TY - JOUR
T1 - On the pricing of Asian options with geometric average of American type with stochastic interest rate
T2 - A stochastic optimal control approach
AU - Martínez-Palacios, María Teresa V.
AU - Hernández-Del-Valle, Adrián
AU - Ortiz-Ramírez, Ambrosio
N1 - Publisher Copyright:
© American Institute of Mathematical Sciences.
PY - 2019
Y1 - 2019
N2 - 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 oating 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.
AB - 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 oating 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.
KW - American-style Asian option pricing
KW - Geometric mean
KW - Optimal stochastic control
KW - Portfolio choice
KW - Stochastic interest rate
UR - http://www.scopus.com/inward/record.url?scp=85062513572&partnerID=8YFLogxK
U2 - 10.3934/jdg.2019004
DO - 10.3934/jdg.2019004
M3 - Artículo
SN - 2164-6074
VL - 6
SP - 53
EP - 64
JO - Journal of Dynamics and Games
JF - Journal of Dynamics and Games
IS - 1
ER -