Regime-switching Jump-diffusion Model Research Articles

Real option pricing under the regime-switching model with jumps on a finite time horizon

We consider an irreversible investment decision problem on a finite time horizon where an instantaneous cash flow process of a firm follows a regime-switching jump-diffusion (RSJD) model. The value of a project can be derived from a partial integro-differential equation (PIDE) and then we obtain a closed-form solution of the PIDE. It is proved that the value of the project converges to the solution on an infinite time horizon problem as the lifetime of the project tends to infinity. Moreover, the value function of a real option with a finite expiration date can be evaluated by solving a linear complementarity problem (LCP) under the RSJD model. We apply a 2-step backward differentiation formula (BDF2 method) combined with an operator splitting method to solve the LCP and then it has the second-order convergence rate in the discrete ℓ2-norm with respect to the time and spatial variables. The stability of the proposed method to solve the LCP is also proved in the discrete ℓ2-norm. Finally, a variety of numerical experiments are carried out to determine the optimal investment time and the numerical results on the finite time horizon are analyzed.

Implicit-explicit Runge–Kutta methods for pricing financial derivatives in state-dependent regime-switching jump-diffusion models

Implicit-explicit Runge–Kutta methods for pricing financial derivatives in state-dependent regime-switching jump-diffusion models

Forward starting options pricing under a regime-switching jump-diffusion model with Wishart stochastic volatility and stochastic interest rate

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application.

RBF-FD based some implicit-explicit methods for pricing option under regime-switching jump-diffusion model with variable coefficients

RBF-FD based some implicit-explicit methods for pricing option under regime-switching jump-diffusion model with variable coefficients

Valuing equity-linked guaranteed minimum death benefits with European-style Asian payoffs under a regime switching jump-diffusion model

This paper presents a novel and efficient approach to pricing equity-linked guaranteed minimum death benefits (GMDB) with European-style geometric Asian and arithmetic Asian payoffs. Our method assumes that the underlying asset process follows a regime-switching Lévy model, which captures the key features of market dynamics in the continuous transition of the economy. To derive the approximate value of GMDB products, we employ the complex Fourier series (CFS) expansion method. Our error analysis demonstrates that this approach exhibits an exponential convergence rate. In our numerical experiments, we compare the CFS approach to other Fourier transform methods and Monte Carlo simulation. The results show that our method outperforms the other approaches in terms of both efficiency and accuracy. This paper contributes to the literature on pricing equity-linked GMDB products by proposing a novel and efficient approach based on regime-switching Lévy models and complex Fourier series expansion. The implications of our results may have significant practical implications for the insurance industry and financial markets.

Error analysis of finite difference scheme for American option pricing under regime-switching with jumps

This paper mainly focuses on evaluating American options under regime-switching jump-diffusion models (Merton’s and Kou’s models). An efficient numerical method is designed for the concerned problems. The problem of American option pricing under regime-switching jump-diffusion models can be described as a free-boundary problem or a complementarity problem with integral and differential terms on an unbounded domain. By analyzing the relation of optimal exercise boundaries among several options, we truncate the solving domain of regime-switching jump-diffusion options, and present reasonable boundary conditions. For the integral terms of the truncated model, a composite trapezoidal formula is applied, which guarantees that the integral discretized matrix is a Toeplitz matrix. Meanwhile, a finite difference scheme is proposed for the resulting system, which leads to a linear complementary problem (LCP) with a unique solution. Moreover, we also prove the stability, monotonicity, and consistency of the discretization scheme and estimate the convergence order. In consideration of the characteristics of the discrete matrix, a projection and contraction method is suggested to solve the discretized LCP. Numerical experiments are carried out to verify the efficiency of the proposed scheme.

Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model

<p style='text-indent:20px;'>This paper is devoted to study the open-loop equilibrium strategy for a mean-variance portfolio problem with investment constraints in a non-Markovian regime-switching jump-diffusion model. Specially, the investment strategies are constrained in a closed convex cone and all coefficients in the model are stochastic processes adapted to the filtration generated by a Markov chain. First, we provide a necessary and sufficient condition for an equilibrium strategy, which involves a system of forward and backward stochastic differential equations (FBSDEs, for short). Second, by solving these FBSDEs, we obtain a feedback representation of the equilibrium strategy. Third, we prove a theorem ensuring the almost everywhere uniqueness of the equilibrium solution. Finally, the results are applied to solve an example of the Markovian regime-switching model.</p>

American options pricing under regime-switching jump-diffusion models with meshfree finite point method

In an incomplete market construction and by no-arbitrage assumption, the American options pricing problem under the jump-diffusion regime-switching process is formulated by a variational form of coupled partial integro-differential equations. In this paper, a valuation algorithm is developed for American options when the dynamics of underlying assets follow the regime-switching jump-diffusion processes. Using the fact that the price of an American option under jump-diffusion regime-switching processes is formulated by a collection of coupled variational partial integro-differential equations with the free boundary characteristic, we combine the moving least-squares approximation with an operator splitting method to treat American constraints. Numerical experiments with American options under three, five, and seven regimes demonstrate the efficiency and effectiveness of our computational scheme for pricing American options under the regime-switching models.

Valuation of chooser options with state-dependent risks

In this study, the underlying stock price dynamics and the forward interest rate processes are, respectively, driven by a Markovian regime-switching jump-diffusion model and a state-dependent Heath-Jarrow-Morton model. Furthermore, we employ the regime-switching Esscher transform to identify a risk-neutral martingale measure, which is used to derive integral expressions on prices of European-style chooser options. Eventually, numerical examples are provided and discussed.

Efficient valuation of guaranteed minimum maturity benefits in regime switching jump diffusion models with surrender risk

We present an efficient valuation approach for guaranteed minimum maturity benefits (GMMBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We allow early surrender of the VA contract and impose surrender charges, which are important in practice to discourage early termination/lapse of the contract. We consider both continuously-monitored and discretely-monitored surrender behaviors before maturity, and utilize an intensity-based framework. Based on the continuous-time Markov chain (CTMC) approximation combined with the Fourier cosine series expansion method, we find that the valuation problem can be solved under a regime-switching jump diffusion framework. Both error analysis and numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Convergence Rate of the High-Order Finite Difference Method for Option Pricing in a Markov Regime-Switching Jump-Diffusion Model

The high-order finite difference method for option pricing is one of the most popular numerical algorithms. Therefore, it is of great significance to study its convergence rate. Based on the relationship between this algorithm and the trinomial tree method, as well as the definition of local remainder estimation, a strict mathematical proof is derived for the convergence rate of the high-order finite difference method for option pricing in a Markov regime-switching jump-diffusion model. The theoretical result shows that the convergence rate of this algorithm is O(Δτ) . Moreover, the results also hold in the case of Brownian motion and jump-diffusion models that are specialized forms of the given model.

Spread Option Pricing in Regime-Switching Jump Diffusion Models

In this paper, we consider the problem of pricing a spread option when the underlying assets follow a bivariate regime-switching jump diffusion model. We exploit an approximation technique which is based on the univariate Fourier transform representation of the option price. The method proves to be computationally very effective with respect to benchmark Monte Carlo estimators and permits the use of several kinds of jump models other than the standard Gaussian setting. As a by-product, the exact price of an Exchange Option may be efficiently computed within this framework.

Second-order IMEX scheme for a system of partial integro-differential equations from Asian option pricing under regime-switching jump-diffusion models

Second-order IMEX scheme for a system of partial integro-differential equations from Asian option pricing under regime-switching jump-diffusion models

Pricing equity-linked death benefits by complex Fourier series expansion in a regime-switching jump diffusion model

In this paper, we consider the valuation problem of equity-linked annuity with guaranteed minimum death benefit (GMDB) by complex Fourier series method under regime-switching jump diffusion models. We show that the price formulas can be expressed by some discounted density functions associated with the residual lifetime random variable. Fourier transforms for discounted density functions are derived, and complex Fourier series expansion method is applied to recover the discounted density functions. Some explicit formulas for computing the GMDB price are given. Finally, we give some numerical results to show effectiveness of our method.

Variance Swap Pricing under Markov-Modulated Jump-Diffusion Model

This paper investigates the pricing of discretely sampled variance swaps under a Markov regime-switching jump-diffusion model. The jump diffusion, as well as other parameters of the underlying stock’s dynamics, is modulated by a Markov chain representing different states of the market. A semi-closed-form pricing formula is derived by applying the generalized Fourier transform method. The counterpart pricing formula for a variance swap with continuous sampling times is also derived and compared with the discrete price to show the improvement of accuracy in our solution. Moreover, a semi-Monte-Carlo simulation is also presented in comparison with the two semi-closed-form pricing formulas. Finally, the effect of incorporating jump and regime switching on the strike price is investigated via numerical analysis.

Fitted Finite Volume Method for Pricing American Options under Regime-Switching Jump-Diffusion Models Based on Penalty Method

Fitted Finite Volume Method for Pricing American Options under Regime-Switching Jump-Diffusion Models Based on Penalty Method

Convergence rates of moving mesh methods for moving boundary partial integro–differential equations from regime-switching jump–diffusion Asian option pricing

This paper studies the convergence rates of moving mesh methods for a system of moving boundary partial integro-differential equations (PIDEs) which arise in the Asian option pricing under the state-dependent regime-switching jump–diffusion models. The value function of the Asian option under the model is governed by a system of two-dimensional PIDEs. In this paper, the two-dimensional PIDEs are recast into a one-dimensional moving boundary problem of the PIDEs. A moving finite difference method (FDM) is proposed to solve the one-dimensional moving boundary problem and the convergence rates are proved. Numerical examples are provided to confirm the theoretical results.

Empirical analysis of SH50ETF and SH50ETF option prices under regime-switching jump-diffusion models

In this paper, we use extensive empirical data sets from Shanghai 50ETF (SH50ETF) and SH50ETF options markets in China to study how regime-switching jump-diffusion models improve goodness of fit and option pricing performance. Firstly, the model parameters are estimated by using maximum likelihood estimation and the numerical analysis indicates that the regime-switching jump-diffusion models outperform a range of other models. Secondly, the analytical option pricing formulae are obtained via the fast Fourier transform and the empirical results using the proposed option pricing formulae are presented. Finally, we find that the calculated option prices are fairly consistent with the actual market prices.

A RBF based finite difference method for option pricing under regime-switching jump-diffusion model

In this article, we presented a method for option pricing problem under regime-switching jump-diffusion models. We have proposed a numerical method for solving a partial integro-differential equation (PIDE) for pricing European option and for solving linear complementarity problem (LCP), to evaluate the price of American options. We use implicit explicit method for time semi discretization, followed by radial basis function based finite difference (RBF-FD) method for spatial discretization to solve PIDE. The proposed method is further extended to solve the LCP by coupling it with operator splitting method. Numerical simulation is done for European and American option to demonstrate efficiency and accuracy of the proposed method.

Stability of numerical methods under the regime-switching jump-diffusion model with variable coefficients

In this paper we introduce three numerical methods to evaluate the prices of European, American, and barrier options under a regime-switching jump-diffusion model (RSJD model) where the volatility and other parameters are considered as variable coefficients. The prices of the European option, which is one of the financial derivatives, are given by a partial integro-differential equation (PIDE) problem and those of the American option are evaluated by solving a linear complementarity problem (LCP). The proposed methods are constructed to avoid the use of any fixed point iteration techniques at each state of the economy and time step. We analyze the stability of the proposed methods with respect to the discrete l2-norm in the time and spatial variables. A variety of numerical experiments are carried out to show the second-order convergence of the three numerical methods under the regime-switching jump-diffusion model with variable coefficients.