scholarly journals Power Exchange Option with a Hybrid Credit Risk under Jump-Diffusion Model

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 53
Author(s):  
Junkee Jeon ◽  
Geonwoo Kim
Keyword(s):  
Credit Risk ◽  
Structural Model ◽  
Risk Model ◽  
Diffusion Processes ◽  
Jump Diffusion ◽  
Reduced Form ◽  
Jump Diffusion Processes ◽  
Power Exchange ◽  
Jump Diffusion Model ◽  
Exchange Option

In this paper, we study the valuation of power exchange options with a correlated hybrid credit risk when the underlying assets follow the jump-diffusion processes. The hybrid credit risk model is constructed using two credit risk models (the reduced-form model and the structural model), and the jump-diffusion processes are proposed based on the assumptions of Merton. We assume that the dynamics of underlying assets have correlated continuous terms as well as idiosyncratic and common jump terms. Under the proposed model, we derive the explicit pricing formula of the power exchange option using the measure change technique with multidimensional Girsanov’s theorem. Finally, the formula is presented as the normal cumulative functions and the infinite sums.

scholarly journals Valuation of Exchange Option with Credit Risk in a Hybrid Model

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2091
Author(s):  
Geonwoo Kim
Keyword(s):  
Credit Risk ◽  
Hybrid Model ◽  
Structural Model ◽  
Risk Model ◽  
Probabilistic Approach ◽  
Option Price ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Reduced Form ◽  
Exchange Option

In this paper, the valuation of the exchange option with credit risk under a hybrid credit risk model is investigated. In order to build the hybrid model, we consider both the reduced-form model and the structural model. We adopt the probabilistic approach to derive the closed-form formula of an exchange option price with credit risk under the proposed model. Specifically, the change of measure technique is used repeatedly, and the pricing formula is provided as the standard normal cumulative distribution functions.

PRICING OF FIRST TOUCH DIGITALS UNDER NORMAL INVERSE GAUSSIAN PROCESSES

2006 ◽  
Vol 09 (06) ◽  
pp. 915-949 ◽  
Author(s):  
OLEG KUDRYAVTSEV ◽  
SERGEI LEVENDORSKIǏ
Keyword(s):  
Diffusion Model ◽  
Diffusion Processes ◽  
Jump Diffusion ◽  
Spot Price ◽  
Inverse Gaussian ◽  
Jump Diffusion Processes ◽  
Double Exponential ◽  
Motion Approximation ◽  
Jump Diffusion Model ◽  
Normal Inverse Gaussian

We calculate prices of first touch digitals under normal inverse Gaussian (NIG) processes, and compare them to prices in the Brownian model and double exponential jump-diffusion model. Numerical results are produced to show that for typical parameters values, the relative error of the Brownian motion approximation to NIG price can be 2–3 dozen percent if the spot price is at the distance 0.05–0.2 from the barrier (normalized to one). A similar effect is observed for approximations by the double exponential jump-diffusion model, if the jump component of the approximation is significant. We show that two jump-diffusion processes can give approximately the same results for European options but essentially different results for first touch digitals and barrier options. A fast approximate pricing formula under NIG is derived.

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 Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes

2007 ◽  
Vol 44 (03) ◽  
pp. 713-731 ◽  
Author(s):  
Pavel V. Gapeev
Keyword(s):  
Optimal Stopping ◽  
Diffusion Processes ◽  
Free Boundary Problems ◽  
Compound Poisson Process ◽  
Jump Diffusion ◽  
Dimensional System ◽  
Closed Form Solutions ◽  
Jump Diffusion Processes ◽  
Lookback Options ◽  
Jump Diffusion Model

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

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.

scholarly journals Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes

2007 ◽  
Vol 44 (3) ◽  
pp. 713-731 ◽  
Author(s):  
Pavel V. Gapeev
Keyword(s):  
Optimal Stopping ◽  
Diffusion Processes ◽  
Free Boundary Problems ◽  
Compound Poisson Process ◽  
Jump Diffusion ◽  
Dimensional System ◽  
Closed Form Solutions ◽  
Jump Diffusion Processes ◽  
Lookback Options ◽  
Jump Diffusion Model

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

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