Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 10, 384-391, 016 doi:10.1311/001.39.10.45 Pricing Binary Opions Based on Fuzzy Number Theory Guojun Yuan chool of Economics and Managemen, Wes Anhui Universiy, Lu an 3701, Anhui, China, E-mail: ygj1010@16.com Absrac Opions pricing model parameers are inherenly imprecise due o flucuaions in he real-world financial marke. Tradiional opion pricing mehods do no accoun for he uncerainy in parameers, bu he fuzzy se heory may be applicable. This paper proposes a cash-or-nohing European call binary opion pricing model based on he hypohesis ha he underlying asse price, risk-free rae of ineres, and volailiy all are uncerain. We presen he fuzzy pricing model of he cash-or-nohing call binary opion under he fuzzy environmen. Two numerical examples presened in he paper illusrae he raionaliy and effeciveness of he fuzzy opion pricing model. Keywords: Binary Opion, Cash-Or-Nohing Opion, Fuzzy Number, Fuzzy Opion Pricing 1. INTRODUCTION An opion is a conrac which gives is holder he righ (bu no he obligaion o buy or sell an underlying asse a a predeermined price on a specified dae. Opions can be caegorized as call opions and pu opions. The call opion gives he holder he righ o buy an underlying asse a a predeermined price; he pu opion gives he holder he righ o sell an underlying asse a a predeermined price. Opions firs appeared in he early 1970 s as a finance innovaion and since hen have developed quickly ino an efficien approach oward risk hedging. In effor o saisfy he increasingly diverse needs of invesors, exoic opions wih a wide array of characerisics have been designed based on he foundaion of he sandard conrac. Exoic opions, pu simply, are opions ha do no share he same characerisics as sandard opions (Wei e al., 015. The pah-dependen opion is a ype of exoic opion ha feaures payoffs relaed o he underlying asse price a mauriy, or o he price pah beween he curren ime and expiraion dae. The binary opion is a pah-dependen opion ha has disconinuous payoffs. The binary opion can be used o hedge and speculae; i is very popular in over-he-couner marke dealings. There are wo main ypes of binary opions: The cash-or-nohing binary opion, and he asse-or-nohing binary opion. The radiional binary opion pricing model only applies o scenarios in which he underlying sock price, risk-free ineres rae, and price volailiy are cerain. In he acual financial marke, precise opion model parameers are no necessarily aainable i is especially challenging (even impossible o precisely deermine he underlying asse price, risk-free rae of ineres, or volailiy. Furher, invesors end o focus on he range wihin which opion prices end o oscillae, while generally neglecing he precise opion price. In shor: The radiional opion pricing model does no accuraely reflec he real-world financial marke. The fuzzy ses heory firs proposed by Zadeh can be uilized o solve problems ha feaure uncerainy (Zadeh, 1965. Ribeiro e al., in a noable example of his, addressed a finance engineering problem by applying he fuzzy se heory (Ribeiro e al., 1999. Based on Black-choles pricing formula, Wu esablished a fuzzy pricing model for European opions (Wu, 004; Wu, 005; Wu, 007. Considering he randomness and fuzziness under uncerain environmens,yoshida derived a novel mehod for valuaing European opion prices by using fuzzy numbers (Yoshida, 003. Lee e al. developed a fuzzy binominal European opion pricing approach in he CRR model and conduced an empirical analysis by aking &P 500 index opions as an example (Lee e al., 005. Thiagarajah e al. buil a European opions pricing model by using adapive fuzzy numbers (Thiagarajah e al., 007, and Xu e al. buil a fuzzy pricing model for European opions in normal jump-diffusion wih uncerain randomness and fuzziness. The pricing model can be considered an exension of he Meron s opion price model (Xu e al., 009. Yen derived a new opion pricing model by inroducing a non-uniform self-selecive coder (Yen, 010. Based on he assumpion ha he underlying asse price, discoun rae, he volailiy and risk-free rae of ineres are all fuzzy numbers, Zhang e al. obained a fuzzy pricing formula and corresponding algorihm for American opions (Zhang e al., 011. There have indeed been many valuable conribuions o he lieraure, bu o dae, here have been relaively few sudies on exoic opions pricing wihin fuzzy environmens apar from hose by Wang e al. (Wang e al., 014 and Zhan (Zhang, 014. Recenly, Thavaneswaran e al. used fuzzy se heory o price asse-or-nohing 384
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 10, 384-391, 016 European binary opions (Thavaneswaran e al., 013, bu only by fuzzifying he mauriy value of he underlying asse price while considering risk-free rae of ineres and volailiy o be cerain, real numbers. In he acual financial marke, daa such as underlying asse price, risk-free ineres rae, and volailiy simply canno be precisely obained. Our primary goal in conducing he presen sudy was o fill hese gaps in knowledge ha is, o build a cash-or-nohing European binary opion pricing model by fuzzifying he underlying asses price, risk-free rae of ineres, and volailiy. A number of oher sudies served as inspiraion for he presen sudy. Carlsson and Fuller used fuzzy numbers o explore he real opion pricing problem and idenified he possibilisic mean value and possibilisic variance of fuzzy numbers simulaneously (Carlsson and Fuller, 001. Thavaneswaran e al. proved he superioriy of fuzzy forecass compared o oher echniques per he opion pricing problem in he GARCH model by applying he weighed possibilisic momens of fuzzy numbers (Thavaneswaran e al., 009. Zmeskal proposed a fuzzy binomial model for pricing American real opions (Zmeskal, 010. Guerra e al. and Chryafis and Papadopoulos sudied similar pricing problems under he fuzzy environmen and uilizing fuzzy numbers (Guerra e al., 011; Chryafis and Papadopoulos, 009. Few researchers have explored binary opion valuaion under fuzzy environmens, however. In his paper, we inroduce a correlaive definiion of fuzzy numbers and heir operaions; we also derive a fuzzy pricing model of he cash-or-nohing binary European call opion. The remainder of his paper is organized as follows. The pricing model of cash-or-nohing binary opion under a sochasic environmen is derived in ecion. In ecion 3, we inroduce he basic characerisics of fuzzy numbers as uilized hroughou his paper. A general fuzzy pricing model of binary opions is discussed in ecion 4, and he numerical experimens we conduced o verify he model s effeciveness are discussed in ecion 5. ecion 6 provides a brief summary and conclusion.. PRICING MODEL OF CAH-OR-NOTHING BINARY OPTION UNDER TOCHATIC ENVIROMENT There have been many noable breakhroughs in opion pricing heory, namely in 1973, when Black and choles esablished he model for valuing European opions. Their pricing mehod can be used o valuae any claim in he Black-choles model: o d r d dw, 0 T, (1 0 W ( r / 0 e ( r is he risk-free rae of ineres, is volailiy, and W is he Brownian moion. Le T be he mauriy dae and [ T0, T ] denoe he final ime inerval; le K denoe he srike price and V (, denoe he price of he binary call opion a ime. The pricing model of he cash-or-nohing European call binary opion a ime can hen be wrien as follows: V V V r rv 0 (3 wih he following boundary condiion: Q T K H( T, T 0 T K (4 K denoes he srike price and Q is a consan. To solve Eqs. (3 and (4 for he cash-or-nohing binary opion, le based on upper ransformaion, because T, x ln (5 K x H ( K H ( 1 H( e 1 H( x. (6 K Thus, Eqs. (4 and (5 can be ranslaed ino he following Cauchy problem: 385
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 10, 384-391, 016 V V V ( r rv 0 x x V( x,0 H ( x (7 imilar o he derivaion of he Black-choles formula, we have r x ( r / V ( x, e N( (8 w 1 x N( x e dw. Because x ln, T K, we have r ( T V (, e QN( d (9 d K ln ( r / ( T T. 3. FUZZY NUMBER This secion provides several correlaive definiions and operaions of fuzzy numbers ha will be uilized hroughou his paper (Zadeh, 1965; Zimmermann, 001. 3.1 Correlaive Definiion Definiion 1. A fuzzy se A in X R, R is he se of real numbers, is a se of ordered pairs A {( x, ( x : x X}, ( x is he membership funcion of x X which maps x X ono he real inerval [0, 1]. Le R denoe a universal se of all real numbers. Then a fuzzy subse A is defined by is membership funcion A : R [0, 1]. The level se of A is defined by A { x ( x }, (0 1 The 0-level se A 0 of A is defined by he closure of he se { x ( x 0}, which is called a normal fuzzy se A if here exiss a x such ha. Definiion. A fuzzy se A n n in R is called a convex fuzzy se, if and only if for any x1, x R and 0 1, ( x (1 x min{ ( x, ( x } (10 A 1 A 1 A Definiion 3. The following condiions mus be saisfied for A o be defined as a fuzzy number: (1 A is a normal and convex fuzzy se; ( Is membership funcion A is upper semi-coninuous; (3 The level se A is bounded for all [0,1]. Zadeh proved ha if A is a fuzzy number, hen A is a convex and compac se (Zadeh, 1996. Tha is, A is a closed inerval, denoed by [, ] A A A and A has he following propery: A A, A B A B, [0,1] (11 The fuzzy se and membership funcion are an exension of he classical se and characerisic funcion. A crisp or usual number is inroduced accordingly. If he membership funcion of A is in he following forma: 1, x m ( x A 0, x m (1 hen A is a crisp or usual number ha is denoed by A 1 { m }. I readily follows ha (1 (1, [0,1], and ha any real number can be considered a crisp or usual number. { m} { m} m A 386
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 10, 384-391, 016 Definiion 4. If he membership funcion of A has he following form: a x 1 a x a, 0 1 a x b ( x A (13 x b 1 b x b, 0 0 oherwise hen he fuzzy number A is a rapezoidal fuzzy number, which has he core [a, b], lef widh, and righ widh. Le A ( a, b,, denoe a rapezoidal fuzzy number, hen i follows ha he level ses of A have he following form: A [ A, A ] [ a (1, b (1 ] [0,1] (14 3. Fuzzy Number Operaion If he binary operaion of wo fuzzy numbers A, B is defined, hen he membership funcion of A, B is as follows: ( z sup min{ ( x, ( x } (15 A B A B {( x, y x y} denoes + or operaion, and can be easily expanded o and operaion. Le A and B be wo fuzzy numbers. Thus, A [ A, A ] and B [ B, B ] ; hen A B, A B, and A B are also fuzzy numbers wih level ses in he following forma: ( A B A B [ A B, A B ], (A B A - B [ A - B, A - B ], ( A B A B [min( A B, A B, A B, A B,max( A B, A B, A B, A B ] for all [0,1]. If B denoes he level se of B wihou zero, hen A / B is also a fuzzy number wih he level se as he following: (A A A A A A A A A A / B [min(,,,,max(,,, ] B B B B B B B B B 4. FUZZY PRICING MODEL OF EUROPEAN CALL BINARY OPTION Based on he assumpion ha he underlying asses price, ineres rae r, and volailiy are all fuzzy numbers and under he operaional principle of fuzzy numbers, he fuzzy paern pricing model of he cash-or-nohing European call binary opion (Eq. (10 is as follows: r 1 (,,,, ( { T V K r e } N ( d (16 d ((ln( / 1 ( r 1 1 / ( 1 { K} {1/ } { T } { T } The underlying asses price K and he ime are boh real (usual numbers, which are denoed by he crisp numbers 1 { K } and 1 { } wih values K and, respecively, so he fuzzy pricing model of a cash-or-nohing European call binary opion a ime is: C V (, T, K, r, (17 C and are fuzzy random variables and [0,1]. The funcion N( x is an increasing funcion, so he level se of N ( d is wrien as follows: 387
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 10, 384-391, 016 of x, so imilarly, he funcion e r 1 { T } ( N ( d { N( x x d } { N( x d x d } [ N( d, N( d ] (18 x e is a decreasing funcion of x and he funcion ln x is an increasing funcion and ln( / 1 { K} have he following level ses: r 1 { T } r ( T r ( T ( e [ e, e ] (19 (ln( / 1 [ln( / K,ln( / K] (0 { K} The righ-end and lef-poins of he closed inerval ( C [( C,( C ] can now be deermined, which makes i easy o calculae he righ-end poin ( C and he lef-end poin ( C of he closed inerval ( C as follows: ( ( r C e T N( d (1 d d ln( / K ( r (1/ ( ( T T ( ( r C T e N( d [0,1] [0,1] ln( / K ( r (1/ ( ( T T ( If he underlying asses price, risk-free rae of ineres r, and volailiy are rapezoidal fuzzy numbers, he specific fuzzy pricing model of he cash-or-nohing European call binary opion can be derived accuraely. If, r, and are all rapezoidal fuzzy numbers and ( 1,,,, r ( r, r,, 1 r r, ( 1,,,, hen heir level se is as follows: [ 1 (1, (1 ], r [ r1 (1 r, r (1 r ], [ (1, (1 ] 1 The fuzzy pricing formula of C can be obained by using he fuzzy paern of he cash-or-nohing call binary opion pricing model (Eq. (17. The level se of he cash-or-nohing call binary opion price C can be wrien in crisp form as follows: and ( C [( C,( C ], [0,1] (3 (4 ( (1 ( ( r C r T e N( d (5 ( 1 (1 ( + ( r C r T e N( d ln(( 1 (1 / K ( r1 (1 r (1/ ( 1 (1 ( T d ( (1 T + ln(( (1 / K ( r (1 r (1/ ( (1 ( T d ( 1 (1 T If 0 and 0, hen, r, and are all inerval numbers. r r The fuzzy pricing formula of C can be obained by using he fuzzy paern of he cash-or-nohing call binary opion pricing model (Eq. (3. The cash-or-nohing call binary opion price C can be given by he following inerval: C [ C, C ] (6 388
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 10, 384-391, 016 and and r ( T C e N( d C e N( d + r1 ( T + ln( 1 / K ( r1 (1/ 1 ( T d T d + ln( / K ( r (1/ ( T. T 1 If 1, r1 r, 1, r 0, and r 0, hen, r, and are all real, crisp numbers. I readily follows ha C C, so assuming ha he parameers, r, and are colleced or recorded exacly, he range of fuzzy opions price will become a real, crisp number; accordingly, he fuzzy opion pricing model reurns ino a radiional pricing model of he cash-or-nohing call binary opion. To his effec, he fuzzy binary opion pricing model proposed here is an exension of he radiional cash-or-nohing binary call opion pricing model. 5. NUMERICAL EXPERIMENT In order o verify effeciveness of he proposed model, we ook similar algorihms proposed by Wu (Wu, 005 and ran wo numerical experimens o compare heir performance. Consider a cash-or-nohing European call binary opion wih he underlying asses price K=30 and T =6(monh. Assume ha he underlying asses price =35, Q=10, he volailiy of underlying asses =0., and he risk-free rae of ineres r=0.05(per annum. uppose ha = 0 and T =0.5. The fuzzy underlying asses price 0, fuzzy risk-free rae of ineres r, and fuzzy volailiy of underlying asses price are all assumed as rapezoidal fuzzy numbers: 0 = (34.7, 35., 1.9,.6, r =(0.047, 0.05, 0.01, 0.014, and = (0.18, 0., 0.05, 0.06, respecively. According o he fuzzy opion pricing model derived above, he curren fuzzy price C 0 of he cash-or-nohing call binary opion can be obained accuraely. Given differen he cash-or-nohing call binary opion prices c, using he algorihm derived by Wu (Wu, 005, he belief degrees ( c are given in Table 1. C0 Table 1. Belief degrees ( c for cash-or-nohing call binary opion prices c C0 C C 0 ( c 3.63 0.871 3.7 0.905 3.81 0.950 3.90 0.9859 4. 1.0000 4.5 0.973 4.63 0.9455 4.7 0.8913 4.85 0.846 Table 1 yields a few ineresing conclusions. If he cash-or-nohing call binary opion price is 3.81, is belief degree is 0.950. Accordingly, if he invesor is saisfied wih he belief degree 0.950, hen he or she can ake his opion for he price of 3.81 for use in he fuure. If he cash-or-nohing call binary opion price is 4., hen is belief degree equals 1.00. This siuaion reflecs he fac ha he cash-or-nohing call binary opion calculaed via he radiional formula is 4. (based on r = 0.05, =0., =35. 0 389
Rev. Téc. Ing. Univ. Zulia. Vol. 39, Nº 10, 384-391, 016 Table. - level closed inervals of he cash-or-nohing call binary opion fuzzy price ( C0 0.90 [3.6975, 4.744] 0.9 [3.7463, 4.6938] 0.94 [3.7948, 4.6453] 0.96 [3.8434, 4.5967] 0.98 [3.8919, 4.5481] 1.00 [3.9405, 4.4995] Table shows he level closed inervals ( C 0 of he cash-or-nohing call binary opion fuzzy price. If assuming =0.96, hen ( C 0 = [3.8434, 4.5967] his means ha he opion price will lie in [3.8434, 4.5967] when belief degree =0.96. I also means ha if he invesor is ineresed a belief degree 0.96, hen he or she may ake any value from he inerval [3.8434, 4.5967] as he opion price for use in he fuure. 6. CONCLUION This paper presened he resuls of our sudy on a novel cash-or-nohing call binary opion pricing mehod based on fuzzy number heory. We firs inroduced some basic conceps and operaions relaed o fuzzy numbers, hen esablished a fuzzy paern opion pricing model. The proposed model was proven feasible and effecive via wo numerical experimens. ACKNOWLEDGMENT This work was suppored by he Key Research Foundaion for Excellen Youh Talens of he Educaion Bureau of Anhui Province, China (Gran No. 013QRW054 ZD and he of cience Projec of Anhui Province, China (Gran No. 1607a00038. REFERENCE Carlsson C., Fuller R. (001 On possibilisic mean value and variance of fuzzy numbers, Fuzzy es and ysems, 1 (, pp.315 36. Chrysafis K., Papadopoulos B. (009 On heoreical pricing of opions wih fuzzy esimaors, Journal of Compuaional and Applied Mahemaics, 3(, pp.55 566. Guerra M. L., orini L., efanini L. (011 Opion price sensiiviies hrough fuzzy numbers, Compuers and Mahemaics wih Applicaions, 61 (3, pp.515 56. Lee C.F., Tzeng G.H., Wang.Y. (005 A fuzzy se approach for generalized CRR model: an empirical analysis of &P 500 index opions, Review Quaniaive Finance and Accouning, 5(3, pp. 55 75. Ribeiro R.A., Zimmermann H.-J., Yager R.R., Kacprzyk J. (1999 of Compuing in Financial Engineering, Physica-Verlag: Heidelberg. Thavaneswaran A., Appadoo.., Frank J. (013 Binary opion pricing using fuzzy numbers, Applied Mahemaics Leers, 6(1, pp.65 7. Thavaneswaran A., Appadoo.., Paseka A. (009 Weighed possibilisic momens of fuzzy numbers wih applicaions o GARCH modeling and opion pricing, Mahemaical and Compuer Modelling, 49(1, pp.35 368. Thiagarajah K., Appadoo.., Thavaneswaran A. (007 Opion valuaion model wih adapive fuzzy numbers, Compuers & Mahemaics wih Applicaion, 53 (3, pp.831 841. Wang X. D., He J. M., Li. W. (014 Compound opion pricing under fuzzy environmen, Journal Applied Mahemaics, 014(1, pp.1 9. Wu H.C. (004 Pricing European opions based on he fuzzy paern of Black choles formula, Compuers & Operaions Research, 31(7, pp.1069 1081. Wu H.C. (005 European opion pricing under fuzzy environmen, Inernaional Journal Inelligen ysems, 0(1, pp.89 10. Wu H.C. (007 Using fuzzy ses heory and Black choles formula o generae pricing boundaries of European opions, Applied Mahemaics & Compuaion, 185(1, pp.136 146. Xu W.D., Wu C.F., Xu W. J., Li H.Y. (009 A jump-diffusion model for opion pricing under fuzzy environmens, Insurance Mahemaics & Economics, 44(3, pp.337 344. 390
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