Projekbereich B Discussion Paper No. B 167 Mean-Variance Hedging for General Claims by Marin Schweizer ) Ocober 199 ) Financial suppor by Deusche Forschungsgemeinschaf, Sonderforschungsbereich 33 a he Universiy of Bonn, is graefully acknowledged. (Annals of Applied Probabiliy 2 (1992), 171 179)
Mean-Variance Hedging for General Claims Marin Schweizer Universiä Bonn Insiu für Angewande Mahemaik Wegelersraße 6 D 53 Bonn 1 Wes Germany Absrac: We consider a hedger wih a mean-variance objecive who faces a random loss a a fixed ime. The size of his loss depends quie generally on wo correlaed asse prices, while only one of hem is available for hedging purposes. We presen a simple soluion of his hedging problem by inroducing he inrinsic value process of a coningen claim. Key words: hedging, mean-variance crierion, coninuous rading, opion valuaion, coningen claims, equivalen maringale measures. AMS 198 subjec classificaion: 6G35, 9A9 Research suppored by Deusche Forschungsgemeinschaf, Sonderforschungsbereich 33
2. Inroducion In his paper, we solve he coninuous-ime hedging problem wih a mean-variance objecive for general coningen claims. A special case of his problem was reaed by Duffie/Richardson (1991) and provided he moivaion for his work. There are wo asses whose prices are boh modelled by exponenial Brownian moions wih ime-dependen random coefficiens. The raes of reurn beween asses are correlaed. A a fixed ime, he hedger faces a random loss which depends in full generaliy on he enire evoluion of boh asse prices. For he purpose of hedging agains his risk, however, only one asse is available. This implies ha markes are incomplee and coningen claims canno be replicaed by rading. The goal of he hedger is o minimize his oal expeced quadraic coss, or equivalenly o maximize his expeced uiliy from erminal wealh for a quadraic uiliy funcion. A precise saemen is given in Secion 1, and he soluion is presened in Secion 3. We remark ha he same argumens would also work for any number N of driving asses wih n hedging asses, where 1 n N. Our approach o his problem follows he mehod of Duffie/Richardson (1991): we show ha he inner produc associaed wih he normal equaions for orhogonal projecion is defined by an ordinary differenial equaion in ime wih an explici soluion. This is done by choosing a suiable racking process for he coningen claim under consideraion. The essenial difference o he above paper lies in wo poins: we are able o solve he hedging problem for a general coningen claim, and we do no have o conjecure he soluion from discree-ime reasoning. In fac, our approach shows ha he naural choice for he opimal racking process is provided by he inrinsic value process associaed o he given coningen claim. This process is defined in erms of he minimal equivalen maringale measure for ha asse price which is used for hedging. Boh of hese conceps are explained in more deail in Secion 2. We conclude he paper in Secion 4 wih a class of examples where explici formulas can be derived, including as a special case he resul of Duffie/Richardson (1991). 1. The problem In his secion, we formulae he general hedging problem and recall he basic approach o solving i. For ease of reference, we use he same noaions as in Duffie/Richardson (1991), subsequenly abbreviaed as D/R. Le (Ω, F, P ) be a probabiliy space wih a wo-dimensional Brownian moion (B, ε) and IF = (F ) T he augmenaion of he filraion generaed by (B, ε), where T > is a fixed ime horizon. Le (µ ), (m ), (σ ), (v ) and (ϱ ) be bounded adaped processes and assume ha (v ) is bounded away from (uniformly in ω), ϱ 1
3 for all and (1.1) ( m ) v T is a deerminisic funcion. Now define a Brownian moion ξ by seing (1.2) ξ = ϱ u db u + 1 ϱ 2 u dε u, T. The asse price processes S and F are hen described by he sochasic differenial equaions (1.3) ds = µ S d + σ S db, S > df = m F d + v F dξ, F >. Since rading in F is possible, F may be used for hedging purposes. A hedging sraegy ϑ is an IF -predicable process saisfying E ϑ 2 ufu 2 du <. Is associaed cumulaive gains process G(ϑ) is given by he sochasic inegral G (ϑ) = ϑ u df u, T. Θ denoes he se of all hedging sraegies. A coningen claim Π is an F T -measurable random variable saisfying (1.4) Π L p (P ) for some p > 2. The hedging problem is hen (1.5) min ϑ Θ E [ (Π + L GT (ϑ) ) 2 ], where L > is a given consan. Inerpreaion. Π can be viewed as a random loss suffered by he hedger a ime T. The consan L plays he role of an iniial cos, and Π + L G T (ϑ) describes he oal loss or coss incurred a ime T. Thus, he hedger s objecive is o minimize his expeced quadraic coss.
4 Remarks. 1) If Π, hen Π can also be inerpreed as a payoff received by he hedger a ime T. His oal final wealh, including gains from rade, is hen given by Π + G T (ϑ). (Of course, his inerpreaion sill holds for a general Π if one is willing o receive possibly negaive payoffs.) Rewriing (1.5) as [ ( min E Π + GT (ϑ) L ) ] 2 ϑ Θ hen yields he inerpreaion of minimizing he expeced quadraic deviaion of he erminal wealh from a fixed arge level L. This is he quesion addressed in D/R for he special case Π = ks T. 2) If we se u(x) = x cx 2 and c = 1 2L, he hedging problem (1.5) can equivalenly be wrien as (1.6) max ϑ Θ E [ u ( G T (ϑ) Π )], i.e., as maximizaion of expeced uiliy from erminal wealh for he quadraic uiliy funcion u. We refer o D/R for oher relaed opimizaion problems. 3) There are several well-known objecions o he preceding crieria (e.g., increasing absolue risk aversion, ranges of nonmonooniciy and possibiliy of negaive wealh). Neverheless, hey have been widely used in pracice and can ofen be helpful as a firs approximaion. Le us now recall he basic approach o solving (1.5). The same Hilber space projecion argumen as in D/R shows ha a hedging sraegy ϑ is opimal if and only if i saisfies (1.7) E [( Π + L G T (ϑ ) ) G T (ϑ) ] = for all ϑ Θ. To obain (1.7), one defines he funcion H() := E [( Z + L G (ϑ ) ) G (ϑ) ], T for a suiable racking process Z wih Z T = Π. Using (1.1), one derives a differenial equaion for H whose unique soluion is H(), hus proving in paricular (1.7). The crucial problem is hen he choice of ϑ and, even more imporanly, he racking process Z. In D/R, he soluion for heir paricular coningen claim was conjecured from discree-ime argumens. We shall see in Secion 3 ha he simples choice for Z is quie generally given by he inrinsic value process V of Π, and his will also yield a simple descripion and inuiive inerpreaion of he opimal sraegy ϑ.
2. Claims and heir inrinsic values 5 In his secion, we recall from Föllmer/Schweizer (1991) and Schweizer (1991) he noions of a minimal maringale measure and of an inrinsic value process. Since his is of ineres in iself, we begin wih a few words of moivaion. In a general formulaion of opion rading problems, one usually sars ou wih a price process X and an opion or coningen claim H. Two key issues are hen he valuaion and he hedging of H by means of a suiable rading sraegy based on X. In his conex, an imporan role is played by he se IP of equivalen maringale measures for X. The elemens of IP are probabiliy measures Q which are equivalen o P (i.e., have he same null ses as P) and under which he price process X becomes a maringale. By no-arbirage-ype argumens, he value process V of a claim H mus have he form V = E Q [H IF ] for some Q IP. If IP is a singleon, his obviously leads o a unique soluion of he valuaion problem. In general, however, IP has many elemens, all of which give rise o a poenial value process. Bu i urns ou ha in many cases, here is a unique minimal equivalen maringale measure P IP. This concep was inroduced in Föllmer/Schweizer (1991) and used in Schweizer (1991) o obain hedging sraegies which are opimal in a cerain sense (differen from he one used here). Furhermore, i is shown in Schweizer (1991) ha he induced valuaion process V = [H IF ] can be viewed E P as he inrinsic value process of he claim H and ha i corresponds o a riskneural approach o opion valuaion. In our presen hedging problem, he price process X o be used for hedging is given by F and he coningen claim H by Π. Clearly, every valuaion process V := E Q [Π IF ] saisfies V T = Π and hus could in principle serve as a racking process Z. Bu as we shall see in he nex secion, he choice of he inrinsic value process V allows us o give a very simple soluion for (1.5). Le us now urn o he acual consrucion of he minimal maringale measure P for F. For his purpose, we wrie he P -semimaringale F in is canonical decomposiion F = F + M + A wih dm = v F dξ, da = m F d. This implies d M = v 2 F 2 d and herefore da = α d M, wih α = m v 2 F. The minimal maringale measure P for F is hen defined by
6 (2.1) d P ( dp := U T := E = exp = exp ) α dm T α u dm u 1 2 m u dξ u 1 v u 2 αu 2 d M u m 2 u v 2 u du ; see Föllmer/Schweizer (1991). For fuure reference, we noe ha (2.2) U T p 1 L p (P ) and U 1 T p 1 L p ( P ); his follows immediaely from he fac ha ( ) m v is bounded. Le us now define he square inegrable P -maringale N by N := 1 ϱ 2 u db u ϱ u dξ u, T. Using Girsanov s heorem and Iô s represenaion heorem, one can easily verify he following facs: (i) Boh F and N are square-inegrable maringales under P. (ii) M and N are orhogonal under P, and every B L 2 (F T, P ) wih zero expecaion is he sum of wo sochasic inegrals wih respec o M and N, respecively. (iii) F and N are orhogonal under P, and every B L 2 (F T, P ) wih zero expecaion is he sum of wo sochasic inegrals wih respec o F and N, respecively. We remark ha (iii) is sraighforward since, due o (1.1), he change of measure from P o P only involves a deerminisic change of drif. Lemma. Suppose Π is a coningen claim saisfying (1.4). decomposiion Then Π admis a (2.3) Π = Π + ϑ u df u + ν u dn u P a.s.,
where ϑ is a hedging sraegy and ν saisfies E 7 [ T inrinsic value process V associaed o Π is given by ν 2 u du ] <. Furhermore, he (2.4) V = E P [Π F ] = Π + ϑ u df u + ν u dn u, T and saisfies sup V L 2 (P ). T Proof. This follows immediaely from (i) (iii), (1.4) and (2.2). q.e.d. 3. The soluion In his secion, we presen he general soluion o he hedging problem (1.5). Since he mehod of proof is he same as in D/R, we shall confine ourselves o a brief ouline of he required seps. Theorem. Le Π be a coningen claim saisfying (1.4) and denoe by V = [Π IF ] is associaed inrinsic value process. E P Le G be he soluion of he sochasic differenial equaion (3.1) dg = Φ(G ) df, G =, where (3.2) Φ(G ) = ϑ + m v 2 F ( V + L G ) and ϑ is aken from (2.3). Then he hedging sraegy ϑ := Φ(G ) solves (1.5). Proof. 1) Using (2.4), (1.4) and (2.2), one can show as in Proer (199) ha (3.1) has a unique soluion saisfying E [ (G ) 2] <. Combining his wih (2.4) sup T hen implies ha ϑ is indeed a hedging sraegy wih gains process G(ϑ ) = G. 2) Fix any hedging sraegy ϑ and define he funcion [( ) ] H() := E V + L G G (ϑ), T. If we can show ha H(), hen H(T ) = will imply he opimaliy of ϑ by (1.7). For his purpose, we apply he produc rule o ( V + L G ) G(ϑ), noe ha all maringale erms have expecaion and use Fubini s heorem o obain H() = [ m 2 E u ( Vu + L G u v 2 u ) ] G u (ϑ) du.
Since ( m v ) 8 is deerminisic by (1.1), his implies H() = m 2 u v 2 u H(u) du and herefore d d H() = m2 v 2 H(). Bu G (ϑ) = implies H() =, and hus we mus have H(). q.e.d. ha Remarks. 1) The preceding proof relies in a crucial way on he assumpion ( ) m v is deerminisic. Clearly, a sochasic mean-variance radeoff for he hedging insrumen F would be more realisic. I would be ineresing o see a soluion of (1.5) in his general case. 2) Recall ha he hedging problem (1.5) can be rewrien as a problem (1.6) of maximizing expeced uiliy from erminal wealh. The corresponding quadraic uiliy funcion u(x) = x 1 2L x2 has absolue risk aversion R a (x) = 1 L x. The opimal sraegy ϑ in (3.2) can herefore be wrien as (3.3) ϑ = ϑ + 1 R a (G V ) m v 2 F. Bu he infiniesimal condiional mean and variance of he hedging insrumen F are given by E[dF F ] = m F d, Var[dF F ] = v 2 F 2 d. Thus (3.3) shows ha ϑ decomposes ino a pure hedging demand ϑ and a second componen represening a demand for mean-variance purposes. Such a decomposiion was already obained by Meron (1973) for general uiliy funcions u, bu under he assumpion of a Markovian srucure for he underlying asses.
9 4. Some explici resuls In his secion, we give explici expressions for V and ϑ in he case where all coefficiens are deerminisic funcions and he claim Π only depends on he erminal values F T and S T of he asse prices. In paricular, we recover as a special case he resul obained by D/R. Suppose in addiion o our sanding assumpions ha he coefficiens (µ ), (m ), (σ ), (v ) and (ϱ ) are all deerminisic. Define γ := σ m ϱ v µ, T and Y := S exp γ u du, T, so ha Y T = S T. Then i is easy o see from (1.3) ha (F, Y ) is a wo-dimensional Markov process. If he claim Π has he form (4.1) Π = g(f T, S T ) for a funcion g saisfying some growh condiions, he corresponding inrinsic value process V is herefore given by (4.2) V = E P [Π F ] = f(f, Y, ) wih (4.3) f(x, y, ) = E [ ( ( g x exp W 1 1 ) ( 2 Var[W 1], y exp W 2 1 ))] 2 Var[W 2], where W 1 and W 2 are joinly normally disribued wih variances T σu 2 du, respecively, and covariance (2.3) in erms of f, we firs apply Iô s lemma: T vu 2 du and T v u σ u ϱ u du. To obain he decomposiion d V = f x df + f y dy + erms of finie variaion. Bu since we know from (2.4) ha d V = ϑ df + ν dn
1 is a coninuous maringale under P, all finie variaion erms mus vanish, and i only remains o express he maringale par (under P ) of Y in erms of F and N. Afer some calculaions, his leads o (4.4) ϑ = f x (F, Y, ) + f y (F, Y, ) Y σϱ, v F ν = f y (F, Y, ) Y σ 1 ϱ 2. By solving (3.1) for G, we can herefore obain he opimal hedging sraegy ϑ from (3.2). Example. Then Le us ake he claim Π = ks T, k >, considered by D/R. f(x, y, ) = E [ k y e W 2 1 2 Var[W 2] ] = k y implies f x =, f y = k and hus The inrinsic value process is ϑ = k Y σϱ v F. (4.5) V = k Y = k S exp and he opimal sraegy is given by γ u du, (4.6) ϑ = 1 ( m F v 2 (L + V G ) + σ ) ϱ v V, where G solves dg = 1 ( m F v 2 (L + V G ) + σ ) ϱ v V df, G =. This is exacly he soluion obained by D/R. In paricular, heir racking process Z coincides (up o a change of sign) wih he inrinsic value process V of Π. Acknowledgemens. I am graeful o a referee for suggesing a decomposiion (3.3) of he opimal sraegy and o Darrell Duffie who encouraged me o expand a shor wrien commen o he presen paper.
11 References D. Duffie and H. R. Richardson (1991), Mean-Variance Hedging in Coninuous-Time, Annals of Applied Probabiliy 1, 1 15 H. Föllmer and M. Schweizer (1991), Hedging of Coningen Claims under Incomplee Informaion, in: M. H. A. Davis and R. J. Ellio (eds.), Applied Sochasic Analysis, Sochasics Monographs, Vol. 5, Gordon and Breach, London/New York, 389 414 R. C. Meron (1973), An Ineremporal Capial Asse Pricing Model, Economerica 41, 867 887 P. Proer (199), Sochasic Inegraion and Differenial Equaions. A New Approach, Springer, New York M. Schweizer (1991), Opion Hedging for Semimaringales, Sochasic Processes and heir Applicaions 37, 339 363 M. Schweizer (199), Hedging and he CAPM, unpublished manuscrip