Quadratic hedging of basis risk

Hulley, Ha dy; McWal e , Thomas A. A icle Quad a ic hedging o basis isk Jou nal o Risk and Financial Managemen P o ided in Coope a ion wi h: MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel Sugges ed Ci a ion: Hulley, Ha dy; McWal e , Thomas A. (2015) : Quad a ic hedging o basis isk, Jou nal o Risk and Financial Managemen , ISSN 1911-8074, MDPI, Basel, Vol. 8, Iss. 1, pp. 83-102, h ps://doi.o g/10.3390/j m8010083 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/178555 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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McWal e 2,3,* 1Uni e si y o Technology Sydney, Finance Discipline G oup, P.O. Box 123, B oadway, NSW 2007, Aus alia 2Depa men o Ac ua ial Science and he A ican Collabo a ion o Quan i a i e & Risk Resea ch, Uni e si y o Cape Town, Rondebosch, 7701, Sou h A ica 3Facul y o Economic and Financial Sciences, Depa men o Finance and In es men Managemen , Uni e si y o Johannesbu g, P.O. Box 524, Auckland Pa k, 2006, Sou h A ica *Au ho o whom co espondence should be add essed; E-Mail: [email p o ec ed]; Tel.: +27-21-650-2474; Fax: +27-21-689-7580. Academic Edi o : Michael McAlee Recei ed: 28 No embe 2014 / Accep ed: 20 Janua y 2015 / Published: 2 Feb ua y 2015 Abs ac : This pape examines a simple basis isk model based on co ela ed geome ic B ownian mo ions. We apply quad a ic c i e ia o minimize basis isk and hedge in an op imal manne . Ini ially, we de i e he Föllme –Schweize decomposi ion o a Eu opean claim. This allows p icing and hedging unde he minimal ma ingale measu e, co esponding o he local isk-minimizing s a egy. Fu he mo e, since he mean- a iance adeo p ocess is de e minis ic in ou se up, he minimal ma ingale- and a iance-op imal ma ingale measu es coincide. Consequen ly, he mean- a iance op imal s a egy is easily cons uc ed. Simple p icing and hedging o mulae o pu and call op ions a e de i ed in e ms o he Black–Scholes o mula. Due o ma ke incomple eness, hese o mulae depend on he d i pa ame e s o he p ocesses. By making a u he equilib ium assump ion, we de i e an app oxima e hedging o mula, which does no equi e knowledge o hese pa ame e s. The hedging s a egies a e es ed using Mon e Ca lo expe imen s, and a e compa ed wi h esul s achie ed using a u ili y maximiza ion app oach. Keywo ds: op ion hedging; incomple e ma ke s; basis isk; local isk minimiza ion; mean- a iance hedging J. Risk Financial Manag. 2015,884 1. In oduc ion When a con ingen claim is w i en on an asse o p ocess ha is no aded, i is na u al o enqui e abou he e ec i eness o hedging wi h a co ela ed secu i y. In his si ua ion he ma ke is incomple e, and he isk ha a ises as a esul o impe ec hedging is known as basis isk. Examples include wea he de i a i es, eal op ions, op ions on illiquid s ocks and op ions on la ge baske s o s ocks. Since no all he isk can be hedged, we a e dealing wi h a ypical incomple e ma ke si ua ion, in which he appe i e o isk mus be speci ied (usually in e ms o a u ili y unc ion). A numbe o au ho s ha e o mula ed he p oblem o hedging basis isk in e ms o he u ili y maximiza ion app oach (see, e.g., Da is [1,2], Hende son [3], Hende son and Hobson [4], Monoyios [5,6] and Za iphopoulou [7]). By con as , we shall conside he applica ion o quad a ic c i e ia. Ou app oach is simila o ha o Schweize [8] (see also Du ie and Richa dson [9]), whe e he applica ion was hedging u u es wi h a co ela ed asse . Fo comp ehensi e e iews on he heo y o quad a ic hedging, he eade is di ec ed o he wo ks o Pham [10], Schweize [11] and McWal e [12]. We now desc ibe he o ganiza ion o his pape . To s a wi h, in Sec ion 2 we p o ide a summa y o he e minology and gene al heo y used h oughou . In pa icula , we discuss he wo quad a ic app oaches o local isk minimiza ion and mean- a iance hedging. Key o he cons uc ion o hedging s a egies is a decomposi ion o he con ingen claim, known as he Föllme –Schweize (FS) decomposi ion. We also b ie ly desc ibe he minimal ma ingale measu e and he a iance-op imal ma ingale measu e. A simple basis isk model comp ising wo co ela ed geome ic B ownian mo ions is speci ied in Sec ion 3. We assume ha i is no possible o ade in he p ocess he claim is w i en on, bu ha he second p ocess is a secu i y a ailable o ade. Since he gene al heo y is de eloped in e ms o discoun ed secu i ies, we speci y he discoun ed dynamics o he wo p ocesses. The FS decomposi ion o he claim is de i ed in Sec ion 4. This is achie ed by exp essing he non- aded p ocess in e ms o he aded secu i y and an o hogonal p ocess. By using a d i -adjus ed ep esen a ion o he non- aded p ocess, i is possible o cons uc he minimal ma ingale measu e. The Feynman–Kac heo em can hen be used o exp ess he discoun ed claim p ice as he solu ion o a pa ial di e en ial equa ion (PDE) bounda y- alue p oblem. In Sec ion 5 we p esen he hedging s a egies o he wo quad a ic app oaches. The FS decomposi ion makes i easy o speci y he locally isk-minimizing s a egy, wi h p ices de e mined by aking expec a ions unde he minimal ma ingale measu e. Fu he mo e, since he mean- a iance adeo p ocess is de e minis ic unde ou model assump ions, he minimal ma ingale measu e and he a iance-op imal ma ingale measu e coincide. The mean- a iance op imal sel - inancing s a egy is hus easily cons uc ed as well. Ha ing ob ained a PDE ep esen a ion o he p ice o he claim and i s hedge pa ame e s, in discoun ed e ms, Sec ion 6 does he same in non-discoun ed e ms by employing a simple ans o ma ion o a iables. Rema kably, he PDE ha eme ges, o bo h local isk minimiza ion and mean- a iance op imiza ion, is he amilia Black–Scholes PDE, wi h a “di idend yield” pa ame e playing a isk-adjus men ole. Consequen ly, bo h app oaches yield classical closed- o m de i a i e p icing o mulae o Eu opean calls and pu s (which is ad an ageous om a compu a ional poin o iew). The J. Risk Financial Manag. 2015,885 hedge a ios o he wo quad a ic c i e ia a e di e en , which e lec s he di e en a i udes o isk hey imply. These esul s a e summa ized in P oposi ion 3. In Sec ion 7 we b ie ly in oduce he u ili y maximiza ion app oach o ou p oblem ha was p oposed by Monoyios [5,6]. In he limi ing case whe e isk is minimized, we obse e ha his hedging algo i hm becomes he local isk-minimizing s a egy. A disad an age o he quad a ic hedging ules, as well as he app oach o Monoyios, is ha hey ely explici ly on d i es ima es o he asse p ice p ocesses. In Sec ion 8 we make an ex a assump ion, based on equilib ium unde he capi al asse p icing model (CAPM), which allows he de i a ion o a “nai e” app oxima ion o he local isk-minimizing s a egy. The ad an age o his nai e s a egy is ha i does no equi e knowledge o he d i pa ame e s. In Sec ion 9 we demons a e he e ec i eness o he quad a ic hedging app oaches nume ically and compa e he esul s wi h hose ob ained using he Monoyios scheme and he nai e s a egy. Finally, Sec ion 10 concludes he pape . 2. Gene al Theo y This sec ion b ie ly in oduces he e minology and heo y equi ed o he emainde o he pape . We shall me ely summa ize he necessa y esul s; he eade is di ec ed o he li e a u e, p ima ily he accoun o Schweize [11], o u he in o ma ion and p oo s. We s a by ixing a ini e ime-ho izon T∈(0,∞)and a il e ed p obabili y space (Ω,F,F,P). All p ocesses a e de ined on his space, exis o e he ime in e al [0, T], and a e adap ed o he il a ion F= (F ) ∈[0,T ], which in u n is assumed o obey he usual condi ions. Fo he sake o simplici y, we ake F0 o be i ial, and se F:= FT. We conside a ic ionless inancial ma ke , wi h a single isky secu i y and a bank accoun , deno ed by Sand B espec i ely. The p ocess Xwill ep esen he discoun ed isky secu i y, i.e.,X:= S/B. Fo he momen , we lea e he dynamics o he p ocesses unspeci ied, excep o say ha he bank accoun is a p edic able p ocess wi h ini e a ia ion, and ha Xis a special semima ingale wi h canonical decomposi ion X=X0+M+A, whe e M∈ M2 0,loc(P)and Ais a p ocess wi h ini e a ia ion. We say ha Xsa is ies he s uc u e condi ion i he e exis s a p edic able p ocess α, such ha A=Rα dhMi and he mean- a iance adeo p ocess b K:= Rα2dhMiis a.s. ini e. In addi ion, we also in oduce a con ingen claim H, which we ake o be an FT-measu able squa e-in eg able andom a iable. Le Θdeno e he amily o p edic able p ocesses φ, such ha he gain p ocess G(φ) := Rφ dX belongs o he space S2(P)o squa e-in eg able semima ingales. A hedging s a egy is a pai o p ocesses (ξ, η), whe e ξ∈Θand ηis an adap ed p ocess, such ha he alue p ocess V(ξ, η) := ξX +η is igh con inuous and squa e-in eg able. The p ocess ξ ep esen s a holding in X, while η ep esen s a holding in he bank accoun . Since we a e dealing wi h an incomple e ma ke , he cos o a con ingen claim is no unique, and is consequen ly p e e ence-dependen . In o de o quan i y he isk o impe ec hedging, a cos p ocess C(ξ, η) := V(ξ, η)−G(ξ)is in oduced. In a comple e ma ke Cis de e minis ic and equal o he p e e ence-independen p ice o he claim— his ollows di ec ly om he ma ingale ep esen a ion esul s o Ha ison and Pliska [13,14]. In an incomple e ma ke Cis a s ochas ic p ocess. Ou aim is o J. Risk Financial Manag. 2015,886 p ice he claim by es ima ing CTa incep ion, and o minimize he isk (i.e., he de ia ion o he hedge po olio om he e minal payo ) by minimizing a sui able quad a ic unc ional o he cos p ocess. We b ie ly ou line he wo app oaches o local isk minimiza ion and mean- a iance hedging. 2.1. Local Risk Minimiza ion Wi h his app oach we conside hose s a egies ha eplica e he con ingen claim Ha ime T; i.e., we insis on he condi ion VT(ξ, η) = Ha.s. (1) Since he ma ke is incomple e, we need o elax he usual comple e ma ke cons ain ha he alue p ocess be sel - inancing. As i happens, he weake no ion o a mean-sel - inancing s a egy—which co esponds o he si ua ion whe e he cos p ocess is a ma ingale—is app op ia e in his con ex . Local isk minimiza ion is a a ia ional concep . In ui i ely, i en ails he ins an aneous minimiza ion o he condi ional a iance o he inc emen s o he cos unc ion Cunde he measu e P. This is implemen ed wi h he in oduc ion o a isk-quo ien , which we do no conside he e— o de ails we e e he eade o he o iginal e e ences (Schweize [11,15,16]). Subjec o ce ain echnical condi ions, i can be shown ha inding he local isk-minimizing s a egy is equi alen o inding a decomposi ion o he claim, known in he li e a u e as he Föllme –Schweize decomposi ion. A claim His said o admi a Föllme -Schweize (FS)decomposi ion i i can be exp essed as H=H0+ZT 0 ξH sdXs+LH Ta.s.,(2) whe e H0∈R,ξH∈Θand LH∈ M2 0(P)is s ongly o hogonal o M. In o de o p o ide a p ecise s a emen o he local isk minimiza ion op imali y esul , we need o in oduce he so-called minimal ma ingale measu e. To simpli y ma e s, we shall assume ha X, and hence also Mand A, a e con inuous. Now, de ine a p ocess b Z, by se ing b Z := E−Z· 0 αsdMs , o all ∈[0, T]. I can be shown (see Schweize [11]) ha b Z∈ M2 0,loc(P), and ha b ZX and b ZL a e P-local ma ingales, o all L∈ M2 0,loc(P)s ongly o hogonal o M. No e ha con inui y o Mand he assump ion o a.s. ini e b Kensu e ha he Doléans exponen ial abo e is s ic ly posi i e. Now suppose, u he mo e, ha b Z∈ M2(P), and de ine a p obabili y measu e b P∼P, by se ing db P dP:= b ZT∈L2(P). Then b Zmay be in e p e ed as he densi y p ocess o b P, in he sense ha db P/dP|F =b Z , o all ∈[0, T]. The p obabili y measu e b P, de ined abo e, is an equi alen local ma ingale measu e (ELMM) o X, and is called he minimal ma ingale measu e. I is minimal in he sense ha , apa om ans o ming Xin o a local ma ingale, i p ese es he emaining s uc u e o he model—in pa icula i p ese es he ma ingale p ope y o all ma ingales s ongly o hogonal o M(see Föllme and Schweize [17] o an ampli ica ion o his poin ). We a e now able o s a e he op imali y esul . J. Risk Financial Manag. 2015,887 Theo em 1. Suppose ha Xis con inuous, and he e o e sa is ies he s uc u e condi ion (see Schweize [11, p. 553] and [18, Theo em 1] o jus i ica ion). Fu he mo e, suppose ha he densi y p ocess o he minimal ma ingale measu e o Xsa is ies b Z∈ M2(P).(3) I Hadmi s an FS decomposi ion (2), hen (b ξ, bη) := (ξH,b V−ξHX)de e mines a (mean-sel - inancing) locally isk-minimizing s a egy o H, whe e he in insic alue p ocess is de ined by se ing b V := E b P[H|F ] = H0+G (ξH) + LH , o all ∈[0, T]. He e Gis he gain p ocess and LHmay be in e p e ed as he unhedged isk. Fu he mo e, a su icien condi ion o (2)and (3)is ha he mean- a iance adeo p ocess, b K, is uni o mly bounded. P oo . See Theo em 3.5 o Schweize [11]. 2.2. Mean-Va iance Hedging In con as o local isk minimiza ion, we now insis ha he hedge po olio be sel - inancing o e he li e o he op ion [0, T). A i s ma u i y, howe e , a p o i o sho all is ealized, so ha condi ion (1) is me . The mean- a iance op imal s a egy is cha ac e ized as ha s a egy o which he p o i o loss a ime Thas he smalles a iance. Mo e p ecisely, he mean- a iance op imal s a egy is he sel - inancing s a egy (e ξ, eη), wi h e ξ=ξ( )and V0(e ξ, eη) = , such ha E(CT− )2=EhH− −GTξ( )2i is minimized o e all  , ξ( )∈R×Θ. The ini ial alue is known as he app oxima ion p ice o H. Rela ed o he p oblem o inding he mean- a iance op imal s a egy is he p oblem o inding he a iance-op imal ma ingale measu e o X. As be o e, we assume ha Xis con inuous and conside he se P2 e(X)o all measu es Q∼P, whe e Qis an ELMM o X, wi h dQ dP∈L2(P). (In he case whe e X is no con inuous, a mo e gene al se o signed ma ingale measu es mus be conside ed—see Sec ion 4 o Schweize [11] o de ails.) Then a measu e in P2 e(X)is called a iance-op imal i i minimizes Va dQ dP=E"dQ dP−12#=E"dQ dP2#−1, o e all Q∈P2 e(X). In gene al, he minimal ma ingale measu e and he a iance op imal ma ingale measu e a e di e en , in which case signi ican e o is equi ed o ind he mean- a iance op imal s a egy (see, e.g., Hea h e al. [19]). Unde ce ain ci cums ances, howe e , he measu es coincide— he ollowing heo em p o ides such an ins ance, and he esul an o m o he mean- a iance op imal s a egy. Theo em 2. Suppose Xis con inuous, and ha b KTis de e minis ic ( hus ensu ing ha he FS decomposi ion (2)exis s). Then he a iance-op imal ma ingale measu e and he minimal ma ingale J. Risk Financial Manag. 2015,888 measu e coincide. Fu he mo e, he mean- a iance op imal s a egy o His he sel - inancing s a egy (e ξ, eη), wi h e ξ=ξ( ), whe e =E b P[H] = H0 and ξ( ) =ξH +α b V − −G ξ( ), o all ∈[0, T]. He e, b Vis he in insic alue p ocess (Theo em 1) and Gis he gain p ocess. I hen easily ollows om he sel - inancing p ope y ha eη = +G ξ( )−ξ( ) X , o all ∈[0, T]. P oo . See Theo ems 4.6 and 4.7 o Schweize [11]. Wi h he ma hema ical equisi es es ablished, we a e now in a posi ion o in oduce he ma ke assump ions and apply he heo y o he p oblem o hedging basis isk. 3. Ma ke Assump ions As in he p e ious sec ion, we ix a ini e ime-ho izon T∈(0,∞)and a s ochas ic basis (Ω,F,F,P), which suppo s wo o hogonal B ownian mo ions Wand W⊥. All p ocesses a e de ined on he abo e s ochas ic basis (in pa icula , hey exis o e he ime in e al [0, T]), and a e adap ed o he il a ion F= (F ) ∈[0,T ], which we ake o be he augmen a ion o he il a ion gene a ed by Wand W⊥. The e o e i ollows ha Fsa is ies he usual condi ions. We speci y a bank accoun p ocess Bas ollows: B := e , o all ∈[0, T], whe e > 0is a cons an sho a e. The p ocess S ep esen s a aded isky asse , while Uis an obse able co ela ed p ocess on which an op ion is w i en. The op ion is Eu opean, wi h ma u i y Tand payo h(UT), o some Bo el-measu able unc ion h:R+→R+. The objec i e is o hedge his claim using he aded asse S, in such a way ha he basis isk is minimized. Since he analysis is ca ied ou using discoun ed asse s, we in oduce wo new p ocesses—X ep esen ing he discoun ed aded asse , and Y ep esen ing he discoun ed non- aded asse —by se ing X:= S Band Y:= U B. Fu he mo e, we assume ha he discoun ed asse s a e d i en by he B ownian mo ions Wand W⊥, as ollows: dX = (µS− )X d +σSX dW ,(4) dY = (µU− )Y d +σUY (ρ dW +p1−ρ2dW⊥ ),(5) J. Risk Financial Manag. 2015,889 o all ∈[0, T], whe e he d i s µU, µS, he ola ili ies σU, σS>0and co ela ion −1≤ρ≤1a e cons an s. We wish o hedge he discoun ed Eu opean claim ¯ h(YT), whe e ¯ h:R+→R+is de ined by ¯ h(x) := e− T h(e T x), o all x∈R+, using he discoun ed aded asse X. Fo con enience, we de ine he Sha pe a ios o he aded asse and he non- aded p ocess as ollows: θS:= µS− σS and θU:= µU− σU . In he case whe e he asse s a e pe ec ly co ela ed (i.e.,ρ= 1), i is well known (see, e.g., Da is [1]) ha he absence o a bi age implies ha hei Sha pe a ios should be equal (i.e.,θU=θS). Unde his condi ion, he (non-discoun ed) p ice o a Eu opean call o pu on he non- aded asse is gi en by BS( , U ,0, σU), whe e BS( , s, q, σ) := δse−q(T− )N(δd1)−Ke− (T− )N(δd2) is he s anda d Black–Scholes o mula, wi h d1:= ln(s/K)+( −q+σ2/2)(T− ) σ√T− and d2:= d1−√T− . He e, δ= 1 o a call and δ=−1 o a pu , while K > 0is he s ike p ice, and q∈Ris a di idend yield pa ame e . Usually he di idend yield applies o he s ock on which he op ion is p iced. No e ha we ha e no modeled a di idend yield in ou basis isk model. We will, howe e , equi e his mo e gene al Black–Scholes o mula la e . Hedging is hen achie ed by holding σUU σSS ∆BS( , U ,0, σU)(6) uni s o he aded asse Sa each ime ∈[0, T], whe e ∆BS( , s, q, σ) := δe−q( −T)N(δd1) is he usual Black–Scholes del a. I will be shown ha he quad a ic hedging app oaches a e consis en wi h his limi ing egime. 4. The Föllme –Schweize Decomposi ion We now de i e he FS decomposi ion o he basis isk model p esen ed in he p e ious sec ion. To s a wi h, no e ha Xsa is ies he s uc u e condi ion, since i s canonical decomposi ion akes he o m X =X0+M +Z 0 αsdhMis,(7) wi h M := Z 0 σSXsdWsand α := µS− σ2 SX ,(8) J. Risk Financial Manag. 2015,890 o all ∈[0, T]. We seek a decomposi ion o he discoun ed claim o he o m ¯ h(YT) =: H=H0+ZT 0 ξH sdXs+LH Ta.s.,(9) whe e H0∈R,ξH∈Θand LH∈ M2 0(P)is s ongly o hogonal o M. Now, by ea anging (4), we ge dW =dX σSX −θSd , o all ∈[0, T]. Subs i u ing his in o (5) yields dY Y = (µU− −ρσUθS)d +σUρ σSX dX +p1−ρ2dW⊥ ,(10) o all ∈[0, T]. We now speci y he d i -adjus ed p ocess e Yas he unique s ong solu ion o he backwa d s ochas ic di e en ial equa ion de Y =e Y dY Y +κ d , o all ∈[0, T], whe e κ:= σU(ρθS−θU),(11) and e YT=YTis he e minal condi ion ( o exis ence and uniqueness see §2 o El Ka oui e al. [20]). A simple calcula ion shows ha e Y =e−κ(T− )Y , which, when subs i u ed in o (10), yields de Y =σUe Y ρ σSX dX +p1−ρ2dW⊥ ,(12) o all ∈[0, T]. We now cons uc he minimal ma ingale measu e o X. By he de ini ion o he minimal ma ingale measu e and he canonical decomposi ion (7), he densi y p ocess o b Pis gi en by b Z =E−Z· 0 αsdMs =E(−θSW) , o all ∈[0, T]. Since Xis a ma ingale unde b P, we can de ine a new p ocess c Was dc W =dW +θSd , o all ∈[0, T]. Since c Wb Zis a ma ingale and c Wb P =c W = , o all ∈[0, T], Lé y’s cha ac e iza ion o B ownian mo ion (see, e.g., Sh e e [21], Thm. 4.6.4, p. 168) in o ms us ha c Wis a B ownian mo ion unde b P. Rew i ing (4) and (12) in e ms o c Wgi es dX =σSX dc W and de Y =σUe Y ρ dc W +p1−ρ2dW⊥ ,(13) J. Risk Financial Manag. 2015,897 Table 3. Summa y s a is ics o hedging e o o he pu op ion wi h pa ame e s gi en by Table 1and ρ= 0.95. S a egy Max Min Mean SD Median Nai e 23.96 −28.98 −0.0283 4.0133 0.2169 Local Risk 23.59 −29.11 0.0072 4.0075 0.3554 Mean a iance 26.08 −30.76 0.0064 3.9726 0.3249 Monoyios O(a4),γ= 0.001 23.60 −29.11 0.0063 4.0075 0.3521 Monoyios O(a5),γ= 0.001 23.60 −29.11 0.0063 4.0075 0.3521 Monoyios O(a4),γ= 0.01 23.70 −29.08 −0.0014 4.0079 0.3230 Monoyios O(a5),γ= 0.01 23.70 −29.08 −0.0014 4.0079 0.3230 Monoyios O(a4),γ= 0.124.71 −28.77 −0.0796 4.0453 0.0315 Monoyios O(a5),γ= 0.124.71 −28.77 −0.0796 4.0453 0.0316 −40 −30 −20 −10 0 10 20 30 0 2 4 6 8x 104 Nai e (ρ=0.85) F equency −25 −20 −15 −10 −5 0 5 10 15 20 0 2 4 6 8x 104 Nai e (ρ=0.95) −40 −30 −20 −10 0 10 20 30 0 2 4 6 8x 104 Local isk (ρ=0.85) F equency −25 −20 −15 −10 −5 0 5 10 15 20 0 2 4 6 8x 104 Local isk (ρ=0.95) −40 −30 −20 −10 0 10 20 30 0 2 4 6 8x 104 Mean− a iance (ρ=0.85) F equency −25 −20 −15 −10 −5 0 5 10 15 20 0 2 4 6 8x 104 Mean− a iance (ρ=0.95) −40 −30 −20 −10 0 10 20 30 0 2 4 6 8x 104 Monoyios (ρ=0.85, γ=0.01) Te minal hedging e o F equency −25 −20 −15 −10 −5 0 5 10 15 20 0 2 4 6 8x 104 Monoyios (ρ=0.95, γ=0.01) Te minal hedging e o Figu e 1. His og ams o he hedging e o s o he pu op ion, based on one million sample pa hs. The app oxima ion p ices we e 8.6564 and 8.8733, co esponding o co ela ion coe icien s o 0.85 and 0.95, espec i ely. J. Risk Financial Manag. 2015,898 The esul s a e encou aging, wi h he local isk-minimizing s a egy pe o ming a leas as well as he Monoyios algo i hm (i.e., highe mean/median p o i and lowe s anda d de ia ion). This is no su p ising, since o small alues o γ he u ili y o mula ion is close o he local isk-minimizing s a egy. The mean- a iance op imal s a egy pe o med sligh ly be e han he o he wo, wi h a s anda d de ia ion ha was 1%–2% lowe . This can be seen as an enhanced peak a ound he mean in he ele an his og ams. One sligh d awback o his me hod is ha i s la ges losses exceeded he la ges losses o he o he me hods. The nai e s a egy pe o med su p isingly well (ce ainly signi ican ly be e han he nai e s a egy in Monoyios’ simula ions). This is due o he ac ha , unde he choice o pa ame e s used, he CAPM equilib ium condi ion is no un easonable. Table 4shows he app oxima ion p ices o a ious alues o he co ela ion coe icien , o bo h pu and call op ions, based on he pa ame e s p esen ed in Table 1. I is in e es ing o no e ha he app oxima ion p ices o he pu a e lowe han he Black–Scholes p ices, while he con e se is ue o he call. One should no in e p e hese p ices as he p emiums cha ged o he op ions, since no all isk is hedged, due o incomple eness. I is he e o e necessa y o es ima e he s anda d de ia ion o he hedging e o , so ha he op ion w i e can cha ge an app op ia e isk p emium. Table 4. Pu and call op ion app oxima ion p ices o a ious alues o ρ. (Fo ρ= 1, hey a e Black–Scholes p ices.) ρPu Call −0.95 5.3127 23.7315 −0.75 5.6321 22.6965 −0.50 6.0493 21.4435 −0.25 6.4870 20.2358 0 6.9451 19.0730 0.25 7.4238 17.9549 0.50 7.9231 16.8812 0.75 8.4428 15.8514 0.95 8.8733 15.0588 1 9.3542 14.2312 Figu e 2shows he app oxima ion p ices and s anda d de ia ions o hedging e o s o he pu , using he nai e, local isk-minimizing and mean- a iance op imal hedging s a egies. Figu e 3shows he same esul s o he call. The s anda d de ia ions we e es ima ed based on Mon e Ca lo samples o 100,000 pa hs. We see om hese g aphs ha hedging basis isk is only iable when he asse s a e highly co ela ed (in absolu e alue), since he e o in hedging inc eases apidly as he co ela ion be ween he asse s dec eases. I is in e es ing o no e ha he nai e s a egy pe o ms e y well o co ela ions close o uni y. As he co ela ion coe icien educes and becomes nega i e, i becomes less and less e ec i e, howe e . This is due o he ac ha , unde he asse pa ame e s used, he CAPM equilib ium assump ion becomes less ealis ic as he co ela ion dec eases. J. Risk Financial Manag. 2015,899 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 App oxima ion p ice and SD o P&Ls s co ela ion coe icien Co ela ion coe icien App oxima ion P ice SD o local isk−minimiza ion SD o mean− a iance SD o nai e Figu e 2. App oxima ion p ice and s anda d de ia ion o hedging e o s. co ela ion, o he pu op ion wi h pa ame e s gi en by Table 1. 10. Conclusions In his pape we used quad a ic c i e ia o de i e simple hedging ules o minimizing basis isk. These ules a e conside ably simple han he hedging ules based on u ili y maximiza ion and pe o m sligh ly be e in e ms o minimizing isk. Thei simplici y is a consequence o he ac ha hey a e based on he Black–Scholes o mula. By con as , he u ili y maximiza ion app oach equi es he use o se ies expansions in o de o sol e he ele an PDEs, which we e o iginally de i ed using a “dis o ion” echnique. I is impo an o no e ha all hedging schemes a e only e ec i e when he aded and non- aded asse s a e highly co ela ed (in absolu e alue), hus con i ming he sobe ing conclusions o Da is [2]. As is usual in incomple e ma ke se ings, he solu ions depend on es ima ing he g ow h a es o he asse s—a ask ecognized o be e y di icul . In o de o add ess his issue, we ha e de i ed a nai e hedging s a egy ha does no depend on he d i pa ame e s. Howe e , his comes wi h an implici d awback— he pe o mance o he s a egy is dependen on how well a CAPM equilib ium condi ion is obeyed. In o de o es ablish i his assump ion is easonable, in he con ex o speci ic eal-wo ld applica ions, u he in es iga ion is equi ed. J. Risk Financial Manag. 2015,8100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 App oxima ion p ice and SD o P&Ls s co ela ion coe icien Co ela ion coe icien App oxima ion P ice SD o local isk−minimiza ion SD o mean− a iance SD o nai e Figu e 3. App oxima ion p ice and s anda d de ia ion o hedging e o s. co ela ion, o he call op ion wi h pa ame e s gi en by Table 1. Acknowledgmen s The au ho s would like o hank h ee anonymous e iewe s o his pape o hei help ul commen s and e isions. Au ho Con ibu ions T.M. concei ed he p oblem, pe o med he analysis and conduc ed compu a ional expe imen s; H.H. and T.M. w o e he pape . Con lic s o In e es The au ho s decla e no con lic s o in e es . J. Risk Financial Manag. 2015,8101 Re e ences 1. Da is, M.H.A. Op ion Valua ion and Hedging wi h Basis Risk. In Sys em Theo y: Modeling, Analysis and Con ol; Dja e is, T.E., Schuck, I.C., Eds.; Kluwe : New Yo k, NY, USA, 1999; pp. 245–254. 2. Da is, M.H.A. Op ion Hedging wi h Basis Risk. In F om S ochas ic Calculus o Ma hema ical Finance; Kabano , Y., Lip se , R., S oyano , J., Eds.; Sp inge : Be lin, Ge many, 2006; pp. 169–187. 3. Hende son, V. Valua ion o Claims on Non aded Asse s Using U ili y Maximiza ion. Ma h. Financ. 2002,12, 351–373. 4. Hende son, V.; Hobson, D.G. Subs i u e Hedging. Risk 2002,15, 71–75. 5. Monoyios, M. Pe o mance o U ili y-Based S a egies o Hedging Basis Risk. Quan . Financ. 2004,4, 245–255. 6. Monoyios, M. Op imal hedging and pa ame e unce ain y. IMA J. Manag. Ma h. 2007,18, 331–351. 7. Za iphopoulou, T. A solu ion app oach o alua ion wi h unhedgeable isks. Financ. S och. 2001,5, 61–82. 8. Schweize , M. Mean-Va iance Hedging o Gene al Claims. Ann. Appl. P obab. 1992,2, 171–179. 9. Du ie, D.; Richa dson, H.R. Mean- a iance Hedging in Con inuous Time. Ann. Appl. P obab. 1991,1, 1–15. 10. Pham, H. On Quad a ic Hedging in Con inuous Time. Ma h. Me hods Ope . Res. 2000,51, 315–339. 11. Schweize , M. A Guided Tou h ough Quad a ic Hedging App oaches. In Op ion P icing, In e es Ra es and Risk Managemen ; Jouini, E., C i ani´ c, J., Musiela, M., Eds.; Camb idge Uni e si y P ess: Camb idge, UK, 2001; Chap e 15, pp. 538–574. 12. McWal e , T.A. Quad a ic C i e ia o Op imal Ma ingale Measu es in Incomple e Ma ke s. M.Sc. Disse a ion, Uni e si y o he Wi wa e s and, Wi wa e s and, Sou h A ica, 2006. h p://hdl.handle.ne /10539/2084 (accessed on 29 Janua y 2015). 13. Ha ison, J.M.; Pliska, S.R. Ma ingales and S ochas ic In eg als in he Theo y o Con inuous T ading. S och. P ocess. Thei Appl. 1981,11, 215–260. 14. Ha ison, J.M.; Pliska, S.R. A S ochas ic Calculus Model o Con inuous T ading: Comple e Ma ke s. S och. P ocess. Thei Appl. 1983,15, 313–316. 15. Schweize , M. Risk-minimali y and o hogonali y o ma ingales. S och. S och. Rep. 1990,30, 123–131. 16. Schweize , M. Op ion hedging o Semima ingales. S och. P ocess. Thei Appl. 1991,37, 339–363. 17. Föllme , H.; Schweize , M. Hedging o Con ingen Claims Unde Incomple e In o ma ion. In Applied S ochas ic Analysis; Da is, M.H.A., Ellio , R.J., Eds.; Go den and B each Science Publishe s: New Yo k, NY, USA, 1991; pp. 389–414. 18. Schweize , M. On he minimal ma ingale measu e and he Föllme -Schweize decomposi ion. S och. Anal. Appl. 1995,13, 573–599. J. Risk Financial Manag. 2015,8102 19. Hea h, D.; Pla en, E.; Schweize , M. A Compa ison o Two Quad a ic App oaches o Hedging in Incomple e Ma ke s. Ma h. Financ. 2001,11, 385–413. 20. El Ka oui, N.; Peng, S.; Quenez, M.C. Backwa d S ochas ic Di e en ial Equa ions in Finance. Ma h. Financ. 1997,7, 1–71. 21. Sh e e, S.E. S ochas ic Calculus o Finance II: Con inuous-Time Models; Sp inge -Ve lag: New Yo k, NY, USA, 2004. 22. Roge s, L.C.G. The elaxed in es o and pa ame e unce ain y. Financ. S och. 2001,5, 131–154. 23. Monoyios, M. U ili y-based alua ion and hedging o basis isk wi h pa ial in o ma ion. Appl. Ma h. Financ. 2010,17, 519–551. 24. Luenbe ge , D.G. In es men Science; Ox o d Uni e si y P ess: New Yo k, NY, USA, 1998. c 2015 by he au ho s; licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion license (h p://c ea i ecommons.o g/licenses/by/4.0/).