CLAIM HEDGING IN AN INCOMPLETE MARKET
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1 Vol 18 No 2 Journal of Systems Science and Complexity Apr 2005 CLAIM HEDGING IN AN INCOMPLETE MARKET SUN Wangui (School of Economics & Management Northwest University Xi an China wans6312@pubxaonlinecom) WANG Chunfeng (School of Management Tianjin University Tianjin China) Abstract In this paper we compare the performance of the optimal attainable payoffs (of a general claim) derived by the variance-optimal approach and the indifference argument under the mean-variance preference in an incomplete market oth payoffs are expressed by the signed variance-optimal martingale measure Our results are applied to the claim hedging under partial information Key words Incomplete market variance-optimal approach indifference argument general claims partial information 1 Introduction As Pham et al [1 wrote in their paper the pricing and hedging of contingent claims in an incomplete market is one of the important questions in modern finance mathematics There are three different approaches to attack this problem: the super-replication method utility-based arguments and the variance-optimal approach This paper focuses on the last two approaches and compares the performance of the optimal attainable payoffs derived by them The variance-optimal hedging problem was studied by Schal [2 Schweizer [3 and Mercurio [4 under a discrete-time and by Pham et al [1 Gourieroux et al [5 under a continuous time In L 2 -framework Schweizer [6 discussed claim pricing In Schweizer s setting we address the variance-optimal hedging problem The optimal attainable payoff for the problem is given Using the indifference argument under mean-variance preferences claim pricing and hedging was studied by Mercurio [4 in a special case where a stock and a bond are traded under a discretetime setting And Schweizer [6 gave a valuation principle a mapping that assigns a number (value) to a random variable (payoff) on a general financial background Moller [7 applied the valuation principles to determine optimal trading strategies in the case of constant riskless interest rate In this paper we shall consider the more general case of stochastic interest rate We then compare the performance of two hedging strategies under the variance-optimal criterion and the mean-variance preference This result extends the conclusion of Mercurio [4 Lastly we apply our results to the hedging of contingent claims under partial information The structure of the paper is as follows Section 2 introduces some notions and results Section 3 focuses on the variance-optimal hedging problem The hedging under mean-variance preferences is discussed in Section 4 In the last section we use our results to the hedging under partial information Received November Revised February *This work is supported in part by National Science Fund for Distinguished Young Scholar No
2 194 SUN WANGUI WANG CHUNFENG Vol 18 2 Approximate Profits and Martingale Measure Let (Ω F P ) be a complete probability space and L 2 (Ω F P ) the space of all squareintegrable real random variables with scalar product X Y = E[XY and norm X = E[X2 A given element L 2 (Ω F P ) satisfies > 0 P as Let M be a linear closed subspace of L 2 (Ω F P ) and set A be {b + m : b R m M} where R is the set of real numbers We denote by π the orthogonal projection in L 2 (Ω F P ) on the linear subspace M the orthogonal complement of M in L 2 (Ω F P ) The pair (M ) represents the financial environment proposed by Schweizer [6 M a linear closed subspace of L 2 (Ω F P ) corresponds to a financial market without frictions An element m of M models the total gains from trade resulting from a self-financing trading strategy with initial capital 0 is interpreted as the final value of some saving account with initial value 1 In particular the case = 1 corresponds to interest rate with 0 b + m(m M) represents a random payoff as the final wealth of some self-financing trading strategy with some initial capital b The next notions and results were given by Schweizer [6 Definition 1 M is said to have no approximate profits in L 2 (Ω F P ) if M It means that one cannot approximate (in the sense of L 2 (Ω F P ) ) the riskless payoff by a self-financing strategy with initial wealth 0 Let M be a linear subspace of L 2 (Ω F P ) admitting no approximate profits in L 2 (Ω F P ) Then every H L 2 (Ω F P ) has a unique decomposition as H = b H + m H + n H (1) with b H R m H M and n H A In particular one has E(n H ) = 0 and E(mn H ) = 0 for all m M Definition 2 A signed (M )-martingale measure is a signed measure Q on (Ω F ) with Q(Ω) = 1 Q P with 1 dq L2 (Ω F P ) and E Q[ m = 0 for all m M A signed (M )-martingale measure P is called -variance-optimal if 1 d P P 1 Lemma 1 Let M be a linear subspace of L 2 (Ω F P ) The signed -variance-optimal (M)-martingale measure P exists if and only if M admits no approximate profits in L 2 (ΩFP ) In this case P is unique and given by dq d P = π() E[π() (2) From Lemma 1 we give the expression of the orthogonal projection by the signed -varianceoptimal (M )-martingale measure P Let M be a linear subspace of L 2 (Ω F P ) Then the orthogonal projection can be expressed by the relation: π() = 1 d P E ( 1 d P ) 2 (3)
3 No 2 CLAIM HEDGING IN AN INCOMPLETE MARKET 195 In fact dividing on both sides of (2) and raising to the expected square one has where E[π() = E[π() 2 Hence ( 1 d E P ) 2 E[π() = (E[π()) 2 E[π() = 1 E ( 1 Substituting in equality (2) we know that relation (3) holds Since π() M and π() A we know that 1 for all m M and n A and d P ) 2 d P π() m + n = E[π() m + n = 0 (4) [ H d E P 1 d = P H = b H (5) The signed variance-optimal martingale measure is an important tool to attack problems of claim hedging and pricing under market incompleteness; see the work of Gourieroux et al [5 Mercurio [4 Schweizer [6 and Pham [8 The relation (3) shows that the orthogonal projection can be expressed by the signed variance-optimal martingale measure In the next two sections we shall use the orthogonal projection to discuss claim hedging under the variance-optimal criterion and the mean-variance preference respectively 3 Hedging in a Variance-Optimal Criterion A variance-optimal criterion for hedging contingent claims in incomplete markets was introduced by Follmer and Sondermann [9 : J 1 (b) = inf E[(H w 0 m) 2 (6) for a fixed claim H L 2 (Ω F P ) and an initial capital w 0 The optimal attainable payoff is denoted by m In this section the explicit expression of the optimal attainable payoff m is given The main results are shown below Theorem 1 Let M be a closed linear subspace in L 2 (Ω F P ) and admit no approximate profits in L 2 (Ω F P ) Then there exists a unique solution to problem (6) with and J 1 (b) = (b H w 0 ) 2 E[π() + n H 2 = m = (b H w 0 )( π()) + m H = where H has decomposition (1) ( E [ H E [( 1 ) d P 2 w0 d P ( [ H d E P ) [ w 0 ) 2 + n H 2 (7) 1 d P E [( H d P ) 2 + m H (8)
4 196 SUN WANGUI WANG CHUNFENG Vol 18 Proof From the decomposition of H one has J 1 (b) = inf E[( H w 0 m ) 2 = inf E[( b H + m H + n H w 0 m ) 2 = inf E[( (b H w 0 ) + m H + n H m ) 2 = inf E[( (b H w 0 )π() + (b H w 0 )( π()) +m H + n H m ) 2 Notice that (b H w 0 )π() (b H w 0 )( π()) + m H and n H are orthogonal to each other which follows from relation (4) and the definition of π() Thus as we get m = (b H w 0 )( π()) + m H J 1 (b) = (b H w 0 ) 2 E[π() + n H 2 And from relation (3) the theorem is established 4 Hedging under the Quadratic Derived Mean-Variance Utility This section focuses on hedging a general contingent claim in L 2 -framework with a meanvariance utility and a stochastic riskless payoff The optimal attainable payoff for γ units of claims H and the corresponding value of the quadratic derived utility function are given Define [ [ Y Y u(y ) := E A Var (9) where A > 0 is a risk aversion parameter superscripts refer to probability P defined by = 2 E[ 2 (10) The quadratic derived utility function (9) was used for pricing contingent claims by Mercurio [4 and Schweizer [6 The following definition of u-indifference price for γ units of claims H was presented by Schweizer [6 Definition 3 h(w 0 γ) R is called a u-indifference price for γ units of claims H if it satisfies sup u((w 0 + h(w 0 γ)) + m γh) = sup u(w 0 + m) (11) where w 0 is an initial wealth The u-indifference price is determined in such a way that the investor is indifferent between the two following policies: (a) selling γ units of H for the price h(w 0 γ) and investing optimally into a trading strategy; (b) investing optimally into a trading strategy without trading the contingent claim Naturally we consider two investment problems: sup u ( (w 0 + h(w 0 γ)) + m γh ) (12)
5 No 2 CLAIM HEDGING IN AN INCOMPLETE MARKET 197 sup u(w 0 + m) (13) Theorem 2 Let M be a closed linear subspace and admit no approximate profits in L 2 (Ω F P ) If H has decomposition (1) then sup u((w 0 + h(w 0 γ)) + m γh) = w 0 + h(γ) γb H + 1 E[( π()) E[π() and the optimal attainable payoff for γ units of H is given by Proof A Var [ γn H (14) m = γm H + 1 E[ 2 ( π()) (15) E[π() sup u((w 0 + h(w 0 γ)) + m γh) { = sup E [ (w 0 + h(w 0 γ)) + m γh A Var [ (w 0 + h(w 0 γ)) + m γh } { = w 0 + h(w 0 γ) + sup E [ m γh A Var [ m γh } From decomposition (1) we get E [ m γh A Var [ m γh = γb H + E [ m A Var [ m A Var [ γn H where m = m γm H Now one has {E [ m A Var [ m } = sup sup {E [ m ut sup b R m M sup {E [ m AE [ m } 2 m M b { = sup AE [( m m M b 1 ) 2 } + b + 1 { = sup A [( ( m M E[ 2 E m b + 1 = A [ (( E[ 2 E b + 1 ) ) 2 π() + b + 1 = A E[π() ( E[ 2 b E[π() ) E[ 2 b + 1 ( ) ( π()) AE [ m b 2 } (16) ( b + 1 ) ) 2 } π() + b E[π() E[ 2 )
6 198 SUN WANGUI WANG CHUNFENG Vol 18 Thus the solutions to problem (16) are given by ( m (b ) = b + 1 ) ( π()) (17) b = 1 ( E[ 2 ) E[π() 1 (18) Therefore from m = m γm H and relations (16) (18) relation (15) holds The proof is completed Taking h(γ) = 0 and H = 0 in Theorem 2 we have the following conclusion Corollary 1 Let M be a closed linear subspace and admit no approximate profits in L 2 (Ω F P) Then sup u(w 0 + m) = w And the optimal attainable payoff is given by y virtue of relations (15) and (20) we have E[( π()) (19) E[π() m 0 = 1 E[ 2 ( π()) (20) E[π() m = γm H + m 0 (21) It means that the attainable payoff m of first selling γ units of H and then choosing an optimal investment strategy with initial capital 0 is a sum of two parts: one part is the attainable part of γm H units of H which is independent of the risk aversion parameter; the other is the attainable payoff m 0 of optimally investing in the financial market with initial capital 0 without trading the contingent claim H In view of relation (20) the solution m 0 to problem (13) can be changed into the form m 0 = 1 c( π()) (22) A where c = E[2 2E[π() is constant Note three points First all investors have the same composition of risky assets We call the composition a risky fund All investors will be indifferent between choosing portfolios from the original market or from the riskless asset and the risky fund Second since m 1 0 is linear of A aggressive investors hold the risky fund more than conservative investors Third Using decomposition = π() + ( π()) we know problem inf E[( m)2 with solution π() Thus the error of replication of is given by ( π()) = π() In the case discussed by Gourieroux Laurent and Pham [5 they referred π() as a numeraire Therefore the strategy of hedging γ units of H under preference (9) contains two parts: one part is the same for all investors; the other part depends upon the risk aversion parameter and has the same composition of risky assets In view of Definition 3 and relations (14) and (19) the following corollary holds Corollary 2 Let M be a linear subspace of L 2 (Ω F P) admitting no approximate profits in L 2 (Ω F P) For each H L 2 (Ω F P) and each γ w 0 R the u-indifference price for γ units of H is given by [ h(w 0 γ) = h(γ) = γe P H + Aγ 2 Var [ n H (23)
7 No 2 CLAIM HEDGING IN AN INCOMPLETE MARKET 199 where H has a decomposition (1) Relation (23) was first obtained by Schweizer [6 Here we prove it true in a different way y Theorem 1 the optimal attainable payoff under the criterion (6) for contingent claim γh with price h(γ) is m = γm H + (γb H h(γ))( π()) And by Theorem 2 the optimal attainable payoff under the quadratic derived utility function (8) for contingent claim γh with price h(γ) is m = γm H + 1 E[ 2 E[(π()) 2 ( π()) In order to compare the performance of attainable payoff m and m define the error of replication of γ > 0 contingent claims H as ε(γ H m) = γh h(γ) m where m {m m } Thus ε(γhm) denotes the discounted error of replication of γ > 0 contingent claims H Proposition 1 Let M be a closed linear subspace in L 2 (Ω F P) and admit no approximate profits in L 2 (Ω F P) Then E [ ε(γ H m ) Var [ ε(γ H m ) E [ ε(γ H m ) Var [ ε(γ H m ) Proof We first calculate the terms E [ ε(γh m ) E [ ε(γhm ) Var [ ε(γh m ) and Var [ ε(γhm ) Similarly E [ ε(γ H m ) [ = E γ H [ γe P H [ = E γ H [ γe P H = γe [ n H Aγ 2 Var [ n H Aγ 2 Var [ n H Aγ 2 Var [ n H m γ mh 1 1 E[(π()) 2 E[(π()) 2 E[ 2 E[(π()) 2 ( π()) E [ ε(γ H m ) = γe [ n H Aγ 2 Var [ n H + Aγ 2 Var [ n H E[(π()) 2 E[(π()) 2 Var [ ε(γ H m ) [ = Var γ nh Aγ2 Var [ n H [( +Var Aγ 2 Var [ n H + 1 π() E[ 2 ) π() E[(π()) 2
8 200 SUN WANGUI WANG CHUNFENG Vol 18 Var [ ε(γ H m ) [ = Var γ nh Aγ2 Var [ n H π() Then we can see the proposition is true The proof is completed The proposition shows that the variance of the discounted error of replication under the criterion (6) is smaller and to be optimal (15) must compensate the higher variance of the discounted error of replication under the quadratic derived utility function (9) with a smaller expectation of the discounted error itself Proposition 1 is the extension of Proposition 37 obtained by Mercurio [4 in an incomplete market that a stock and a bond are traded under a discrete-time setting 5 Hedging under Partial Information There are several recent papers dealing with partial information in finance The utility maximization problem as stock prices are only observed was studied by Lakner [10 Pham and Quenez [11 and Zohar [12 by martingale duality methods Also under partial observation hedging problems are examined by Di Masi et al [13 Schweizer [14 and Pham [8 Now we apply the indifference argument to claim hedging and pricing under partial information Let = 1 (the interest rate is 0) and the market considered in this part be complete in full information Then there is a decomposition L 2 (Ω F P) = R + M where R represents the set of capitals allocated in saving account at expired time of contingent claims and M denotes the set of investment opportunities (or attainable payoff) in full information with M L 2 (Ω F P) and is a closed linear subspace Let G F be a σ-subalgebra In the market we cannot observe all information F but information G In this case the set of attainable contingent claims is set A G = R + M G where A G = L 2 (Ω G P) M G = {E[f G : f M } Consider the optimal problem J G (w 0 ) := sup u((w 0 + h(w 0 γ)) + m γh) (24) G The solution to problem (24) is expressed by m G Let H L2 (Ω F P) with H = b H G + m H G + n H G b H G R m H G M G and n H G A G And let H G = E[H G Then From Theorem 2 and Corollary 2 where [ H G = b H G + mh G nh G = H H G b H G = E H d P G [ m G (H = γ G E J G (w 0 ) = w H d P G π(1) = ) + 1 d P G E[( d PG )2 E[1 π(1) E[π(1) 1 (1 π(1)) E[π(1) the initial wealth w 0 And the u-indifference price for the set of attainable payoff under partial information G is [ h(w 0 γ) = h(γ) = γe P H + Aγ 2 Var [ H H G
9 No 2 CLAIM HEDGING IN AN INCOMPLETE MARKET 201 References [1 H Pham T Rheinlander and M Schweizer Mean-variance hedging for continuous processes: New proofs and examples Finance and Stochastics : [2 M Schal On quadratic cost criteria for option hedging Mathematics of Operations Research : [3 M Schweizer Variance-optimal hedging in discrete time Mathematics of Operations Research : 1 32 [4 F Mercurio Claim pricing and hedging under market incompleteness and mean-variance preferences European Journal of Operational Research : [5 C Gourieroux J P Laurent and H Pham Mean-variance Hedging and Numeraire Mathematical Finance : [6 M Schweizer From actuarial to financial valuation principles Insurance: Mathematics & Economics : [7 T Moller On transformations of actuarial valuation principles Insurance: Mathematics & Economics : [8 H Pham Mean-variance hedging for partially observed drift processes International J of Theoretical and Applied Finance : [9 H Follmer and D Sondermann Hedging of non-redundant contingent claims In Contributions to Mathematical Economics(ed by A Mas-Colell and W Hilderbrand) Amsterdam North Holland [10 P Lakner Optimal trading strategy for an investor: The case of partial information Stochastic Processes and Their Applications : [11 H Pham M C Quenez Optimal portfolio in partially observed stochastic volatility models Annals Applied Probability : [12 H Zohar Maximizing the probability of achieving a goal in the case of partially observed drift process Inter J Theo Appl Finance : [13 G Di Masi E Platen W Runggaldier Hedging of options under discrete observations on assets with stochastic volatilities Seminar on Stoch Anal Rand Fields Appli(ed by M Dozzi and F Russo) : [14 M Schweizer Risk minimizing hedging strategies under restricted information Mathematical Finance :
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