Optimal portfolio and consumption for a Markovian regime-switching jump-diffusion process

2021 ◽  
Vol 63 ◽  
pp. 308-332
Author(s):  
Caibin Zhang ◽  
Zhibin Liang ◽  
Kam Chuen Yuen
Keyword(s):  
Diffusion Process ◽  
Diffusion Model ◽  
Regime Switching ◽  
Jump Diffusion ◽  
Optimal Portfolio ◽  
Dynamic Programming Principle ◽  
Pure Diffusion ◽  
Special Cases ◽  
Jump Diffusion Process ◽  
Jump Diffusion Model

We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases.   doi:10.1017/S1446181121000122

OPTIMAL PORTFOLIO AND CONSUMPTION FOR A MARKOVIAN REGIME-SWITCHING JUMP-DIFFUSION PROCESS

2021 ◽  
pp. 1-25
Author(s):  
CAIBIN ZHANG ◽  
ZHIBIN LIANG ◽  
KAM CHUEN YUEN
Keyword(s):  
Diffusion Process ◽  
Diffusion Model ◽  
Regime Switching ◽  
Jump Diffusion ◽  
Optimal Portfolio ◽  
Dynamic Programming Principle ◽  
Pure Diffusion ◽  
Special Cases ◽  
Jump Diffusion Process ◽  
Jump Diffusion Model

Abstract We consider the optimal portfolio and consumption problem for a jump-diffusion process with regime switching. Under the criterion of maximizing the expected discounted total utility of consumption, two methods, namely, the dynamic programming principle and the stochastic maximum principle, are used to obtain the optimal result for the general objective function, which is the solution to a system of partial differential equations. Furthermore, we investigate the power utility as a specific example and analyse the existence and uniqueness of the optimal solution. Under the constraints of no-short-selling and nonnegative consumption, closed-form expressions for the optimal strategy and the value function are derived. Besides, some comparisons between the optimal results for the jump-diffusion model and the pure diffusion model are carried out. Finally, we discuss our optimal results in some special cases.

Option Pricing in a Jump-Diffusion Model with Regime Switching

2009 ◽  
Vol 39 (2) ◽  
pp. 515-539 ◽  
Author(s):  
Fei Lung Yuen ◽  
Hailiang Yang
Keyword(s):  
Diffusion Model ◽  
Regime Switching ◽  
Jump Diffusion ◽  
Mathematical Finance ◽  
Extended Model ◽  
Exotic Options ◽  
Switching Model ◽  
Jump Diffusion Model ◽  
Trinomial Tree ◽  
Tree Method

AbstractNowadays, the regime switching model has become a popular model in mathematical finance and actuarial science. The market is not complete when the model has regime switching. Thus, pricing the regime switching risk is an important issue. In Naik (1993), a jump diffusion model with two regimes is studied. In this paper, we extend the model of Naik (1993) to a multi-regime case. We present a trinomial tree method to price options in the extended model. Our results show that the trinomial tree method in this paper is an effective method; it is very fast and easy to implement. Compared with the existing methodologies, the proposed method has an obvious advantage when one needs to price exotic options and the number of regime states is large. Various numerical examples are presented to illustrate the ideas and methodologies.

scholarly journals Optimal Strategies for an Ambiguity-Averse Insurer under a Jump-Diffusion Model and Defaultable Risk

2020 ◽  
Vol 2020 ◽  
pp. 1-26 ◽  
Author(s):  
Man Li ◽  
Yingchun Deng ◽  
Ya Huang ◽  
Hui Ou
Keyword(s):  
Diffusion Process ◽  
Default Risk ◽  
Optimal Strategies ◽  
Jump Diffusion ◽  
Corporate Bond ◽  
Value Functions ◽  
Case Scenario ◽  
Worst Case ◽  
Special Cases ◽  
Jump Diffusion Model

In this paper, we consider a robust optimal investment-reinsurance problem with a default risk. The ambiguity-averse insurer (AAI) may carry out transactions on a risk-free asset, a stock, and a defaultable corporate bond. The stock’s price is described by a jump-diffusion process, and both the jump intensity and the distribution of jump amplitude are uncertain, i.e., the jump is ambiguous. The AAI’s surplus process is assumed to follow an approximate diffusion process. In particular, the reinsurance premium is calculated according to the generalized mean-variance premium principle, and the reinsurance type has to follow a self-reinsurance function. In performing dynamic programming, both the predefault case and the postdefault case are analyzed, and the optimal strategies and the corresponding value functions are derived under the worst-case scenario. Moreover, we give a detailed proof of the verification theorem and give some special cases and numerical examples to illustrate our theoretical results.

scholarly journals An Optimal Portfolio Problem of DC Pension with Input-Delay and Jump-Diffusion Process

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Weixiang Xu ◽  
Jinggui Gao
Keyword(s):  
Diffusion Process ◽  
Control Problem ◽  
Jump Diffusion ◽  
Investment Strategy ◽  
Optimal Portfolio ◽  
Input Delay ◽  
Programming Approach ◽  
Time Interval ◽  
Stochastic Dynamic ◽  
Jump Diffusion Process

In this paper, an optimal portfolio control problem of DC pension is studied where the time interval between the implementation of investment behavior and its effectiveness (hereafter input-delay) is particularly focused. There are two assets available for investment: a risk-free cash bond and a risky stock with a jump-diffusion process. And the wealth process of the pension fund is modeled as a stochastic delay differential equation. To secure a comfortable retirement life for pension members and also avoid excessive risk, the fund managers in this paper aim to minimize the expected value of quadratic deviations between the actual terminal fund scale and a preset terminal target. By applying the stochastic dynamic programming approach and the match method, the optimal portfolio control problem is solved and the closed-form solution is obtained. In addition, an algorithm is developed to calculate the numerical solution of the optimal strategy. Finally, we have performed a sensitivity analysis to explore how the managers’ preset terminal target, the length of input-delay, and the jump intensity of risky assets affect the optimal investment strategy.

scholarly journals Pricing Participating Products under a Generalized Jump-Diffusion Model

2008 ◽  
Vol 2008 ◽  
pp. 1-30 ◽  
Author(s):  
Tak Kuen Siu ◽  
John W. Lau ◽  
Hailiang Yang
Keyword(s):  
Diffusion Model ◽  
Life Insurance ◽  
Regime Switching ◽  
Jump Diffusion ◽  
Martingale Measure ◽  
Practical Implementation ◽  
Switching Models ◽  
Jump Diffusion Model ◽  
Equivalent Martingale ◽  
Markovian Regime Switching

We propose a model for valuing participating life insurance products under a generalized jump-diffusion model with a Markov-switching compensator. It also nests a number of important and popular models in finance, including the classes of jump-diffusion models and Markovian regime-switching models. The Esscher transform is employed to determine an equivalent martingale measure. Simulation experiments are conducted to illustrate the practical implementation of the model and to highlight some features that can be obtained from our model.

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