Tak Kuen Siu

Bond pricing under a Markovian regime-switching jump-augmented Vasicek model via stochastic flows

In this article, we shall explore the state of art of stochastic flows to derive an exponential affine form of the bond price when the short rate process is governed by a Markovian regime-switching jump-diffusion version of the Vasicek model. We provide the flexibility that the market parameters, including the mean-reversion level, the volatility rate and the intensity of the jump component switch over time according to a continuous-time, finite-state Markov chain. The states of the chain may be interpreted as different states of an economy or different stages of a business cycle. We shall provide a representation for the exponential affine form of the bond price in terms of fundamental matrix solutions of linear matrix differential equations.

A stochastic differential game for optimal investment of an insurer with regime switching

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.

A Stochastic Maximum Principle for a Markov Regime-Switching Jump-Diffusion Model and Its Application to Finance

This paper develops a sufficient stochastic maximum principle for a stochastic optimal control problem, where the state process is governed by a continuous-time Markov regime-switching jump-diffusion model. We also establish the relationship between the stochastic maximum principle and the dynamic programming principle in a Markovian case. Applications of the stochastic maximum principle to the mean-variance portfolio selection problem are discussed.

Optimal portfolios with regime switching and value-at-risk constraint

We consider the optimal portfolio selection problem subject to a maximum value-at-Risk (MVaR) constraint when the price dynamics of the risky asset are governed by a Markov-modulated geometric Brownian motion (GBM). Here, the market parameters including the market interest rate of a bank account, the appreciation rate and the volatility of the risky asset switch over time according to a continuous-time Markov chain, whose states are interpreted as the states of an economy. The MVaR is defined as the maximum value of the VaRs of the portfolio in a short time duration over different states of the chain. We formulate the problem as a constrained utility maximization problem over a finite time horizon. By utilizing the dynamic programming principle, we shall first derive a regime-switching Hamilton-Jacobi-Bellman (HJB) equation and then a system of coupled HJB equations. We shall employ an efficient numerical method to solve the system of coupled HJB equations for the optimal constrained portfolio. We shall provide numerical results for the sensitivity analysis of the optimal portfolio, the optimal consumption and the VaR level with respect to model parameters. These results are also used to investigating the effect of the switching regimes.

Bayesian Risk Measures for Derivatives via Random Esscher Transform

Abstract This paper proposes a model for measuring risks for derivatives that is easy to implement and satisfies a set of four coherent properties introduced in Artzner et al. (1999). We construct our model within the context of Gerber-Shiu’s option-pricing framework. A new concept, namely Bayesian Esscher scenarios, which extends the concept of generalized scenarios, is introduced via a random Esscher transform. Our risk measure involves the use of the risk-neutral Bayesian Esscher scenario for pricing and a family of real-world Bayesian Esscher scenarios for risk measurement. Closed-form expressions for our risk measure can be obtained in some special cases.

Option Pricing and Filtering with Hidden Markov-Modulated Pure-Jump Processes

Abstract This article discusses the pricing of derivatives in a continuous-time, hidden Markov-modulated, pure-jump asset price model. The hidden Markov chain modulating the pure-jump asset price model describes the evolution of the hidden state of an economy over time. The market model is incomplete. We employ a version of the Esscher transform to select a price kernel for valuation. We derive a valuation formula for European options using a Fourier transform and the correlation theorem. This formula depends on the hidden Markov chain. It is then estimated using a robust filter of the chain.

A distributed decision making model for risk management of virtual enterprise

Risk management in a Virtual Enterprise (VE) is an important issue due to its agility and diversity of its members and its distributed characteristics. In this paper, we develop a risk management model for the VE. More specifically, we introduce a Distributed Decision Making (DDM) model for risk management of the VE. The model has two levels, namely, the top model and the base model, which describe the decision processes of the owner and the partners of the VE, respectively. It can be regarded as a combination of both the top-down and bottom-up approaches for risk management of the VE. Here we focus on the case of symmetry information between the owner and the partners. A Particle Swarm Optimization (PSO) algorithm is then designed to solve the resulting optimization problem. The result shows that the proposed algorithm is effective and the two-level model can help improve the description of the relationship between the owner and the partners, which is helpful to reduce the risk of the VE.

OPTION PRICING FOR GARCH MODELS WITH MARKOV SWITCHING

In this paper we develop a method for pricing derivatives under a Markov switching version of the Heston-Nandi GARCH (1, 1) model by using a well known tool from actuarial science, namely the Esscher transform. We suppose that the dynamics of the GARCH process switch over time according to one of the regimes described by the states of an observable Markov chain process. By augmenting the conditional Esscher transform with the observable Markov switching process, a Markov switching conditional Esscher transform (MSCET) is developed to identify a martingale measure for option valuation in the incomplete market described by our model. We provide an alternative approach for the derivation of an analytical option valuation formula under the Markov switching Heston-Nandi GARCH (1, 1) model. The use of the MSCET can be justified by considering a utility maximization problem with respect to a power utility function associated with the Markov switching risk-averse parameters.

On pricing barrier options with regime switching

We consider the valuation of both European-style and American-style barrier options in a Markovian, regime-switching, Black-Scholes-Merton economy, where the price process of an underlying risky asset is governed by a Markovian, regime-switching, geometric Brownian motion. Both the probabilistic and partial differential equation (PDE), approaches are used to price the barrier options. For the probabilistic approach to value a European-style barrier option, we employ the fundamental matrix solution and the Fourier transform space to derive a (semi)-analytical solution. The PDE approach is employed to value an American barrier option, where we obtain a system of free-boundary, coupled PDEs and an analytical quadratic approximation to the price by solving the free-boundary problem.

OPTIMAL MIXED IMPULSE-EQUITY INSURANCE CONTROL PROBLEM WITH REINSURANCE ∗

We investigate an optimal financing and dividend control problem of an insurance company facing fixed and proportional transaction costs. The goal of the company is to maximize the expected present value of future dividends after deduction of the equity issuance until the time of bankruptcy. We formulate the problem as a mixed classical-impulse control and discuss the problem using the HJB dynamic programming approach. A viscosity solution is considered and its uniqueness is established. We also give results for the regularity and structure of the value function and the optimal policy of the control problem.