A no-arbitrage theorem for uncertain stock model. (English) Zbl 1449.91149
Summary: Stock model is used to describe the evolution of stock price in the form of differential equations. In early years, the stock price was assumed to follow a stochastic differential equation driven by a Brownian motion, and some famous models such as Black-Scholes stock model and Black-Karasinski stock model were widely used. This paper assumes that the stock price follows an uncertain differential equation driven by Liu process rather than Brownian motion, and accepts Liu’s stock model to simulate the uncertain market. Then this paper proves a no-arbitrage determinant theorem for Liu’s stock model and presents a sufficient and necessary condition for no-arbitrage. Finally, some examples are given to illustrate the usefulness of the no-arbitrage determinant theorem.
References:
| [1] | Black, F; Karasinski, P, Bond and option pricing when short-term rates are lognormal, Financial Analysts Journal, 47, 52-59, (1991) · doi:10.2469/faj.v47.n4.52 |
| [2] | Black, F; Scholes, M, The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654, (1973) · Zbl 1092.91524 · doi:10.1086/260062 |
| [3] | Chen, X; Liu, B, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9, 69-81, (2010) · Zbl 1196.34005 · doi:10.1007/s10700-010-9073-2 |
| [4] | Chen, X, American option pricing formula for uncertain financial market, International Journal of Operations Research, 8, 32-37, (2011) · Zbl 1468.91164 |
| [5] | Chen, X; Gao, J, Uncertain term structure model of interest rate, Soft Computing, 17, 597-604, (2013) · Zbl 1264.91128 · doi:10.1007/s00500-012-0927-0 |
| [6] | Liu, B. (2007). Uncertainty Theory (2nd ed.). Berlin: Springer. · Zbl 1141.28001 |
| [7] | Liu, B, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2, 3-16, (2008) |
| [8] | Liu, B, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3, 3-10, (2009) |
| [9] | Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer. · doi:10.1007/978-3-642-13959-8 |
| [10] | Liu, B. (2013). Toward uncertain finance theory. Journal of Uncertainty Analysis and Applications, 1, Article 1. |
| [11] | Liu, B. (2014). Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and Decision Making, 13(3), 259-271. · Zbl 1428.60008 |
| [12] | Liu, Y. H. (2012). An analytic method for solving uncertain differential equations. Journal of Uncertain Systems, \(6\)(4), 244-249. |
| [13] | Liu, Y. H., Chen, X., & Ralescu, D. A. (2012). Uncertain currency model and currency option pricing. Technical Report. http://www.orsc.edu.cn/online/091010.pdf. Accessed 1 April 2014. |
| [14] | Merton, R, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4, 141-183, (1973) · Zbl 1257.91043 · doi:10.2307/3003143 |
| [15] | Peng, J; Yao, K, A new option pricing model for stocks in uncertainty markets, International Journal of Operations Research, 8, 18-26, (2011) · Zbl 1468.91175 |
| [16] | Qin, Z; Gao, X, Fractional Liu process with application to finance, Mathematical and Computer Modelling, 50, 1538-1543, (2009) · Zbl 1185.60039 · doi:10.1016/j.mcm.2009.08.031 |
| [17] | Samuelson, P, Rational theory of warrant pricing, Industrial Management Review, 6, 13-31, (1965) |
| [18] | Yao, K; Gao, J; Gao, Y, Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making, 12, 3-13, (2013) · Zbl 1412.60104 · doi:10.1007/s10700-012-9139-4 |
| [19] | Yao, K; Chen, X, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 25, 825-832, (2013) · Zbl 1291.65025 |
| [20] | Yao, K. (2013a). Extreme values and integral of solution of uncertain differential equation. Journal of Uncertainty Analysis and Applications, 1, Article 2. |
| [21] | Yao, K. (2013b). A type of nonlinear uncertain differential equations with analytic solution. Journal of Uncertainty Analysis and Applications, 1, Article 8. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
