlðsÞ ds ¼ x þ bðsÞ d CðsÞ þ bðsÞ½F ðs; p bðsÞ^ p ðsÞrðsÞ dW l ðsÞ; 0 6 t 6 T . bðtÞX x;^p;C ðtÞ þ 0
0
0
So bðtÞX
^ x;^ p;C
¼xþ
ðtÞ
Z
Z
t
bðsÞf ðs; lÞ ds þ
0
Z
t
b bðsÞ d CðsÞ þ
0
Z
t
^Þ p ^ ðsÞlðsÞ ds bðsÞ½f ðs; lÞ þ F ðs; p
0
t
bðsÞ^ p ðsÞrðsÞ dW l ðsÞ;
0 6 t 6 T.
0
The stochastic integral on the right-hand side is a Pl-martingale by virtue of the Burkholder–Davis–Gundy inequalities [4, Theorem 3.3.28], since we have Z T 12 12 Z T 2 2 l 0 2 0 E kbðsÞ^ p ðsÞrðsÞk dl 6 E Z l ðT ÞE kbðsÞ^ p ðsÞrðsÞk dl < 1. 0
For any l 2 D, we have ^Þ p ^ ðsÞlðsÞ P 0. f ðs; lÞ þ F ðs; p
0
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W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
From the optional sampling theorem, we obtain Z s Z s Z t ^ ^ þ ^Þ p ^ ðsÞlðsÞ ds bðsÞf ðs; lÞ ds þ bðsÞ dCðsÞ bðsÞ½f ðs; lÞ þ F ðs; p x P El bðsÞX x;^p;C ðsÞ 0 0 0 Z s l bðsÞf ðs; pÞ ds P E bðsÞBðsÞ 0
for every s 2 C and l 2 D. Therefore, Z s bðsÞf ðs; lÞ ds ¼ V . x P sup sup El bðsÞBðsÞ l2D s2C
0
So hup P V . The following is the proof of hup 6 V. From the Doob–Meyer decomposition and the Martingale representation theory, we can represent the supermartingle bðÞX ðÞ of Lemma 1 Z t bðtÞX ðtÞ ¼ V þ M l ðtÞ Al ðtÞ ¼ V þ wl ðsÞ dW l ðsÞ Al ðtÞ; 0 6 t 6 T 0
RT for every l 2 D. Here wl : [0, T] · X ! Rd is F-progressively measurable with 0 kwl ðtÞk2 dt < þ1 a.s. and Al : [0, T] · X ! [0, +1) is F-adapted, natural increasing, with right continuous paths and Al(0) = 0. Observe that for any l 2 D and k 2 D, we have Z t Z t wl ðsÞ dW k ðsÞ Al ðtÞ þ wl ðsÞr1 ðsÞðkðsÞ lðsÞÞ ds; 0 6 t 6 T . bðtÞX ðtÞ ¼ V þ 0
0
Compare this with bðtÞX ðtÞ ¼ V þ
Z
t
wk ðsÞ dW k ðsÞ Ak ðtÞ; 0
we obtain the two processes wl ðtÞ wk ðtÞ , hðtÞ ¼ bðtÞ bðtÞ and Z
t
b1 ðsÞ dAl ðsÞ 0
¼
Z
t
½F ðs; h ðsÞr1 ðsÞÞ h ðsÞr1 ðsÞlðsÞ ds
0
Z
t 1
b ðsÞ dAk ðsÞ 0
Z
t
b ½F ðs; h ðsÞr1 ðsÞÞ h ðsÞr1 ðsÞkðsÞ ds , CðtÞ;
0
b ¼ do not depend on l 2 D. In particular, putting l(Æ) = F(Æ, h*(Æ)r1(Æ)), we have CðtÞ b Hence, CðÞ is an increasing adapted RCLL process. We also have Z T Z T 1fX ðtÞ¼0g khðtÞk2 dt ¼ b2 ðtÞ1fX ðtÞ¼0g dhM l iðtÞ ¼ 0; a.s. 0
Rt 0
b1 ðsÞ dAl ðsÞ; 0 6 t 6 T .
0
^ : ½0; T X ! Rd , from equations (12.1) and (12.3), p. 365 in [6]. Define F-progressively measurable process p ^ðÞ , ðr ðÞÞ1 hðÞ; p RT pðtÞk2 dt < 1. So we have 0 k^ Z t Z t Z t b ^ðsÞÞ p ^ ðsÞlðsÞ ds þ bðsÞ d CðsÞ bðsÞ½F ðs; p bðsÞ^ p ðsÞrðsÞ dW l ðsÞ bðtÞX ðtÞ ¼ V 0
0
0
for every l 2 D. In particular, for l(Æ) 0, (6) gives that Z t Z t Z t b ^ðsÞÞ ds þ bðsÞ d CðsÞ bðsÞF ðs; p bðsÞ^ p ðsÞrðsÞ dW 0 ðsÞ; bðtÞX ðtÞ ¼ V 0
0
0
0 6 t 6 T;
ð6Þ
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
603
b b 2 A0 ðV Þ and X V ;^p;C^ ðsÞ ¼ X ðsÞ P BðsÞ 8s 2 C. So this shows X V ;^p; C ðÞ ¼ X ðÞ. Since X ðÞ P BðÞ P 0, then ð^ p; CÞ hup(K) 6 V < 1. The proof of the theorem is completed. h
6. Examples The following examples motivate the analysis in the present paper and illustrate the connection with the familiar market model. For purpose of comparison, we start from the standard complete-market setting. Standard setting: In this case F(t, p) 0 and 0 if l ¼ 0; f ðt; lÞ ¼ 1 otherwise. Therefore D ¼ 0 for all ðt; xÞ 2 ½0; T X. This is the setting examined by Karatzas [2,3]. Proportional transaction costs setting: In this case F ðt; pÞ ¼
d X ðmi pþ i ðtÞ þ ni pi ðtÞÞ i¼1
and f ðt; lÞ ¼
0 if l 2 ½m1 ; n1 ½md ; nd ; 1 otherwise.
Therefore D ¼ ½m1 ; n1 ½md ; nd ;
for all ðt; xÞ 2 ½0; T X.
This is the setting examined by Wang Bo and Meng Qingxin [7].
Acknowledgement Research is supported by The Natural Science Foundation of Zhejiang Province under grant Y605478.
References [1] [2] [3] [4] [5] [6]
Bensoussan A. On the theory of option pricing. Acta Appl Math 1984;2:139–58. Karatzas I. On the pricing of American option. Appl Math Optim 1988;17:37–60. Karatzas I, Kou S. Hedging American contingent claims with constrained portfolios. Finance Stochast 1998:215–58. Karatzas I, Shreve SE. Brownian motion and stochastic calculus. 2nd ed. New York: Springer; 1991. Karoui NEl, Peng S, Quenez QC. Bachward stochastic differential equations in finance. Math Finance 1997;7:1–71. Meyer PA. Un cours sur les integrales stochastiques. Lecture notes in mathematics, 511. Berlin, Heidelberg: Springer; 1976. pp. 245–398. [7] Wang B, Meng Q. Hedging American contingent claims with constrained portfolios under proportional transaction costs. Chaos, Solitons & Fractals 2005;23:1153–62.
Hedging American contingent claims with arbitrage costs Wang Bo b
a,*
, Meng Qingxin
b
a Wenzhou Medical College, Wenzhou 325035, China Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Accepted 2 November 2005
Abstract In a continuous-time market model, the wealth process have an arbitrage costs. we give a representation for the upper hedging prices hup of American contingent claims. Furthermore, we give some example of the arbitrage costs. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The valuation for American contingent claims, due to Bensoussan [1] and Karatzas [2], is extended to deal with constraints on portfolio choice, included incomplete markets and borrowing/short-selling constraints, or with different interest rats for borrowing and lending. Karatzas and Kou [3] obtained the hedging price of American contingent claims with constrained portfolios in 1998. Furthermore, Wang Bo and Meng Qingxin [7] obtained the hedging price of American contingent claims with constrained portfolios under proportional transaction costs. In this paper, we consider the wealth process have an arbitrage cost. The arbitrage cost is a lipschitz function, which can be transaction costs or the costs due to the different interest rats for borrowing and lending, and so on. In this model the upper hedging price hup is obtained. In the last we give some example of the arbitrage cost.
2. The model We consider a financial market with d + 1 assets. One of these asset, called the bond, has price S0(t) given by dS 0 ðtÞ ¼ S 0 ðtÞrðtÞ dt; 0 6 t 6 T ; S 0 ð0Þ ¼ 1. The remaining d assets are subject to systematic risk; we refer to them as stock, and assume that the price per share Si(t) of the ith satisfies the equation:
*
Corresponding author. E-mail addresses: [email protected] (W. Bo), [email protected] (M. Qingxin).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.007
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
599
" # 8 d > < dS ðtÞ ¼ S ðtÞ b ðtÞ dt þ P r ðtÞ dW ðtÞ ; i i i ij j j¼1 > : S i ð0Þ ¼ si 2 ð0; 1Þ for every i = 1, . . . , d. Here W = {(W1(t), . . . , Wd(t))*, 0 6 t 6 T} is a d-dimensional standard Brownian motion on a complete probability space ðX; F; P Þ, endowed with a filtration F ¼ fFðtÞ; 0 6 t 6 T g which is the P-augmentation of the natural filtration generated by W. The scalar processes r(Æ), b(Æ) = (b1(Æ), . . . , bd(Æ))* and r(Æ) = (rij(Æ))16i,j6d are bounded uniformly in (t, w) 2 [0, T] · X and progressively measurable with respect to F. Furthermore, r(t) is assumed to be invertible, with r1(t, x) bounded uniformly in (t, w) 2 [0, T] · X. We define the relative risk process of the market hðtÞ , r1 ðtÞðbðtÞ rðtÞ~ 1Þ; then h(t) is bounded and F-progressively measurable; thus Z t Z 1 t h ðsÞ dW ðsÞ khðsÞk2 ds ; 0 6 t 6 T Z 0 ðtÞ , exp 2 0 0 is a martingale, and W 0 ðtÞ , W ðtÞ þ
Z
t
hðsÞ ds;
06t6T
0
is a Brownian motion under the probability measure P 0 ðAÞ , E½Z 0 ðT Þ1A ;
A 2 FðT Þ
by the Girsanov theorem. Definition 1 RT (i) An F-progressively measurable process p : [0, T] · X ! Rd with 0 kpðtÞk2 dt < 1 a.s. is called portfolio process. (ii) An F-adapted process C : [0, T] · X ! [0, 1) with increasing, right continuous paths and C(0) = 0, C(T) < 1 a.s., is called cumulative consumption process. In this Market, the wealth process corresponding to a given portfolio/consumption (p, C) and initial capital X(0) = x satisfies ! " # d d d X X X pi ðtÞ rðtÞ dt þ pi ðtÞ bi ðtÞ dt þ rij ðtÞ dW j ðtÞ F ðt; pÞ dt dCðtÞ; ð1Þ dX ðtÞ ¼ X ðtÞ i¼1
i¼1
j¼1
here F(t, p) may expressed as an additional cost, F : [0, T] · X · Rd ! R is a P BðRd Þ progressively measurable proR T be 2 cess with E 0 F ðt; x; 0Þ dt < 1 and jF ðt; x; yÞ F ðt; x; y 0 Þj 6 kjy y 0 j. We rewrite (1) dbðtÞX ðtÞ ¼ bðtÞ dCðtÞ þ bðtÞp ðtÞrðtÞ dW 0 ðtÞ bðtÞF ðt; pÞ dt; where
Z t bðtÞ , exp rðsÞ ds ;
ð2Þ
0 6 t 6 T.
0
Finally, let us notice that the solution of (4) is given by Z t Z t Z t bðsÞ dCðsÞ þ bðsÞF ðp; sÞ ds ¼ x þ bðsÞp ðsÞrðsÞ dW 0 ðsÞ. bðtÞX ðtÞ þ 0
0
ð3Þ
0
Definition 2. We say that a portfolio/consumption process pair (p, C) is admissible for the initial wealth x, if the pair p(Æ), C(Æ) obeys the conditions of Definition 1.
600
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
The set of admissible pair (p, C) will be denoted by A0(x). Here and in the sequel, we denote Cs,t the class of F-stopping times s : X ! [s, t] for 0 6 s 6 t 6 T and let C , C0,T.
3. American contingent claims and constraint on portfolio choice Definition 3. An American contingent claim (ACC) is an F-adapted process B : [0, T] · X ! (0, 1) with continuous sample paths and E sup ðbðtÞBðtÞÞ2 < þ1. 06t6T
Let us consider now the following situation: at time t = 0, the buyer and the seller enter into an agreement. The seller agrees to provide the buyer with the random payment B(s(x), x) at time t = s(x) where s 2 f and at the disposal of the buyer. What is the amount of money x P 0 that the buyer should pay at time t = 0, in return for this commitment? Define ^
b 2 A0 ðxÞ s.t. X x;^p;C ðsÞ P BðsÞ 8s 2 Cg. U , fx P 0j9ð^ p; CÞ We introduce the upper hedging price hup , inffx P 0jx 2 Ug. In the next two sections, we shall consider the upper hedging price hup of ACC.
4. The drift F(t, x, Æ) In this section we study the case of a drift random field F. We make the following assumptions. Assumption 1. The function y ! F(t, x, y) is convex on Rd, for every (t, x) 2 [0, T] · X. Following Karoui et al. [5], we introduce for every fixed (t, x) 2 [0, T] · X the dual f ðt; x; lÞ , sup½lT y F ðt; x; yÞ;
l 2 Rd
y2Rd
of the convex function F(t, x, Æ), as well as its effective domain O , fðt; x; lÞ 2 ½0; T X Rd jf ðt; x; lÞ < 1g. As in Karoui et al. [5], one can show that each (t, x)-section of O is include in a bounded set Q in Rd, independent of (t,Rx). Let us also introduce the class D of F-progressively measurable processes l(Æ) : [0, T] · X ! Q which satisfies T E 0 f 2 ðt; lðtÞÞ dt < 1. Lemma 1. For any p(Æ) 2 A0(x), there exists a process l(Æ) 2 D such that f ðt; lðtÞÞ ¼ l ðtÞpðtÞ F ðt; pðtÞÞ;
0 6 t 6 T;
hold almost surely. This result is proved in Karoui et al. [5]. 5. The upper hedging For any given pair of processes l(Æ) 2 D, let us introduce the exponential martingale Z t Z 1 t 1 Z l ðtÞ , exp ðr1 ðsÞlðsÞÞ dW 0 ðsÞ kr ðsÞlðsÞk2 ds 2 0 0 as well as the probability measure P l ðAÞ , E½Z l ðT Þ1A ;
A 2 FT
ð4Þ
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
601
under which process Z
W l ðtÞ , W 0 ðtÞ
t
r1 ðsÞlðsÞ ds
0
is Brownian motion. we also denote by El the expectation with respect to the probability measure of Pl. Then, we can rewrite (3) for X(Æ) = Xx,p,C(Æ) equivalently as Z t Z t Z t bðsÞ dCðsÞ þ bðsÞp ðsÞrðsÞ dW l ðsÞ bðsÞ½F ðs; pÞ p ðsÞlðsÞ ds; 0 6 t 6 T bðtÞX ðtÞ ¼ x 0
0
ð5Þ
0
for every l 2 D. Define
Z s U l ð0Þ , sup El bðsÞBðsÞ bðsÞf ðs; lÞ ds ; s2C 0 Z T 1 b l ðtÞ , bðsÞf ðs; lÞ dsjFðtÞ ; X ess sup El bðsÞBðsÞ bðtÞ s2Ct;T s V , sup U l ð0Þ. l2D
b l ðT Þ ¼ BðT Þ, a.s. Finally, we introduce the process b l ð0Þ ¼ U l ð0Þ; X Clearly X b l ðtÞ. X ðtÞ , ess sup X l2D
Clearly that X ð0Þ ¼ V , X ðT Þ ¼ B. Then we have the following lemma. Lemma 2. The F-adapted process X ðÞ can be considered in its RCLL modification and bðÞX ðÞ is a Pl—supermartingale for every l 2 D. The proof is similar to A.2 lemma of Karatzas [3]. Theorem 1. The upper hedging price hup is given by Z T l bðsÞf ðs; lÞ ds . hup ¼ V ¼ sup sup E bðsÞBðsÞ l2D s2C
s
b 2 A0 ðV Þ such that X V ;^p;C ðsÞ ¼ X ðsÞ P BðsÞ holds almost surely for Furthermore, if V < 1, there exists a pair ð^ p; CÞ "s 2 C. ^
Proof. First we shall prove the inequality hup P V. If hup = 1, the inequality is obvious; if not, the set U is nonempty. b 2 A0 ðxÞ, s.t. X x;^p;C^ ðsÞ P BðsÞ 8s 2 C. With x P 0 an arbitrary element of U, there exists ð^ p; CÞ From (5), we have Z t Z t Z t ^ b ^Þ p ^ ðsÞlðsÞ ds ¼ x þ bðsÞ d CðsÞ þ bðsÞ½F ðs; p bðsÞ^ p ðsÞrðsÞ dW l ðsÞ; 0 6 t 6 T . bðtÞX x;^p;C ðtÞ þ 0
0
0
So bðtÞX
^ x;^ p;C
¼xþ
ðtÞ
Z
Z
t
bðsÞf ðs; lÞ ds þ
0
Z
t
b bðsÞ d CðsÞ þ
0
Z
t
^Þ p ^ ðsÞlðsÞ ds bðsÞ½f ðs; lÞ þ F ðs; p
0
t
bðsÞ^ p ðsÞrðsÞ dW l ðsÞ;
0 6 t 6 T.
0
The stochastic integral on the right-hand side is a Pl-martingale by virtue of the Burkholder–Davis–Gundy inequalities [4, Theorem 3.3.28], since we have Z T 12 12 Z T 2 2 l 0 2 0 E kbðsÞ^ p ðsÞrðsÞk dl 6 E Z l ðT ÞE kbðsÞ^ p ðsÞrðsÞk dl < 1. 0
For any l 2 D, we have ^Þ p ^ ðsÞlðsÞ P 0. f ðs; lÞ þ F ðs; p
0
602
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
From the optional sampling theorem, we obtain Z s Z s Z t ^ ^ þ ^Þ p ^ ðsÞlðsÞ ds bðsÞf ðs; lÞ ds þ bðsÞ dCðsÞ bðsÞ½f ðs; lÞ þ F ðs; p x P El bðsÞX x;^p;C ðsÞ 0 0 0 Z s l bðsÞf ðs; pÞ ds P E bðsÞBðsÞ 0
for every s 2 C and l 2 D. Therefore, Z s bðsÞf ðs; lÞ ds ¼ V . x P sup sup El bðsÞBðsÞ l2D s2C
0
So hup P V . The following is the proof of hup 6 V. From the Doob–Meyer decomposition and the Martingale representation theory, we can represent the supermartingle bðÞX ðÞ of Lemma 1 Z t bðtÞX ðtÞ ¼ V þ M l ðtÞ Al ðtÞ ¼ V þ wl ðsÞ dW l ðsÞ Al ðtÞ; 0 6 t 6 T 0
RT for every l 2 D. Here wl : [0, T] · X ! Rd is F-progressively measurable with 0 kwl ðtÞk2 dt < þ1 a.s. and Al : [0, T] · X ! [0, +1) is F-adapted, natural increasing, with right continuous paths and Al(0) = 0. Observe that for any l 2 D and k 2 D, we have Z t Z t wl ðsÞ dW k ðsÞ Al ðtÞ þ wl ðsÞr1 ðsÞðkðsÞ lðsÞÞ ds; 0 6 t 6 T . bðtÞX ðtÞ ¼ V þ 0
0
Compare this with bðtÞX ðtÞ ¼ V þ
Z
t
wk ðsÞ dW k ðsÞ Ak ðtÞ; 0
we obtain the two processes wl ðtÞ wk ðtÞ , hðtÞ ¼ bðtÞ bðtÞ and Z
t
b1 ðsÞ dAl ðsÞ 0
¼
Z
t
½F ðs; h ðsÞr1 ðsÞÞ h ðsÞr1 ðsÞlðsÞ ds
0
Z
t 1
b ðsÞ dAk ðsÞ 0
Z
t
b ½F ðs; h ðsÞr1 ðsÞÞ h ðsÞr1 ðsÞkðsÞ ds , CðtÞ;
0
b ¼ do not depend on l 2 D. In particular, putting l(Æ) = F(Æ, h*(Æ)r1(Æ)), we have CðtÞ b Hence, CðÞ is an increasing adapted RCLL process. We also have Z T Z T 1fX ðtÞ¼0g khðtÞk2 dt ¼ b2 ðtÞ1fX ðtÞ¼0g dhM l iðtÞ ¼ 0; a.s. 0
Rt 0
b1 ðsÞ dAl ðsÞ; 0 6 t 6 T .
0
^ : ½0; T X ! Rd , from equations (12.1) and (12.3), p. 365 in [6]. Define F-progressively measurable process p ^ðÞ , ðr ðÞÞ1 hðÞ; p RT pðtÞk2 dt < 1. So we have 0 k^ Z t Z t Z t b ^ðsÞÞ p ^ ðsÞlðsÞ ds þ bðsÞ d CðsÞ bðsÞ½F ðs; p bðsÞ^ p ðsÞrðsÞ dW l ðsÞ bðtÞX ðtÞ ¼ V 0
0
0
for every l 2 D. In particular, for l(Æ) 0, (6) gives that Z t Z t Z t b ^ðsÞÞ ds þ bðsÞ d CðsÞ bðsÞF ðs; p bðsÞ^ p ðsÞrðsÞ dW 0 ðsÞ; bðtÞX ðtÞ ¼ V 0
0
0
0 6 t 6 T;
ð6Þ
W. Bo, M. Qingxin / Chaos, Solitons and Fractals 32 (2007) 598–603
603
b b 2 A0 ðV Þ and X V ;^p;C^ ðsÞ ¼ X ðsÞ P BðsÞ 8s 2 C. So this shows X V ;^p; C ðÞ ¼ X ðÞ. Since X ðÞ P BðÞ P 0, then ð^ p; CÞ hup(K) 6 V < 1. The proof of the theorem is completed. h
6. Examples The following examples motivate the analysis in the present paper and illustrate the connection with the familiar market model. For purpose of comparison, we start from the standard complete-market setting. Standard setting: In this case F(t, p) 0 and 0 if l ¼ 0; f ðt; lÞ ¼ 1 otherwise. Therefore D ¼ 0 for all ðt; xÞ 2 ½0; T X. This is the setting examined by Karatzas [2,3]. Proportional transaction costs setting: In this case F ðt; pÞ ¼
d X ðmi pþ i ðtÞ þ ni pi ðtÞÞ i¼1
and f ðt; lÞ ¼
0 if l 2 ½m1 ; n1 ½md ; nd ; 1 otherwise.
Therefore D ¼ ½m1 ; n1 ½md ; nd ;
for all ðt; xÞ 2 ½0; T X.
This is the setting examined by Wang Bo and Meng Qingxin [7].
Acknowledgement Research is supported by The Natural Science Foundation of Zhejiang Province under grant Y605478.
References [1] [2] [3] [4] [5] [6]
Bensoussan A. On the theory of option pricing. Acta Appl Math 1984;2:139–58. Karatzas I. On the pricing of American option. Appl Math Optim 1988;17:37–60. Karatzas I, Kou S. Hedging American contingent claims with constrained portfolios. Finance Stochast 1998:215–58. Karatzas I, Shreve SE. Brownian motion and stochastic calculus. 2nd ed. New York: Springer; 1991. Karoui NEl, Peng S, Quenez QC. Bachward stochastic differential equations in finance. Math Finance 1997;7:1–71. Meyer PA. Un cours sur les integrales stochastiques. Lecture notes in mathematics, 511. Berlin, Heidelberg: Springer; 1976. pp. 245–398. [7] Wang B, Meng Q. Hedging American contingent claims with constrained portfolios under proportional transaction costs. Chaos, Solitons & Fractals 2005;23:1153–62.