Optimal portfolio in a continuous-time self-exciting threshold model
Hui Meng Fei Lung Yuen Tak Kuen Siu Hailiang Yang
This paper discusses an optimal portfolio selection problem in a continuous-time economy, where the price dynamics of a risky asset are governed by a continuous-time self-exciting threshold model. This model provides a way to describe the effect of regime switching on price dynamics via the self-exciting threshold principle. Its main advantage is to incorporate the regime switching effect without introducing an additional source of uncertainty. A martingale approach is used to discuss the problem. Analytical solutions are derived in some special cases. Numerical examples are given to illustrate the regime-switching effect described by the proposed model.
keywords: logarithmic utility. self-exciting threshold model Portfolio selection power utility regime switching
Optimal reinsurance and investment strategy with two piece utility function
Lv Chen Hailiang Yang

This paper studies optimal reinsurance and investment strategies that maximize expected utility of the terminal wealth for an insurer in a stochastic market. The insurer's preference is represented by a two-piece utility function which can be regarded as a generalization of traditional concave utility functions. We employ martingale approach and convex optimization method to transform the dynamic maximization problem into an equivalent static optimization problem. By solving the optimization problem, we derive explicit expressions of the optimal reinsurance and investment strategy and the optimal wealth process.

keywords: Two piece utility function martingale technique optimal control investment proportional reinsurance
American type geometric step options
Xiaoyu Xing Hailiang Yang
The step option is a special contact whose value decreases gradually in proportional to the spending time outside a barrier of the asset price. European step options were introduced and studied by Linetsky [11] and Davydov et al. [2]. This paper considers American step options, including perpetual case and finite expiration time case. In perpetual case, we find that the optimal exercise time is the first crossing time of the optimal level. The closed price formula for perpetual step option could be derived through Feynman-Kac formula. As for the latter, we present a system of variational inequalities satisfied by the option price. Using the explicit finite difference method we could get the numerical option price.
keywords: finite difference method Feynman-Kac formula Geometric step option variational inequality. optimal exercise level
Optimal investment-consumption strategy in a discrete-time model with regime switching
Ka Chun Cheung Hailiang Yang
This paper analyzes the investment-consumption problem of a risk averse investor in discrete-time model. We assume that the return of a risky asset depends on the economic environments and that the economic environments are ranked and described using a Markov chain with an absorbing state which represents the bankruptcy state. We formulate the investor's decision as an optimal stochastic control problem. We show that the optimal investment strategy is the same as that in Cheung and Yang [5], and a closed form expression of the optimal consumption strategy has been obtained. In addition, we investigate the impact of economic environment regime on the optimal strategy. We employ some tools in stochastic orders to obtain the properties of the optimal strategy.
keywords: Second-order stochastic dominance Bankruptcy risk Bellman equation Dynamic programming Optimal investment-consumption strategy Stochastically monotone Recovery rate.
Option pricing under threshold autoregressive models by threshold Esscher transform
Tak Kuen Siu Howell Tong Hailiang Yang
This paper develops a valuation model for options under the class of self-exciting threshold autoregressive (SETAR) models and their variants for the price dynamics of the underlying asset using the self-exciting threshold autoregressive Esscher transform (SETARET). In particular, we focus on the first generation SETAR models first proposed by Tong (1977, 1978) and later developed in Tong (1980, 1983) and Tong and Lim (1980), and the second generation models, including the SETAR-GARCH model proposed in Tong (1990) and the double-threshold autoregressive heteroskedastic time series model (DTARCH) proposed by Li and Li (1996). The class of SETAR-GARCH models has the advantage of modelling the non-linearity of the conditional first moment and the varying conditional second moment of the financial time series. We adopt the SETARET to identify an equivalent martingale measure for option valuation in the incomplete market described by the discrete-time SETAR models. We are able to justify our choice of probability measure by the SETARET by considering the self-exciting threshold dynamic utility maximization. Simulation studies will be conducted to investigate the impacts of the threshold effect in the conditional mean described by the first generation model and that in the conditional variance described by the second generation model on the qualitative behaviors of the option prices as the strike price varies.
keywords: Setar models infinitely divisible distributions Dtarch models Setaret Setar-Garch models Option valuation threshold dynamic utility maximization.
Optimal financing and dividend strategies in a dual model with proportional costs
Dingjun Yao Hailiang Yang Rongming Wang
We consider the optimal control problem with dividend payments and issuance of equity in a dual risk model. Such a model might be appropriate for a company that specializes in inventions and discoveries, which pays costs continuously and has occasional profits. Assuming proportional transaction costs, we aim at finding optimal strategy which maximizes the expected present value of the dividends payout minus the discounted costs of issuing new equity before bankruptcy. By adopting some of the techniques and methodologies in L$\phi$kka and Zervos (2008), we construct two categories of suboptimal models, one is the ordinary dual model without issuance of equity, the other one assumes that, by issuing new equity, the company never goes bankrupt. We identify the value functions and the optimal strategies corresponding to the suboptimal models in two different cases. For exponentially distributed jump sizes, closed-form solutions are obtained.
keywords: Hamilton-Jacobi-Bellman equation. The dual risk model dividend payment optimal strategy proportional transaction costs equity issuance
Heung Wong Hailiang Yang Xian Zhou
This special issue is based on, but not limited to, contributions from invited speakers of the International Workshop in Financial Mathematics and Statistics held at the Hong Kong Polytechnic University, on December 16, 2004. The workshop was well attended by experts in the field all over the world.
    The issue aims to look at leading-edge research on the interface between derivatives, insurance, securities and quantitative finance. As financial mathematics and statistics are two essential components in these four areas, the issue, as we hope, will give the readers a survey of the important tools of mathematics and statistics being used in the modern financial institutions.
    In this special issue, 7 papers are included. The papers cover mathematical finance topics, such as option pricing, interest models and stochastic volatility; topics in risk management, such as Value at Risk, liquidity risk management; and actuarial science topics, such as ruin theory.
    The papers in the issue were selected with a view towards readers coming from finance, actuarial science, mathematics or statistics. Hopefully this is a first step to provide a platform for people who are interested in the interplay among theory and practice of these disciplines.
keywords: optimization. Management
Numerical methods for dividend optimization using regime-switching jump-diffusion models
Zhuo Jin George Yin Hailiang Yang
This work develops numerical methods for finding optimal dividend policies to maximize the expected present value of dividend payout, where the surplus follows a regime-switching jump diffusion model and the switching is represented by a continuous-time Markov chain. To approximate the optimal dividend policies or optimal controls, we use Markov chain approximation techniques to construct a discrete-time controlled Markov chain with two components. Under simple conditions, we prove the convergence of the approximation sequence to the surplus process and the convergence of the approximation to the value function. Several examples are provided to demonstrate the performance of the algorithms.
keywords: stochastic control. Jump diffusion regime switching dividend policy

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