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A finite element method for pricing of continuous-installment options under a Markov-modulated model: existence, uniqueness, and stability of solutions

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Abstract

In this paper, we study the existence, uniqueness, and stability of solutions to the pricing problem of European continuous-installment options under the regime-switching model based on a numerical approach. For this, we consider a two-state continuous-time Markov chain for the regime-switching model and a one-dimensional finite element method for the numerical scheme. Under our proposed model and the installment option feature for the option holder to continue paying installments until maturity and receive payoff or to stop installments and terminate the contract, the valuation problem has been formulated as coupled partial differential equations (CPDE) with free boundaries. For this problem, we obtained some appropriate assumptions for the model parameters to prove that the pricing problem has unique solutions under the regime-switching model. We also illustrated some proven theorems to show the stability of solutions with a numerical approach. Finally, some numerical examples are considered to show the performance of the obtained theoretical results through numerical implementations.

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References

  • Alobaidi G, Mallier R (2006) Installment options close to expiry. J Appl Math Stoch Anal 2006:1–9

    Article  MathSciNet  Google Scholar 

  • Alobaidi G, Mallier R, Deakin A (2004) Laplace transforms and installment options. Math Models Methods Appl Sci 14:1167–1189

    Article  MathSciNet  Google Scholar 

  • Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–654

    Article  MathSciNet  Google Scholar 

  • Buffington J, Elliott RJ (2002) American options with regime switching. Int J Theor Appl Financ 5:497–514

    Article  MathSciNet  Google Scholar 

  • Chance DM (2011) Essays in derivatives: risk-transfer tools and topics made easy. Wiley

    Google Scholar 

  • Choi SY, Jeon J, Yoon JH (2020) Analytic valuation of European continuous-installment barrier options. J Comput Appl Math 363:392–412

    Article  MathSciNet  Google Scholar 

  • Ciurlia P (2011) Valuation of European continuous-installment options. Comput Math Appl 62(6):2518–2534

    Article  MathSciNet  Google Scholar 

  • Deng G (2013) Pricing American continuous-installment put option in a jump-diffusion model. In: Proceedings of the 32nd Chinese Control Conference. IEEE, pp 8289–8294

  • Dinçer H, Kou G, Olgu Akdeniz O, Yüksel S (2021) Fintech investments in European banks: a hybrid its fuzzy multidimensional decision-making approach. Financ Innov 7(1):39

    Article  PubMed  PubMed Central  Google Scholar 

  • Dixit AK, Pindyck RS (1994) Investment under uncertainty. Princeton University Press

    Book  Google Scholar 

  • François P (2005) Pricing ASX installment warrants under Garch H. Ben-Ameur, M. Breton. Les Cahiers du GERAD ISSN 711:2440

    Google Scholar 

  • Goard J, AbaOud M (2022) Pricing European and American installment options. Mathematics 10(19):3494

    Article  Google Scholar 

  • Gong X, Zhang W, Xu W, Li Z (2022) Uncertainty index and stock volatility prediction: evidence from international markets. Financ Innov 8(1):1–44

    Article  Google Scholar 

  • Griebsch S, Kuhn C, Wystup U (2007) Installment options: a closed-form solution and the limiting case. Mathematical Control Theory and Finance. Springer, Berlin, Heidelberg

    Google Scholar 

  • Heidari S (2023) Numerical approach for coupled systems resulting from pricing of derivatives: modeling and pricing of installment options. Math Methods Appl Sci 46(4):4663–4675

    Article  MathSciNet  Google Scholar 

  • Heidari S, Azari H (2017a) A front-fixing finite element method for pricing American options under regime-switching jump-diffusion models. Comput Appl Math 37:3691–3707

    Article  MathSciNet  Google Scholar 

  • Heidari S, Azari H (2017b) Pricing American options under multi-states: a radial basis collocation approach. SeMA J 75:365–378

    Article  MathSciNet  Google Scholar 

  • Holmes A, Yang H, Zhang S (2012) A front-fixing finite element method for the valuation of American options with regime switching. Int J Comput Math IJCM 89(1–18):06

    MathSciNet  Google Scholar 

  • Jeon J, Kim G (2019) Pricing European continuous-installment strangle options. N Am J Econ Financ 50:101049

    Article  Google Scholar 

  • Jeon J, Kim G (2022) Pricing European continuous-installment currency options with mean-reversion. N Am J Econ Financ 59:101605

    Article  Google Scholar 

  • Kimura T (2010) Valuing continuous-installment options. Eur J Oper Res 201(1):222–230

    Article  MathSciNet  Google Scholar 

  • MacRae CD (2008) The employee stock option: an installment option. Available at SSRN 1286928

  • Merton RC (1973) Theory of rational option pricing. Bell J Econ Manag Sci 4:141–183

    Article  MathSciNet  Google Scholar 

  • Miglio E, Sgarra C (2011) A finite element discretization method for option pricing with the bates model. SeMA J 55(1):23–40

    Article  MathSciNet  Google Scholar 

  • Oosterlee CW (2003) On multigrid for linear complementarity problems with application to American-style options. Electron Trans Numer Anal 15(1):165–185

    MathSciNet  Google Scholar 

  • Schachermayer W, Davis M, Tompkins R (2000) Pricing no-arbitrage bounds and robust hedging of installment options. Quant Financ 1:597–610

    Google Scholar 

  • Schachermayer W, Davis M, Tompkins R (2004) The evaluation of venture capital as an instalment option: valuing real options using real options. Real options. Springer, pp 77–96

    Google Scholar 

  • Tie Li G, Kou Y. Peng, Yu Philip S (2021) An integrated cluster detection, optimization, and interpretation approach for financial data. IEEE Trans Cybern 52(12):13848–13861

    Google Scholar 

  • Yang H (2010) A numerical analysis of American options with regime switching. J Sci Comput 44(1):69–91

    Article  MathSciNet  Google Scholar 

  • Yang Z, Yi F (2009) Valuation of European installment put option: variational inequality approach. Commun Contemp Math 11(02):279–307

    Article  MathSciNet  Google Scholar 

  • Yi F, Yang Z, Wang X (2008) A variational inequality arising from European installment call options pricing. SIAM J Math Anal 40:306–326

    Article  MathSciNet  Google Scholar 

  • Yüksel S, Kou G, Dinçer H (2022) Inventive problem-solving map of innovative carbon emission strategies for solar energy-based transportation investment projects. Appl Energy 311:118680

    Article  Google Scholar 

  • Zhang W, Zhou ZQ, Li J, Xiong X (2022) Government intervention model based on behavioral heterogeneity for China’s stock market. Financ Innov 8(1):1–19

    Google Scholar 

  • Zhu SP, Pasricha P, He XJ (2022) A closed-form pricing formula for European options in an illiquid asset market. Financ Innov 8(1):30

    Article  Google Scholar 

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Acknowledgements

This research was supported by Shahid Beheshti University.

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  1. Department of Actuarial Science, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    Saghar Heidari

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  1. Saghar Heidari
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Heidari, S. A finite element method for pricing of continuous-installment options under a Markov-modulated model: existence, uniqueness, and stability of solutions. Soft Comput 28, 3341–3351 (2024). https://doi.org/10.1007/s00500-023-08601-2

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