Asian option pricing problems of uncertain mean-reverting stock model

Transcription

1 Soft Comput 18 : FOCUS Aian option pricing problem of uncertain mean-reverting tock model Yiyao Sun 1 Kai Yao 1, Jichang Dong 1, Publihed online: 5 February 17 Springer-Verlag Berlin Heidelberg 17 Abtract An Aian option i a pecial type of option contract which reduce the volatility inherent in the option becaue of the averaging feature, o it i one of the mot actively exotic option traded in today financial derivative market. A an application of the uncertain proce in the field of finance, the uncertain finance aume that the aet price follow an uncertain differential equation. In thi paper, Aian option are propoed in the uncertain financial market baed on a mean-reverting tock model and their pricing formula are derived. In addition, ome numerical algorithm are deigned to compute the price of the Aian option on the bai of the pricing formula. Keyword Uncertainty theory Uncertain differential equation Stock model Aian option 1 Introduction Bachelier 19 made the earliet exploration of the tock price behavior in hi doctoral thei, in which he regarded the up and down of the tock price a a tochatic motion, and the equation he obtained i exactly imilar to the one ued to decribe the Brownian motion. In the year of 19, the Brownian motion wa given a trict mathematical decrip- Communicated by Y. Ni. B Jichang Dong jcdonglc@uca.ac.cn 1 School of Economic and Management, Univeity of Chinee Academy of Science, Beijing 119, China Key Laboratory of Big Data Mining and Knowledge Management, Chinee Academy of Science, Beijing 119, China tion by Wiener 19 for the fit time. A the tock price may take negative value under the aumption of Brownian motion, geometric Brownian motion opened a new chapter of tochatic finance. Since the tock option wa accepted by inveto a a rik management tool, Black and Schole 197 contricted the famou Black Schole formula for determining the European option price of a tock in 197. he European option and American option are two mot common form of traditional option, and the Aian option wa fitly introduced in 1987 a a pecial type of option contract. Different from the European option and American option, the payoff of an Aian option i determined by the average of underlying aet price over a certain period of the option contract. Becaue Aian option can avoid the rik brought by market manipulation cloe to maturity date, they have become one of the mot popular exotic option. And the approache to the problem of valuing Aian option have been tudied by many reearche on the bai of Black Schole formula. For example, Kemna and Vot 199 ued the Monte Carlo imulation with variance reduction element a the pricing trategy of Aian option in 199. In the ame year, the fat Fourier tranform technique were advocated by Carverhill and Clewlow 199. Roge and Shi 1995 formulated a partial differential equation to model the price of the Aian option. It i undeniable that the Black Schole model ha been widely ued in financial field, but many chola init that it i inconitent with the actual ituation of financial market becaue the ditribution of the underlying aet i different from the normal probability ditribution which i the baic aumption of Black Schole model. Becaue of the information aymmetry, inveto cannot get enough data to olve the problem of invetment option, and they are more willing to make their own deciion depending on previou experience or take the advice of expert. So the belief degree

2 5584 Y. Sun et al. play a ignificant role in making deciion for financial activitie. In order to decribe the belief degree rationally, Liu 7 founded an uncertainty theory in 7 which i a branch of axiomatic mathematic for modeling the uncertain behavio of human being. Nowaday, the uncertain theory ha been applied to many area uch a game theory Gao 1; Gao et al. 17; Yang and Gao 17, project management Yang and Gao 16; Ni and Zhao 17, aet pricing Guo and Gao 17 and ocial network Ni et al. 17; Ni 17. Baed on the aumption that the fluctuation of tock price follow a geometric Liu proce, Liu 9 tarted the pioneer work of uncertain finance in which he built an uncertain tock model and derived it European option pricing formula in 9. Afterward, many chola committed themelve to tudy the financial problem under the framework of uncertainty theory. For example, Chen 11 invetigated the pricing method for American option baed on Liu tock model in 11. Chen et al. 1 propoed an uncertain tock model with dividend. Zhang and Liu 14 invetigated the geometric Aian option pricing problem Shen and Zhu 16. Sun and Chen 15 derived the Aian option pricing formula for uncertain financial market. Conidering the influence of udden facto on the tock price, Ji and Zhou 15 propoed an uncertain tock model with both poitive and negative jump and applied it to the European option pricing formula. Sun and Su 16 took long-term facto into conideration and preented an uncertain mean-reverting tock model with floating interet rate. A an application, they tudied the pricing problem of European option and American option baed on the model. Option pricing problem are the focu point of modern finance. Aian option are widely concerned by inveto for their price advantage and lower rik compared with traditional option. In thi paper, we dicu the pricing problem of Aian option in the uncertain financial market baed on the mean-reverting tock model with floating interet rate. he ret of thi paper i organized a follow. In Sect., we recall ome ignificant concept in uncertainty theory, for example, uncertain variable and uncertain differential equation. hen we introduce in Sect. the mean-reverting tock model with floating interet rate that wa previouly propoed by Sun and Su 16. Subequently, we derive the Aian option pricing formula baed on the model and deign ome numerical algorithm to calculate the Aian option price in Sect. 4 and 5, repectively. Finally, we make ome concluion in Sect. 6. Preliminary In thi ection, we introduce ome baic definition and theorem about the uncertain variable and the uncertain differential equation..1 Uncertain variable Definition 1 Liu 7 LetL be a σ -algebra on a nonempty et Ɣ. AetfunctionM: L [, 1] i called an uncertain meaure if it atifie the following axiom: Axio: Normality Axiom M{Ɣ} =1 for the univeal et Ɣ. Axiom : Duality Axiom M{ } +M{ c }=1for any event. Axiom : Subadditivity Axiom For every countable equence of event 1,,...,wehave { } M i M{ i }. he triplet Ɣ, L, M i called an uncertainty pace. In addition, Liu 9 defined a product uncertain meaure a the following axiom in order to decribe the et function M on the product σ -algebra L. Axiom 4: Let Ɣ k, L k, M k be uncertainty pace for k = 1,,... he product uncertain meaure M i an uncertain meaure atifying { } M k = M k { k } k=1 k=1 where k are arbitrarily choen event from L k for k = 1,,..., repectively. An uncertain variable Liu 7 i a meaurable function ξ from an uncertainty pace Ɣ, L, M to the et of real numbe. he uncertainty ditribution of an uncertain variable ξ i defined by x = M{ξ x}, x R for any real number x. If the uncertainty ditribution x i a continuou and trictly increaing function with repect to x at which < x <1, and lim x =, lim x x = 1, x + then it i called regular. In thi cae, the invee function 1 i called the invee uncertainty ditribution of ξ. Definition Liu 9 he uncertain variable ξ 1,ξ,..., ξ n are aid to be independent if { n } M ξ i B i = n M{ξ i B i } for any Borel et B 1, B,...,B n of real numbe.

3 Aian option pricing problem of uncertain mean-reverting tock model 5585 heore Liu 1 Let ξ 1,ξ,...,ξ n be independent uncertain variable with regular uncertainty ditribution 1,,..., n, repectively. If the function f i trictly increaing with repect to x 1, x,...,x m and trictly decreaing with repect to x m+1, x m+,...,x n, then ξ = f ξ 1,ξ,...,ξ n i an uncertain variable with an invee uncertainty ditribution 1 = f 1 1,..., 1 m, 1 m+1 1,..., 1 1. n Definition Liu 7 he expected value of an uncertain variable ξ i defined by + E[ξ] = M{ξ x}dx M{ξ x}dx provided that at leat one of the two integral exit. heorem Liu 7 Let ξ be an uncertain variable with an uncertainty ditribution. If it expected value exit, then + E[ξ] = 1 xdx xdx. heorem Liu 1 Let ξ be an uncertain variable with a regular uncertainty ditribution. If it expected value exit, then E[ξ] = 1 d.. Uncertain differential equation An uncertain proce, a a equence of uncertain variable indexed by the time, i ued to model the evolution of uncertain phenomena. Definition 4 Liu 9 An uncertain proce C t i called a canonical Liu proce if i C = and almot all ample path are Lipchitz continuou, ii C t ha tationary and independent increment, iii every increment C +t C i a normal uncertain variable with an uncertainty ditribution t x = π x exp. t Definition 5 Liu 8 Suppoe that C t i a canonical Liu proce, and f and g are two real function. hen dx t = f t, X t dt + gt, X t dc t 1 i called an uncertain differential equation. heorem 4 Chen and Liu 1 Let u 1t, u t,v 1t,v t be integrable uncertain procee. hen the linear uncertain differential equation dx t = u 1t X t + u t dt + v 1t X t + v t dc t ha a olution X t = U t V t where U t = exp V t = X + u 1 d + u U d + v 1 dc, v dc. U Definition 6 Yao and Chen 1he -path <<1 of an uncertain differential equation dx t = f t, X t dt + gt, X t dc t with an initial value X i a determinitic function Xt with repect to t that olve the correponding equation dx t = f t, X t dt + gt, X t 1 dt, X = X where 1 i the invee uncertainty ditribution of tandard normal uncertain variable, i.e., 1 = π ln 1, <<1. heorem 5 Yao and Chen 1 Aume that X t and X t are the olution and -path of the uncertain differential equation dx t = f t, X t dt + gt, X t dc t, repectively. hen M{X t Xt, t} =, M{X t > Xt, t} =1. heorem 6 Yao and Chen 1 Let X t and Xt be the olution and -path of the uncertain differential equation dx t = f t, X t dt + gt, X t dc t,

4 5586 Y. Sun et al. repectively. hen the olution X t ha an invee uncertainty ditribution 1 = X t. heorem 7 Yao 1 Let X t and Xt be the olution and -path of the uncertain differential equation dx t = f t, X t dt + gt, X t dc t, repectively. hen for a trictly increaing function Jx,the upremum Y t = up JX t Y t = up JX, t and the time integral proce Z t = Z t = JX d JX d. Conveely, for a trictly decreaing function Jx, the upremum Y t = up JX t Y t = up JX 1, t and the time integral proce Z t = JX d to x 1, x,...,x m and trictly decreaing with repect to x m+1, x m+,...,x n, then the uncertain proce X t = f X 1t, X t,...,x nt X t = f X1t,...,X mt, X m+1,t 1 1,...,Xnt. Mean-reverting uncertain tock model he uncertain differential equation are commonly ued in financial market. Sun and Su 16 aumed that both the interet rate r t and the tock price X t fluctuate around ome fixed value in the long term and preented an uncertain mean-reverting tock model with floating interet rate a below, { drt = a 1 r t dt + σ 1 dc 1t dx t = m a X t dt + σ dc t where, m, a 1, a,σ 1 and σ are ome poitive real numbe with a 1, a =, a 1 and σ 1 are the log-drift and log-diffuion of the interet, repectively, a and σ are the log-drift and log-diffuion of the tock price, repectively, and C 1t and C t are two independent canonical Liu procee. By olving the uncertain differential equation, we have r t = a 1 + exp a 1 t r a 1 + σ 1 expa 1 a 1 tdc 1 X t = m a + exp a t X m a t + σ expa a tdc. According to heorem 5, the-path of r t and X t are the olution of ordinary differential equation { dr t = a 1 r t dt + σ 1 1 dt = m a X t dt + σ 1 4 dt dx t where 1 = π ln 1, Z t = JX 1 d. heorem 8 Yao 15 Aume that X 1t, X t,...,x nt are ome independent uncertain procee derived from the olution of ome uncertain differential equation. If the function f x 1, x,...,x n i trictly increaing with repect i.e., rt = r exp a 1 t + + σ 1 a 1 a 1 π ln 1 exp a 1 t, 1 5

5 Aian option pricing problem of uncertain mean-reverting tock model 5587 Xt = X exp a t m + + σ a a π ln 1 exp a t. 6 1 In the next two ection, we introduce the Aian call option and Aian put option, and derive ome formula to calculate the price of thee option for the tock model. 4 Aian call option pricing formula An Aian call option i a financial contract whoe payoff depend on the average price of the tock over a certain period of time in which the invetor believe that the tock price will have a certain level of raie. And the payoff of the invetor at the expiration time with a triking price K i X d K. Let f c repreent the price of the contract. hen [ ] f c = E r d X d K. heorem 9 he price of the Aian call option with a trike price K and an expiration data for the tock model i f c = where r 1 1 d X d K d = r exp a σ 1 a 1 a 1 π ln 1 1 exp a 1, X = X exp a m + + σ a a π ln 1 exp a. 1 Proof Since r t and X t are uncertain procee with -path repreented by Eq. 5 and 6, it follow from heorem 7 that the -path of the time integral are r d, r d, X d X d, repectively. Furthermore, the -path of the dicount rate r d i 1 d. Since the uncertain proce which repreent the payoff of the invetor at the expiration date t X d K t t X t d K, the preent value t r d X d K t t 1 d X t d K according to heorem 8. A a reult, we have f c = 1 d X d K d according to heorem and 6. he theorem i proved. According to heorem 9, the algorithm to calculate the Aian call option price of the tock model i deigned a below. Step : Chooe two large numbe N and M according to the deired preciion degree. Set i = i/n and = j /M, i = 1,,...,N 1, j = 1,,...,M. Step 1: Set i =. Step : Set i i + 1. Step : Se =. Step 4: Se j + 1. Step 5: Calculate the floating interet rate and the tock price

6 5588 Y. Sun et al. f c Fig. 1 Aian call option price f c with repect to expiration time in Example 1 r 1 i = r exp a σ 1 a 1 a 1 π ln 1 i 1 exp a 1, i = X exp a m + + σ a a π ln i 1 exp a. 1 i If j < M, then return to Step 4. Step 6: Calculate the dicount rate r 1 i d exp M r 1 i. Step 7: Calculate the arithmetic average of the tock price within the period 1 d 1 M Step 8: Calculate the poitive deviation between the arithmetic average of the tock price within the period and the triking price K d K = max, 1 M Step 9: Calculate r 1 i 1 d d K. If i < N 1, then return to Step. Step 1: Calculate the price of Aian call option f c 1 N 1 N 1 r 1 i d K. d K. Example 1 Aume the paramete of the interet rate are r =., =.1, a 1 =.5 and σ 1 =.4, and the paramete of the tock price are X = 4, m = 6, a = 1 and σ =. hen the price of an Aian call option with a triking price K = 5 and an expiration time = 1i f c = A Fig. 1 how, the price f c i an increaing function with repect to the expiration time when the other paramete remain unchanged. Example Aume the paramete of the interet rate are r =., =.1, a 1 =.5 and σ 1 =.4, and the paramete of the tock price are X = 4, m = 6, a = 1 and σ =. hen the price of an Aian call option with a triking price K = and an expiration time = 1i f c = A Fig. how, the price f c i a decreaing function with repect to the triking price K when the other paramete remain unchanged. Example Aume the paramete of the interet rate are r =., =.1, a 1 =.5 and σ 1 =.4, and the paramete of the tock price are X = 4, m = 6, a = 1 and σ =. hen the price of an Aian call option with a triking price K = and an expiration time = 1i f c = A Fig. how, the price f c i a decreaing function with repect to when the other paramete remain unchanged. 5 Aian put option pricing formula An Aian put option i a financial contract whoe payoff depend on the average price of the tock over a certain period of time in which the invetor believe that the tock price will have a certain level of decline. And the payoff of the invetor at the expiration time with a triking price K i K 1 + X d.

7 Aian option pricing problem of uncertain mean-reverting tock model 5589 f c K Fig. Aian call option price f c with repect to triking price K in Example f c Fig. Aian call option price f c with repect to in Example Let f p repreent the price of the contract. hen [ f p = E r d K 1 ] X d. heore he price of the Aian put option with a trike price K and an expiration data for the tock model i f p = where d K 1 X d d = r exp a σ 1 a 1 a 1 X = X exp a m + + σ a a π ln π ln 1 exp a 1, 1 1 exp a. 1 Proof Since r t and X t are uncertain procee with -path repreented by Eq. 5 and 6, it follow from heorem 7 that the -path of the time integral are r d, r d, X d X d, repectively. Furthermore, the -path of the dicount rate r d i 1 d. Since the uncertain proce which repreent the payoff of the invetor at the expiration date K 1 X d t K 1 + X d 1, t

8 559 Y. Sun et al. f p Fig. 4 Aian put option price f p with repect to expiration time in Example 4 the preent value of the payoff r d K 1 X d t Step 6: Calculate the dicount rate r i d exp M r i. 1 d K 1 t X 1 d according to heorem 8. A a reult, we have f p = = 1 d K 1 r d K 1 X 1 d d X d d according to heorem and 6. he theorem i proved. Regarding the numerical algorithm for calculating the Aian put option price of the tock model baed on heorem 1, the fit five tep are the ame a that for the Aian call option price, and the ret tep are a follow. Step 5: Calculate the floating interet rate and the tock price r i = r exp a σ 1 a 1 a 1 π ln i 1 exp a 1, 1 i = X exp a m + + σ a a π ln i 1 exp a. 1 i If j < M, then return to Step 4. Step 7: Calculate the arithmetic average of the tock price within the period 1 d 1 M Step 8: Calculate the poitive deviation between the arithmetic average of the tock price within the period and the triking price K K 1 d = max, K 1 M Step 9: Calculate r i d K 1 d. If i < N 1, then return to Step. Step 1: Calculate the price of Aian call option f c 1 N 1 K 1 N 1 r i d d.. Example 4 Aume the paramete of the interet rate are r =., =.1, a 1 =.5 and σ 1 =.4, and the paramete of the tock price are X = 4, m = 6, a = 1 and σ =. hen the price of an Aian put option with a triking price K = 5 and an expiration time = 1i f p = A Fig. 4 how, the price f p i an increaing

9 Aian option pricing problem of uncertain mean-reverting tock model 5591 f p K Fig. 5 Aian put option price f p with repect to triking price K in Example 5 fp Fig. 6 Aian put option price f p with repect to in Example 6 function with repect to the expiration time when the other paramete remain unchanged. Example 5 Aume the paramete of the interet rate are r =., =.1, a 1 =.5 and σ 1 =.4, and the paramete of the tock price are X = 4, m = 6, a = 1 and σ =. hen the price of an Aian put option with a triking price K = and an expiration time = 1i f p =.. A Fig. 5 how, the price f p i an increaing function with repect to the triking price K when the other paramete remain unchanged. Example 6 Aume the paramete of the interet rate are r =., =.1, a 1 =.5 and σ 1 =.4, and the paramete of the tock price are X = 4, m = 6, a = 1 and σ =. hen the price of an Aian put option with a triking price K = and an expiration time = 1i f p =.. A Fig. 6 how, the price f p i a decreaing function with repect to when the other paramete remain unchanged. 6 Concluion In thi paper, the Aian option pricing problem of an uncertain mean-reverting tock model with floating interet rate were invetigated, and ome pricing formula were derived. o calculate the price numerically, ome algorithm were deigned, and numerical experiment were performed to illutrate the efficiency of thee algorithm. In addition, the relationhip between the price and ome paramete wa alo dicued baed on the numerical experiment. Acknowledgement hi tudy wa funded by the National Natural Science Foundation of China Grant No. 6146, 7151 and and the Open Project of Key Laboratory of Big Data Mining and Knowledge Management, Chinee Academy of Science. Compliance with ethical tandard Conflict of interet he autho declare that they have no conflict of interet. Ethical approval hi article doe not contain any tudie with human participant performed by any of the autho. Clarification hi work wa carried out in collaboration between all autho. All autho read and approved the final manucript. Reference Bachelier L 19 héorie de la péculation. Ann Sci L École Normale Supérieure 17:1 86 Black F, Schole M 197 he pricing of option and corporate liabilitie. J Polit Econ 81: Carverhill A, Clewlow L 199 Flexible convolution. In: From Black Schole to Black Hole, Rik Book, London pp

10 559 Y. Sun et al. Chen X 11 American option pricing formula for uncertain financial market. Int J Oper Re 8: 7 Chen X, Liu B 1 Exitence and uniquene theorem for uncertain differential equation. Fuzzy Optim Deci Mak 91:69 81 Chen X, Liu Y, Ralecu DA 1 Uncertain tock model with periodic dividend. Fuzzy Optim Deci Mak 11:111 Gao JW 1 Uncertain bimatrix game with application. Fuzzy Optim Deci Mak 11:65 78 Gao JW, Yang XF, Liu D 17 Uncertain hapley value of coalitional game with application to upply chain alliance. Appl Soft Comput. doi:1.116/j.aoc Guo C, Gao JW 17 Optimal dealer pricing under tranaction uncertainty. J Intell Manuf. doi:1.17/ Ji XY, Zhou J 15 Option pricing for an uncertain tock model with jump. Soft Comput 1911: 9 Kemna A, Vot A 199 A pricing method for option baed on average aet value. J Bank Financ 14:11 19 Liu B 7 Uncertain heory, nd edn. Springer, Berlin Liu B 8 Fuzzy proce, hybrid proce and uncertain proce. J Uncertain Syt 1: 16 Liu B 9 Some reearch problem in uncertainty theory. J Uncertain Syt 1: 1 Liu B 1 Uncertainty theory: a branch of mathematic for modeling human uncertainty. Springer, Berlin Ni YD 17 Sequential eeding to optimize influence diffuion in a ocial network. Appl Soft Comput. doi:1.116/j.aoc Ni YD, Zhao ZJ 17 wo-agent cheduling problem under fuzzy environment. J Intell Manuf 8: Ni YD, Ning L, Ke H, Ji XY 17 Modeling and minimizing information ditortion in information diffuion through a ocial network. Soft Comput. doi:1.17/ Roge L, Shi Z 1995 he value of an Aian option. J Appl Probab : Shen JY, Zhu YG 16 Chance-contrained model for uncertain job hop cheduling problem. Soft Comput 6:8 91 Sun JJ, Chen XW 15 Aian option pricing formula for uncertain financial market. J Uncertain Anal Appl :11 Sun YY, Su Y 16 Mean-reverting tock model with floating interet rate in uncertain environment. Fuzzy Optim Deci Mak. doi:1. 17/ Wiener N 19 Differential pace. J Math Phy 1: Yang XF, Gao JW 16 Linear quadratic uncertain differential game with application to reource extraction problem. IEEE ran Fuzzy Syt 44: Yang XF, Gao JW 17 Bayeian equilibria for uncertain bimatrix game with aymmetric information. J Intell Manuf. doi:1.17/ Yao K 1 Extreme value and integral of olution of uncertain differential equation. J Uncertain Anal Appl 1: Yao K 15 Uncertain contour proce and it application in tock model with floating interet rate. Fuzzy Optim Deci Mak 144:99 44 Yao K, Chen X 1 A numerical method for olving uncertain differential equation. J Intell Fuzzy Syt 5:85 8 Zhang ZQ, Liu WQ 14 Geometric average Aian option pricing for uncertain financial market. J Uncertain Syt 84:17