scholarly journals DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

2019 ◽  
Vol 34 (2) ◽  
pp. 235-257
Author(s):  
Peter Spreij ◽  
Jaap Storm
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Regime Switching ◽  
Limit Theorems ◽  
Counting Process ◽  
Limit Behavior ◽  
Diffusion Approximations ◽  
Default Rate ◽  
Markov Modulated ◽  
Rapid Switching

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

HEDGING OPTIONS IN A DOUBLY MARKOV-MODULATED FINANCIAL MARKET VIA STOCHASTIC FLOWS

2019 ◽  
Vol 22 (08) ◽  
pp. 1950047 ◽  
Author(s):  
TAK KUEN SIU ◽  
ROBERT J. ELLIOTT
Keyword(s):  
Markov Chain ◽  
Financial Market ◽  
Continuous Time ◽  
Regime Switching ◽  
Stochastic Flows ◽  
Finite State ◽  
Markov Modulated ◽  
Appreciation Rate ◽  
The Interest Rate ◽  
Over Time

The hedging of a European-style contingent claim is studied in a continuous-time doubly Markov-modulated financial market, where the interest rate of a bond is modulated by an observable, continuous-time, finite-state, Markov chain and the appreciation rate of a risky share is modulated by a continuous-time, finite-state, hidden Markov chain. The first chain describes the evolution of credit ratings of the bond over time while the second chain models the evolution of the hidden state of an underlying economy over time. Stochastic flows of diffeomorphisms are used to derive some hedge quantities, or Greeks, for the claim. A mixed filter-based and regime-switching Black–Scholes partial differential equation is obtained governing the price of the claim. It will be shown that the delta hedge ratio process obtained from stochastic flows is a risk-minimizing, admissible mean-self-financing portfolio process. Both the first-order and second-order Greeks will be considered.

scholarly journals Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

2021 ◽  
Vol 53 (1) ◽  
pp. 1-29
Author(s):  
Ari Arapostathis ◽  
Guodong Pang ◽  
Yi Zheng
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Regime Switching ◽  
Convergence Rates ◽  
Death Process ◽  
Ergodic Properties ◽  
State Diffusion ◽  
Markov Modulated ◽  
Steady State Diffusion ◽  
Birth Death

AbstractWe study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.

scholarly journals Pricing Options with Credit Risk in Markovian Regime-Switching Markets

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jinzhi Li ◽  
Shixia Ma
Keyword(s):  
Markov Chain ◽  
Credit Risk ◽  
Regime Switching ◽  
Stock Price ◽  
Arrival Rate ◽  
Reduced Form ◽  
European Option ◽  
Pricing Options ◽  
Markov Modulated ◽  
The Interest Rate

This paper investigates the valuation of European option with credit risk in a reduced form model when the stock price is driven by the so-called Markov-modulated jump-diffusion process, in which the arrival rate of rare events and the volatility rate of stock are controlled by a continuous-time Markov chain. We also assume that the interest rate and the default intensity follow the Vasicek models whose parameters are governed by the same Markov chain. We study the pricing of European option and present numerical illustrations.

PRICING CREDIT DERIVATIVES IN A MARKOV-MODULATED REDUCED-FORM MODEL

2013 ◽  
Vol 16 (04) ◽  
pp. 1350018 ◽  
Author(s):  
TAMAL BANERJEE ◽  
MRINAL K. GHOSH ◽  
SRIKANTH K. IYER
Keyword(s):  
Markov Chain ◽  
Regime Switching ◽  
Credit Rating ◽  
Credit Derivatives ◽  
Reduced Form ◽  
New Family ◽  
Default Intensity ◽  
High Volatility ◽  
Markov Modulated ◽  
The Impact

Numerous incidents in the financial world have exposed the need for the design and analysis of models for correlated default timings. Some models have been studied in this regard which can capture the feedback in case of a major credit event. We extend the research in the same direction by proposing a new family of models having the feedback phenomena and capturing the effects of regime switching economy on the market. The regime switching economy is modeled by a continuous time Markov chain. The Markov chain may also be interpreted to represent the credit rating of the firm whose bond we seek to price. We model the default intensity in a pool of firms using the Markov chain and a risk factor process. We price some single-name and multi-name credit derivatives in terms of certain transforms of the default and loss processes. These transforms can be calculated explicitly in case the default intensity is modeled as a linear function of a conditionally affine jump diffusion process. In such a case, under suitable technical conditions, the price of credit derivatives are obtained as solutions to a system of ODEs with weak coupling, subject to appropriate terminal conditions. Solving the system of ODEs numerically, we analyze the credit derivative spreads and compare their behavior with the nonswitching counterparts. We show that our model can easily incorporate the effects of business cycle. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low floating interest rate, high default intensity rate, and high volatility. We also model the effects of firm restructuring on the credit spread, in case of a default.

scholarly journals Diffusion approximations for randomly arriving expert opinions in a financial market with Gaussian drift

2021 ◽  
Vol 58 (1) ◽  
pp. 197-216 ◽  
Author(s):  
Jörn Sass ◽  
Dorothee Westphal ◽  
Ralf Wunderlich
Keyword(s):  
Stock Returns ◽  
Financial Market ◽  
Discrete Time ◽  
High Frequency ◽  
Utility Maximization ◽  
Limit Theorems ◽  
Approximate Solutions ◽  
Diffusion Approximations ◽  
Expert Opinions ◽  
Arrival Intensity

AbstractThis paper investigates a financial market where stock returns depend on an unobservable Gaussian mean reverting drift process. Information on the drift is obtained from returns and randomly arriving discrete-time expert opinions. Drift estimates are based on Kalman filter techniques. We study the asymptotic behavior of the filter for high-frequency experts with variances that grow linearly with the arrival intensity. The derived limit theorems state that the information provided by discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. These diffusion approximations are extremely helpful for deriving simplified approximate solutions of utility maximization problems.

Continuous-time mean-variance portfolio selection with regime-switching financial market: Time-consistent solution

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ishak Alia ◽  
Farid Chighoub
Keyword(s):  
Portfolio Selection ◽  
Regime Switching ◽  
Optimal Time ◽  
Uniqueness Results ◽  
State Dependent ◽  
Consistent Solution ◽  
Game Theoretic ◽  
Markov Modulated ◽  
Mean Variance Portfolio ◽  
Mean Variance

Abstract This paper studies optimal time-consistent strategies for the mean-variance portfolio selection problem. Especially, we assume that the price processes of risky stocks are described by regime-switching SDEs. We consider a Markov-modulated state-dependent risk aversion and we formulate the problem in the game theoretic framework. Then, by solving a flow of forward-backward stochastic differential equations, an explicit representation as well as uniqueness results of an equilibrium solution are obtained.

scholarly journals An Analytic Approach for Pricing American Options with Regime Switching

2021 ◽  
Vol 14 (5) ◽  
pp. 188
Author(s):  
Leunglung Chan ◽  
Song-Ping Zhu
Keyword(s):  
Regime Switching ◽  
American Options ◽  
Option Price ◽  
Switching Model ◽  
Homotopy Analysis ◽  
State Regime ◽  
Markov Modulated ◽  
Appreciation Rate ◽  
The Interest Rate ◽  
Hidden States

This paper investigates the American option price in a two-state regime-switching model. The dynamics of underlying are driven by a Markov-modulated Geometric Wiener process. That means the interest rate, the appreciation rate, and the volatility of underlying rely on hidden states of the economy which can be interpreted in terms of Markov chains. By means of the homotopy analysis method, an explicit formula for pricing two-state regime-switching American options is presented.

Partially observable semi-Markov reward processes

1993 ◽  
Vol 30 (3) ◽  
pp. 548-560 ◽  
Author(s):  
Yasushi Masuda
Keyword(s):  
Markov Process ◽  
Counting Process ◽  
Probabilistic Interpretation ◽  
Real Domain ◽  
Semi Markov Process ◽  
Reward Process ◽  
Markov Reward ◽  
Partially Observable ◽  
Joint Behavior

The main objective of this paper is to investigate the conditional behavior of the multivariate reward process given the number of certain signals where the underlying system is described by a semi-Markov process and the signal is defined by a counting process. To this end, we study the joint behavior of the multivariate reward process and the multivariate counting process in detail. We derive transform results as well as the corresponding real domain expressions, thus providing clear probabilistic interpretation.

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