Quadrinomial trees with stochastic volatility to value real options


  • Freddy H. Marín-Sánchez Mathematical Sciences, EAFIT University, Medellin, Colombia
  • Julián A. Pareja-Vasseur Finance, EAFIT University, Medellin, Colombia
  • Diego Manzur Economics and Finance, EAFIT University, Medellin, Colombia


Real options, Stochastic volatility, Diffusion processes, GARCH-diffusion, Quadrinomial numerical method


Purpose. The purpose of this article is to propose a detailed methodology to estimate, model and incorporate the non-constant volatility onto a numerical tree scheme, to evaluate a real option, using a quadrinomial multiplicative recombination.

Design/methodology/approach. This article uses the multiplicative quadrinomial tree numerical method with non-constant volatility, based on stochastic differential equations of the GARCH-diffusion type to value real options when the volatility is stochastic.

Findings. Findings showed that in the proposed method with volatility tends to zero, the multiplicative binomial traditional method is a particular case, and results are comparable between these methodologies, as well as to the exact solution offered by the Black–Scholes model.

Originality/value. The originality of this paper lies in try to model the implicit (conditional) market volatility to assess, based on that, a real option using a quadrinomial tree, including into this valuation the stochastic volatility of the underlying asset. The main contribution is the formal derivation of a risk-neutral valuation as well as the market risk premium associated with volatility, verifying this condition via numerical test on simulated and real data, showing that our proposal is consistent with Black and Scholes formula and multiplicative binomial trees method.

DOI: https://doi.org/10.1108/JEFAS-08-2020-0306


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Aı¨t-Sahalia, Y. and Kimmel, R. (2007), “Maximum likelihood estimation of stochastic volatility models”, Journal of Financial Economics, Vol. 83 No. 2, pp. 413-452, doi: 10.1016/j.jfineco.2005.10.006.

Álvarez Echeverría, F., López Sarabia, P. and Venegas-Martínez, F. (2012), “Financial valuation of investment projects in new technologies with real options”, Contaduría y Administración, Vol. 57 No. 3, pp. 115-145, doi: 10.22201/fca.24488410e.2012.400.

Aziz, P.A. (2017), “The implementation of real options theory for economic evaluation in oil and gas field project: case studies in Indonesia”, International Journal of Applied Engineering Research, Vol. 12 No. 24, pp. 15759-15771.

Barone-Adesi, G., Rasmussen, H. and Ravanelli, C. (2005), “An option pricing formula for the GARCH diffusion model”, Computational Statistics and Data Analysis, Vol. 49 No. 2, pp. 287-310, doi: 10.1016/j.csda.2004.05.014.

Black, F. and Scholes, M. (1973), “The pricing of options and corporate liabilities”, Journal of Political Economy, Vol. 81 No. 3, pp. 637-654, doi: 10.1086/260062.

Bollerslev, T. (1986), “Generalized autoregressive conditional heteroskedasticity”, Journal of Econometrics, Vol. 31 No. 3, pp. 307-327, doi: 10.1016/0304-4076(86)90063-1.

Brandão, L., Dyer, J.S. and Hahn, W.J. (2012), “Volatility estimation for stochastic project value models”, European Journal of Operational Research, Vol. 220 No. 3, pp. 642-648, doi: 10.1016/j.ejor.2012.01.059.

Chang, C.C. and Fu, H.C. (2009), “A binomial option pricing model under stochastic volatility and jump”, Canadian Journal of Administrative Sciences, Vol. 18 No. 3, pp. 192-203, doi: 10.1111/j.1936-4490.2001.tb00255.x.

Chesney, M. and Scott, L. (1989), “Pricing European currency options: a comparison of the modified black-scholes model and a random variance model”, The Journal of Financial and Quantitative Analysis, Vol. 24 No. 3, p. 267, doi: 10.2307/2330812.

Chourdakis, K. and Dotsis, G. (2011), “Maximum likelihood estimation of non-affine volatility processes”, Journal of Empirical Finance, Vol. 18 No. 3, pp. 533-545, doi: 10.1016/j.jempfin.2010.10.006.

Christoffersen, P., Jacobs, K. and Mimouni, K. (2010), “Volatility dynamics for the S&P500: evidence from realized volatility, daily returns, and option prices”, Review of Financial Studies, Vol. 23 No. 8, pp. 3141-3189, doi: 10.1093/rfs/hhq032.

Cobb, B.R. and Charnes, J.M. (2004), “Real options volatility estimation with correlated inputs”, The Engineering Economist, Vol. 49 No. 2, pp. 119-137, doi: 10.1080/00137910490453392.

Copeland, T. and Antikarov, V. (2003), Real Options: A Practitioner's Guide, New edition, New York, NY.

Cox, J.C., Ross, S.A. and Rubinstein, M. (1979), “Option pricing: a simplified approach”, Journal of Financial Economics, Vol. 7 No. 3, pp. 229-263, doi: 10.1016/0304-405X(79)90015-1.

Damodaran, A. (2019), Equity Risk Premiums (ERP): Determinants, Estimation and Implications – the 2019 Edition, pp. 1-135.

Dixit, A.K. and Pindyck, R.S. (1994), Investment under Uncertainty, Princeton University Press, Princeton: NJ.

Drost, F.C. and Werker, B.J.M. (1996), “Closing the GARCH gap: continuous time GARCH modeling”, Journal of Econometrics, Vol. 74 No. 1, pp. 31-57, doi: 10.1016/0304-4076(95)01750-X.

Duan, J.C. (1996), “A unified theory of option pricing under stochastic volatility-from GARCH to diffusion”, available at: https://rmi.nus.edu.sg/duanjc/index_files/files/opm_sv.pdf.

Duan, J.C. (1997), “Augmented GARCH (p,q) process and its diffusion limit”, Journal of Econometrics, Vol. 79, pp. 97-172, doi: 10.1016/S0304-4076(97)00009-2.

Fernández Castaño, H. (2007), “EGARCH: an asymmetric model for estimating the volatility of financial series”, Revista Ingenierías Universidad de Medellín, Vol. 9 No. 16, pp. 49-60, available at: http://www.scielo.org.co/pdf/rium/v9n16/v9n16a05.pdf.

Figà-Talamanca, G. (2009), “Testing volatility autocorrelation in the constant elasticity of variance stochastic volatility model”, Computational Statistics and Data Analysis, Vol. 53 No. 6, pp. 2201-2218, doi: 10.1016/j.csda.2008.08.024.

Florescu, I. and Viens, F.G. (2008), “Stochastic volatility: option pricing using a multinomial recombining tree”, Applied Mathematical Finance, Vol. 15 No. 2, pp. 151-181, doi: 10.1080/13504860701596745.

Godinho, P.M.C. (2006), “Monte Carlo estimation of project volatility for real options analysis”, Journal of Applied Finance, Vol. 16 No. 1, pp. 1-34, available at: http://gemf.fe.uc.pt/workingpapers/pdf/2006/gemf06_01.pdf.

Grajales Correa, C.A. and Pérez Ramírez, F.O. (2008), “Discrete and continuous models to estimate the probability density of stochastic volatility of financial series yields”, Cuadernos de Administración, Vol. 21 No. 36, pp. 113-132, available at: http://www.scielo.org.co/pdf/cadm/v21n36/v21n36a06.pdf.

Herath, H.S.B. and Park, C.S. (2002), “Multi-stage capital investment opportunities as compound real options”, The Engineering Economist, Vol. 47 No. 1, pp. 1-27, doi: 10.1080/00137910208965021.

Heston, S.L. (1993), “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Review of Financial Studies, Vol. 6 No. 2, pp. 327-343, doi: 10.1093/rfs/6.2.327.

Hilliard, J.E. and Schwartz, A. (1996), “Binomial option pricing under stochastic volatility and correlated state variables”, Journal of Derivatives, Vol. 4 No. 1, pp. 23-39.

Hull, J. (2003), “Solutions manual options, futures, and other derivatives”, in Options, Futures, and Other Derivatives.

Hull, J. and White, A. (1987), “The Pricing of options on assets with stochastic volatilities”, The Journal of Finance, Vol. 42 No. 2, pp. 281-300, doi: 10.1111/j.1540-6261.1987.tb02568.x.

Jones, C.S. (2003), “The dynamics of stochastic volatility: evidence from underlying and options markets”, Journal of Econometrics, Vol. 116 Nos 1-2, pp. 181-224, doi: 10.1016/S0304-4076(03)00107-6.

Kaeck, A. and Alexander, C. (2012), “Volatility dynamics for the S&P 500: further evidence from non-affine, multi-factor jump diffusions”, Journal of Banking and Finance, Vol. 36 No. 11, pp. 3110-3121, doi: 10.1016/j.jbankfin.2012.07.012.

Keswani, A. and Shackleton, M.B. (2006), “How real option disinvestment flexibility augments project NPV”, European Journal of Operational Research, Vol. 168 No. 1, pp. 240-252, doi: 10.1016/j.ejor.2004.02.028.

Kim, J.S. and Ryu, D. (2015), “Are the KOSPI 200 implied volatilities useful in value-at-risk models?”, Emerging Markets Review, Elsevier B.V., Vol. 22, pp. 43-64, doi: 10.1016/j.ememar.2014.11.001.

Lari-Lavassani, A., Simchi, M. and Ware, A. (2001), “A discrete valuation of swing options”, Canadian Applied Mathematics Quarterly, Vol. 9 No. 1, pp. 37-74, doi: 10.1216/camq/1050519917.

Lewis, N. and Spurlock, D. (2004), “Volatility estimation of forecasted project returns for real options analysis”, American Society for Engineering Management 2004 National Conference, pp. 1-10, available at: https://scholarworks.bridgeport.edu/xmlui/handle/123456789/772.

Mao, X. (1997), in Chichester (Ed.), Stochastic Differential Equations and Applications, Horwood Publishing.

Marín Sánchez, F. (2010), “Binomial trees for evaluating options on processes derived from the autonomous stochastic differential equation”, Ingeniería y Ciencia, Vol. 6 No. 12, pp. 145-170, available at: http://www.scielo.org.co/pdf/ince/v6n12/v6n12a07.pdf.

Moretto, E., Pasquali, S. and Trivellato, B. (2010), “Derivative evaluation using recombining trees under stochastic volatility”, Advances and Applications in Statistical Sciences, Vol. 1 No. 2, pp. 453-480.

Mun, J. (2002), Real Options Analysis: Tools and Techniques for Valuing Strategic Investments and Decisions, John Wiley & Sons.

Nelson, D.B. (1990a), “ARCH models as diffusion approximations”, Journal of Econometrics, Vol. 45 Nos 1-2, pp. 7-38, doi: 10.1016/0304-4076(90)90092-8.

Nelson, D.B. (1990b), “Stationarity and persistence in the GARCH(1,1) model”, Econometric Theory, Vol. 6 No. 3, pp. 318-334, doi: 10.1017/S0266466600005296.

Pareja Vasseur, J. and Cadavid Pérez, C. (2016), “Valuation of pharmaceutical patents through real options: certainty equivalent and utility function”, Contaduría y Administración, Vol. 61 No. 4, pp. 794-814, doi: 10.1016/j.cya.2016.06.004.

Pareja-Vasseur, J.A. and Marín-Sánchez, F.H. (2019), “Quadrinomial trees to value options in stochastic volatility models”, The Journal of Derivatives, Vol. 27 No. 1, pp. 49-66, doi: 10.3905/jod.2019.1.076.

Pareja-Vasseur, J.A., Marin-Sanchez, F.H. and Tuesta Reategui, V. (2020), “GARCH-type volatility in the multiplicative quadrinomial tree method: an application to real options”, Contaduría y Administración, Vol. 66 No. 2, pp. 1-25, doi: 10.22201/fca.24488410e.2021.2331.

Peng, B. and Peng, F. (2016), “Pricing maximum-minimum bidirectional options in trinomial CEV model”, Journal of Economics, Finance and Administrative Science, Vol. 21 No. 41, pp. 50-55, doi: 10.1016/j.jefas.2016.06.001.

Plienpanich, T., Sattayatham, P. and Thao, T.H. (2009), “Fractional integrated GARCH diffusion limit models”, Journal of the Korean Statistical Society, Vol. 38 No. 3, pp. 231-238, doi: 10.1016/j.jkss.2008.10.003.

Ritchken, P. and Trevor, R. (1999), “Pricing options under generalized GARCH and stochastic volatility processes”, The Journal of Finance, Vol. 54 No. 1, pp. 377-402, doi: 10.1111/0022-1082.00109.

Rogers, J. (2002), Strategy, Value and Risk: The Real Options approachReconciling Innovation, Strategy and Value Management, Palgrave Macmillan, Ney York, available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=

Sabet, A.H. and Heaney, R. (2017), “Real options and the value of oil and gas firms: an empirical analysis”, Journal of Commodity Markets, Elsevier B.V., Vol. 6, September 2015, pp. 50-65, doi: 10.1016/j.jcomm.2017.05.001.

Scott, L.O. (1987), “Option pricing when the variance changes randomly: theory, estimation, and an application”, The Journal of Financial and Quantitative Analysis, Vol. 22 No. 4, p. 419, doi: 10.2307/2330793.

Siña, M. and Guzmán, J.I. (2019), “Real option valuation of open pit mines with two processing methods”, Journal of Commodity Markets, Elsevier B.V., Vol. 13, May 2018, pp. 30-39, doi: 10.1016/j.jcomm.2018.05.003.

Smith, J.E. (2005), “Alternative approaches for solving real-options problems”, Decision Analysis, Vol. 2 No. 2, pp. 89-102, doi: 10.1287/deca.1050.0041.

Stein, E.M. and Stein, J.C. (1991), “Stock price distributions with stochastic volatility: an analytic approach”, Review of Financial Studies, Vol. 4 No. 4, pp. 727-752, doi: 10.1093/rfs/4.4.727.

Trigeorgis, L. (1996), R,eal Options: Managerial Flexibility and Strategy in Resource Allocation, The MIT Press, Cambridge, Massachusetts.

Valencia Herrera, H. and Martínez Gándara, E.E. (2009), “Relationship between uncertainty and investment in Mexico, a real options approach”, Revista de Administración, Finanzas y Economía, Vol. 3 No. 2, pp. 70-90, available at: http://alejandria.ccm.itesm.mx/egap/documentos/2009V3A11Valencia-Gandara.pdf.

Vellekoop, M. and Nieuwenhuis, H. (2009), “A tree-based method to price American options in the Heston model”, The Journal of Computational Finance, Vol. 13 No. 1, pp. 1-21, doi: 10.21314/jcf.2009.197.

Wiggins, J.B. (1987), “Option values under stochastic volatility: theory and empirical estimates”, Journal of Financial Economics, Vol. 19 No. 2, pp. 351-372, doi: 10.1016/0304-405X(87)90009-2.

Wilmott, P. (1998), Derivatives. The Theory and Practice of Financial Engineering, Wiley Publishing, Chichester.

Wu, X. and Zhou, H. (2016), “Garch difussion model, iVIX, and volatility risk premium”, Economic Computation and Economic Cybernetics Studies and Research, Vol. 50 No. 1, pp. 327-342.

Wu, X.Y., Ma, C.Q. and Wang, S.Y. (2012), “Warrant pricing under GARCH diffusion model”, Economic Modelling, Vol. 29 No. 6, pp. 2237-2244, doi: 10.1016/j.econmod.2012.06.020.

Wu, X., Yang, W., Ma, C. and Zhao, X. (2014), “American option pricing under GARCH diffusion model: an empirical study”, Journal of Systems Science and Complexity, Vol. 27 No. 1, pp. 193-207, doi: 10.1007/s11424-014-3279-2.

Wu, X., Zhou, H. and Wang, S. (2018), “Estimation of market prices of risks in the G.A.R.C.H. diffusion model”, Economic Research-Ekonomska Istraživanja, Vol. 31 No. 1, pp. 15-36, doi: 10.1080/1331677X.2017.1421989.

Wu, X., Wang, X. and Wang, H. (2020), “Forecasting stock market volatility using implied volatility: evidence from extended realized EGARCH-MIDAS model”, Applied Economics Letters, Routledge, Vol. 28 No. 11, pp. 915-920, doi: 10.1080/13504851.2020.1785617.




How to Cite

Marín-Sánchez, F. H., Pareja-Vasseur, J. A., & Manzur, D. (2021). Quadrinomial trees with stochastic volatility to value real options. Journal of Economics, Finance and Administrative Science, 26(52), 282–299. Retrieved from https://revistas.esan.edu.pe/index.php/jefas/article/view/562