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

doi:10.3934/jdg.2019004

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2019, Volume 6, Issue 1: 53-64. Doi: 10.3934/jdg.2019004

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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

* Corresponding author: María Teresa V. Martínez-Palacios
* Corresponding author: María Teresa V. Martínez-Palacios

Received: September 2018

Revised: November 2018

Published: January 2019

We thank the anonymous referee for his useful suggestions.

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Abstract

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.

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Mathematics Subject Classification: Primary: 91G10, 91G20, 91G30, 93E20; Secondary: 26E60.

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