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, Ambrosio Ortiz-Ramírez
2019 Journal of Dynamics & Games  
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
more » ... ption 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. 2010 Mathematics Subject Classification. Primary: 91G10, 91G20, 91G30, 93E20; Secondary: 26E60. Key words and phrases. Portfolio choice, American-style Asian option pricing, stochastic interest rate, optimal stochastic control, geometric mean. We thank the anonymous referee for his useful suggestions. * Corresponding author: María Teresa V. Martínez-Palacios. 53 54 MARTÍNEZ-PALACIOS, HERNÁNDEZ-DEL-VALLE AND ORTIZ-RAMÍREZ
doi:10.3934/jdg.2019004 fatcat:r5hp64erhngaxbdiupjn56mu7e