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.

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.

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 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.

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 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 Is the Jump-Diffusion Model a Good Solution for Credit Risk Modeling? The Case of Convertible Bonds

2019 ◽  
Author(s):  
Tim Xiao
Keyword(s):  
Credit Risk ◽  
Diffusion Model ◽  
Stock Price ◽  
Jump Diffusion ◽  
Convertible Bonds ◽  
Reduced Form ◽  
Risk Modeling ◽  
Jump Diffusion Model ◽  
Credit Risk Modeling ◽  
Arbitrage Strategy

This paper argues that the reduced-form jump diffusion model may not be appropriate for credit risk modeling. To correctly value hybrid defaultable financial instruments, e.g., convertible bonds, we present a new framework that relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.

scholarly journals Is the Jump-Diffusion Model a Good Solution for Credit Risk Modeling? The Case of Convertible Bonds

2019 ◽  
Author(s):  
Tim Xiao
Keyword(s):  
Credit Risk ◽  
Diffusion Model ◽  
Stock Price ◽  
Jump Diffusion ◽  
Convertible Bonds ◽  
Reduced Form ◽  
Risk Modeling ◽  
Jump Diffusion Model ◽  
Credit Risk Modeling ◽  
Arbitrage Strategy

This paper argues that the reduced-form jump diffusion model may not be appropriate for credit risk modeling. To correctly value hybrid defaultable financial instruments, e.g., convertible bonds, we present a new framework that relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.

INTERBANK CREDIT RISK MODELING WITH SELF-EXCITING JUMP PROCESSES

2020 ◽  
Vol 23 (06) ◽  
pp. 2050039
Author(s):  
CHARLES GUY NJIKE LEUNGA ◽  
DONATIEN HAINAUT
Keyword(s):  
Credit Risk ◽  
Interest Rates ◽  
Arrival Rate ◽  
Jump Processes ◽  
Reduced Form ◽  
Rate Theory ◽  
Risk Modeling ◽  
Interbank Market ◽  
The Impact ◽  
Swap Rates

The credit crunch of 2007 caused major changes in the market of interbank rates making the existing interest rate theory inconsistent. This paper puts forward one way to reconcile practice and theory by modifying the arbitrage-free condition. In this framework, the forward Libor rate is no longer considered as a risk-free rate and the credit and liquidity risks within the interbank market partly drive its dynamics. In a similar manner to the multiple-curve approach, we model the evolution of default-free rates, assimilated to overnight interest swap rates, and the default times of an interbank market segment determined by its tenor. For each segment, we use the reduced form approach to model the arrival rate of defaults with a self-exciting jump-diffusion process. Then, we deduce the dynamics of the spot forward Libor rates and provide closed-form approximation pricing formulae for options on forward Libor rates and swap rates. Even in a context of negative interest rates and compared to other forms of intensity processes such as a CIR, the self-excitation property allows a better understanding of the spread OIS-IRS and provides information about the interbank credit risk. Furthermore, our framework enables to parse the impact of the interbank credit risk on forward Libor as well as on interest rates derivatives like caps, floors, and swaptions.

scholarly journals Is the Jump-Diffusion Model a Good Solution for Credit Risk Modeling? The Case of Convertible Bonds

2019 ◽  
Author(s):  
Tim Xiao
Keyword(s):  
Credit Risk ◽  
Diffusion Model ◽  
Stock Price ◽  
Jump Diffusion ◽  
Convertible Bonds ◽  
Reduced Form ◽  
Risk Modeling ◽  
Jump Diffusion Model ◽  
Credit Risk Modeling ◽  
Arbitrage Strategy

This paper argues that the reduced-form jump diffusion model may not be appropriate for credit risk modeling. To correctly value hybrid defaultable financial instruments, e.g., convertible bonds, we present a new framework that relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.

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