Markov-modulated Geometric Brownian Motion Research Articles

An exact and explicit formula for pricing lookback options with regime switching

<p style='text-indent:20px;'>This paper investigates the pricing of European-style lookback options when the price dynamics of the underlying risky asset are assumed to follow a Markov-modulated Geometric Brownian motion; that is, the appreciation rate and the volatility of the underlying risky asset depend on states of the economy described by a continuous-time Markov chain process. We derive an exact, explicit and closed-form solution for European-style lookback options in a two-state regime switching model.</p>

The Valuation of a Guaranteed Minimum Maturity Benefit under a Regime-Switching Framework

Global insurance markets have become more sophisticated in recent times in response to the evolving needs of populations that tend to live longer. Policy holders desire the benefits of longevity/mortality protection while taking advantage of investment growth opportunities in equity markets. As a result, insurers incorporate payment guarantees in new insurance products, known as equity-linked contracts, whose values are dependent on prices of risky assets. A guaranteed minimum maturity benefit (GMMB) is now common in many equity-linked contracts. We develop an integrated pricing framework for a GMMB focusing on segregated fund contracts. More specifically, we construct hidden Markov models (HMMs) for a stock index, interest rate, and mortality rate. The dependence between these risk factors is characterized explicitly. We assume that the stock index follows a Markov-modulated geometric Brownian motion and the interest and mortality rates have Markov-modulated affine dynamics. A series of measure changes is employed to obtain a semi-closed-form solution for the GMMB price. A Fourier transform method is applied to numerically approximate the prices more efficiently. Recursive HMM filtering is used in our model calibration. Numerical investigations in our article demonstrate the accuracy of GMMB prices and an extensive analysis is included to systematically examine how risk factors affect the value of a GMMB.

Joint dynamic modeling and option pricing in incomplete derivative-security market

In this study, we evaluate the option prices on two assets under stochastic interest rates when the stochastic process that underlying asset prices follow is depending on a correlated bivariate Markov-modulated geometric Brownian motion model with jump risks. More specifically, we conduct the joint dynamic modeling by identifying two independent compound Poisson processes with the log-normal jump sizes to describe both individual jumps and systematic cojumps. Facilitating the cojumping behavior this way with the time-inhomogeneity of the volatility, option pricing expressions are readily obtainable since the Gerber–Siu’s approach is employed to determine a pricing kernel. The empirical results and numerical illustrations are provided to show the impact of cojumps and stochastic volatilities on option prices.

Optimal investment–consumption strategy with liability and regime switching model under Value-at-Risk constraint

This paper considers the optimal investment–consumption problem with liability subject to a maximum Value-at-Risk (denoted by MVaR) constraint. The model also contains regime-switching market modes, whose states are interpreted as the states of the economy. In each state of economy state, we constrain a VaR value for the portfolio in a short time duration, and MVaR is defined as the maximum value of the VaRs in all economy states. We suppose that both the price dynamics of the risky asset and the liability value process are governed by a Markov-modulated geometric Brownian motion. With the objective of maximizing the discounted utility of consumption, we obtain a system of HJB equations corresponding to the economy states by using the dynamic programming principle. Then, by adopting the techniques of Chen et al. (2010), we get explicit expressions of value functions. Moreover, with the help of Lagrange multiplier method, we derive the optimal investment and the optimal consumption. Finally, a numerical example is investigated, and the effects of many parameters on the optimal investment, on the optimal consumption and on VaR value are studied. Furthermore, we also explore how VaR value affects the optimal investment and the optimal consumption.

Saddlepoint approximations to option price in a regime-switching model

In this paper we consider the saddlepoint approximation for the valuation of a European-style call option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. The standard option pricing procedure under this model becomes problematic as the occupation time of chains for a given state cannot be evaluated easily. In the case of two-state Markov chains, we present an explicit analytic formula of the cumulant generating function (CGF). When the process has more than two states, an approximate formula of the CGF is provided. We adopt a splitting method to reduce the complexity of computing the matrix exponential function. Then we use these CGFs to develop the use of the saddlepoint approximations. The numerical results show that the saddlepoint approximation is an efficient and reliable approach for option pricing under a multi-state regime-switching model.

Nonzero-Sum Stochastic Differential Portfolio Games under a Markovian Regime Switching Model

We consider a nonzero-sum stochastic differential portfolio game problem in a continuous-time Markov regime switching environment when the price dynamics of the risky assets are governed by a Markov-modulated geometric Brownian motion (GBM). The market parameters, including the bank interest rate and the appreciation and volatility rates of the risky assets, switch over time according to a continuous-time Markov chain. We formulate the nonzero-sum stochastic differential portfolio game problem as two utility maximization problems of the sum process between two investors’ terminal wealth. We derive a pair of regime switching Hamilton-Jacobi-Bellman (HJB) equations and two systems of coupled HJB equations at different regimes. We obtain explicit optimal portfolio strategies and Feynman-Kac representations of the two value functions. Furthermore, we solve the system of coupled HJB equations explicitly in a special case where there are only two states in the Markov chain. Finally we provide comparative statics and numerical simulation analysis of optimal portfolio strategies and investigate the impact of regime switching on optimal portfolio strategies.

Portfolio Optimization in the Financial Market with Correlated Returns Under Constraints, Transaction Costs and Different Rates for Borrowing and Lending

In this work, we consider the optimal portfolio selection problem under hard constraints on trading amounts, transaction costs and different rates for borrowing and lending when the dynamics of the risky asset returns are governed by a discrete-time approximation of the Markov-modulated geometric Brownian motion. The states of Markov chain are interpreted as the states of an economy. The problem is stated as a dynamic tracking problem of a reference portfolio with desired return. Our approach is tested on a set of a real data from Russian Stock Exchange MICEX.

An explicit analytic formula for pricing barrier options with regime switching

This paper investigates the valuation of a European-style barrier option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. An explicit analytic solution in infinite series form for the price of a European-style barrier option in a two-state regime is presented.

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.

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two-stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.

A PDE approach for risk measures for derivatives with regime switching

This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.

Option pricing and Esscher transform under regime switching

We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).

Fair valuation of participating policies with surrender options and regime switching

We consider the fair valuation of a participating life insurance policy with surrender options when the market values of the asset are modelled by Markov-modulated Geometric Brownian Motion (GBM). We reduce the dimension of the optimal stopping problem for the policy by changing probability measures. We also provide a decomposition result for the value of the policy. The Barone–Adesi–Whaley approximation has been employed to approximate the solution of the free boundary problem for the policy by second-order piecewise linear ordinary differential equations (ODEs). The fair valuation of participating perpetual American contracts are also considered.