## Convergence rate of regime-switching trees.(English)Zbl 1358.41009

Summary: Considering a general class of regime-switching geometric random walks and a broad class of piecewise twice differentiable payoff functions, we show that convergence of option prices occurs at a speed of order $$\mathcal{O}(n^{- \beta})$$, where $$\beta = 1/2$$ when the payoff is discontinuous and $$\beta = 1$$ otherwise.

### MSC:

 41A25 Rate of convergence, degree of approximation 65C50 Other computational problems in probability (MSC2010) 65C20 Probabilistic models, generic numerical methods in probability and statistics
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### References:

 [1] Zhu, S. P.; Badran, A.; Lu, X., A new exact solution for pricing European options in a two-state regime-switching economy, Comput. Math. Appl., 64, 8, 2744-2755, (2012) · Zbl 1268.91170 [2] Liu, R. H., A new tree method for pricing financial derivatives in a regime-switching mean-reverting model, Nonlinear Anal. RWA, 13, 6, 2609-2621, (2012) · Zbl 1254.91726 [3] Liu, R. H.; Zhao, J. L., A lattice method for option pricing with two underlying assets in the regime-switching model, J. Comput. Appl. Math., 250, 96-106, (2013) · Zbl 1285.91143 [4] Buffington, J.; Elliott, R. J., American options with regime switching, Int. J. Theor. Appl. Finance, 5, 05, 497-514, (2002) · Zbl 1107.91325 [5] Fuh, C. D.; Ho, K. W.R.; Hu, I.; Wang, R. H., Option pricing with Markov switching, J. Data Sci., 10, 3, 483-509, (2012) [6] Guo, X., Information and option pricings, Quant. Finance, 1, 1, 38-44, (2001) [7] Hardy, Mary R., A regime-switching model of long-term stock returns, N. Am. Actuar. J., 5, 2, 41-53, (2001) · Zbl 1083.62530 [8] Hardy, Mary, Investment guarantees: modeling and risk management for equity-linked life insurance, vol. 215, (2003), John Wiley & Sons · Zbl 1092.91042 [9] Li, M. Y.L.; Lin, H. W.W., Examining the volatility of Taiwan stock index returns via a three-volatility-regime Markov-switching arch model, Rev. Quant. Finance Account., 21, 2, 123-139, (2003) [10] Hobbes, G.; Lam, F.; Loudon, G. F., Regime shifts in the stock-bond relation in Australia, Rev. Pac. Basin Financ. Markets Policies, 10, 01, 81-99, (2007) [11] Nishina, K.; Maghrebi, N.; Holmes, M. J., Nonlinear adjustments of volatility expectations to forecast errors: evidence from Markov-regime switches in implied volatility, Rev. Pac. Basin Financ. Markets Policies, 15, 03, (2012) [12] Costabile, M.; Leccadito, A.; Massabó, I.; Russo, E., A reduced lattice model for option pricing under regime-switching, Rev. Quant. Finance Account., 42, 4, 667-690, (2014) [13] Bollen, N. P.B., Valuing options in regime-switching models, J. Derivatives, 6, 1, 38-49, (1998) [14] Khaliq, A. Q.M.; Liu, R. H., New numerical scheme for pricing American option with regime-switching, Int. J. Theor. Appl. Finance, 12, 03, 319-340, (2009) · Zbl 1204.91127 [15] Liu, R. H., Regime-switching recombining tree for option pricing, Int. J. Theor. Appl. Finance, 13, 03, 479-499, (2010) · Zbl 1233.91284 [16] Yuen, F. L.; Yang, H., Option pricing with regime switching by trinomial tree method, J. Comput. Appl. Math., 233, 8, 1821-1833, (2010) · Zbl 1181.91315 [17] J.H. Yoon, U.J. Choi, B.H. Lim, B.G. Jang, A lattice method for lookback options with regime-switching volatility, Available at SSRN 1523634 (2011). [18] Yuen, F. L.; Siu, T. K.; Yang, H., Option valuation by a self-exciting threshold binomial model, Math. Comput. Modelling, 58, 1, 28-37, (2013) · Zbl 1297.91139 [19] Liu, R. H.; Nguyen, D., A tree approach to options pricing under regime-switching jump diffusion models, Int. J. Comput. Math., 92, 12, 2575-2595, (2015) · Zbl 1335.91106 [20] Ma, J.; Zhu, T., Convergence rates of trinomial tree methods for option pricing under regime-switching models, Appl. Math. Lett., 39, 13-18, (2015) · Zbl 1320.91158 [21] J. Ma, T. Zhu, Erratum to “convergence rates of trinomial tree methods for option pricing under regime-switching models”, available at ResearchGate 268631955 (2015). · Zbl 1320.91158 [22] Leduc, G., Option convergence rate with geometric random walks approximations, Stoch. Anal. Appl., 34, 5, 767-791, (2016) · Zbl 1410.91456 [23] Naik, V., Option valuation and hedging strategies with jumps in the volatility of asset returns, J. Finance, 48, 5, 1969-1984, (1993) [24] Leduc, G., A European option general first-order error formula, ANZIAM J., 54, 4, 248-272, (2013) · Zbl 1282.91337
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