Hulley, Ha dy; McWal e , Thomas A.
A icle
Quad a ic hedging o basis isk
Jou nal o Risk and Financial Managemen
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Sugges ed Ci a ion: Hulley, Ha dy; McWal e , Thomas A. (2015) : Quad a ic hedging o basis isk,
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Risk and Financial
Managemen
ISSN 1911-8074
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A icle
Quad a ic Hedging o Basis Risk
Ha dy Hulley 1and Thomas A. 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
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