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The optimal retirement decision is an optimal stopping problem when retirement is irreversible. We investigate the optimal consumption, investment, and retirement decisions when the mean return of a risky asset is unobservable and is estimated by filtering from historical prices. To ensure nonnegativity of the consumption rate and the borrowing constraints on the wealth process of the representative agent, we conduct our analysis using a duality approach. We link the dual problem to American option pricing with stochastic volatility and prove that the duality gap is closed. We then apply our theory to a hidden Markov model for regime-switching mean return with Bayesian learning. We fully characterize the existence and uniqueness of variational inequality in the dual optimal stopping problem, as well as the free boundary of the problem. An asymptotic closed-form solution is derived for optimal retirement timing by small-scale perturbation. We discuss the potential applications of the results to other partial-information settings.

We study a finite horizon optimal contracting problem with limited commitment. A risk-neutral principal enters into an insurance contract with a risk-averse agent who receives a stochastic income stream and cannot commit to keeping the contract. We consider a general concave utility function and a general process. We use the dual approach and the Lagrangian method to solve our optimization problem by transforming the dual problem into an infinite series of optimal stopping problems. We derive the optimal contract by representing the optimal intermediate and terminal payments from the principal to the agent in a closed-form. We show that the contract begins with a low level of payment to the agent and ratchets up the payment if the stochastic income of the agent rises above a pre-specified threshold level. In particular, if the agent’s income follows a geometric Brownian motion, the threshold level is a deterministic decreasing function of time. We also show that the final payment depends on the history of the agent’s income and the sale value of the production facility at the final time .

We study the consumption and portfolio selection problem of a finitely lived agent who derives utility from the stock of durable goods. We show that the agent’s effective relative risk aversion implied by the optimal portfolio tends to decline and approaches zero, as the planning horizon approaches, whereas the agent exhibits constant effective relative risk aversion in the infinite horizon model of Hindy and Huang (1993). The existence of the stock of durable goods acts as buffer stock and induces the highly risk-tolerant attitude. We approach this problem using successive transformations. We transform our problem by applying an isomorphism proposed by Schroder and Skiadas (2002) to a singular control problem involving the choice of a monotone increasing consumption process. We next transform the problem into a dual singular control problem using the dual martingale method. We then transform the dual singular control problem into an optimal stopping problem. We analyze the variational inequality arising from the optimal stopping problem and provide an integral representation of optimal strategies.

In this paper, we study the problem for pricing of American better-of option on two assets. Due to two correlated underlying assets and early-exercise feature which requires two free boundaries to be determined for the option price, this problem is a complex. We propose a new and efficient approach to solve this problem. Mellin transform methods are mainly used to find the pricing formula, and explicit formula for the option price is derived as an integral equation representation. The formula has two free boundaries which are represented by the coupled integral equations. We propose the numerical scheme based on recursive integration method to implement the integral equations and show that our approach with the proposed numerical scheme is accurate and efficient in computing the prices. In addition, we illustrate significant movements on the option prices and two free boundaries with respect to the selected parameters.

In this study, we generalize the results of Arun (The Merton problem with a drawdown constraint on consumption. Working paper, 2013) on the optimal consumption and investment problem of an infinitely lived agent who does not accept her consumption falling below a fixed proportion of her historically highest level, the so-called drawdown constraint on consumption. We extend the results to a general class of utility functions. We use the martingale method to study the dual problem, which involves the choice of a maximum consumption process. The dual problem can be formulated as a two-dimensional singular control problem, with the free boundary depending on a state variable of the maximum process. We establish the duality theorem and provide semi-closed form solutions regarding the optimal strategies. To highlight our methodology, we present some special cases of utility functions that do not allow for the dimension reduction considered in Arun (2013).

In this paper we investigate the optimal consumption and investment problem with stochastic borrowing constraints for a finitely lived agent. To be specific, she faces a credit limit which is a constant fraction of the present value of her stochastic labor income at each time. By using the martingale approach and transformation into an infinite series of optimal stopping problems which has the same characteristic as finding the optimal exercise time of an American option. We recover the value function by establishing a duality relationship and obtain the integral equation representation solution for the optimal consumption and portfolio strategies. Moreover, we provide some numerical illustrations for optimal consumption and investment policies.

In this paper we study the consumption and portfolio selection problem of a finitely-lived economic agent with an early retirement option, that is, the agent can choose her/his early retirement time before a mandatory retirement time. Based on the theoretical results in Yang and Koo (Math Oper Res, 43(4):1378–1404, 2018), we derive an integral equation satisfied by the optimal retirement boundary or free boundary using the Mellin transform technique. We also derive integral equation representations for the optimal consumption-portfolio strategies and the optimal wealth process. By using the recursive integration method, we obtain the numerical solutions for the integral equations and discuss economic implications for the optimal retirement strategies by using numerical solutions.

In this paper we consider a general optimal consumption and portfolio selection problem of a finitely-lived agent whose consumption rate process is subject to time-varying subsistence consumption constraints. That is, her consumption rate should be greater than or equal to some convex, non-decreasing and continuous function of time t. Using martingale duality approach and Feynman–Kac formula, we derive the partial differential equation of the Cauchy problem satisfied by the dual value function. We use the integral transform method for solving this Cauchy problem to obtain the general optimal policies in an explicit form. With constant relative risk aversion and constant absolute risk aversion utility functions we illustrate some numerical results of the optimal policies.

This study examines the valuation of European continuous-installment barrier options with dividend payments, under the classical Black–Scholes model. By using Mellin transform techniques, we derive the analytic integral solutions of European continuous-installment options with two types of barriers, so that the integral representations are decomposed into the value of the European barrier option and the premium of the early exercise installment. In the similar manner, the pricing of European continuous-installment barrier option which includes a rebate provision is investigated. Additionally, we investigate the effect of the free boundaries of European installment barrier options against barrier levels. Finally, by comparing the recursive integration method with the existing finite difference method – which is, in turn, based on the trinomial tree method as a benchmark – we analyze the accuracy and efficiency of the recursive integration method with respect to continuous-installment barrier options.

We study the partial differential equation (PDE) approach for efficient and accurate valuation of a variable annuity (VA) contract with a surrender option. Specifically, using the Laplace–Carson Transform (LCT), we derive an analytic pricing formula for a VA contract with a surrender option which is formulated as a PDE with an optimal surrender boundary. To demonstrate the efficiency and accuracy of our approach, we show that our pricing formula efficiently provides the exact value of a VA contract. Moreover, we compare the performance of three numerical methods for Laplace inversion to find the most efficient method. Among these methods, we found that the Gaver–Stehfest method is the most efficient pricing for a VA contract with a surrender option.

This paper studies the valuation of the American call-option under the Heston model in two regimes, i.e., fast-mean reverting and slow-mean reverting regimes. In the case of the European-style option under the Heston model, a closed-form solution for one-dimensional integration can be derived. However, in the case of the American-style option, it is impossible to obtain a general analytic integral equation for the price. By using singular and regular perturbation techniques introduced by Fouque et al. (Multiscale stochastic volatility for equity, interest-rate and credit derivative, Cambridge University Press, Cambridge, 2011) and the maturity randomization method introduced by Carr (Rev Financ Stud 11:597–626, 1998), we provide an approximate analytic solution of the American call-option and describe a numerical scheme to evaluate the value of this solution. Numerical results show that our method is accurate and efficient compared to the finite-difference method and the Longstaff and Schwartz (Rev Financ Stud 14(1):113–147, 2001) method.

In this paper we investigate an investment problem with partial reversibility proposed by Abel and Eberly [4] in a finite horizon. In this model, a firm can purchase capital at a given price and sell capital at a lower price. This problem can be categorized into a singular control problem and can be formulated as a Hamilton–Jacobi–Bellman(HJB) equation. Based on theoretical results in [10] and the Mellin transform techniques, we derive the coupled integral equations satisfied by the optimal investment and disinvestment boundaries, respectively. By using the recursive integration method, we solve numerically the integral equations and present the optimal investment boundary and disinvestment boundary.

This paper studies an optimal consumption and portfolio strategy with a negative wealth constraint for a finitely-lived economic agent. That is, we allow the individual agent to borrow partially against her/his future labor income during a finite time horizon. We derive the associated Hamilton–Jacobi–Bellman equation and use the Mellin transform to obtain the integral equation representation satisfied by the free boundary. Moreover, we derive an analytic representation for the optimal consumption, wealth, and portfolio, and provide some numerical implications for the optimal strategies.

In this paper we propose a duality approach to solving an optimal consumption-portfolio selection problem in which an economic agent can choose (irreversible) the preference change time in finite horizon. We assume that after preference change, the agent's coefficient of relative risk aversion becomes higher than before. We use the martingale method and study the variational inequality or free boundary problem arising from the dual problem. Based on partial differential equation (PDE) theory, we analytically characterize the variational inequality and establish the duality relationship. Specifically, we prove that the agent will change his/her preference as soon as the agent's wealth reaches the threshold of wealth as a function of time.