A convex duality approach for pricing contingent claims under partial information and short selling constraints
aa r X i v : . [ q -f i n . M F ] F e b A convex duality approach for pricingcontingent claims under partial information andshort selling constraints
Kristina Rognlien Dahl ∗ February 28, 2019
Abstract
We consider the pricing problem facing a seller of a contingent claim.We assume that this seller has some general level of partial information,and that he is not allowed to sell short in certain assets. This pricing prob-lem, which is our primal problem, is a constrained stochastic optimizationproblem. We derive a dual to this problem by using the conjugate du-ality theory introduced by Rockafellar. Furthermore, we give conditionsfor strong duality to hold. This gives a characterization of the price ofthe claim involving martingale- and super-martingale conditions on theoptional projection of the price processes.
Keywords:
Convex duality. Mathematical finance. Pricing. Partialinformation.
AMS subject classification: ∗ Department of Mathematics, University of Oslo. [email protected]. The research lead-ing to these results has received funding from the European Research Council under the Euro-pean Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreementno. 228087.
This paper analyzes an optimization problem from mathematical finance usingconjugate duality. We consider the pricing problem of a seller of a contingentclaim B in a discrete time, arbitrary scenario space setting. The seller has ageneral level of partial information, and is subject to short selling constraints.The seller’s (stochastic) optimization problem is to find the minimum price ofthe claim such that she, by investing in a self-financing portfolio, has no risk oflosing money at the terminal time T . The price processes are only assumed tobe non-negative, stochastic processes, so the framework is model independent(in this sense).The main contribution of the paper is a characterization of the dual of theseller’s price of the claim B as a Q -expectation of the claim, where Q is amixed martingale- and super-martingale measure with respect to the conditionalexpectation of the price process, see Theorem 3.1. To the best of our knowledge,this is a new result. The mix of martingale- and super-martingale measure isdue to the presence of short selling constraints on some of the assets, while theconditional expectation is due to the seller’s partial information. The optimalvalue of this dual problem is an upper bound of the seller’s price. To provethis characterization, we use a conjugate duality technique. This techniqueis different from what is common in the mathematical finance literature, andresults in (fairly) brief proofs. Moreover, it does not rely on the reduction toa one-period model. This feature makes it possible to solve the optimizationproblem even though it contains partial information.Conjugate duality (also called convex duality), which is used to analyze theseller’s problem, is a general framework for studying and solving optimizationproblems. This framework was introduced by Rockafellar [28], see Appendix Afor a brief summary. For a further treatment of conjugate duality and its rolein stochastic optimization, see Shapiro et al. [33].Some of the main features of this paper are:2ricing contingent claims by convex duality • We have a completely general filtration representing partial information.This is in contrast to for instance Kabanov and Stricker [9], where delayedinformation is used. • The use of conjugate duality is a general approach and it provides anefficient way of deriving the dual of the seller’s pricing problem, withoutreduction to a one-period model. • Since we use discrete time, general price processes are considered.The use of conjugate duality in mathematical finance is a fairly recent de-velopment. Over the last few years, Pennanen has done some pioneering workin this area, see Pennanen [19], [20], [21] as well as Pennanen and Perkkiö [22].King [13] and King and Korf [14] have also worked on the connection betweenconjugate duality and mathematical finance. Duality theory in a broader senseis at the core of mathematical financial theory. Various kinds of duality, such aslinear programming duality, Lagrange duality and the bipolar theorem, are usedin many areas of finance. For instance, Pinar [24], [25] applies Lagrange dual-ity to derive dual representations for contingent claim pricing using a gain-losscriterion. In the setting of the present paper, this Lagrange duality approach isequivalent to our conjugate duality method. However, conjugate duality has theadvantage that it can be generalized to a continuous time setting as well. In par-ticular, duality theory (typically, in infinite dimensions) is used in utility max-imization, hedging, analyzing convex risk measures, consumption and invest-ment problems and optimal stopping. Kramkov and Schachermayer [16], [17],Karatzas and Shreve [12] and Pham [23] consider duality in utility maximizationproblems. The books by Karatzas and Shreve [12] and Pham [23] also considerduality in hedging. Pliska [26] uses linear programming duality in arbitrage-related problems. Frittelli and Rozassa Gianin [7] apply conjugate duality toconvex risk measures. Also, Rogers considers many applications of duality inmathematical finance, for instance in consumption, investment and hedgingproblems, see Rogers [31] as well as Klein and Rogers [15]. Rogers also derives3ricing contingent claims by convex dualitya pure dual method for solving optimal stopping problems, see Rogers [32].For more on replication of claims under short selling constraints, see Cvitani`cand Karatzas [3], Föllmer and Kramkov [6], Jounini and Kallal [8], Karatzas andKou [11], Karatzas and Shreve [12] and Pulido [27].Kabanov and Stricker [9] derive a version of the Dalang-Morton-Willingertheorem under delayed information. They do this by generalizing a proof ofthe no-arbitrage criteria from Kabanov et al. [10]. Their result is related to ourpricing result Theorem 4.3, in the sense that it involves martingale conditions onthe optional projection of the price processes. However, in contrast to Kabanovand Stricker [9], we have completely general partial information (i.e., it does notneed to be delayed information). Moreover, we consider pricing of claims, notarbitrage problems like in [9]. We also have short-selling constraints and ourmethods, in particular the use of conjugate duality, are different than those in[9]. Bouchard [2] and De Valliére et al. [4] also consider no arbitrage conditionsunder partial information, but with transaction cost, and without short-sellingconstraints like we do.The rest of the paper is organized as follows: Section 2 introduces the finan-cial market model and analyzes the seller’s optimization problem by derivinga dual problem using conjugate duality. Section 3 consists of our main theo-rem with proof, and gives an alternative characterization of the dual problem.In Section 4 it is shown that there is no duality gap in the case without bor-rowing or short-selling. By combining this with the previous results, we find acharacterization of the seller’s price involving martingale- and super-martingaleconditions. We also give a numerical example to illustrate the results. Finally,Section 5 concludes, and poses some open questions for further research.4ricing contingent claims by convex duality
We model the financial market as follows. There is a given probability space (Ω , F , P ) consisting of a scenario space Ω , a σ -algebra F on Ω and a probabilitymeasure P on the measurable space (Ω , F ) . The financial market consists of N + 1 assets: N risky assets (stocks) and one non-risky asset (a bond). The as-sets each have a (not identically equal zero) stochastic price process S n ( t, ω ) , n =0 , , . . . , N , for ω ∈ Ω and t ∈ { , , . . . , T } where T < ∞ , and S denotes theprice process of the bond. We denote by S ( t, ω ) := ( S ( t, ω ) , S ( t, ω ) , . . . , S N ( t, ω )) ,the vector in R N +1 consisting of the price processes of all the assets. We assumethat S ( t, ω ) := 1 for all t ∈ { , , . . . , T } , ω ∈ Ω , so the market is discounted.Let ( F t ) Tt =0 be a filtration corresponding to full information in the market. Weassume that the price process S is adapted to this filtration. For more on asimilar kind of framework, see Øksendal [18].Associated with each seller in the market there is a filtration ( G t ) t := ( G t ) Tt =0 ,where G = {∅ , Ω } and G T = F . The filtration represents the development ofthe information available to the seller. The assumptions on G and G T implythat at time the seller knows nothing, while at time T the true world scenariois revealed. We assume that G t ⊆ F t for all t = 0 , , . . . , T . This means thatthe seller only has partial information, in contrast to Kabanov and Stricker [9],where they use delayed information. By considering a general partial informa-tion, we include for instance the possibility of unobserved/hidden processes forthe seller.Let H n ( t, ω ) , n = 0 , , · · · , N be the number of units of asset number n theseller has at time t ∈ { , , . . . , T − } in scenario ω ∈ Ω . Then, the seller choosesa trading strategy H ( t, ω ) := ( H ( t, ω ) , H ( t, ω ) , · · · , H N ( t, ω )) H ( t ) is G t -measurable for all t ∈{ , , . . . , T − } . Hence, the trading strategy process ( H ( t )) t ∈{ , ,...,T − } is ( G t ) t -adapted. Let the space of all such ( G t ) t -adapted trading strategies H bedenoted by H G .We consider the pricing problem of a seller of a non-negative F -measurablecontingent claim B ( B is non-negative without loss of generality by translation).Let I ⊆ { , , . . . , N } be a subset of the risky assets, and let I = { , , . . . , N }\ I (i.e., the compliment of I ). The seller is not allowed to short sell in riskyasset S j , where j ∈ I . Also, we assume that there is no arbitrage w.r.t. ( G t ) t .Let ∆ H ( t ) := H ( t ) − H ( t − . The seller’s optimization problem is: inf { v,H } v subject to ( i ) S ( T ) · H ( T − ≥ B a.s. , ( ii ) S ( t ) · ∆ H ( t ) = 0 for ≤ t ≤ T − , a.s. , ( iii ) H j ( t ) ≥ for ≤ t ≤ T − , a.s. , j ∈ I ( iv ) S (0) · H (0) ≤ v, (1)where v ∈ R and H is ( G t ) t -adapted. Note that the inequality ( iii ) is the noshort-selling constraint. Hence, the seller’s problem is: Minimize the price v of the claim B such that the seller is able to pay B at time T (constraint ( i ) )from investments in a self-financing (constraint ( ii ) ), adapted (w.r.t. the partialinformation) portfolio that costs less than or equal to v at time (constraint ( iv ) ). In addition, the trading strategy cannot involve selling short in assets S j , j ∈ I (constraint ( iii ) ). Note that if there is a ( G t ) t -arbitrage, problem (1) isunbounded. Also, the absence of arbitrage under the full information filtration ( F t ) t implies absence of arbitrage under the partial information ( G t ) t .Note that problem (1) is an infinite linear programming problem, i.e. theproblem is linear with infinitely many constraints and variables. For more on6ricing contingent claims by convex dualityinfinite programming, see for instance Anderson and Nash [1] and for a nu-merical method, see e.g. Devolder et al. [5]. However, if Ω is finite, (1) is alinear programming problem. In this case, the problem can be solved numeri-cally using the simplex algorithm or an interior point method, see for exampleVanderbei [35]We will rewrite problem (1) in a way suitable for determining its dual.Clearly, one can remove constraint ( iv ) , and instead minimize over S (0) · H (0) .Also, since there is no ( G t ) t -arbitrage, it suffices to minimize over the portfoliossuch that S (0) · H (0) ≥ . Then, the pricing problem is a minimization prob-lem with four types of constraints ( S (0) · H (0) ≥ is the fourth type). Now,the problem can be rewritten so it fits the conjugate duality framework (seeAppendix A for a general presentation of conjugate duality or Rockafellar [28]).Let | I | denote the number of elements in I , that is the number of assets theseller is not allowed to short-sell in. Let p ∈ [1 , ∞ ) and the perturbation space U be defined by U := { u = ( γ, ( w t ) T − t =1 , ( x ( j ) t ) T − t =0 , j ∈ I , z ) : u ∈ L p (Ω , F , P : R ( | I | +1) T +1 ) } . Define (for notational convenience) w := ( w t ) T − t =1 and x ( j ) := ( x ( j ) t ) T − t =0 .Let Y := U ∗ = L q (Ω , F , P : R ( | I | +1) T +1 ) , the dual space of U , where p + q = 1 . Note that y := ( y , ( y t ) T − t =1 , ( ξ ( j ) t ) T − t =0 ,j ∈ I , y ) ∈ Y has componentscorresponding to u ∈ U . Note also that u consists of four types of variables, γ, w, ( x ( j ) ) j ∈ I , and z . Each of these variables correspond to a constraint typein the rewritten minimization problem. The same will hold for the dual variable y . Consider the pairing of U and Y using the bilinear form h u, y i = E [ u · y ] . Choose the perturbation function F : H G × U → R (again, see Appendix Afor more on perturbation functions) in the following way:( i ) If B − S ( T ) · H ( T − ≤ γ a.s., S ( t ) · ∆ H ( t ) = w t for all t ∈ { , . . . , T − } .1 Two Lemmas Pricing contingent claims by convex dualitya.s., − H j ( t ) ≤ x ( j ) t for all t ∈ { , . . . , T − } , j ∈ I a.s., S (0) · H (0) ≥ z ,then let F ( H, u ) := S (0) · H (0) .( ii ) Otherwise, let F ( H, u ) := ∞ .The corresponding Lagrange function is K ( H, y ) = S (0) · H (0) + E [ y ( B − S ( T ) · H ( T − P T − t =1 E [ y t S ( t ) · ∆ H ( t )] − P j ∈ I P T − t =0 E [ ξ jt H j ( t )] − E [ y S (0) · H (0)] if y , ξ jt , y ≥ a.s. for all t ∈ { , . . . , T − } and K ( H, y ) = −∞ otherwise. Wecan now determine the (conjugate) dual problem to the primal problem (1). Bycollecting terms for each H i ( t ) , the dual objective function is g ( y ) := inf { H : ( G t ) t − adapted } K ( H, y )= E [ y B ] + P i ∈ I inf H i (0) { E [ H i (0) { S i (0)(1 − y ) − y S i (1) } ] } + P j ∈ I inf H j (0) { E [ H j { S j (0)(1 − y ) − y S (1) − ξ ( j )0 } ] } + P T − t =1 (cid:0) P i ∈ I inf H i ( t ) { E [ H i ( t )( y t S i ( t ) − y t +12 S i ( t + 1))] } + P j ∈ I inf H j ( t ) { E [ H j ( t )( y t S j ( t ) − y t +12 S j ( t + 1) − ξ ( j ) t )] } (cid:1) + P i ∈ I inf H i ( T − { E [ H i ( T − − y S i ( T ) + y T − S i ( T − } + P j ∈ I inf H j ( T − { E [ H j ( T − − y S j ( T ) + y T − S j ( T − − ξ ( j ) T − )] } . (2) This section consists of two lemmas needed in the following presentation. Weinclude the proofs for completeness.
Lemma 2.1
Let f be any random variable w.r.t. (Ω , F , P ) and let G be a sub- σ -algebra of F . Let X denote the set of all G -measurable random variables.Then inf { g ∈X } E [ f g ] > −∞ if and only if R A f dP = 0 for all A ∈ G . .1 Two Lemmas Pricing contingent claims by convex duality
Proof. ⇒ : Assume there exists A ∈ G such that R A f dP = K = 0 . Define g ( ω ) := M for all ω ∈ A , where M is a constant, and g ( ω ) := 0 for all ω ∈ Ω \ A .The result follows by letting M → + / − ∞⇐ : Prove the result for simple functions. The Lemma follows by an approx-imation argument. (cid:3) In the next lemma the notation is the same as in Lemma 2.1:
Lemma 2.2 inf { g ∈X } E [ f g ] > −∞ implies that inf { g ∈X } E [ f g ] = 0 .Proof. Follows by observing that inf { g ∈X } E [ f g ] ≤ ( g = 0 is feasible) andthe definition of the infimum. (cid:3) By combining Lemma 2.2 with Lemma 2.1, it follows that inf { g ∈X } E [ f g ] = 0 if and only if R A f dP = 0 for all A ∈ G .There exists a feasible dual solution if and only if all the infima in equa-tion (2) are greater than −∞ . To derive the dual problem, we consider each ofthese minimization problems separately and use the comment after Lemma 2.1and Lemma 2.2. We also use that since ξ ( j ) t ≥ a.e. for all t ∈ { , , . . . , T − } and j ∈ I , then R A ξ ( j ) t dP ≥ for all A ∈ G t for all t . Also, from the deriveddual feasibility conditions, it is sufficient to only maximize over solutions where y = 0 P -a.e. Note that such a solution exists, because we have assumed that9ricing contingent claims by convex dualitythere is no ( G t ) t -arbitrage. Hence, the dual problem is sup { y ∈ Y : y ≥ } E [ y B ] s.t. ( i ) R A S i (0) dP = R A y S i (1) dP ∀ A ∈ G , ( i ) ∗ R A S j (0) dP ≥ R A y S j (1) dP ∀ A ∈ G , ( ii ) R A y t S i ( t ) dP = R A y t +12 S i ( t + 1) dP ∀ A ∈ G t , t = 1 , . . . , T − , ( ii ) ∗ R A S j ( t ) y t dP ≥ R A y t +12 S j ( t + 1) dP ∀ A ∈ G t , t = 1 , . . . , T − , ( iii ) R A y T − S i ( T − dP = R A y S i ( T ) dP ∀ A ∈ G T − , ( iii ) ∗ R A y T − S j ( T − dP ≥ R A y S j ( T ) dP ∀ A ∈ G T − (3)where the equality constraints ( i ) , ( ii ) and ( iii ) hold for i ∈ I and the inequalityconstraints ( i ) ∗ , ( ii ) ∗ and ( iii ) ∗ hold for j ∈ I . Note that the dual feasibilityconditions come in pairs, where the only difference is whether there is = (shortselling allowed) or ≥ (short selling not allowed).This dual problem (3) is, like the primal problem (1), an infinite linear pro-gramming problem. As before, if Ω is finite, it is a regular linear programmingproblem which can be solved using the simplex algorithm or an interior pointmethod. However, this version of the dual problem is not significantly simplerto solve than the original problem. Therefore, we will rewrite problem (3) in amore interpretable form, which in some cases is more attractive to solve thanthe primal problem. In this section, we will show our main theorem, Theorem 3.1, which states thatthe dual problem (3) is equivalent to another problem involving martingale- andsuper-martingale conditions on the optional projection of the price process.In the following, let ¯ M aI ( S, G ) be the set of probability measures Q on (Ω , F ) that are absolutely continuous w.r.t. P and are such that the price processes S i i ∈ I satisfy E Q [ S i ( t + k ) |G t ] = E Q [ S i ( t ) |G t ] , while for j ∈ I they satisfy E Q [ S j ( t + k ) |G t ] ≤ E Q [ S j ( t ) |G t ] for k ≥ and t ∈ , , . . . , T − k , i.e. Q is amixed martingale and super-martingale measure for the optional projection ofthe price process. Theorem 3.1
The dual problem (3) is equivalent to the following optimizationproblem. sup Q ∈ ¯ M aI ( S, G ) E Q [ B ] . (4) Proof.
First, assume there exists a Q ∈ ¯ M aI ( S, G ) , i.e., a feasible solutionto problem (4). We want to show that there is a corresponding feasible solutionto problem (3).Define y := dQdP (the Radon-Nikodym derivative of Q w.r.t. P , see Shilling [34]),and y t := E [ y |F t ] for t = 0 , , . . . , T − . We prove that y , y t satisfy the dualfeasibility conditions of problem (3). • ( iii ) ∗ : From the definition of conditional expectation, it suffices to prove Z A E [ y S j ( T ) |G T − ] dP ≤ Z A y T − S ( T − dP for all A ∈ G T − , j ∈ I . In particular, it suffices to prove E [ y S j ( T ) |G T − ] ≤ E [ y T − S j ( T − |G T − ] P -a.e . By the definition of y T − , this is equivalent to E [ y S j ( T ) |G T − ] ≤ E [ E [ y |F T − ] S j ( T − |G T − ] P -a.e . Since S j ( T − is F T − -measurable, the inequality above is the same as E [ y S j ( T ) |G T − ] ≤ E [ E [ y S j ( T − |F T − ] |G T − ] P -a.e . E [ y S j ( T ) |G T − ] ≤ E [ y S j ( T − |G T − ] P -a.e . By change of measure under conditional expectation, it is enough to show E [ y |G T − ] E Q [ S j ( T ) |G T − ] ≤ E [ y |G T − ] E Q [ S j ( T − |G T − ] . This holds, because y ≥ P -a.e. and Q ∈ ¯ M aI ( S, G ) . • ( ii ) ∗ : First, we prove this for t = T − . Note that for all A ∈ G T − , j ∈ I R A y T − S j ( T − dP = R A E [ y T − S j ( T − |G T − ] dP = R A E [ E [ y |F T − ] S j ( T − |G T − ] dP = R A E [ E [ y S j ( T − |F T − ] |G T − ] dP = R A E [ y S j ( T − |G T − ] dP Hence, from the definition of conditional expectation and change of mea-sure under conditional expectation, it suffices to prove E [ y T − S j ( T − |G T − ] ≥ E [ y S j ( T − |G T − ]= E [ y |G T − ] E Q [ S j ( T − |G T − ] . (5)But, by the definition of y ( T − , the tower property and change of measureunder conditional expectation E [ y T − S j ( T − |G T − ] = E [ y S j ( T − |G T − ] = E [ y |G T − ] E Q [ S j ( T − |G T − ] . (6)By combining equation (5) and (6), it suffices to prove that E [ y |G T − ] E Q [ S j ( T − |G T − ] ≥ E [ y |G T − ] E Q [ S j ( T − |G T − ] . y ≥ P-a.e. and Q ∈ ¯ M aI ( S, G ) . Similarly, one canshow ( ii ) ∗ for t = 1 , . . . , T − . • ( i ) ∗ : Recall that G = {∅ , Ω } . The inequality is trivially true for A = ∅ .Hence, it only remains to check that E [ y S j (1)] ≤ E [ S j (0)] = S j (0) for j ∈ I . Note that E [ y S j (1)] = E [ y S j (1) |G ]= E [ E [ y |F ] S j (1) |G ]= E [ E [ y S j (1) |F ] |G ]= E [ y S j (1) |G ]= E Q [ S j (1)]= E Q [ S j (1) |G (0)] ≤ S j (0) where the second equality follows from the definition of y and the in-equality follows from Q ∈ ¯ M aI ( S, G ) . Hence, ( i ) ∗ holds as well. • The equality conditions ( i ) , ( ii ) and ( iii ) follow from the same kind ofarguments, based on the definition of ¯ M aI ( S, G ) and change of measureunder conditional expectation.Hence, any Q ∈ ¯ M aI ( S, G ) corresponds to a feasible dual solution, i.e. sat-isfies the constraints of the dual problem (3).Conversely, assume there exists a feasible dual solution y ≥ , ( y t ) T − t =1 ofproblem (3).Define Q ( F ) := R F y dP for all F ∈ F . This defines a probability measuresince y ≥ , and one can assume that E [ y ] = 1 since the dual problem (3) isinvariant under translation. The remaining part of the proof is to show that Q ∈ ¯ M aI ( S, G ) , (7)13ricing contingent claims by convex dualityi.e., that the dual feasibility conditions of problem (3) correspond to the condi-tions for being in ¯ M aI ( S, G ) . We divide this into several claims, which we thenprove. Claim 1: E Q [ S i ( T ) |G T − ] = E Q [ S i ( T − |G T − ] for i ∈ I . Proof of Claim 1:
From the definition of conditional expectation, equation ( iii ) in problem (3) is equivalent to E [ y S i ( T ) |G T − ] = E [ y T − S i ( T − |G T − ] .From change of measure under conditional expectation E [ y S i ( T ) |G T − ] = E [ y |G T − ] E Q [ S i ( T ) |G T − ] (8)and E [ y T − S i ( T − |G T − ] = E [ y T − |G T − ] E Q [ S i ( T − |G T − ] . (9)By combining equations (8) and (9), ( iii ) is equivalent to E [ y |G T − ] E Q [ S i ( T ) |G T − ] = E [ y t |G T − ] E Q [ S i ( T − |G T − ] . By considering equation ( iii ) for the bond and using that the market isnormalized (by assumption), Z A y dP = Z A y T − dP for all A ∈ G T − . (10)From the definition of conditional expectation, this implies that E [ y T − |G T − ] = E [ y |G T − ] . Since y > a.e., E Q [ S i ( T ) |G T − ] = E [ S i ( T − |G T − ] . This provesClaim 1. Claim 2: E Q [ S i ( t + k ) |G t ] = E Q [ S i ( t ) |G t ] for k ∈ N , i ∈ I . Proof of Claim 2:
Let i ∈ I . First, one can show by induction that E Q [ S i ( T ) |G t ] = E Q [ S i ( t ) |G t ] for all t ≤ T, i ∈ I , using Claim 1. Also byan inductive argument (for i ∈ I ), this can be generalized to Claim 3. Claim 3: E Q [ S j ( T ) |G T − ] ≤ E Q [ S j ( T − |G T − ] for j ∈ I .14ricing contingent claims by convex duality Proof of Claim 3:
To prove E Q [ S j ( T ) |G T − ] ≤ E Q [ S j ( T − |G T − ] for j ∈ I ,we use ( iii ) ∗ , an argument similar to that used to show Claim 1, and Claim 2. Claim 4: E Q [ S j ( T ) |G t ] ≤ E [ S j ( t ) |G t ] for all t ≤ T and j ∈ I . Proof of Claim 4:
Let j ∈ I . To show that E Q [ S j ( T ) |G t ] ≤ E [ S j ( t ) |G t ] forall t ≤ T : Note that from equation ( ii ) ∗ of problem (3) for t + 1 , t + 2 , . . . , T − ,it follows that R A y t S j ( t ) dP ≥ R A y t +12 S j ( t + 1) dP ∀ A ∈ G t ≥ R A y t +22 S j ( t + 2) dP ∀ A ∈ G t +1 , in particular ∀ A ∈ G t ≥ . . . ≥ R A y T − S j ( T − dP ∀ A ∈ G t ≥ R A y S j ( T ) dP ∀ A ∈ G t where the final inequality uses ( iii ) ∗ from problem (3). Hence, by the definitionof conditional expectation and change of measure under conditional expectation Z A E [ y t |G t ] E Q [ S j ( t ) |G t ] dP ≥ Z A E [ y |G t ] E Q [ S j ( T ) |G t ] dP From equation ( ii ) for the bond, we know that E [ y t |G t ] = E [ y |G t ] (see theargument related to equation (10)), so Z A { E [ y t |G t ]( E Q [ S j ( t ) |G t ] − E Q [ S j ( T ) |G t ]) } dP ≥ ∀ A ∈ G t . (11)If y t ( A ) ≥ , but not identically equal a.e., this implies Claim 4, i.e.: E Q [ S j ( t ) |G t ]( A ) ≥ E Q [ S j ( T ) |G t ]( A ) for A ∈ G t . If y t ( A ) = 0 a.e., then Q ( A ) = 0 , so E Q [ S j ( T ) |G t ]( A ) = 0 by convention. Hence,since the price processes are non-negative, E Q [ S j ( t ) |G t ] ≥ E Q [ S j ( T ) |G t ] . Thisproves Claim 4. 15ricing contingent claims by convex duality Claim 5: E Q [ S j ( t + k ) |G t ] ≤ E Q [ S j ( t ) |G t ] for k ∈ N , j ∈ I . Proof of Claim 5:
For A ∈ G t and j ∈ I , R A y t S j ( t ) dP ≥ R A y t + k S j ( t + k ) dP = R A E [ y t + k S j ( t + k ) |G t ] dP = R A E [ y t + k |G t ] E Q [ S j ( t + k ) |G t ] dP = R A E [ y |G t ] E Q [ S j ( t + k ) |G t ] dP where the first inequality follows from ( ii ) ∗ (from problem (3)) iterated and thethird equality from E [ y t + k |G t ] = E [ y |G t ] (see the proof of Claim 4). Hence,by the definition of conditional expectation and since E [ y t |G t ] = E [ y |G t ] ≥ (because y ≥ ) Z A { E [ y |G t ]( E Q [ S j ( t ) |G t ] − E Q [ S j ( t + k ) |G t ] } dP ≥ for all A ∈ G t . By a similar argument as for equation (11), Claim 5 holds, i.e., E Q [ S j ( t + k ) |G t ] ≤ E Q [ S j ( t ) |G t ] for all k ∈ N , j ∈ I . By combining these claims, we see that Q ∈ ¯ M aI ( S, G ) , and the theoremfollows. (cid:3) The version of the dual problem (4) is attractive because of its connectionto martingale measures, which are an essential part of mathematical financeliterature, see e.g. Karatzas and Shreve [12] and Øksendal [18]. Another nicefeature of the formulation (4) is that when one has found the set ¯ M aI ( S, G ) ,solving the problem for each new claim B may be fairly simple (depending onstructure of ¯ M aI ( S, G ) ) since the set does not depend on the claim. In contrast,the primal problem (1) must be solved from scratch whenever one considers anew claim B . 16ricing contingent claims by convex duality Remark 3.2
Note that Theorem 3.1 has some similarities with Theorem 1 inKabanov and Stricker [9]. However, we consider the pricing problem of a con-tingent claim instead of the no-arbitrage criterion, which is the topic of [9].Moreover, we have short-selling constraints, which [9] do not have. Also, wehave a general level of partial information (not necessarily delayed) and thetechniques we use, in particular the use of convex duality, are different.Kabanov and Stricker [9] also comment that, to their knowledge, their proofof the partial information Dalang-Morton-Willinger theorem is the only one thatdoes not reduce the problem to a one-step model. Our technique, using convexduality, does not rely on reduction to a one-period model either. So (to the bestof our knowledge), our method of proof must be a new way to avoid reductionto one-period in discrete time models.
The main goal of this section is to prove that there is no duality gap, i.e., thatthe value of the primal problem (1) is equal to the value of the dual problem (4).This can be done using the following theorem from Pennanen and Perkkiö [22](see Theorem 9 in [22]). In order to prove strong duality, we also assume that I = { , , . . . , N } , i.e. that no short-selling or borrowing is allowed.We use the same notation as in Section 2, and consider the value function ϕ ( · ) as defined in Appendix A. In the following theorem, H is a stochastic processwith N + 1 components at each time t ∈ { , , . . . , T − } and H G denotes thefamily of all stochastic processes that are adapted to the filtration ( G t ) t . Also, F ∞ is the recession function of F , defined by F ∞ ( H ( ω ) , , ω ) := sup λ> F ( λH ( ω ) + ¯ H ( ω ) , ¯ y ( ω ) , ω ) − F ( ¯ H ( ω ) , ¯ y ( ω ) , ω ) λ (12)(which is independent of ¯ H, ¯ y ). Then, we have the following theorem: Theorem 4.1 (Theorem 9, Pennanen and Perkkio [22]) Assume there exists y ∈ Y and m ∈ L (Ω , F , P ) such that for P -a.e. ω ∈ Ω , F ( H, u ) ≥ u · y + m a.s. for all ( H, u ) ∈ R T ( N +1) × R ( | I | +1) T +1 , (13) where ( · ) denotes Euclidean inner product. Assume also that A := { H ∈ H G : F ∞ ( H, ≤ P -a.s. } is a linear space. Then, the value function ϕ ( u ) is lower semi-continuous on U and the infimum of the primal problem is attained for all u ∈ U . For the proof of Theorem 4.1, see [22]. Theorem 4.1 gives conditions forthe value function ϕ (see Appendix A) to be lower semi-continuous. Hence,from Theorem A.2, if these conditions hold, there is no duality gap since ϕ ( · ) isconvex (because the perturbation function F was chosen to be convex). Remark 4.2
Note that there is a minor difference between the frameworks ofRockafellar [28] and Pennanen and Perkkiö [22]. In the latter it is assumedthat the perturbation function F is a so-called convex normal integrand. How-ever, from Example 1 in Pennanen [19] and Example 14.29 in Rockafellar andWets [30], it follows that our choice of F is in fact a convex normal integrand. The following theorem states that there is no duality gap and characterizesthe seller’s price of the contingent claim.
Theorem 4.3
Consider the setting of this paper, and assume that there is noarbitrage with respect to ( G t ) t . If the seller of the claim B has information ( G t ) t and no short selling or borrowing is allowed, she will offer the claim at the price β := sup Q ∈ ¯ M a ( S, G ) E Q [ B ] . (14) where ¯ M a ( S, G ) is the set the set of probability measures Q on (Ω , F ) that are absolutely continuous w.r.t. P and are such that the price processes satisfy E Q [ S j ( t + k ) |G t ] ≤ E Q [ S j ( t ) |G t ] for k ≥ and t ∈ , , . . . , T − k .Proof. We apply Theorem 4.1 in order to show that there is no duality gapfor our pricing problem: • We first show that the set A is a linear space. We compute F ∞ ( H ( ω ) , , ω ) by choosing ¯ y = 0 and ¯ H to be the portfolio that starts with ω ∈ Ω B ( ω ) units of the bond and just follows the market development (without anytrading) until the terminal time. Then, we find that A = { H : G -adapted , H ( t ) ≥ ∀ t, S ( t ) · ∆ H ( t ) = 0 ,S ( T ) · H ( T − ≥ , S (0) · H (0) ≤ } = { } , where the final equality holds since we assume that there is no arbitragew.r.t. the filtration ( G t ) t . Hence, A = { } , which is a (trivial) linear space.Hence the first condition of Theorem 4.1 is satisfied. • To check the other assumption of the theorem, choose y = (0 , (0) t , (0) t , − ∈ L q (Ω , F , P : R ( | I | +1) T +1 ) , where represents the -function. Also, choose m ( w ) = − for all ω ∈ Ω .Then m ∈ L (Ω , F , P ) . Then, given ( H, u ) ∈ R T ( N +1) × R ( | I | +1) T +1 : F ( H, u ) ≥ S (0) · H (0) (from the definition of F ) ≥ − z (from the definition of F ) = u · y ( ω ) + m ( ω ) (from the choice of y and m ) . This proves that the conditions of Theorem 4.1 are satisfied. Therefore, thereis no duality gap, so the seller’s price of the contingent claim is19ricing contingent claims by convex duality sup Q ∈ ¯ M a ( S, G ) E Q [ B ] . (cid:3) Remark 4.4
We remark that Proposition 4.1 in Föllmer and Kramkov [6] givesan expression for the seller’s price of a claim with super-martingale conditionson the price process. However, we do not consider the same problem as [6], sincewe have partial information. The presence of partial information results in adifferent type of martingale measure (we get a martingale- and super-martingalemeasure on the optional projection of the price process) than in the paper [6].
In order to prove the strong duality in Theorem 4.3, we have assumed thatno short-selling or borrowing is allowed. This is necessary in order for the space A to be a linear space, as required by the strong duality characterization inTheorem 4.1. However, we have not been able to find a numerical examplewhere there actually is a duality gap. In the finite Ω (i.e. linear programming)case, there will be no duality gap even when borrowing or short selling is allowed.Hence, if there exists an example of a duality gap, it must be in the infinite Ω case.This leads one to believe that it may be possible to close the duality gapin general. However, we have not found a way to achieve this through ourconvex analysis of the problem. Another option is to try to close the dualitygap in the short selling case by analyzing the primal problem using Lagrangeduality, see e.g. Pinar [24], [25]. However, as this methodology is equivalentto our convex duality approach in the discrete time setting, it seems likely thatone will run into a similar problem with linearity. This is an open problem forfurther research. Example 4.5
We illustrate the previous results by considering a simple nu-merical example. Although the results of this paper hold when Ω is an arbitrary set, we consider a situation where Ω is finite. This simplifies the intuition andallows for illustration via scenario trees.Consider times t = 0 , , , Ω := { ω , ω , . . . , ω } and a market with twoassets: one bank account S and one risky asset S . Assume that the market isdiscounted, so S ( t, ω ) = 1 for all times t and all ω ∈ Ω . Let S ( t, ω ) := X ( t, ω ) + ξ ( t, ω ) , i.e. the price of the risky asset is composed of two other processes, X and ξ .The seller does not observe these two processes, only the prices. The followingscenario trees show the development of the processes X and ξ , as well as the pricedevelopment observed by the seller. Note that we only display the informationneeded in the following calculations. ✉ ✉✉ ✉✉✉✉✉ (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅✦✦✦✦❛❛❛❛✧✧✧✧❜❜❜❜ Ω = { ω , ω , . . . , ω } ξ = 3 , { ω , ω , ω } ξ = 5 , { ω , ω } ω ω ω ω ω q q q t = 0 t = 1 t = T = 2 Figure 1: The process ξ Full information in this market corresponds to observing both processes X and ξ , i.e. the full information filtration ( F t ) t is the sigma algebra generated by X and ξ , σ ( X, ξ ) . However, the filtration observed by the seller ( G t ) t , generatedby the price processes, is (strictly) smaller than the full information filtration.For instance, if you observe that ξ (1) = 3 and X (1) = 4 , you know that therealized scenario is ω . However, this is not possible to determine only throughobservation of the price process S . Hence, this is an example of a model with ✉ ✉✉ ✉✉✉✉✉ (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅✦✦✦✦❛❛❛❛✧✧✧✧❜❜❜❜ Ω = { ω , ω , . . . , ω } X = 4 , { ω , ω } X = 2 , { ω , ω , ω } ω ω ω ω ω q q q t = 0 t = 1 t = T = 2 Figure 2: The process X ✉ ✉✉✉ ✉✉✉✉✉ (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅✦✦✦✦❛❛❛❛✏✏✏✏❜❜❜❜ S = 6 , Ω = { ω , ω , . . . , ω } S = 7 , { ω , ω } S = 9 , { ω , ω } S = 5 , { ω } S = 3 , ω S = 8 , ω S = 9 , ω S = 7 , ω S = 4 , ω q q q t = 0 t = 1 t = T = 2 Figure 3: The price process S hidden processes, which is a kind of partial information that is not delayed in-formation.Assume that the seller is not allowed to short-sell. In this case, the seller’sproblem (1) is to solve the following minimization problem w.r.t. v and all ( G t ) t -adapted trading strategies H : inf v,H v s.t. H (1 , ω ) + H (1 , ω ) ≥ B ( ω )9 H (1 , ω ) + H (1 , ω ) ≥ B ( ω )5 H (1 , ω ) + H (1 , ω ) ≥ B ( ω )8 H (1 , ω ) + H (1 , ω ) ≥ B ( ω )4 H (1 , ω ) + H (1 , ω ) ≥ B ( ω ) H (1 , ω ) − H (0) + 7 (cid:0) H (1 , ω − H (0)) (cid:1) = 0 H (1 , ω ) − H (0) + 5 (cid:0) H (1 , ω ) − H (0)) (cid:1) = 0 H (1 , ω ) − H (0) + 9 (cid:0) H (1 , ω ) − H (0)) (cid:1) = 0 H j (0) ≥ , H j (1 , ω i ) ≥ for j = 0 , , i = 1 , , H (0) + 6 H (0) ≤ v (15) where H (1 , ω ) = H (1 , ω ) and H (1 , ω ) = H (1 , ω ) due to H being ( G t ) t -adapted. This is a linear programming problem which can be solved using thesimplex algorithm. Note that the simplex algorithm is a duality method, whichleads to a dual problem equivalent to the one we have derived in Section 3. Anadvantage with solving this problem directly is that we get the trading strategy H explicitly. However, a downside with solving problem (15) directly is that foreach new claim B , the problem must be solved from scratch. This is not the casewhen solving the dual problem instead. From Theorem 4.3, the dual problem forthe price of the claim is: sup Q ∈ ¯ M a ( S, G ) E Q [ B ] (16) where ¯ M a ( S, G ) is the set of absolutely continuous probability measures makingthe price process S a ( G t ) t -conditional super-martingale (and S , but this is trivial since the price processes are discounted). In order to solve this problem,we must find the set ¯ M a ( S, G ) . By using the definition of ¯ M a ( S, G ) , we geta system of linear inequalities to solve. By solving these, using for exampleFourier Motzkin elimination, we find that ¯ M a ( S, G ) = { Q = ( q , q , . . . , q ) : 0 ≤ q ≤ , ≤ q ≤ − q , ≤ q ≤ − q − q , ≤ q ≤ − q − q − q and q = 1 − q − q − q − q } (17) Hence, given some claim B , one can solve the problem (16) for the set in (17) in order to find the seller’s price. When one would like to find prices forseveral claims B , B , . . . , B m , solving the dual problem is simpler than solvingthe primal LP problem since the set ¯ M a ( S, G ) is the same for all the claims. In this paper, we have shown how convex duality can be used to obtain pricingresults for a seller of a claim who has partial information and is facing shortselling constraints in a discrete time financial market model. This gives newresults, which are summarized in Theorem 3.1 and Theorem 4.3.It seems natural that these results can be generalized to a model with con-tinuous time, possibly using a discrete time approximation. However, this maybe quite technical.
A Conjugate duality and paired spaces
Conjugate duality theory (also called convex duality), introduced by Rockafel-lar [28], provides a method for solving very general optimization problems viadual problems.Let X be a linear space, and let f : X → R be a function. The minimizationproblem min x ∈ X f ( x ) is called the primal problem , denoted ( P ) . In order to24ricing contingent claims by convex dualityapply the conjugate duality method to the primal problem, we consider anabstract optimization problem min x ∈ X F ( x, u ) where F : X × U → R is afunction such that F ( x,
0) = f ( x ) , U is a linear space and u ∈ U is a parameterchosen depending on the particular problem at hand. The function F is calledthe perturbation function . We would like to choose ( F, U ) such that F is aclosed, jointly convex function of x and u .Corresponding to this problem, one defines the optimal value function ϕ ( u ) := inf x ∈ X F ( x, u ) , u ∈ U. (18)Note that if the perturbation function F is jointly convex, then the optimalvalue function ϕ ( · ) is convex as well.A pairing of two linear spaces X and V is a real-valued bilinear form h· , ·i on X × V . Assume there is a pairing between the spaces X and V . A topologyon X is compatible with the pairing if it is a locally convex topology suchthat the linear function h· , v i is continuous, and any continuous linear functionon X can be written in this form for some v ∈ V . A compatible topologyon V is defined similarly. The spaces X and V are paired spaces if there isa pairing between X and V and the two spaces have compatible topologieswith respect to the pairing. An example is the spaces X = L p (Ω , F, P ) and V = L q (Ω , F, P ) , where p + q = 1 . These spaces are paired via the bilinearform h x, v i = R Ω x ( s ) v ( s ) dP ( s ) .In the following, let X be paired with another linear space V , and U pairedwith the linear space Y . The choice of pairings may be important in appli-cations. Define the Lagrange function K : X × Y → ¯ R to be K ( x, y ) :=inf { F ( x, u ) + h u, y i : u ∈ U } . The following Theorem A.1 is from Rockafel-lar [28] (see Theorem 6 in [28]). Theorem A.1
The Lagrange function K is closed, concave in y ∈ Y for each x ∈ X , and if F ( x, u ) is closed and convex in uf ( x ) = sup y ∈ Y K ( x, y ) . (19)For the proof of this theorem, see Rockafellar [28]. Motivated by Theorem A.1,we define the dual problem of ( P ) , ( D ) max y ∈ Y g ( y ) where g ( y ) := inf x ∈ X K ( x, y ) .One reason why problem ( D ) is called the dual of the primal problem ( P ) is that, from equation (19), problem ( D ) gives a lower bound on problem ( P ) .This is called weak duality . Sometimes, one can prove that the primal and dualproblems have the same optimal value. If this is the case, we say that there is no duality gap and that strong duality holds . The next theorem (see Theorem7 in Rockafellar [28]) is important: Theorem A.2
The function g in ( D ) is closed and concave. Also sup y ∈ Y g ( y ) = cl ( co ( ϕ ))(0) and inf x ∈ X f ( x ) = ϕ (0) . (where cl and co denote respectively the closure and the convex hull of a func-tion, see Rockafellar [29]). For the proof, see Rockafellar [28]. Theorem A.2implies that if the value function ϕ is convex, the lower semi-continuity of ϕ isa sufficient condition for the absence of a duality gap .26 EFERENCES
Pricing contingent claims by convex duality
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