A New Stock Model for Option Pricing in Uncertain Environment

Document Type: Research Paper


1 Institute of Uncertain Systems, Huanggang Normal University, Hubei 438000, China

2 School of Mathematics and Statistics, Huazhong Normal University, Hubei 430079, China


The option-pricing problem is always an important part in modern finance. Assuming that the stock diffusion is a constant, some literature has introduced many stock models and given corresponding option pricing formulas within the framework of the uncertainty theory. In this paper, we propose a new stock model with uncertain stock diffusion for uncertain markets. Some option pricing formulas on the proposed uncertain stock model are derived and a numerical calculation is illustrated.


bibitem{Ba:Tdls} L. Bachelier, {it Theorie de la speculation}, Paris Doctoral Dissertation in Mathematics, 1900.
%bibitem{BaXi:Paojd} E. Bayraktar and H. Xing, {it Pricing asian options for jump diffusion}, Mathematical Finance, {bf 21}textbf{(1)}(2011), 117--143.
%bibitem{BlKa:Bopwstral} F. Black and P. Karasinski, {it Bond and option pricing when short-term rates are lognormal}, Financial Analysts Journal, {bf 47}textbf{(4)}(1991), 52--59.
bibitem{BlSc:Tpocl} F. Black and M. Scholes, {it The pricing of options and corporate liabilities}, Journal of Political Economy, {bf 81}textbf{(3)} (1973), 637--654.
%bibitem{BuEs:Psoufs} J. J. Buckley and E. Eslamt, {it Pricing stock options using fuzzy sets}, Iranian Journal of Fuzzy Systems, {bf 4}textbf{(2)}(2007), 1--14.
%bibitem{CaRb:Afarov} C. Carlssona and F. Robert, {it A fuzzy approach to real option valuation}, Fuzzy Sets and Systems, {bf 139}textbf{(2)}(2003), 297--312.
bibitem{Ch:Aopfufm} X. Chen, {it American option pricing formula for uncertain financial market}, International Journal of Operations Research, {bf 8}textbf{(2)} (2011), 32--37.
bibitem{ChDa:Mepuv} X. Chen and W. Dai, {it Maximum entropy principle for uncertain variables}, International Journal of Fuzzy Systems, {bf 13}textbf{(3)} (2011), 232--236.
bibitem{ChLi:Eutude} X. Chen and B. Liu, {it Existence and uniqueness theorem for uncertain differential equations}, Fuzzy Optimization and Decision Making, {bf 9}textbf{(1)} (2010), 69--81.
bibitem{CoRoRu:Opasa} C. Cox, A. Ross and M. Rubinstein, {it Option pricing: a simplified approach}, Journal of Financial Economics, {bf 7}textbf{(3)} (1979), 229--263.
bibitem{Ga:Spcum} X. Gao, {it Some properties of continuous uncertain measure}, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, {bf 17}textbf{(3)} (2009), 419--426.
bibitem{GaGa:Ansmcop} J. Gao and X. Gao, {it A new stock model for credibilistic option pricing}, Journal of Uncertain Systems, {bf 2}textbf{(4)} (2008), 243--247.
%bibitem{He:Acfsosvabco} L. Heston, {it A closed-form solution for options with stochastic volatility with applications to bond and currency options}, The Review of Financial Studies, {bf 6}textbf{(2)}(1993), 327--343.
bibitem{HuWh:Tpoasv} J. Hull and A. White, {it The pricing of options on assets with stochastic volatility}, Journal of Finance, {bf 42}textbf{(2)} (1987), 281--300.
%bibitem{LeTzWa:Anafstbsopm} C. Lee, G. Tzeng and S. Wang, {it A new application of fuzzy set theory to the Black-Scholes option pricing model}, Expert Systems with Applications, {bf 29}textbf{(2)}(2005), 330--342.
%bibitem{LiRaTa:Ftecmma} X. Li, D. Ralescu and T. Tang, {it Fuzzy train energy consumption minimization model and algorithm}, Iranian Journal of Fuzzy Systems, {bf 8}textbf{(4)}(2011), 77--91.
bibitem{Li:Ut} B. Liu, {it Uncertainty theory}, Springer-Verlag, Berlin, 2004.
bibitem{Li:Ut2} B. Liu, {it Uncertainty theory}, 2nd ed., Springer-Verlag, Berlin, 2007.
bibitem{Li:Fphpup} B. Liu, {it Fuzzy process, hybrid process and uncertain process}, Journal of Uncertain Systems, {bf 2}textbf{(1)} (2008), 3--16.
bibitem{Li:Srput} B. Liu, {it Some research problems in uncertainty theory}, Journal of Uncertain Systems, {bf 3}textbf{(1)} (2009), 3--10.
bibitem{Li:Tpup} B. Liu, {it Theory and practice of uncertain programming}, 2nd ed., Springer-Verlag, Berlin, 2009.
bibitem{Li:Utabmmhu} B. Liu, {it Uncertainty theory: a branch of mathematics for modeling human uncertainty}, Springer-Verlag, Berlin, 2010.
bibitem{Li:Evtupairm} B. Liu, {it Extreme value theorems of uncertain process with application to insurance risk model}, Soft Computing, {bf 17}textbf{(4)} (2013), 549--556.
bibitem{LiHa:Evfuv} Y. Liu and M. Ha, {it Expected value of function of uncertain variables}, Journal of Uncertain Systems, {bf 4}textbf{(3)} (2010), 181--186.
bibitem{Me:Trop} R. Merton, {it Theory of rational option pricing}, The Bell Journal of Economics and Management Science, {bf 4}textbf{(1)} (1973), 141--183.
bibitem{Me:Opwusrad} R. Merton, {it Option pricing when underlying stock returns are discontinuous}, Journal of Financial Economics, {bf 3}textbf{(1-2)} (1976), 125--154.
bibitem{Pe:Agsmfm} J. Peng, {it A general stock model for fuzzy markets}, Journal of Uncertain Systems, {bf 2}textbf{(4)} (2008), 248--254.
bibitem{PeYa:Anopmsum} J. Peng and K. Yao, {it A new option pricing model for stocks in uncertainty markets}, International Journal of Operations Research, {bf 8}textbf{(2)} (2011), 18--26.
bibitem{QiLi:Opfffm} Z. Qin and X. Li, {it Option pricing formula for fuzzy financial market}, Journal of Uncertain Systems, {bf 2}textbf{(1)} (2008), 17--21.
%bibitem{QiLi:Eptsieoffm} Z. Qin and X. Li, {it Expected payoff of trading strategies involving European options for fuzzy financial market}, Iranian Journal of Fuzzy Systems, {bf 8}textbf{(3)}(2011), 81--94.
bibitem{StSt:Spdsvaaa} E. M. Stein and J. C. Stein, {it Stock price distributions with stochastic volatility: an analytic approach}, Review of Financial Studies, {bf 4}textbf{(4)} (1991), 727--752.
bibitem{Yo:Sctus} C. You, {it Some convergence theorems of uncertain sequences}, Mathematical and Computer Modelling, {bf 49}textbf{(3-4)} (2009), 482--487.
bibitem{Za:Fs} L. A. Zadeh, {it Fuzzy sets}, Information and Control, {bf 8}textbf{(3)} (1965), 338--353.