A n e w pr oposit ion on t h e m a r t in ga le r e pr e se n t a t ion t h e or e m a n d on t h e a ppr ox im a t e h e dgin g of t in ge n t cla im in m e a n - va r ia n ce cr it e r ion A Fa r be r , N gu ye n V.H a n d Vu on g Q.H I n t his work we revisit t he problem of t he hedging of cont ingent claim using m ean- square crit erion We prove t hat in incom plet e m arket , som e probabilit y m easure can be Q ~ P ident ified so t hat { Sn } becom es { Fn } - m art ingale under Q This is in fact a new proposit ion on t he m art ingale represent at ion t heorem The new result s also ident ify a weight funct ion t hat serves t o be an approxim at ion t o t he Radon- Nikodým derivat ive of t he unique neut ral m art ingale m easure Q JEL Classificat ions: G12; G13 Keywords: Mart ingale represent at ion t heorem ; Hedging; Cont ingent claim ; Mean- variance CEB Working Paper N° 06/ 004 April 2006 Université Libre de Bruxelles – Solvay Business School – Centre Emile Bernheim ULB CP 145/01 50, avenue F.D Roosevelt 1050 Brussels – BELGIUM e-mail: ceb@admin.ulb.ac.be Tel : +32 (0)2/650.48.64 Fax : +32 (0)2/650.41.88 A new proposition on the martingale representation theorem and on the approximate hedging of contingent claim in mean-variance criterion André FARBER*, NGUYEN Van Huu†, and VUONG Quan Hoang‡ April, 2006 Abstract: In this work we revisit the problem of the hedging of contingent claim using mean-square criterion We prove that in incomplete market, some probability measure Q ~ P can be identified so that {S n } becomes {Fn } -martingale under Q This is in fact a new proposition on the martingale representation theorem The new results also identify a weight function that serves to be an approximation to the Radon-Nikodým derivative of the unique neutral martingale measure Q JEL Classification: G12; G13 Keywords: Martingale representation theorem; Hedging; Contingent claim; Mean-variance Introduction The activity of a stock market takes place usually in discrete time Unfortunately such markets with discrete time are incomplete, so the traditional pricing and hedging of contingent claim are usually not applicable The purpose of this work is to propose a simple method for hedging a contingent claim or an option in mean-variance criterion (a) Let ^S n , n 0,1, , N `, S n R d , be a sequence of discounted stock prices defined on a probability space ^:, F , P` , and ^Fn , n 0,1, , N ` be a sequence of sigma-algebras of information available up to the time n (b) An ^Fn ` -measurable random variable H is called a contingent claim that in the case of a standard call option H max(S n K ,0) (c) The portfolio J {J n , n 1,2, .N } with J n (J n1 , J n2 , J nj ) , where J nj is the number of securities of type j kept by the investor in the time interval [n 1, n) J n is F( n 1) - measurable (based on the information available up to the time n ) Thus, ^J n ` is said to be predictable (d) Suppose that 'S n S n S n 1 , H L2 (P ) , (e) Gn (J ) n ¦ J k 'S k is gain with J k 'S k k d ¦J j k VS kj j The traditional problem is to find constant c and J * {J n , n 1,2, .N } , such that Centre Emile Bernheim, Université Libre de Bruxelles, 21 Ave F.D Roosevelt, 1050-Bruxelles, Belgium Vietnam National University, Hanoi, 334 Nguyen Trai, Hanoi, Vietnam ‡ Centre Emile Bernheim, Université Libre de Bruxelles, 21 Ave F.D Roosevelt, 1050-Bruxelles, Belgium † E P ^H c G N (J )` o (1.1) Definition (J n* ) (J n* (c)) minimizes the expectation in (1.1) is called an optimal strategy in the mean square criterion corresponding to initial capital c Problem (1.1) has been investigated in a number of works such as Föllmer and Schweiser (1991), Schweiser (1995, 1996), Schäl (1994), and Nechaev (1998) However, the solution for (1.1) has been very complicated as {S n } is not a {Fn } -martingale under P When {S n } is {Fn } -martingale under some measure Q ~ P , we can find c, J such that: EQ ^H c G N (J )` o (1.2) The solution of this problem may be simple enough, and the construction of an optimal strategy is much easier in practice We notice that if L N dQ / dP then EQ ^H c G N (J )` ^ E P ^H c G N (J )` LN ` (1.3) can be considered a weighted expectation under P of H c G N with the weight L N This is similar to the pricing of asset based on a neutral martingale measure In this work we give a solution of the problem (1.3) and a martingale representation theorem in the case of discrete time Defining the optimal portfolio Let Q be a probability measure such that Q is equivalent to P, and under Q, {S N , n 1,2, , N } is a martingale, then En ( X ) Theorem If ^S n , n EQ ( X FN ), H N H, Hn En ( H ) 0,1, , N `, S n R d is a ^Fn ` Q then EQ ( H H G N (J * )) EQ ( H c G N (J )) , (2.1) Where J n* E n 1{( 'H n 'S n )[var('S n )] 1 } E n 1{( H'S n )[var('S n )] 1 } P a.s with the convention that 0/0 = Proof We shall prove the theorem only for the case d H N H 'H 'H N and E n 1 ( 'H N J n 'S n ) (2.2) We note that: E n 1 ( 'H N ) 2J n E n 1 ( 'H N 'S n ) J n2 E n 1 ( 'S n ) This expression takes the minimum value when J n Furthermore, we have: J n* EQ ( H N c G N (J )) N ½ EQ ® H c ¦ ('H N J n 'S n )¾ n 1 ¯ ¿ 2 N N ( H c) ¦ EQ {'H n J n 'S n }2 ( H c) E Q ¦ E n 1{'H n J n 'S n }2 n 1 n 1 N ( H c) EQ ¦ E n 1{'H n J n* 'S n }2 n 1 E Q {H N H G N (J * )}2 The martingale representation theorem Theorem Let {H n , n 0,1,} , {S n , n same probability space ^:, F , P` , F 0,1, } be random variables defined on the V ( S , , S n ) We denote ( S , P ) a set of the probability measure Q such that Q ~ P , and that ^S n ` is ^FnS `-martingale under Q S n f Thus, if F FnS , H n , S n L2 (P) and if ^H n ` is also a martingale under Q , we have: n n H n H ¦ J k 'S k C n , a.s (3.1) k where ^C n ` is ^FnS `- Q -martingale orthogonal to the martingale ^S n ` , that is 0,1,2, , whereas ^J n ` is ^FnS `-predictable E n 1 ^'C n 'S n ` 0, n n H n H ¦ J k 'S k : H G N (J ), P - a.s (3.2) k for all n finite iff the set ( S , P ) consists of only one element Remark By the fundamental theorem of mathematical finance, a stock market has no arbitrage opportunity and is complete iff ( S , P ) consists of only one element and in this case we have (3.2) with J being defined by (2.2) Furthermore, in this case the conditional probability distribution of ^S n ` given ^FnS1 ` concentrates at d points of R d (see [2]) Examples Example Let us consider a stock with the discounted price S at t , S1 at t where: 32 S with prob p1 ° S1 ® S with prob p p1 , p , p3 ! 0, p1 p p3 ° S with prob p ¯2 1, Suppose that there is an option on the above stock with the maturity at t and with strike price K S We shall show that there are several probability measures Q ~ P such that under Q , S , S1 is a martingale, or equivalently E Q ( 'S1 ) In fact, suppose that Q is a probability measure such that under Q , S1 takes the values of S , S , S , with the positive probabilities q1 , q , q , respectively, then: E Q ( 'S ) S ( q1 q ) / Therefore, Q is defined by q1 ,1 2q1 , q1 , q1 q1 q3 In the above market, the payoff of the option is: H ( S1 K ) ('S1 ) max('S1 ,0) Apparently, it is feasible to construct an optimal portfolio with: EQ ( H'S1 ) J* EQ ('S1 ) 2 q1 S EQ ( H ) Example A semi-continuous market model, which is discrete in time, but continuous in state Now, let us consider a financial market with two assets: (a) A risk-less asset {Bn , n 0,1, , N } which exhibits a dynamics given by (4.1): n ẵ expđƯ rk ắ,0 rk k ¿ (b) A risky asset {S n , n 0,1, , N } given by the following dynamics Bn (4.1) ẵ ưn (4.2) S expđƯ [ P ( S k 1 ) V ( S k 1 ) g k ]¾ ¿ ¯k where {g n , n 0,1,, N } is a sequence of an NIID ~ N (0,1) random variable It follows directly from (4.2) that S n S n 1 exp^P ( S n 1 ) V ( S n 1 ) g n `, (4.3) P ( S n1 ) a ( S n 1 ) V ( S n 1 ) / 2, Sn with S given, and a ( x), V ( x) being some functions defined on [0, f ) ~ The discounted price of risky asset S n S n / Bn is: ½ ưn S expđƯ [ P ( S k 1 ) rk V ( S k 1 ) g k ]¾ ¿ ¯k We now find a martingale measure Q for this model ~ Sn (4.4) It is easy to see that E P {exp(Og k )} exp(O2 / 2) , for g k ~ N (0,1) , hence Đ n Ãẵ E đexpă Ư E k ( S k 1 ) g k E k2 ( S k 1 ) áắ âk Thus, putting Đ n à Ln expă Ư E k ( S k 1 ) g k E k2 ( S k 1 ) ¸, n 1, , N , âk > @ > ~ @ (4.5) (4.6) ~ and if S n / S n 1 is a measure such that dQ addition, we see that ~ ~ S n / S n 1 L N dP , then Q is also a probability measure In exp^P ( S n 1 ) rn V ( S n 1 ) g n ` (4.7) Denoting by E $ , E expectations corresponding to P, Q , and E n () ^ ` E () FnS , then choosing En a ( S n 1 ) rn , V ( S n1 ) (4.8) ~ ~ ½ ~ ~ ½ ư~ ẵ then E n 1 đS n / S n 1 ắ E $ đ Ln S n / S n 1 FnS ¾ / Ln 1 , which implies đ S n ắ is a martingale ¯ ¿ ¿ ¯ ¿ under Q Also, under Q , S n can be represented in the form Sn ^ S n 1 exp P * ( S n 1 ) V ( S n 1 ) g n* ` (4.9) Where P * ( S n 1 ) rn V ( S n 1 ) / 2, g n* E n g n is a Gaussian N (0,1) It is not easy to show the structure of ( S , P ) for this model We can choose the probability measure Q or the weight function L N to find the optimal portfolio Remark The models (4.1), (4.2) are that of discretization of the following diffusion model Let us consider a financial market with continuous time of two assets : ưt ẵ (a) A risk-less asset: Bt exp đ r (u )du ¾ , and ¯o ¿ (b) A risky asset: dS t S t a ( S t )dt V ( S t )dWt , S is given, or St t ưt ẵ exp đ a ( S u ) V ( S u ) / du ³ V ( S u )dWu ¾, d t d T ¯o ¿ > @ (4.10) Putting P (S ) a( S ) V ( S ) / and dividing [0, T ] into N equal intervals {0, ',2', , N'} , where N large, it follows from (4.10),(4.11) that n' ° n' ½° S n' S ( n 1) ' exp® ³ P ( S u )du ³ V ( S u )dWu ¾ °¯( n 1) ' °¿ ( n 1) ' # S ( n 1) ' exp^P ( S ( n 1) ' )' V ( S ( n 1) ' )[Wn' W( 1) ' ]` ^ # S ( n 1) ' exp P ( S ( n 1) ' )' V ( S ( n 1) ' )'1 / g n (4.11) T / ' sufficiently ` where ^g n , n 1, , N ` is a sequence of the NIID ~ N (0,1) random variables Thus, we obtain the model: S n' S ( n 1) ' exp^P ( S ( n 1) ' )' V ( S ( n 1) ' )'1 / g n ` Similarly we have Bn B( n 1) exp^'rn ` (4.12) (4.13) According to (4.10) ,the discounted price of the stock S t is t t ½ S n S t / Bt S exp ®³ >P ( S u ) ru @du ³ V ( S u )dWu ¾ ¯0 ¿ ~ ½ The unique probability measure Q under which ®S t , Ft S , Q ¾ is a martingale is defined by ¯ ¿ ~ T T ½ exp ®³ E u dWu 12 ³ E u2 du ¾ : LT (Z ) , ¯0 ¿ dQ / dP Ft S where E s (a( S s ) rs ) / V ( S s ) , and under Q, then: Wt* (4.14) t Wt ³ E u du is a Wiener process It is obvious that LT can be approximated by: where > @ LN ẵ ưN expđƯ E k '1 / g k 'E k2 / ¾ ¿ ¯k (4.15) En a(S (4.16) ( n 1) ' ) rn' / V ( S ( n 1) ' ) Therefore, the weight function (4.14) is an approximation to a Radon-Nikodým derivative of the unique neutral martingale measure Q to P , where Q can be used to price such derivatives Further problems to be investigated We realize that further problems that could arise in these models are the following: 1) We have to show that for the weight function (4.15) EQ ( H H G N (J * )) o as N o f or ' o 2) Which neutral martingale measure Q is the nearest one with the subjective measure P in the semi-continuous model? REFERENCES [1] Föllmer H., and Schweiser M “Hedging of contingent claim under incomplete information,” Applied Stochastic Analysis, Ed by M Davis, and R Elliot London: Gordan & Breach, 1991, pp.389-414 [2] Jacod J., Shiryaev A.N “Local martingales and the fundamental asset pricing theorem in the discrete case,” Finance Stochastics 2, pp.259-272 [3] Nechaev M.L “On mean-variance hedging,” Proceeding of Workshop on Mathematical Finance, May 18-19, 1998 Institut Franco-Russe Liapunov, Ed by A Shiryaev and A Sulem [4] Nguyen V.H., and Tran T.N “On a generalized Cox-Ross-Rubinstein option market model,” Acta Mathematica Vietnamica, 26(2), 2001, pp 187-204 [5] Schweiser M “Variance-optimal hedging in discrete time,” Mathematics of Operations Research, 20(1), 1995, pp 1-32 [6] Schweiser M “Approximation pricing and the variance-optimal martingale measure,” Annals of Probability, 24(1), 1996, pp 206-236 [7] Schäl M “On quadratic cost criteria for option hedging,” Mathematics of Operations Research, 19(1), 1994, pp 131-141 .. .A new proposition on the martingale representation theorem and on the approximate hedging of contingent claim in mean- variance criterion André FARBER*, NGUYEN Van Huu†, and VUONG Quan Hoang‡... approximation to the Radon-Nikodým derivative of the unique neutral martingale measure Q JEL Classification: G12; G13 Keywords: Martingale representation theorem; Hedging; Contingent claim; Mean- variance. .. that {S n } becomes {Fn } -martingale under Q This is in fact a new proposition on the martingale representation theorem The new results also identify a weight function that serves to be an approximation