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On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach
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 |
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.
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. |
[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. |
[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. |
[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. |
[5] |
A. Dassios and J. Nagaradjasarma,
The square-root process and Asian options, Quant. Finance, 6 (2006), 337-347.
doi: 10.1080/14697680600724775. |
[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. |
[7] |
A. Eydeland and K. Wolyniec, Energy and Power Risk Management, Hoboken, NJ: John Wiley & Sons, 2003. |
[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.
|
[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. |
[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. |
[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. |
[12] |
N. Hakansson,
Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
R. C. Merton, Continuous-time Finance, Rev. Ed., Oxford, U.K.: Basil Blackwell, 1990, 1992. |
[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. |
[22] |
F. Venegas-Martínez, Riesgos financieros y económicos, productos derivados y decisiones económicas bajo incertidumbre, Segunda edición, Cengage, México, 2008. |
[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. |
[24] |
M. Yor,
On some exponential functionals of Brownian motion, Adv. in Appl. Probab., 24 (1992), 509-531.
doi: 10.2307/1427477. |
[25] |
M. Yor,
Sur certaines fonctionnelles exponentielles du mouvement brownien réel, J. Appl. Probab., 29 (1992b), 202-208.
doi: 10.2307/3214805. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
A. Dassios and J. Nagaradjasarma,
The square-root process and Asian options, Quant. Finance, 6 (2006), 337-347.
doi: 10.1080/14697680600724775. |
[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. |
[7] |
A. Eydeland and K. Wolyniec, Energy and Power Risk Management, Hoboken, NJ: John Wiley & Sons, 2003. |
[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.
|
[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. |
[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. |
[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. |
[12] |
N. Hakansson,
Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
R. C. Merton, Continuous-time Finance, Rev. Ed., Oxford, U.K.: Basil Blackwell, 1990, 1992. |
[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. |
[22] |
F. Venegas-Martínez, Riesgos financieros y económicos, productos derivados y decisiones económicas bajo incertidumbre, Segunda edición, Cengage, México, 2008. |
[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. |
[24] |
M. Yor,
On some exponential functionals of Brownian motion, Adv. in Appl. Probab., 24 (1992), 509-531.
doi: 10.2307/1427477. |
[25] |
M. Yor,
Sur certaines fonctionnelles exponentielles du mouvement brownien réel, J. Appl. Probab., 29 (1992b), 202-208.
doi: 10.2307/3214805. |
[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. |
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