Lookback option pricing for regime-switching jump diffusion models
Zhuo Jin Linyi Qian
In this paper, we will introduce a numerical method to price the European lookback floating strike put options where the underlying asset price is modeled by a generalized regime-switching jump diffusion process. In the Markov regime-switching model, the option value is a solution of a coupled system of nonlinear integro-differential partial differential equations. Due to the complexity of regime-switching model, the jump process involved, and the nonlinearity, closed-form solutions are virtually impossible to obtain. We use Markov chain approximation techniques to construct a discrete-time Markov chain to approximate the option value. Convergence of the approximation algorithms is proved. Examples are presented to demonstrate the applicability of the numerical methods.
keywords: floating strike Lookback option Markov chain approximation.
Optimal liability ratio and dividend payment strategies under catastrophic risk
Linyi Qian Lyu Chen Zhuo Jin Rongming Wang

This paper investigates the optimal strategies for liability management and dividend payment in an insurance company. The surplus process is jointly determined by the reinsurance policies, liability levels, future claims and unanticipated shocks. The decision maker aims to maximize the total expected discounted utility of dividend payment in infinite time horizon. To describe the extreme scenarios when catastrophic events occur, a jump-diffusion Cox-Ingersoll-Ross process is adopted to capture the substantial claim rate hikes. Using dynamic programming principle, the value function is the solution of a second-order integro-differential Hamilton-Jacobi-Bellman equation. The subsolution--supersolution method is used to verify the existence of classical solutions of the Hamilton-Jacobi-Bellman equation. The optimal liability ratio and dividend payment strategies are obtained explicitly in the cases where the utility functions are logarithm and power functions. A numerical example is provided to illustrate the methodologies and some interesting economic insights.

keywords: Optimal strategy nonlinear partial differential equation dynamic programming dividend payment catastrophic risk
Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model
Linyi Qian Wei Wang Rongming Wang
This paper extends the model in Riesner (2007) to a Markov modulated Lévy process. The parameters of the Lévy process switch over time according to the different states of an economy, which is described by a finite-state continuous time Markov chain. Employing the local risk minimization method, we find an optimal hedging strategy for a general payment process. Finally, we give an example for single unit-linked insurance contracts with guarantee to display the specific locally risk-minimizing hedging strategy.
keywords: Unit-linked life insurance regime switching locally risk-minimizing strategy. Lévy process
Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model
Wei Wang Linyi Qian Xiaonan Su
In this paper, we consider pricing and hedging of catastrophe equity put options under a Markov-modulated jump diffusion process with a Markov switching compensator. We assume that the risk free interest rate, the appreciation rate and the volatility of the risky asset depend on a finite-state Markov chain. We investigate the pricing of catastrophe equity put options and obtain the explicit pricing formulas. A numerical analysis is provided to illustrate the effect of regime switching on the price of catastrophe equity put options. In the end, since the market which we consider is not complete, we also provide an optimal hedging strategy by using the local risk minimization method.
keywords: option pricing. Markov-modulated Local risk minimization

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