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.

PARAMETER ESTIMATION FOR A REGIME-SWITCHING MEAN-REVERTING MODEL WITH JUMPS

2005 ◽  
Vol 08 (06) ◽  
pp. 791-806 ◽  
Author(s):  
PING WU ◽  
ROBERT J. ELLIOTT
Keyword(s):  
Parameter Estimation ◽  
Markov Chain ◽  
Continuous Time ◽  
Regime Switching ◽  
Energy Markets ◽  
Finite Dimensional ◽  
The Em Algorithm ◽  
Finite State ◽  
The Mean ◽  
The Interest Rate

In this paper we propose a type of mean reverting model with jumps, where the mean reverting level changes according to a continuous time, finite state Markov chain. This model could be applied to the interest rate and energy markets. We apply filtering techniques and obtain finite dimensional filters for the unobservable state of the Markov chain based on observations of the mean reverting diffusion. Various auxiliary filters are developed that allow us to estimate the parameters of the Markov chain by the EM algorithm. A simulation study is done for a concrete example.

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.

scholarly journals Integration by Parts and Martingale Representation for a Markov Chain

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Tak Kuen Siu
Keyword(s):  
Markov Chain ◽  
Financial Market ◽  
Continuous Time ◽  
Contingent Claims ◽  
Jump Processes ◽  
Stochastic Integrals ◽  
Integration By Parts ◽  
Martingale Representation ◽  
Finite State ◽  
Change Of Measures

Integration-by-parts formulas for functions of fundamental jump processes relating to a continuous-time, finite-state Markov chain are derived using Bismut's change of measures approach to Malliavin calculus. New expressions for the integrands in stochastic integrals corresponding to representations of martingales for the fundamental jump processes are derived using the integration-by-parts formulas. These results are then applied to hedge contingent claims in a Markov chain financial market, which provides a practical motivation for the developments of the integration-by-parts formulas and the martingale representations.

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.

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.

On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions

1996 ◽  
Vol 33 (3) ◽  
pp. 640-653 ◽  
Author(s):  
Tobias Rydén
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Poisson Processes ◽  
Hitting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Minimum Number ◽  
Markov Modulated ◽  
Two Parameters

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Multi-valued backward stochastic differential equations with regime switching

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ruijuan Deng ◽  
Yong Ren
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Regime Switching ◽  
Original System ◽  
Backward Stochastic Differential Equations ◽  
Lower Semi ◽  
Finite State ◽  
Uniqueness Of The Solution ◽  
Continuous Convex Function

AbstractThe paper considers a class of multi-valued backward stochastic differential equations with subdifferential of a lower semi-continuous convex function with regime switching, whose generator is a continuous-time Markov chain with a finite state space. Firstly, we get the existence and uniqueness of the solution by the penalization method. Secondly, we prove that the solution of the original system is weakly convergent. Finally, we give an application to the homogenization of a class of multi-valued PDEs with Markov chain.

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 On Markov Modulated Mean-Reverting Price-Difference Models

2008 ◽  
Vol 13 ◽  
pp. 55
Author(s):  
W. P. Malcom ◽  
Lakhdar Aggoun ◽  
Mohamed Al-Lawati
Keyword(s):  
Stochastic Model ◽  
Markov Processes ◽  
Continuous Time ◽  
Price Difference ◽  
Time Dynamics ◽  
Price Process ◽  
Finite State ◽  
The Mean ◽  
New Feature ◽  
Markov Modulated

In this paper we develop a stochastic model incorporating a double-Markov modulated mean-reversion model. Unlike a price process the basis process X can take positive or negative values. This model is based on an explicit discretisation of the corresponding continuous time dynamics. The new feature in our model is that we suppose the mean reverting level in our dynamics as well as the noise coefficient can change according to the states of some finite-state Markov processes which could be the economy and some other unseen random phenomenon.  

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