Markov-modulated Jump-diffusion Model Research Articles

Pricing vulnerable options under cross-asset markov-modulated jump-diffusion dynamics

In this study, the dynamics of the underlying asset price and the counterparty's asset price are governed by a cross-asset Markov-modulated jump-diffusion model that captures the time-inhomogeneity and systematic cojumps. Additionally, the forward interest rate processes are driven by a Markov-modulated Heath–Jarrow–Morton model to depict stochastic volatilities. Under an incomplete-market setting, we apply the Markov-modulated Esscher transform technique to determine a risk-neutral martingale measure. After determining the Markov-modulated Esscher parameters, we obtain an integral expression on the prices of vulnerable European-style Black–Scholes options. The numerical illustrations indicate that the findings are consistent with Klein (1996) and contribute to the extant literature on cojumping impacts on vulnerable option prices.

Markov modulated jump-diffusions for currency options when regime switching risk is priced

In the current literature, regime-switching risk is NOT priced in the Markov-modulated jump-diffusion models for currency options. We therefore develop a hidden Markov-modulated jump-diffusion model under the regime-switching economy where the regime-switching risk is priced. In the model, the dynamics of the spot foreign exchange rate captures both the rare events and the time-inhomogeneity in the fluctuating currency market. In particular, the rare events are described by a compound Poisson process with log-normal jump amplitude, and the time-varying rates are formulated by a continuous-time finite-state Markov chain. Unlike previous research, the proposed model can price regime-switching risk, in addition to diffusion risk and jump risk, based on the Esscher transform conditional on a single initial regime of economy. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the currency option prices does not seem significant in contradictory to the findings made by Siu and Yang [Siu, TK and H Yang (2009). Option Pricing When The Regime-Switching Risk is priced. Acta Mathematicae Applicatae Sinica, English Series, Vol. 25, No. 3, pp. 369–388].

Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales

In this paper, we consider the valuation of vulnerable options under a Markov-modulated jump-diffusion model, where the option writer's asset value is subject to price pressure from other financial institutions due to distressed selling. A change of numeraire technique, proposed by Geman et al. [ 14 ], is employed to obtain a semi-analytical pricing formula for an vulnerable European option in the presence of regime switching effect. The method is numerically implemented using the multinomial approach in Costabile et al. [ 6 ]. We study the impacts of distressed selling and regime switching on the European option prices via numerical experiments.

Valuation and hedging strategy of currency options under regime-switching jump-diffusion model

The main purpose of this thesis is in analyzing and empirically simulating risk minimizing European foreign exchange option pricing and hedging strategy when the spot foreign exchange rate is governed by a Markov-modulated jump-diffusion model. The domestic and foreign money market interest rates, the drift and the volatility of the exchange rate dynamics all depend on a continuous-time hidden Markov chain which can be interpreted as the states of a macro-economy. In this paper, we will provide a practical lognormal diffusion dynamic of the spot foreign exchange rate for market practitioners. We employing the minimal martingale measure to demonstrate a system of coupled partial-differential-integral equations satisfied by the currency option price and attain the corresponding hedging schemes and the residual risk. Numerical simulations of the double exponential jump diffusion regime-switching model are used to illustrate the different effects of the various parameters on currency option prices.

The extension from independence to dependence between jump frequency and jump size in Markov-modulated jump diffusion models

We set out in this study to investigate the relationship between jump frequency and jump size for the 30 component stocks of the Dow Jones Industrial Average (DJIA) index, extending the Markov-modulated jump diffusion model from independence to dependence between jump frequency and jump size. We propose an estimation method for the parameters of the Markov-modulated jump diffusion model based upon dependence between jump frequency and size, with our results indicating that when abnormal events occur, the Markov-modulated jump diffusion models with both state-independent jump sizes (MJMI) and state-dependent jump sizes (MJMD) outperform the pure jump diffusion (JD) model in terms of capturing the risks of jump frequency and jump size. Based upon Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC), our results further indicate that for 23 of the component stocks, the MJMD model may be better suited, as compared to the MJMI model. Finally, our empirical observations reveal that the behavior of jump risks in the stock markets, including jump frequency and jump size, is not independent, since these phenomena are found to coincide during both financial crisis periods and stock market crashes, with the largest jump size risks, during certain periods, being accompanied by either systematic or idiosyncratic risks.

Risk-minimizing pricing and hedging foreign currency options under regime-switching jump-diffusion models

ABSTRACTThis article mainly investigates risk-minimizing European currency option pricing and hedging strategy when the spot foreign exchange rate is driven by a Markov-modulated jump-diffusion model. We suppose the domestic and foreign money market floating interest rates, the drift, and the volatility of the exchange rate dynamics all depend on the state of the economy, which is modeled by a continuous-time hidden Markov chain. The model considered in this article will provide market practitioners with flexibility in characterizing the dynamics of the spot foreign exchange rate. Using the minimal martingale measure, we obtain a system of coupled partial-differential-integral equations satisfied by the currency option price and find the corresponding hedging strategies and the residual risk. According to simulation of currency option prices in the special case of double exponential jump-diffusion regime-switching model, we further discuss and show the effects of the parameters on the prices.

Pricing credit-risky bonds and spread options modelling credit-spread term structures with two-dimensional Markov-modulated jump-diffusion

The relationship between company hazard rates and the business cycle becomes more apparent after a financial crisis. To address this relationship, a regime-switching process with an intensity function is adopted in this paper. In addition, the dynamics of both interest rates and asset values are modelled with a Markov-modulated jump-diffusion model, and a 2-factor hazard rate model is also considered. Based on this more suitable model setting, a closed-form model of pricing risky bonds is derived. The difference in yield between a risky bond and risk-free zero coupon bond is used to model a term structure of credit spreads (CSs) from which a closed-form pricing model of a call option on CSs is obtained. In addition, the degree to which the explicit regime shift affects CSs and credit-risky bond prices is numerically examined using three forward-rate functions under various business-cycle patterns.

Pricing Equity-indexed Annuities When Discrete Dividends Follow a Markov-Modulated Jump Diffusion Model

The Equity-Indexed annuities (EIAs) provide investors not only a minimum rate of return but also an opportunity to gain a potential profit that is linked to the performance of an equity index, typically SP 500. These properties make EIAs become very popular in the market and receive considerable attention in the actuarial literature. In this article, we investigate the Equity-Indexed annuities valuation when the discrete dividend payments follow a Markov-modulated jump diffusion model. Using the generalized Itô formula, we obtain the explicit solution for dividend payments of the model. From Dividend Discount theory, which implies that the stock price should be equal to the net present value of all its future dividend payments, the stock price process is then deduced. Within this framework, closed-form solution to the point-to-point EIA pricing problem is derived via the characteristic function of the occupation times.

Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model

In this paper, we consider pricing and hedging of catastrophe equity put options undera Markov-modulated jump diffusion process with a Markov switching compensator. We assume that the risk free interest rate,the appreciation rate and the volatility of the risky asset depend on a finite-state Markov chain. We investigate the pricing of catastrophe equity put options and obtain the explicit pricing formulas. A numerical analysis is provided toillustrate the effect of regime switching on the price of catastrophe equity put options. In the end, since the market which we consider is not complete, we also provide an optimal hedging strategy by using the local risk minimization method.

A tale of two regimes: Theory and empirical evidence for a Markov-modulated jump diffusion model of equity returns and derivative pricing implications

We provide closed-form solutions for a continuous time, Markov-modulated jump diffusion model in a general equilibrium framework for options prices under a variety of jump diffusion specifications. We further demonstrate that the two-state model provides the leptokurtic return features, volatility smile, and volatility clustering observed empirically for the Dow Jones Industrial Average (DJIA) and its component stocks. Using 10years of stock return data, we confirm the existence of jump intensity switching and clustering, illustrate transition probabilities, and verify superior empirical fit over competing Poisson-style models.

A radial basis collocation method for pricing American options under regime-switching jump-diffusion models

The Markovian regime-switching paradigm has become one of the prevailing models in mathematical finance. It is now widely known that under the regime-switching model, the market is incomplete and so the option valuation problem in this framework will be a challenging task of considerable importance for market practitioners and academia. Our concern here is to solve the pricing problem for American options in a Markov-modulated jump-diffusion model, based on a meshfree approach using radial basis functions. In this respect, we solve a set of coupled partial integro-differential equations with the free boundary feature by expanding the solution vector in terms of radial basis functions and then collocating the resulting system of equations at some pre-specified points. This method exhibits a superlinear order of convergence in space and a linear order in time and also has an acceptable speed in comparison with some existing methods. We will compare our results with some recently proposed approaches.

Risk-minimizing option pricing under a Markov-modulated jump-diffusion model with stochastic volatility

In this paper, we deal with the pricing of European style options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump diffusion with stochastic volatility. We investigate the Radon–Nikodym derivative for the minimal martingale measure and a partial differential equation approach for pricing European options. An optimal hedging strategy in terms of local risk minimization is obtained.

Pricing Vulnerable Options Under a Markov-Modulated Regime Switching Model

In this article, we consider the pricing of vulnerable European options when the dynamic of the risky assets are governed by Markov-modulated Geometric Brownian Motions. The regime switching Esscher transform is employed to determine an equivalent martingale measure. In particular, we also provide analytical pricing formulas of vulnerable European options under a Markov-modulated jump-diffusion model.

Pricing Risky Debts Under a Markov-modudated Merton Model with Completely Random Measures

We consider the pricing of both fixed rate and floating rate risky debts when the value of a firm is governed by a Markov-modulated generalized jump-diffusion model with the jump component described by a completely random measure process with a Markov-switching compensator; that is, the compensator switches over time according to the states of an economy modelled by a continuous-time Markov chain. We shall employ the well-known tool in actuarial science, namely, the Esscher transform, to determine the price of the risky debts. We shall investigate consequences for the prices of the risky debts of various parametric specifications of the jump component. Sensitivity analysis for the prices of the risky debts with respect to various model parameters will be conducted. We also compare the pricing results obtained from our model with those from the celebrated Merton jump-diffusion model to illustrate the effect of correlated jump times and sizes on the prices of the debts.

Pricing Options Under a Generalized Markov-Modulated Jump-Diffusion Model

We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.