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January  2019, 6(1): 53-64. doi: 10.3934/jdg.2019004

On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach

María Teresa V. Martínez-Palacios , , Adrián Hernández-Del-Valle and  Ambrosio Ortiz-Ramírez

Escuela Superior de Economía, Instituto Politécnico Nacional, Plan de Agua Prieta no. 66, Col. Plutarco Elías Calles, Delegación Miguel Hidalgo, Ciudad de México, C.P. 11340, México

* Corresponding author: María Teresa V. Martínez-Palacios

Received  September 2018 Revised  November 2018 Published  January 2019

Fund Project: We thank the anonymous referee for his useful suggestions.

In this work, through stochastic optimal control in continuous time the optimal decision making in consumption and investment is modeled by a rational economic agent, representative of an economy, who is a consumer and an investor adverse to risk; this in a finite time horizon of stochastic length. The assumptions of the model are: a consumption function of HARA type, a representative company that has a stochastic production process, the agent invests in a stock and an American-style Asian put option with floating strike equal to the geometric average subscribed on the stock, both modeled by controlled Markovian processes; as well as the investment of a principal in a bank account. The model is solved with dynamic programming in continuous time, particularly the Hamilton-Jacobi-Bellman PDE is obtained, and a function in separable variables is proposed as a solution to set the optimal trajectories of consumption and investment. In the solution analysis is determined: in equilibrium, the process of short interest rate that is driven by a square root process with reversion to the mean; and through a system of differential equations of risk premiums, a PDE is deduced equivalent to the Black-Scholes-Merton but to value an American-style Asian put option.

Keywords: Portfolio choice, American-style Asian option pricing, stochastic interest rate, optimal stochastic control, geometric mean.
Mathematics Subject Classification: Primary: 91G10, 91G20, 91G30, 93E20; Secondary: 26E60.
Citation: María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004
References:
[1]

H. Ben-Ameur, M. Breton and P. L'Ecuyer, A dynamic programming procedure for pricing american-style asian options, Management Science, 48 (2002), 625-643. Retrieved from http://www.jstor.org/stable/822502 doi: 10.1287/mnsc.48.5.625.7803.  Google Scholar

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D. Brigo and F. Mercurio, Interest Rate Models: Theory and Practice, With smile, inflation, and credit, (2$^{nd}$ ed.), Springer Finance. Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-34604-3.  Google Scholar

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F. Chen and F. Weiyin, Optimal control of markovian switching systems with applications to portfolio decisions under inflation, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 439-458.  doi: 10.1016/S0252-9602(15)60014-5.  Google Scholar

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J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar

[5]

A. Dassios and J. Nagaradjasarma, The square-root process and Asian options, Quant. Finance, 6 (2006), 337-347.  doi: 10.1080/14697680600724775.  Google Scholar

[6]

K. D. Dingeç, H. Sak and W. Hörmann, Variance reduction for Asian options under a general model framework, Rev. Finance, 19 (2015), 907-949.  doi: 10.1093/rof/rfu005.  Google Scholar

[7]

A. Eydeland and K. Wolyniec, Energy and Power Risk Management, Hoboken, NJ: John Wiley & Sons, 2003. Google Scholar

[8]

V. Fanelli, L. Maddalena and S. Musti, Asian options pricing in the day-ahead electricity market, Sustainable Cities and Society, 27 (2016), 196-202.   Google Scholar

[9]

A. Farhadi, G. H. Erjaee and M. Salehi, Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications, Comput. Math. Appl., 73 (2017), 2066-2075.  doi: 10.1016/j.camwa.2017.02.031.  Google Scholar

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S. Gounden and J. G. O'Hara, An analytic formula for the price of an American-style Asian option of floating strike type, Appl. Math. Comput., 217 (2010), 2923-2936.  doi: 10.1016/j.amc.2010.08.025.  Google Scholar

[11]

P. Guasoni and S. Robertson, Optimal importance sampling with explicit formulas in continuous time, Finance Stoch., 12 (2008), 1-19.  doi: 10.1007/s00780-007-0053-5.  Google Scholar

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N. Hakansson, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607.   Google Scholar

[13]

B. Jourdain and M. Sbai, Exact retrospective Monte Carlo computation of arithmetic average Asian options, Monte Carlo Methods Appl., 13 (2007), 135-171.  doi: 10.1515/mcma.2007.008.  Google Scholar

[14]

A. G. Z. Kemna and A. C. F. Vorst, A pricing method for options based on average asset values, Journal of Banking and Finance, 14 (1990), 113-129.  doi: 10.1016/0378-4266(90)90039-5.  Google Scholar

[15]

B. Kim and I.-S. Wee, Pricing of geometric Asian options under Heston's stochastic volatility model, Quant. Finance, 14 (2014), 1795-1809.  doi: 10.1080/14697688.2011.596844.  Google Scholar

[16]

D. M. Marcozzi, Optimal control of ultradiffusion processes with application to mathematical finance, Int. J. Comput. Math., 92 (2015), 296-318.  doi: 10.1080/00207160.2014.890714.  Google Scholar

[17]

M. T. Martínez-Palacios, A. Ortiz-Ramírez and J. F. Martínez-Sánchez, Valuación de opciones asiáticas con precio de ejercicio flotante igual a la media aritmética: Un enfoque de control óptimo estocástico, Revista Mexicana de Economía y Finanzas (REMEF), 12 (2017), 389-404.  doi: 10.21919/remef.v12i4.240.  Google Scholar

[18]

M. T. Martínez-Palacios, J. F. Martínez-Sánchez and F. Venegas-Martínez, Consumption and portfolio decisions of a rational agent that has access to an American put option on an underlying asset with stochastic volatility, Int. J. Pure Appl. Math., 102 (2015), 711-732.  doi: 10.12732/ijpam.v102i4.10.  Google Scholar

[19]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[20]

R. C. Merton, Continuous-time Finance, Rev. Ed., Oxford, U.K.: Basil Blackwell, 1990, 1992. Google Scholar

[21]

E. Russo and A. Staino, On pricing Asian options under stochastic volatility, The Journal of Derivatives, 23 (2016), 7-19.  doi: 10.3905/jod.2016.23.4.007.  Google Scholar

[22]

F. Venegas-Martínez, Riesgos financieros y económicos, productos derivados y decisiones económicas bajo incertidumbre, Segunda edición, Cengage, México, 2008. Google Scholar

[23]

W. Yan, Optimal portfolio of continuous-time mean-variance model with futures and options, Optimal Control Appl. Methods, 39 (2018), 1220-1242.  doi: 10.1002/oca.2404.  Google Scholar

[24]

M. Yor, On some exponential functionals of Brownian motion, Adv. in Appl. Probab., 24 (1992), 509-531.  doi: 10.2307/1427477.  Google Scholar

[25]

M. Yor, Sur certaines fonctionnelles exponentielles du mouvement brownien réel, J. Appl. Probab., 29 (1992b), 202-208.  doi: 10.2307/3214805.  Google Scholar

[26]

L. Weiping and C. Su, Pricing and hedging of arithmetic Asian options via the Edgeworth series expansion approach, Journal of Finance and Data Science, 2 (2016), 1-25.  doi: 10.1016/j.jfds.2016.01.001.  Google Scholar

show all references

References:
[1]

H. Ben-Ameur, M. Breton and P. L'Ecuyer, A dynamic programming procedure for pricing american-style asian options, Management Science, 48 (2002), 625-643. Retrieved from http://www.jstor.org/stable/822502 doi: 10.1287/mnsc.48.5.625.7803.  Google Scholar

[2]

D. Brigo and F. Mercurio, Interest Rate Models: Theory and Practice, With smile, inflation, and credit, (2$^{nd}$ ed.), Springer Finance. Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-34604-3.  Google Scholar

[3]

F. Chen and F. Weiyin, Optimal control of markovian switching systems with applications to portfolio decisions under inflation, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 439-458.  doi: 10.1016/S0252-9602(15)60014-5.  Google Scholar

[4]

J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar

[5]

A. Dassios and J. Nagaradjasarma, The square-root process and Asian options, Quant. Finance, 6 (2006), 337-347.  doi: 10.1080/14697680600724775.  Google Scholar

[6]

K. D. Dingeç, H. Sak and W. Hörmann, Variance reduction for Asian options under a general model framework, Rev. Finance, 19 (2015), 907-949.  doi: 10.1093/rof/rfu005.  Google Scholar

[7]

A. Eydeland and K. Wolyniec, Energy and Power Risk Management, Hoboken, NJ: John Wiley & Sons, 2003. Google Scholar

[8]

V. Fanelli, L. Maddalena and S. Musti, Asian options pricing in the day-ahead electricity market, Sustainable Cities and Society, 27 (2016), 196-202.   Google Scholar

[9]

A. Farhadi, G. H. Erjaee and M. Salehi, Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications, Comput. Math. Appl., 73 (2017), 2066-2075.  doi: 10.1016/j.camwa.2017.02.031.  Google Scholar

[10]

S. Gounden and J. G. O'Hara, An analytic formula for the price of an American-style Asian option of floating strike type, Appl. Math. Comput., 217 (2010), 2923-2936.  doi: 10.1016/j.amc.2010.08.025.  Google Scholar

[11]

P. Guasoni and S. Robertson, Optimal importance sampling with explicit formulas in continuous time, Finance Stoch., 12 (2008), 1-19.  doi: 10.1007/s00780-007-0053-5.  Google Scholar

[12]

N. Hakansson, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607.   Google Scholar

[13]

B. Jourdain and M. Sbai, Exact retrospective Monte Carlo computation of arithmetic average Asian options, Monte Carlo Methods Appl., 13 (2007), 135-171.  doi: 10.1515/mcma.2007.008.  Google Scholar

[14]

A. G. Z. Kemna and A. C. F. Vorst, A pricing method for options based on average asset values, Journal of Banking and Finance, 14 (1990), 113-129.  doi: 10.1016/0378-4266(90)90039-5.  Google Scholar

[15]

B. Kim and I.-S. Wee, Pricing of geometric Asian options under Heston's stochastic volatility model, Quant. Finance, 14 (2014), 1795-1809.  doi: 10.1080/14697688.2011.596844.  Google Scholar

[16]

D. M. Marcozzi, Optimal control of ultradiffusion processes with application to mathematical finance, Int. J. Comput. Math., 92 (2015), 296-318.  doi: 10.1080/00207160.2014.890714.  Google Scholar

[17]

M. T. Martínez-Palacios, A. Ortiz-Ramírez and J. F. Martínez-Sánchez, Valuación de opciones asiáticas con precio de ejercicio flotante igual a la media aritmética: Un enfoque de control óptimo estocástico, Revista Mexicana de Economía y Finanzas (REMEF), 12 (2017), 389-404.  doi: 10.21919/remef.v12i4.240.  Google Scholar

[18]

M. T. Martínez-Palacios, J. F. Martínez-Sánchez and F. Venegas-Martínez, Consumption and portfolio decisions of a rational agent that has access to an American put option on an underlying asset with stochastic volatility, Int. J. Pure Appl. Math., 102 (2015), 711-732.  doi: 10.12732/ijpam.v102i4.10.  Google Scholar

[19]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[20]

R. C. Merton, Continuous-time Finance, Rev. Ed., Oxford, U.K.: Basil Blackwell, 1990, 1992. Google Scholar

[21]

E. Russo and A. Staino, On pricing Asian options under stochastic volatility, The Journal of Derivatives, 23 (2016), 7-19.  doi: 10.3905/jod.2016.23.4.007.  Google Scholar

[22]

F. Venegas-Martínez, Riesgos financieros y económicos, productos derivados y decisiones económicas bajo incertidumbre, Segunda edición, Cengage, México, 2008. Google Scholar

[23]

W. Yan, Optimal portfolio of continuous-time mean-variance model with futures and options, Optimal Control Appl. Methods, 39 (2018), 1220-1242.  doi: 10.1002/oca.2404.  Google Scholar

[24]

M. Yor, On some exponential functionals of Brownian motion, Adv. in Appl. Probab., 24 (1992), 509-531.  doi: 10.2307/1427477.  Google Scholar

[25]

M. Yor, Sur certaines fonctionnelles exponentielles du mouvement brownien réel, J. Appl. Probab., 29 (1992b), 202-208.  doi: 10.2307/3214805.  Google Scholar

[26]

L. Weiping and C. Su, Pricing and hedging of arithmetic Asian options via the Edgeworth series expansion approach, Journal of Finance and Data Science, 2 (2016), 1-25.  doi: 10.1016/j.jfds.2016.01.001.  Google Scholar

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