Market Application of the Fuzzy-Stochastic Approach in the Heston Option Pricing Model *

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1 JEL Classfcaon: D8, C3, G3 Keywords: fuzzy numbers, sochasc volaly, rsk-neural measure, opon prcng Marke Applcaon of he Fuzzy-Sochasc Approach n he Heson Opon Prcng Model * Ganna FIGÀ-TALAMANCA Deparmen of Economcs, Fnance and Sascs, Unversy of Peruga, Ialy Mara Leza GUERRA Deparmen Maemaes, Unversy of Bologna, Ialy correspondng auhor (mleza.guerra@unbo.) Lucano STEFANINI Deparmen of Economcs, Socey and Polcs, Unversy of Urbno, Ialy Absrac The presen sudy analyzes he exra nsghs ha opon prcng models may acheve when uncerany abou parameers s modeled hrough fuzzy numbers. Specfcally, we consder he Heson sochasc volaly model, whch assumes ha sock prce changes and her nsananeous varance evolve as a bvarae, possbly correlaed, dffusve process. The orgnal Heson model provdes a quas-closed formula for he prcng of several dervave producs such as European opons. By applyng he fuzzy exenson prncple, we generalze he model o he case of fuzzy parameers; gven her shape we are able o derve he membershp of he fuzzy prce of a European opon. Fnally, o undersand he exen o whch our approach mgh be useful n pracce, we gve a numercal llusraon of our procedure on he S&P 500 and VIX ndexes. As a by-produc of our research, a smple esmaon mehod s nroduced o oban (crsp) parameers n he Heson model under he rsk-neural measure and appled n he sequel of he paper o oban alernave shapes for he fuzzy parameers of he model.. Inroducon In recen research several sudes have been developed n order o handle, n a proper way, he nrnsc uncerany ever presen n fnancal and economc models. Many auhors, such as Hrynewcz (00) and Zadeh (008), have argued ha he mahemacal heory of fuzzy numbers s he correc descrpon of vagueness and mprecson; n Zadeh (008) s also underlned ha fuzzness exss n many felds, especally n human scences such as economcs, and he applcaon of fuzzy mahemacs can provde rgorous resuls. The orgn of he mahemacal heory of fuzzy numbers s essenally due o Zadeh (Zadeh, 965), bu many resuls n hs feld were acheved by Dubos and Prade (980, 993). Mos of he fnancal models n dervave prcng heory descrbe marke uncerany hrough he sochasc evoluon of he prce of he underlyng asses, where some consan parameers are assgned. We are confden ha exra value may be added o hese sochasc models by only consderng he uncerany of he evoluon and he vagueness of he parameers nvolved, whch may be modeled hrough fuzzy numbers. By assumng ha parameers are fuzzy, s possble o reflec n he shape of her membershp funcon boh obecve feaures and personal belefs abou he behavor of he parameers hemselves. * Ths research was parally suppored by Naonal Proec PRIN (008JNWWBP_004): Models and Fuzzy Calculus for Economc and Fnancal Decsons, fnanced by he Ialan Mnsry of Unversy. 6 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

2 We follow he sream of research nroduced by Zmeškal, who frs nroduced he fuzzy-sochasc approach and so-called hybrd models (see, for nsance, Zmeškal, 00, 00). The frs approach o he fnancal applcaons of such models s gven n Guerra, Sorn, and Sefann (0), where he advanages of he fuzzy-sochasc approach are nvesgaed n he Black and Scholes envronmen. In he quoed paper, he key parameers of he model are he volaly, he rsk-free neres rae, and he prce of he underlyng asse. I s observed by he auhors ha he fuzzy prce for he call opon seems no o be heavly nfluenced by he shape of he fuzzy volaly para-meer. Marke praconers, however, clam ha volaly behavor s crucal n he marke; a cenral ssue s he unsasfacory hypohess of consan volaly hrough me and s nconssency wh sylzed facs observed n fnancal daa. In order o generalze he Black and Scholes consan volaly assumpon, a vas leraure s devoed o volaly modelng and volaly forecasng. The movaon for hs paper s based on our convcon ha, gven he relevance of he volaly parameer/varable n he marke, he possble use of fuzzy heory n volaly modelng deserves deeper nvesgaon. Volaly models usually assume ha volaly s self a sochasc process; among ohers, he Heson model (Heson, 993) s grealy apprecaed due o he avalably of a closed formula for he prce of European opons and oher dervaves. In hs paper we generalze he Heson seng by assumng ha he parameers are modeled as fuzzy numbers and we show how her vagueness affecs he fnal opon prce, whch s also obaned as a fuzzy number. The analyss s performed for several choces of he shapes of he membershp funcons of he parameers. In Fgà- Talamanca and Guerra (009) and Fgà-Talamanca, Guerra, Sorn, and Sefann (00), we presen prelmnary sudes on he same argumen. Sochasc volaly modelng n a fuzzy scenaro has been prevously addressed n he leraure. For nsance, n Thavaneswaran, Thagaraah, and Appadoo (007), he dea of fuzzy parameers s nroduced n a dscree sochasc seng and fuzzy numbers are used o ncorporae volaly varably. In Hung (009) and Thavaneswaran, Appadoo, and Paseka (009), Generalzed Auoregresson Condonal Heeroskedascy (GARCH) dscree models are analyzed n a fuzzy conex. In he former conrbuon he auhor modfes he hreshold values for aposve/ /negave nformaon dsncon wh a fuzzy rule and many emprcal nvesgaons are repored n order o valdae he mehod. In he laer one, he auhors sudy he cenered momens and kuross for aclass of FCA (Fuzzy Coeffcen Auoregressve) and FCV (Fuzzy Coeffcen Volaly) models. Fnally, n Swshchuk, Ware, and L (008), he auhors also nvesgae he Heson model and oban a fuzzy opon prce n he same conex as ours; however, hey assume ha he nsananeous (local) volaly s self a fuzzy number and derve s membershp by ransformng he probably dsrbuon of he nsananeous varance process o s possbly dsrbuon hrough he mehod descrbed n Dubos, Prade, and Sandr (993). The non-lnear fuzzy PDE s hen used o prce European opons. On he conrary, we rean he orgnal dynamcs specfcaon for he nsananeous varance and assgn a membershp funcon o he model parameers; Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 63

3 he fuzzy prce for he opon s hen obaned by applyng he fuzzy exenson prncple. Noe ha fuzzy opon prces may prove useful n real markes wh frcons; n fac, he range of he suppor (or a dfferen α-cu) of he fuzzy opon prce may be consdered as a measure of he bd-ask spread. Anoher conrbuon of hs work s he dea of applyng a rollng esmaon mehod for crsp model parameers n order o desgn fuzzy parameer shapes whch are conssen wh emprcal marke observaons. We pon ou ha esmaon of he Heson model s sll a subec of ongong research and has smulaed a debae on wheher o use sock or dervave marke daa. So, as a by-produc of our research, we also nroduce a smple mehod o derve he Heson model parameers, based on boh sock and dervaves nformaon. More precsely, he suggesed procedure s based on on observaons of he sock ndex and of s volaly ndex (he value of whch depends on he prce of raded opons on he ndex self). In our numercal llusraon he S&P 500 sock ndex and he VIX volaly ndex are consdered. Whle we apply he rollng approach o he Heson model, and consderng our esmaon mehod, s worh nong ha he same dea may be appled o any model as well as for any esmaon mehod. The paper s dvded no fve secons. Afer he nroducon, n Secon we gve a bref descrpon of he Heson model and we nroduce he basc elemens of he heory of fuzzy numbers. The hrd secon s devoed o he nroducon of our smple esmaon procedure for Heson (crsp) parameers and o descrbe how s used for he emprcal consrucon of several possble suppors and shapes for he fuzzy parameers. In Secon 4 a numercal expermen s oulned n order o analyze how fuzzy opon prces are obaned n our seng. A fnal secon, devoed o concludng remarks, underlnes he relevance of he resuls obaned and races some pahs for possble fuure research.. The Heson Model The Heson model (Heson, 993) s a benchmark among sochasc volaly models due o he avalably of a closed formula for he prce of several dervave secures. In parcular, he prce a me of a European call opon wh maury T and srke prce X s gven by he followng formula: rt ( ) C ( T, X) S Xe () where r s he marke rsk-free rae, assumed o reman consan unl maury. The quanes and represen he probably ha he opon wll be exercsed a maury wh respec o dfferen probably measures and are obaned hrough nverson of a Fourer ransform. More precsely, for =, we have wh uln X e F ln S, v, T ; u Re du 0 u F ( x, vt, u ; ) exp A ( T u, ) B ( T v ) ux 64 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

4 where Defnng T, he exac expressons of he funcons A and a ge A(, u) ru b cu d ln c g d b cu d e B (, u) c ge d b cu d g g( u) b cu d d B are and d d ( u) cu b c u b b c a Hence, we have C ( T, X) C, X, S, r,,,, c, () where,,c are he parameers appearng n he on dynamcs of he prce (a me ) of he underlyng sock and s local varance V. Such dynamcs are specfed, under he so-called rsk-neural probably measure, by he followng bdmensonal sochasc equaon: d S VdB log dv V d c VdW where (B,W) s a possbly correlaed Brownan moon wh db, dw Heson, 993, for furher deals). The nsananeous varance by a mean-reverng process where parameer V d S (see s hus modeled represens he long-run mean varance, s he speed of reurn o he long-run mean, and c s he so-called volaly of volaly parameer.. Fuzzy Numbers and he Exenson Prncple In order o descrbe he mehodology appled, we recall some prelmnares abou fuzzy numbers. A fuzzy number core u a, aa, L,R s usually specfed by s a R and a membershp funcon :R [0,], wh suppor n [ a, a ] Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 65

5 defned as L( x) f a x a ( x) R( x) f a x a for x R 0 oherwse where L(x) s an ncreasng funcon wh L( a ) 0, L( a) and R( x) s a decreasng funcon wh R( a), R( a ) 0. Funcons L(.) and R(.) are he lef and rgh shape funcons of u, and hey are assumed o be dfferenable. For values of [0,], he α-cus are defned o be he compac nervals [ u] { x ( x) }, whch are nesed closed nervals. For our purposes s more convenen o specfy fuzzy parameers wh he Lower-Upper (LU) represenaon nroduced n Sefann, Sorn, and Guerra (006); we brefly recall ha an LU-fuzzy number u s deermned by any par u u, u of funcons u, u : 0, R, defnng he end-pons of he α-cus, sasfyng some condons, and possessng non-empy and compac α-cus of he form [ u] u, u R. In he LU represenaon he suppor of u s he nerval u0, u 0 he core s u, u, and he lower and upper branches u (.) and u (.) are connuous nverble funcons defnng he membershp funcon u (.) as wo connuous branches, he lef beng he ncreasng nverse of u (.) on u0, u and he rgh he decreasng nverse of u (.) on u, u 0. The wo monoonc branches u and u are paramerzed over a decomposon of he nerval 0, no N sub-nervals, for =,,,N. For each decomposon, 4(N+) parameers are requred: ;,,, 0,,..., u u u u u sasfyng he followng condons: u0 u... un un un... u0 (daa) u 0, u 0 (slopes) The smples represenaon s obaned on he rval decomposon of he nerval [0,], wh N = (whou nernal pons) and 0 0,. In hs smple case, u can be represened by a vecor of egh componens u ( u, u, u, u ; u, u, u, u ) where u 0, u 0, u, u are used for he lower branch u, and u 0, u 0, u, u he upper branch u. N for 66 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

6 When managng funcons of real varables, he fuzzy exenson has o be he resul of correc applcaon of he exenson prncple. Gven a funcon y f x, x,..., xn of n real varables x, x,..., x n, s fuzzy exenson s obaned o evaluae heeffec of uncerany on he modeled by he correspondng fuzzy number values of u,.e., for each level by he nerval u,, u, x x for ha level. If v f u u u,,..., n gvng he possble denoes he fuzzy exenson of a con- nuous funcon f n n varables, for each level he resulng nerval v, v represens he propagaon of uncerany from he varables o he varable y. In parcular, f he uncerany on he orgnal varables s also modeled by lnear fuzzy numbers, he obaned v s sll a fuzzy number sarng from a sngle value (a level ) o he mos unceran nerval (a level 0 ), bu loses he lneary propery. I follows ha paramerc represenaon s also necessary when he npu varables are rangular fuzzy numbers n order o apply he exenson prncple and o represen he non-lnear oupu fuzzy numbers. In pracce, o oban he fuzzy exenson of f o normal upper sem-connuous fuzzy nervals, we have o compue he α-cus [ v, v ] of v, defned as he mages of he α-cus of ( u, u,..., un ) ha are hen obaned by solvng he followng boxconsraned opmzaon problems for [0,] : v mn f x, x,..., xn xk uk,, u k,, k,..., n ( EP) : v max f x, x,..., xn xk uk,, u k,, k,..., n Only n smple cases can he opmzaon problems above be solved analycally. In general, he soluon s dffcul and compuaonally expensve o fnd, as, for each [0,], he global soluons of wo non-lnear programmng problems are requred..3 The Fuzzy Heson Prcng Formula I s worh nong ha he opon value obaned hrough he above-menoned prcng formula () can only be compued when we have complee nformaon on he parameers. Ths s hardly he case when dealng wh real daa applcaons. Of course, he parameers are no known and should be esmaed; furhermore, as already noed n he nroducon, many procedures have been nroduced n he sascal and fnancal leraure for he esmaon of Heson model parameers and s a subec of debae wheher he esmaes should be obaned by usng pure sascal mehods based only on a me seres of he sock prce (consderng he nsananeous varance V as a laen varable) or calbraon mehods based essenally on marke opon prces (some hghlghs can be found n Fgà-Talamanca and Guerra, 006). In addon, some auhors have suggesed usng proxes for process V, such as he realzed varance (see, for nsance, Bollerslev and Zhou, 00). I s hard o esablsh he bes soluon, and even hough we were able o choose one of he mehods, sascal x Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 67

7 esmaon s mprecse by defnon, no o menon ha parameer esmaes change over me. In Benhamou, Gobe, and Mr (009), an effor s made o exend he Heson model o he case of me-dependen parameers, bu hs drascally reduces he analycal racably of he model. To cope wh he mprecson of he parameers, whch s he fnal am of hs analyss, reanng boh he parsmony and analycal racably characerzng he Heson model, we generalze he model by assumng he fundamenal parameers o be fuzzy numbers. More precsely, we se,,,,, L, R L,R c c c, cc,,,, L,R c and,, and we assgn he correspondng membershp funcons ( ), ( ), ( c ), ( ), ( ). As soon as nformaon on he parameers s modeled by fuzzy numbers, he value of he call opon n () also becomes fuzzy and may be represened by s membershp as well as by s α-cus C, C for all degrees of possbly. The maxmum uncerany corresponds o he suppor a 0. Ths membershp funcon and he correspondng α-cus are obaned by applyng he fuzzy exenson of funcon C ( T, X ) n () n a rgorous way. The shape of he membershp obaned for he opon prce serves as a measure of mprecson propagaon from he parameers o he call opon value. I s worh nong ha C ( T, X ) s hghly non-lnear n he parameers, makng he compuaon of he correspondng fuzzy-valued funcon a rcky sep. To reduce he compuaonal burden we mplemen he mulple populaon dfferenal evoluon (DE) algorhm, whch s desgned o compue he values and he slopes of he LU represenaon of he fuzzy exensons of (). 3. Dervaon of he Fuzzy Suppor for he Model Parameers To oban he fuzzy suppor of he model parameers and properly assgn he membershps of he parameers we proceed n wo seps. Frs, we nroduce a smple esmaon echnque for he Heson model parameer, under rsk-neural dynamcs; hen, we buld he fuzzy suppor by applyng he esmaon procedure on rollng (movng) wndows of he daa. Our procedure provdes esmaes for he rskneural parameers; s based on observaons for he sock ndex prce and for he volaly ndex value. The underlyng dea s o read he volaly ndex as a proxy for he negraed volaly under he rsk-neural probably. Ths laer approach s a sascal alernave o he calbraon of parameers obaned hrough he mnmzaon of he dsance beween heorecal and marke opon prces (see Con and Kokholm, 009, and Sepp, 008). We deal he procedure by consderng he S&P 500 sock marke ndex and s volaly ndex, he VIX. The VIX s a volaly ndex compued on he bass of prces for opons on he S&P 500 ndex (SPX opons). Denoe 68 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

8 K F (3) T T r ( T ), ; e Q K T T K T K 0 where T s he me o maury of all consdered opons, level a me (derved from ndex opon prces), K0 F s he forward ndex F, s he frs srke below s he srke prce of he -h ou-of-he money opon and K K K /, s he rsk free rae a me for abond wh maury T, and QK, T ; he mdpon of he bd-ask values for he opon wh srke K. he VIX a me s gven by he followng formula: VIX 365 T T NT 30 T 30 N T T NT N T N T N T K T r s Then, he value of.e., he VIX value s fnally obaned as a weghed mean of he oupu of formula (3) for values T and T correspondng o near and nex erm maury. The opons o be consdered n he calculaon of he VIX are ou-of-hemoney calls and pus wh non-zero bd prces, cenered around he a-he-money srke prce K 0. Full deals on he VIX compuaon are avalable n he VIX whe paper The CBOE Volaly Index, 009. As menoned above, we use he VIX volaly ndex as a proxy for he square roo of he expecaon (under he rsk-neural measure) of he annualzed negraed varance over a one-monh horzon. Specfcally, f we defne he negraed varance process I (, ) as we assume ha I (, ) Vds Q VIX E I 30 /00, 365 Hence, he value of he square VIX/00 ndex a me s assumed o be he sample observaon, under he rsk-neural measure, of he process I (, ) wh The heursc movaon for hs choce s gven n Sepp (008). A dealed analyss of hs assumpon s beyond he scope of our work; however, he neresed reader can fnd a heorecal usfcaon n Con and Kokholm (009). In addon, we ake advanage of he fac ha he momens of he negraed varance under he Heson dynamcs specfcaon can be derved as a funcon of he parameers. We know from he resuls n he leraure (see Fgà-Talamanca, 009, for nsance) ha I (, ) s a saonary process for a fxed value of, for whch s Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 69

9 (, ) EI(0, ) E I e ( h) Cov I (, ), I ( h, ) Var V0 e, h where expecaons are compued wh respec o he rsk-neural probably measure. Hence Cov I (, ), I (, ) Cov I (, ), I ( h, ) e ( h) Gven m observaons of he VIX ndex, namely, VIX, VIX,..., VIX m, and defnng ˆ VIX I 00 we hus sugges he followng esmaes, respecvely, for he long-run mean and mean reverson speed: ˆ* * ˆ m m Iˆ ˆ I, log h Iˆ, h where Iˆ, h Cov Iˆ,, Iˆ h, of I ˆ a lag h. Once from whch ˆ*, * ˆ are compued we can esmae Var V0 ^ e Iˆ, Var V ^ s he emprcal auocovarance funcon 0 * ˆ ˆ ˆ* * ˆ, ˆ* e Var V I 0 ( ) Snce, under he Heson model assumpons, Var V c, we may com- pue parameer ĉ as cˆ * ^ ˆ Var V ˆ* 0 0 by wrng Parameer ρ s smply derved as he sample correlaon beween he S&P 500 ndex excess reurns and he I ˆ me-seres: 70 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

10 where ˆ m, m, SP ˆ ˆ SPI I SP ˆ ˆ SP I I SP m ˆ I m m m Iˆ are he sample means of he S&P 500 excess reurns and of he proxy for he negraed varance, respecvely. 3. Choce of he Fuzzy Suppor The suppor of he fuzzy parameers may be consruced when a daase of n observaons s avalable by rollng he esmaon procedure on movng wndows of lengh m so ha k = n m esmaes are avalable for each parameer. In our numercal llusraon we are gven a daase for he S&P 500 ndex and s volaly ndex (he VIX Index) from January 990 o Ocober, 00 for a oal number of n = 5,44 daly observaons. The esmaon s repeaed k = 3,500 mes for (overlappng) movng wndows of lengh m =,744 of he VIX ndex; he esmaed values are repored n Fgure for each parameer of he model. Consderng he me span, he esmaes show good sably performance. Of course, he mos volale of he parameers s he correlaon, whch s he only esmae relyng on boh he S&P 500 reurn and he VIX seres. Gven hese oucomes we sugges several possble ways o oban he fuzzy suppor for each parameer; n all cases he crsp value s gven by he medan esmaed value. We consder lnear fuzzy numbers wh he suppor havng he exremes obaned accordng o a percenage varaon p of he crsp value. If we denoe by x he medan of a generc parameer, he suppor s he closed nerval [ x( p), xx, ( p)] and s, by consrucon, a cenered nerval on x. In Fgure he hsogram s repored for each of he four parameers. I s clear from he fgures ha he esmaed values are asymmerc wh respec o her medan value. In order o ake no accoun hs asymmery we use he emprcal dsrbuon of he parameers for an alernave defnon of he parameer suppor. We herefore defne he fuzzy suppor of parameer x as [ lx ( ), hx ( )], where lx ( ) s a low percenle (l%) and h(x) s a hgh percenle (h%) of he esmaed values for x. Typcal examples would be l = 0 and h = 90 or l = 5 and h = 75. As a naural generalzaon we also consder rapezodal suppors of he ype lx ( ), m( x), m( x), h( x) consderng percenles around he medan (m = 45, m = 55). Noe ha hs approach o he consrucon of marke daa conssen fuzzy shapes may be used n dfferen model sengs as well as for dfferen esmaon procedures. SP Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 7

11 Fgure S&P500 Index and VIX volaly: Parameers Esmaed Values from he Begnnng of 000 o he End 005 (from op lef, clockwse we have κ, θ, c, ρ) Fgure S&P500 Index and VIX volaly: Hsograms of Parameers Esmaed Values from he Begnnng of 000 o he End 005 (from op lef, clockwse we have κ, θ, c, ρ) 4. Numercal Illusraon We are gven a daase of he VIX ndex values from January 990 o Ocober, 00 for aoal number of n = 5,44 daly observaons. In order o derve he suppor of he fuzzy parameers he esmaon procedure descrbed above s repeaed k = 3,500 mes for (overlappng) movng wndows of lengh m =,744. From he 3,500 values obaned we derve he percenles repored n Table. 7 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

12 Table ˆ* * ˆ Mn h h h h h h h Max ĉ ˆ In our expermen, denong by x he me seres of esmaed d values for he parameer x and by q x, n s n-h percenle, he followng specfcaons for he suppor of each fuzzy parameer x are developed: x x x x0 x, 0.0, x0 x.80, provdng a symmerc rangular fuzzy number ( p 0.8 ); x x x, x0 x 0.60, x0 x.40, provdng a symmerc rangular fuzzy number ( p 0.4 ); x0 q x,0, x 0 q x,90, x x x, provdng an asymmerc rangular fuzzy number; x0 q x,5, x 0 q x,75, x x x, provdng an asymmerc rangular fuzzy number;, x0 q x,0, x 0 q x,90 provdng a rapezodal fuzzy number; x q x,45, x q x,55,, x0 q x,5, x 0 q x,75 provdng a rapezodal fuzzy number. x q x,45, x q x,55, In Fgure 3 we plo, as an example, he sx dfferen shapes for parameer accordng o he defnons gven for he fuzzy suppor n he above ls. I s worh nong ha he emprcal dsrbuon of s very skewed; whle he hgher exremes of he suppor are smlar for cases, 3, 5 and, 4, 6, he same s no rue for he lower exremes. Smlar plos, wh dfferen numerc values, can be derved for he oher parameers. 4. The Fuzzy Opon Prce and s Membershp Funcon Accordng o he seleced shape for he fuzzy parameers we compue he fuzzy prces for European opons by applyng he exenson prncple o he opon prcng formula as llusraed n Secon 3. We llusrae he resuls by consderng hree dfferen srkes, whch are aken as examples of ou-of-he-money (OTM), a-he-money (ATM), and n-he-money (ITM) opons, and for wo dfferen maures ( and Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 73

13 Fgure 3 Fuzzy Shapes of Parameer κ Accordng o Several Defnons of he Fuzzy Suppor Fgure 4 Fuzzy Prce for Opons on he S&P500 Index Traded on Ocober, h, 00: Symmerc versus Asymmerc Trangular Fuzzy Parameers 3 monhs). In Fgure 4, from lef o rgh, he membershp for fuzzy opon prces s repored for he examples of OTM, ATM, and ITM opons, respecvely; he op graphs refer o maury T = monh and he boom graphs o T = 3 monhs. In parcular, opon prces are compued sarng from symmerc rangular fuzzy parameers wh p 0.8 (dash-doed) and p 0.4 (doed) as well as for asymmerc rangular fuzzy parameers wh exremes fxed a he 0h and 90h percenles (dashed) and 5h and 75h percenles (sold). 74 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

14 Fgure 5 Fuzzy Prce for Opons on he S&P500 Index Traded on Ocober, h, 00: Trangular versus Trapezodal Fuzzy Parameers In Fgure 5, from lef o rgh, fuzzy opon prces are repored for he examples of OTM, ATM, and ITM opons, respecvely; he op graphs refer o maury T = monh and he boom graphs o T = 3 monhs. Opon prces are compued n hs case assumng asymmerc rangular fuzzy parameers wh exremes fxed a he 0h and 90h percenles (dashed) and 5h and 75h percenles (sold) as well as her rapezodal generalzaon consderng he 45h and he 55h percenles as exremes for he -cu of he parameers fuzzy shape (dashdoed and doed, respecvely). From Fgure 4 we undersand how he call prces vary wh he choce of symmerc or asymmerc rangular fuzzy parameer (Cases and agans Cases 3 and 4), whle Fgure 5 shows he varaon of he call prces wh he choce of rangular or rapezodal asymmerc fuzzy parameer (Cases 3 and 4 agans Cases 5 and 6). I s worh nong ha sarng from lnear or symmerc fuzzy parameers does no gve lnear or symmerc opon prces; hs s due, of course, o he hgh non-lneary of he Heson opon prcng formula wh respec o he underlyng model parameers. To ge a beer dea of how opon prces vary agans he srke prce, we plo n Fgures 6 o 8 he membershp (shape) funcons for he consdered call opon prces agans all he avalable srke prces (denoed by K) on he dae of neres (Ocober, 00), for maures T =,, 3 monhs, respecvely. The plos correspond o a choce of asymmerc rangular fuzzy parameers wh he 0h and 90h percenles as he exremes of her suppor (Case 3). I s clear from he fgures ha no only do prces ncrease wh he me o maury for each fxed srke, bu also ha he membershp α-cus for fxed levels of possbly, become larger and larger wh he me o maury of he opons. In addon, he α-cus ranges ncrease wh he srke prce and are more asym- Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 75

15 Fgure 6 The Fuzzy Smle for Opons wh Maury T = Monh Fuzzy Membershps for he Opon Prces Is Ploed Agans he Srkes Prces Traded on Ocober, h, 00 Fgure 7 The Fuzzy Smle for Opons wh Maury T = Monh Fuzzy Membershps for he Opon Prces Is Ploed Agans he Srkes Prces Traded on Ocober, h, 00 merc for lower srke prces. Smlar graphs can be obaned for he oher shapes of he fuzzy parameers, leadng o analogous commens. In all he above fgures he maxmum possbly value ( ) corresponds o he crsp opon prce under he Heson model wh crsp parameers. Noe ha n he dervave marke, raded opons are no quoed wh a sngle prce bu wh a bd prce and an ask prce. The dfference beween hese prces s called he bd-ask spread 76 Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

16 Fgure 8 The Fuzzy Smle for Opons wh Maury T = 3 Monh Fuzzy Membershps for he Opon Prces Is Ploed Agans he Srkes Prces Traded on Ocober, h, 00 and, n sandard opon prcng models, s consdered a frcon n he marke. In fac, a sandard assumpon n prcng models s ha he prce of a fnancal asse s unque; hs problem s usually overcome by consderng he mdpon of he bd-ask range as he unque prce. In our generalzed Heson framework, wh fuzzy parameers, we are gven a range for opon prces (for each α-cu wh ) whch may properly represen he bd-ask spread. 5. Concluson Our analyss s nended o cope only wh he uncerany arsng from he random evoluon of a sock prce and he vagueness of nformaon on he assgned parameers. Ths fuzzy-sochasc approach s developed here whn he framework of he Heson sochasc volaly model by formally represenng he vagueness of he parameers hrough he mahemacal heory of fuzzy numbers. The Heson model provdes a closed formula for heprce of European opons when he parameers are known. By assgnng dfferen membershp funcons o he fuzzy parameers we oban, by rgorous applcaon of he exenson prncple, he membershp funcon of he fuzzy opon value. I s worh nong ha, due o he hgh non-lneary of he Heson opon prcng formula wh respec o he parameers, lnear shapes for he membershp funcon of he npu parameers propagae o non-lnear shapes for fuzzy opon prces. Ths observaon movaes he paramerc represenaon of he fuzzy numbers nvolved. In he cenral par of he paper we sugges a consrucve mehod for deermnng he shape of he fuzzy parameers so ha her membershps are properly assgned o be conssen wh he emprcal observaons n he sock and dervaves markes. Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 77

17 Our analyss s developed hrough several expermens n whch a dfferen shape for he fuzzy parameers s assgned. We hen gve a numercal llusraon on a daase for he S&P 500 ndex and he VIX volaly ndex. The oucomes of our sudy seem encouragng and mgh prove useful n real markes, especally where ransacon coss, such as dfferen bd-ask prces, are ndeed allowed; n hs case here s no unque prce for a raded asse, bu nsead here s a bd prce and an ask prce, whose range s called he bd-ask spread. Under suable assumpons on he shape and he suppor of he fuzzy parameers, he exremes of he suppor (or of a specfc α-cu) of he fuzzy opon prce, obaned usng our approach, may be used o represen he bd-ask spread for he opon self. The nex sep for our research s o undersand he sascal heorecal properes of he on fuzzy-sochasc Heson model and possbly o apply our approach for rsk measuremen purposes. As a by-produc of hs sudy we nroduce a way o derve parameers n he Heson model by usng a momen-machng esmaon procedure based on on observaons of a sock marke ndex and s volaly ndex, whch served as a proxy for he negraed volaly. The proposed esmaon mehod, hough novel and very smple, s no he man focus of he paper. Of course, we wll nvesgae hs mehod furher n fuure research by dervng s heorecal properes and by comparng s performance wh ha of exsng procedures. REFERENCES Benhamou E, Gobe E, Mr M (009): Tme Dependen Heson Model. Avalable a SSRN: hp://ssrn.com/absrac= Bollerslev T, Zhou H (00): Esmang sochasc volaly dffuson usng condonal momens of negraed volaly. Journal of Economercs, 09: Con R, Kokholm T (009): A Conssen Prcng Model for Index Opons and Volaly Dervaves. Mahemacal Fnance, DOI: 0./ x. Dubos D, Prade H (980): Fuzzy Ses and Sysems: Theory and Applcaons. Academc Press, New York. Dubos D, Prade H, Sandr S (993): On Possbly/Probably Transformaons. In: Lowen R, Roubens M. (Eds.): Fuzzy Logc. Kluwer Academc Publshers, Dordrech, pp. 03. Dubos D, Prade H (Eds.) (000): Fundamenals of Fuzzy Ses, The Handbooks of Fuzzy Ses Seres. Kluwer, Boson. Fgà-Talamanca G (009): Tesng volaly auocorrelaon n he consan elascy of varance sochasc volaly model. Compuaonal Sascs and Daa Analyss, 30():0 8. Fgà-Talamanca G, Guerra ML (006): Fng prces wh a complee model. Journal of Bankng and Fnance, 30: Fgà-Talamanca G, Guerra ML(009): Fuzzy opon value wh sochasc volaly models. ISDA 009 9h Inernaonal Conference on Inellgen Sysems Desgn and Applcaons, ar. no , pp Fgà-Talamanca G, Guerra ML, Sorn L, Sefann L (00): Unceran parameers as fuzzy numbers n opon prcng models. Proceedngs of he 47h EWGFM meeng, Prague, pp Guerra ML, Sorn L, Sefann L (0): Opon prce sensves hrough fuzzy numbers. Compuers and Mahemacs wh Applcaons, 6: Heson SL (993): A Closed-Form Soluon for Opons wh Sochasc Volaly wh Applcaons o Bond and Currency Opons. Revew of Fnancal Sudes, 6(): Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no.

18 Hrynewcz O (00): On on modellng of random uncerany and fuzzy mprecson. In: Mar K e al. (Eds.): Copng wh Uncerany. Sprnger-Verlag, pp. 38. Hung JC (009): A Fuzzy Asymmerc GARCH model appled o sock markes. Informaon Scences, 79: Sepp A (008): Prcng Opons on Realzed Varance n he Heson Model wh Jumps n Reurns and Volaly. The Journal of Compuaonal Fnance, (4): Sefann L, Sorn L, Guerra ML (006): Paramerc Represenaons of Fuzzy Numbers and Applcaons o Fuzzy Calculus. Fuzzy Ses and Sysems, 57(8): Swshchuk A, Ware A, L H (008): Opon Prcng wh Sochasc Volaly Usng Fuzzy Ses Theory. hp://norhernfnance.org, 8. Thavaneswaran A, Thagaraah K, Appadoo SS (007): Fuzzy coeffcen volaly (FCV) models wh applcaons. Mahemacal and Compuer Modellng, 45: Thavaneswaran A, Appadoo SS, Paseka A (009): Weghed possblsc momens of fuzzy numbers wh applcaons o GARCH modelng and opon prcng. Informaon Scences, 79: The CBOE Volaly Index VIX, 009. Avalable a hp:// Zadeh LA (965): Fuzzy Ses. Informaon and Conrol, 8: Zadeh LA (008): Is here a need for fuzzy logc? Informaon Scences, 78: Zmeškal Z (00): Applcaon of he fuzzy-sochasc mehodology o apprasng he frm value as a European call opon. European Journal of Operaonal Research, 35: Zmeškal Z (00): Generalsed sof bnomal Amercan real opon prcng model (fuzzy-sochasc approach). European Journal of Operaonal Research, 07: Fnance a úvěr-czech Journal of Economcs and Fnance, 6, 0, no. 79