European option pricing with transaction costs in Lévy jump environment. (English) Zbl 1406.91448
Summary: The European option pricing problem with transaction costs is investigated for a risky asset price model with Lévy jump. By the aid of arbitrage pricing theory and the generalized Itô formula (which includes Poisson jump), the explicit solution to the risk asset price model is given. According to arbitrage-free principle, we first discretize the continuous-time model. Then, in each small time interval, the transaction costs are introduced. By using the \(\Delta\)-hedging strategy, the explicit solutions of the European options pricing formula with transaction costs are given for the risky asset price model with Lévy jump.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 60G51 | Processes with independent increments; Lévy processes |
References:
| [1] | Leland, E., Option pricing and replication with transactions costs, The Journal of Finance, 40, 5, 1283-1301 (1985) · doi:10.1111/j.1540-6261.1985.tb02383.x |
| [2] | Black, F.; Scholes, M., The pricing of options and corporate liabilities, The Journal of Political Economy, 81, 3, 637-654 (1973) · Zbl 1092.91524 · doi:10.1086/260062 |
| [3] | Cox, J. C.; Ross, S. A., The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3, 1-2, 145-166 (1976) · doi:10.1016/0304-405X(76)90023-4 |
| [4] | Scott, L. O., Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: applications of fourier inversion methods, Mathematical Finance, 7, 4, 413-426 (1997) · Zbl 1020.91030 · doi:10.1111/1467-9965.00039 |
| [5] | Davis, M.; Norman, A., Portfolio selection with transaction costs, Mathematics of Operations Research, 15, 4, 676-713 (1990) · Zbl 0717.90007 · doi:10.1287/moor.15.4.676 |
| [6] | Elliott, R.; Osakwe, C., Option pricing for pure jump processes with Markov switching compensators, Finance and Stochastics, 10, 2, 250-275 (2006) · Zbl 1101.91034 · doi:10.1007/s00780-006-0004-6 |
| [7] | George, M.; Perrakis, S., Stochastic dominance bounds on derivatives prices in a multiperiod economy with proportional transaction costs, Journal of Economic Dynamics and Control, 26, 7-8, 1323-1352 (2002) · Zbl 1131.91332 · doi:10.1016/S0165-1889(01)00047-1 |
| [8] | Boyle, P. P., Option replication indiscrete time with transaction costs, The Journal of Finance, 47, 271-293 (1992) · doi:10.1111/j.1540-6261.1992.tb03986.x |
| [9] | Li, Q., Option pricing with transaction costs [M.S. thesis] (2007), Hubei, China: Huazhong University of Science and Technology, Hubei, China |
| [10] | Merton, R. C., Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4, 1, 141-183 (1973) · Zbl 1257.91043 |
| [11] | Merton, R. C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 1-2, 125-144 (1976) · Zbl 1131.91344 |
| [12] | Bailey, W.; Stulz, R. M., The pricing of stock index options in a general equilibrium model, The Journal of Financial and Quantitative Analysis, 24, 1, 1-12 (1989) · doi:10.2307/2330744 |
| [13] | Wei, G. L.; Wang, L. C.; Han, F., A gain-scheduled approach to fault-tolerant control for discrete-time stochastic delayed systems with randomly occurring actuator faults, Systems Science & Control Engineering, 1, 1, 82-90 (2013) · doi:10.1080/21642583.2013.842509 |
| [14] | Ahmada, H.; Namerikawa, T., Extended Kalman filter-based mobile robot localization with intermittent measurements, Systems Science & Control Engineering, 1, 1, 113-126 (2013) · doi:10.1080/21642583.2013.864249 |
| [15] | Hu, J.; Wang, Z.; Shen, B.; Gao, H., Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements, International Journal of Control, 86, 4, 650-663 (2013) · Zbl 1278.93269 · doi:10.1080/00207179.2012.756149 |
| [16] | El-Tawil, K.; Abou Jaoude, A., Stochastic and nonlinear-based prognostic model, Systems Science & Control Engineering, 1, 1, 66-81 (2013) · doi:10.1080/21642583.2013.850754 |
| [17] | Darouach, M.; Chadli, M., Admissibility and control of switched discrete-time singular systems, Systems Science & Control Engineering, 1, 1, 43-51 (2013) · doi:10.1080/21642583.2013.832642 |
| [18] | Balasubramaniam, P.; Senthilkumar, T., Delay-dependent robust stabilization and \(H_\infty\) control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays, International Journal of Automation and Computing, 10, 1, 18-31 (2013) · doi:10.1007/s11633-013-0692-2 |
| [19] | Zhou, P.; Wang, Y.; Wang, Q.; Chen, J.; Duan, D., Stability and passivity analysis for Lur’e singular systems with Markovian switching, International Journal of Automation and Computing, 10, 1, 79-84 (2013) · doi:10.1007/s11633-013-0699-8 |
| [20] | Shen, B.; Wang, Z.; Hung, Y. S., Distributed \(H_\infty \)-consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case, Automatica, 46, 10, 1682-1688 (2010) · Zbl 1204.93122 · doi:10.1016/j.automatica.2010.06.025 |
| [21] | Shen, B.; Wang, Z.; Liu, X., Sampled-data synchronization control of dynamical networks with stochastic sampling, IEEE Transactions on Automatic Control, 57, 10, 2644-2650 (2012) · Zbl 1369.93047 · doi:10.1109/TAC.2012.2190179 |
| [22] | Tang, B.; Zeng, Q.; He, D.; Zhang, Y., Random stabilization of sampled-data control systems with nonuniform sampling, International Journal of Automation and Computing, 9, 5, 492-500 (2012) · doi:10.1007/s11633-012-0672-y |
| [23] | Shen, B.; Wang, Z.; Liu, X., A stochastic sampled-data approach to distributed \(H_\infty\) Filtering in sensor networks, IEEE Transactions on Circuits and Systems I, 58, 9, 2237-2246 (2011) · Zbl 1468.93070 · doi:10.1109/TCSI.2011.2112594 |
| [24] | Liang, X.; Hou, M.; Duan, G., Output feedback stabilization of switched stochastic nonlinear systems under arbitrary switchings, International Journal of Automation and Computing, 10, 6, 571-577 (2013) · doi:10.1007/s11633-013-0755-4 |
| [25] | Wang, Z.; Shen, B.; Liu, X., \(H_\infty\) filtering with randomly occurring sensor saturations and missing measurements, Automatica, 48, 3, 556-562 (2012) · Zbl 1244.93162 · doi:10.1016/j.automatica.2012.01.008 |
| [26] | Stettner, L., Option pricing in the CRR model with proportional transaction costs: a cone transformation approach, Applicationes Mathematicae, 24, 4, 475-514 (1997) · Zbl 1043.91511 |
| [27] | Monoyios, M., Option pricing with transaction costs using a Markov chain approximation, Journal of Economic Dynamics and Control, 28, 5, 889-913 (2004) · Zbl 1179.91244 · doi:10.1016/S0165-1889(03)00059-9 |
| [28] | Amster, P.; Averbuj, C. G.; Mariani, M. C., Stationary solutions for two nonlinear Black-Scholes type equations, Applied Numerical Mathematics, 47, 3-4, 275-280 (2003) · Zbl 1063.91026 · doi:10.1016/S0168-9274(03)00070-9 |
| [29] | Wu, Q., European options with transaction costs, Journal of Tongji University, 37, 6 (2008) |
| [30] | Aase, K. K., Contingent claims valuation when the security price is a combination of an Itô process and a random point process, Stochastic Processes and their Applications, 28, 2, 185-220 (1988) · Zbl 0645.90009 · doi:10.1016/0304-4149(88)90096-8 |
| [31] | Chan, T., Pricing contingent claims on stocks diven by Lévy processes, The Annals of Applied Probability, 9, 2, 504-528 (1999) · Zbl 1054.91033 |
| [32] | Yan, H., Pricing option and EIAs when discrete dividends follow a Markov-modulated jump diffusion model [M.S. thesis] (2012), Shanghai, China: Donghua University, Shanghai, China |
| [33] | Cont, R.; Tankov, P., Financial Modelling with Jump Processes (2004), Boca Raton, Fla, USA: Chapman & Hall, Boca Raton, Fla, USA · Zbl 1052.91043 |
| [34] | Shreve, S., Stochastic Analysis for Finance (2004), Berlin, Germany: Springer, Berlin, Germany |
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