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 , 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.
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.
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.
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  and Davydov et al. . 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.
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
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.
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.
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
This paper deals with the optimal liability and dividend strategies for an insurance company in Markov regime-switching models. The objective is to maximize the total expected discounted utility of dividend payment in the infinite time horizon in the logarithm and power utility cases, respectively. The switching process, which is interpreted by a hidden Markov chain, is not completely observable. By using the technique of the Wonham filter, the partially observed system is converted to a completely observed one and the necessary information is recovered. The upper-lower solution method is used to show the existence of classical solution of the associated second-order nonlinear Hamilton-Jacobi-Bellman equation in the two-regime case. The explicit solution of the value function is derived and the corresponding optimal dividend policies and liability ratios are obtained. In the multi-regime case, a general setting of the Wonham filter is presented, and the value function is proved to be a viscosity solution of the associated system of Hamilton-Jacobi-Bellman equations.