A game theoretic approach to option valuation under Markovian regime-switching models

In this paper, we consider a game theoretic approach to option valuation under Markovian regime-switching models, namely, a Markovian regime-switching geometric Brownian motion (GBM) and a Markovian regime-switching jump-diffusion model. In particular, we consider a stochastic differential game with two players, namely, the representative agent and the market. The representative agent has a power utility function and the market is a “fictitious” player of the game. We also explore and strengthen the connection between an equivalent martingale measure for option valuation selected by an equilibrium state of the stochastic differential game and that arising from a regime switching version of the Esscher transform. When the stock price process is governed by a Markovian regime-switching GBM, the pricing measures chosen by the two approaches coincide. When the stock price process is governed by a Markovian regime-switching jump-diffusion model, we identify the condition under which the pricing measures selected by the two approaches are identical.     

Continuous-time mean–variance portfolio selection with liability and regime switching

A continuous-time mean–variance model for individual investors with stochastic liability in a Markovian regime switching financial market, is investigated as a generalization of the model of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482]. We assume that the risky stock’s price is governed by a Markovian regime-switching geometric Brownian motion, and the liability follows a Markovian regime-switching Brownian motion with drift, respectively. The evolution of appreciation rates, volatility rates and the interest rates are modulated by the Markov chain, and the Markov switching diffusion is assumed to be independent of the underlying Brownian motion. The correlation between the risky asset and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. Using the Lagrange multiplier technique and the linear-quadratic control technique, we get the expressions of the optimal portfolio and the mean–variance efficient frontier in closed forms. Further, the results of our special case without liability is consistent with those results of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482].     

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.     

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 comparison of regime-switching temperature modeling approaches for applications in weather derivatives

A comparison of regime-switching approaches to modeling the stochastic behavior of temperature with an aim to the valuation of temperature-based weather options is presented. Four models are developed. Three of these are two-state Markov regime-switching models and the other is a single-regime model. The regime-switching models are generated from a combination of different underlying processes for the stochastic component of temperature. In Model 1, one regime is governed by a mean-reverting process and the other by a Brownian motion. In Model 2, each regime is governed by a Brownian motion. In Model 3, each regime is governed by a mean-reverting process in which the mean and speed of the mean-reversion remain the same, but only the volatility switches between the states. Model 4 is a single-regime model, where the temperature dynamics are governed by a single mean-reverting process. All four models are utilized to determine the expected heating degree days (HDD) and cooling degree days (CDD), which play a crucial role in the valuation of weather options. A four-year temperature dataset from Toronto, Canada, is used for the analysis. Results demonstrate that Model 1 captures the temperature dynamics more accurately than the other three models. Model 1 is then used to price the monthly call options based on a range of strike HDD.     

A version of Pitman's 2M — X theorem for geometric Brownian motions

We show that geometric Brownian motion with parameter μ, i.e., the exponential of linear Brownian motion with drift μ, divided by its quadratic variation process is a diffusion process. Taking logarithms and an appropriate scaling limit, we recover the Rogers-Pitman extension to Brownian motion with drift of Pitman's representation theorem for the three-dimensional Bessel process. Time inversion and generalized inverse Gaussian distributions play crucial roles in our proofs.     

Dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps

In this paper, the dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps is considered. Besides a standard geometric Brownian motion, another two driving processes are taken into account: a stationary Poisson point process and a continuous time finite-state Markov chain. Firstly, we establish sufficient conditions for persistence in the mean of the disease. Then we obtain sufficient conditions for extinction of the disease. In addition, we also establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.     

Martingale Representation and Admissible Portfolio Process with Regime Switching

We discuss the existence of an admissible investment strategy for any given consumption rate process in a Markov, regime-switching Black–Scholes–Merton economy. A martingale representation for a double martingale generated by the Brownian motion and the Markov chain is used to establish the existence of the admissible investment strategy. We also employ the martingale representation to prove the attainability of a European contingent claim in the regime-switching environment under a pricing kernel specified by the Esscher transform based on the Laplace cumulant process.     

Asian option as a fixed-point

We characterize the price of an Asian option, a financial contract, as a fixed-point of a non-linear operator. In recent years, there has been interest in incorporating changes of regime into the parameters describing the evolution of the underlying asset price, namely the interest rate and the volatility, to model sudden exogenous events in the economy. Asian options are particularly interesting because the payoff depends on the integrated asset price. We study the case of both floating- and fixed-strike Asian call options with arithmetic averaging when the asset follows a regime-switching geometric Brownian motion with coefficients that depend on a Markov chain. The typical approach to finding the value of a financial option is to solve an associated system of coupled partial differential equations. Alternatively, we propose an iterative procedure that converges to the value of this contract with geometric rate using a classical fixed-point theorem.     

A Recursive Algorithm for Selling at the Ultimate Maximum in Regime-Switching Models

We propose a recursive algorithm for the numerical computation of the optimal value function \(\inf _{t\le \tau \le T} \mathbb {E} \left [\sup _{0\le s\le T } Y_{s} / Y_{\tau } \left | {\mathcal F}_{t}\right .\right ] \) over the stopping times τ with respect to the filtration of a geometric Brownian motion Y _t with Markovian regime switching. This method allows us to determine the boundary functions of the optimal stopping set when no associated Volterra integral equation is available. It applies in particular when regime-switching drifts have mixed signs, in which case the boundary functions may not be monotone.     

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In this paper, we extend the previous Markov-modulated reflected Brownian motion model discussed in [1] to a Markov-modulated reflected jump diffusion process, where the jump component is described as a Markov-modulated compound Poisson process. We compute the joint stationary distribution of the bivariate Markov jump process. An abstract example with two states is given to illustrate how the stationary equation described as a system of ordinary integro-differential equations is solved by choosing appropriate boundary conditions. As a special case, we also give the sationary distribution for this Markov jump process but without Markovian regime-switching.     

标的资产服从几何分数布朗运动的期权定价

本文在M ogens B ladt和T ina H av iid R ydberg无市场假设,仅利用价格过程的实际概率的期权保险精算定价模型的基础上,得出了标的资产服从几何分数布朗运动的欧式期权定价公式,并说明了几何布朗运动是本文的一种特殊情况.     

Exponential functionals of Lévy processes and variable annuity guaranteed benefits

Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black–Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian motion does not adequately capture features of market equity returns. One popular alternative for modeling equity returns consists in replacing the geometric Brownian motion by an exponential of a Lévy process. In this paper we use this latter model to study variable annuity guaranteed benefits and to compute explicitly the distribution of certain exponential functionals.     

Conditioned stochastic differential equations: theory, examples and application to finance

We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard Brownian motion by its first hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a “quantitative” value to this weak information.     

Nearly-Optimal Asset Allocation in Hybrid Stock Investment Models

This work develops a class of stock-investment models that are hybrid in nature and involve continuous dynamics and discrete-event interventions. In lieu of the usual geometric Brownian motion formulation, hybrid geometric Brownian motion models are proposed, in which both the expected return and the volatility depend on a finite-state Markov chain. Our objective is to find nearly-optimal asset allocation strategies so as to maximize the expected returns. The use of the Markov chain stems from the motivation of capturing the market trends as well as various economic factors. To incorporate these economic factors into the models, the underlying Markov chain inevitably has a large state space. To reduce the complexity, a hierarchical approach is suggested, which leads to singularly-perturbed switching diffusion processes. By aggregating the states of the Markov chains in each weakly irreducible class into a single state, limit switching diffusion processes are obtained. Using such asymptotic properties, nearly-optimal asset allocation policies are developed.     

Ruin probability for Lévy risk process compounded by geometric Brownian motion

In this paper, we assume that the surplus of an insurer follows a Lévy risk process and the insurer would invest its surplus in a risky asset, whose prices are modeled by a geometric Brownian motion. It is shown that the ruin probabilities (by a jump or by oscillation) of the resulting surplus process satisfy certain integro-differential equations.      

Ultracontractivity for Brownian motion with Markov switching

Consider the transition density functions for Brownian motion with two-state Markov switching. The characteristic functions for transition density functions are presented. Then, we show that the semigroup-associated Brownian motion with Markov switching is ultracontractive. And an explicit time-dependent upper bound for heat kernels are presented. Moreover, we prove that the Dirichlet form associated Brownian motion with Markov switching satisfies the Nash inequality.     

Linear-quadratic optimal control under non-Markovian switching

We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which turned out to be a backward stochastic differential equation driven by the Brownian motion and by the random measure associated with the marked point process.     

“政策市”下的中国企业实物投资机理研究

在我国市场经济发展的过程中,某些具有不确定性和投资不可逆性的产业市场容易出现投资过热问题.这时政府往往频繁出台一系列宏观调控政策以规范市场,但效果不佳.为此从研究该类市场的投资主体—企业的投资决策机理的角度出发,通过引入实物期权的理论,给出了在完善和平稳发展的市场条件下及在政府政策冲击的市场条件下,价格分别服从单纯的几何布朗运动及混合的几何布朗运动/泊松跳跃过程的企业最佳投资规则及其临界价格,并进行了比较,结果表明:“政策市”下的企业投资更富“冒进性”.     

Static Hedging with Uncertain Quantity and Departure from the Cost-of-Carry Valuation

In this paper, we are concerned with the optimal hedge ratio under quantity risk as well as discrepancies between the futures market price and its theoretical valuation according to the cost- of-carry model. Assuming a geometric Brownian motion for forecasting process, we model mispricing as a specific noise corn poncnt in the dynamics of filturcs market prices, based on which the optimal hedging strategy is calculated. Finally, we illustrate optimal strategy and its properties by numerical examples.