Default Barrier Research Articles

Value of tax benefits of a dynamic debt policy with regime switching

ABSTRACTThe article presents a method for valuation of stochastic future income with three barriers: a default barrier, a pre-default barrier and a refinancing barrier. Between the pre-default and default barriers, there is an ongoing cost of financial distress. In this framework, we derive state-dependent present value factors that can be applied to problems of valuation of firms, optimal financial structure and mitigation of agency conflicts between managers and investors. We illustrate our method with an analysis of the value of tax benefits of a dynamic debt policy.

Pricing vulnerable options with stochastic default barriers

In this paper, we investigate the pricing issue of vulnerable options by assuming that the dynamics of all assets are governed by jump-diffusion processes with common factors in both continuous process and jump process components. Moreover, assume credit default event occurs when the value of the counterparty’s assets falls below the default barrier, which is stochastically affected by common factors as well. In the proposed framework, we derive a closed-form formula for vulnerable options and illustrate the impacts of stochastic barriers on option prices. Additionally, the U-shape curve appears when we investigate option prices against the volatility of default barriers.

Bank Contingent Capital: Valuation and the Role of Market Discipline

This paper develops a structural model to evaluate contingent capital notes (CCN) of Basel III under alternative regulatory closure rules. Our dynamic model has a fixed default barrier and at specific discrete time points an additional higher default barrier depending on the closure threshold. The closed-form expressions of CCN and subordinated debts (SD) in the simple Merton model are presented to understand the convex relationship between the price and capital ratio trigger of CCN and to examine the effects of closure rules on CCN and SD through their derivatives’ properties. Our numerical results in the more general model show that a lax closure rule increases the price of SD and distorts the risk information of issuing banks, but not so for CCN. The policy implications are that CCN are more effective than SD in terms of enhancing market discipline because the price/yield information of CCN is more sensitive to the issuing bank’s risk than SD and will not be distorted by regulatory closure rules.

CAPITAL STRUCTURE AND TAX CONVEXITY WHEN THE MATURITY OF DEBT IS FINITE

We examine the impact of tax convexity on the optimal leverage, the optimal default barrier, and the credit spread. Contrary to previous papers, debt maturity in our setting is finite. We find that maturity plays a crucial role to fully capture the effect of tax convexity. Our results show that tax convexity has a substantial impact on the optimal default boundary when the maturity is intermediate. We argue that, in the presence of tax convexity, equityholders inject cash into the firm because they weigh the value surplus they achieve from delaying default against the loss they incur, given the absolute priority rule, when the firm fails to fully redeem the debt’s principal at maturity. We also find that the longer is the maturity, the stronger is the negative impact of tax convexity on the optimal leverage.

Multiple Risk Factor Dependence Structures: Copulas and Related Properties

Copulas have become an important tool in the modern best practice Enterprise Risk Management, often supplanting other approaches to modelling stochastic dependence. However, choosing the 'right' copula is not an easy task, and the temptation to prefer a tractable rather than a meaningful candidate from the encompassing copulas toolbox is strong. The ubiquitous applications of the Gaussian copula is just one illuminating example.Speaking generally, a 'good' copula should conform to the problem at hand, allow for asymmetry in the domain of deffinition and exhibit some extent of tail dependence. In this paper we introduce and study a new class of Multiple Risk Factor (MRF) copula functions, which we show are exactly such. Namely, the MRF copulas (1) arise from a number of meaningful default risk specifcations with stochastic default barriers, (2) are in general non-exchangeable and (3) possess a variety of tail dependences. That being said, the MRF copulas turn out to be surprisingly tractable analytically.

Structural Credit Risk Model Incorporating the First Hitting Condition of Random Default Barrier

Structural Credit Risk Model Incorporating the First Hitting Condition of Random Default Barrier

General properties of solutions to inhomogeneous Black–Scholes equations with discontinuous maturity payoffs

We provide representations of solutions to terminal value problems of inhomogeneous Black–Scholes equations and study such general properties as min–max estimates, gradient estimates, monotonicity and convexity of the solutions with respect to the stock price variable, which are important for financial security pricing. In particular, we focus on finding representation of the gradient (with respect to the stock price variable) of solutions to the terminal value problems with discontinuous terminal payoffs or inhomogeneous terms. Such terminal value problems are often encountered in pricing problems of compound-like options such as Bermudan options or defaultable bonds with discrete default barrier, default intensity and endogenous default recovery. Our results can be used in pricing real defaultable bonds under consideration of existence of discrete coupons or taxes on coupons.

Building the Model

This paper introduces an advanced structural credit risk model. It is built upon the Black-Cox model with the addition of a jump-diffusion driving process, a dynamic capital structure that targets an optimal gearing, a flexible liability maturity structure, a multi-rate tax regime treating government as a stakeholder, and an endogenously calculated default barrier.

Time Donating Barrier for efficient task scheduling in competitive multicore systems

Nowadays, co-locating multithreaded applications on a multicore system has increasingly become a common case in cloud data centers, where multiple threads generally compete for computing resources. These competitive environments may suffer problems of system throughput and fairness caused by barrier operations in multithreaded applications. This is because most implementations of the barrier synchronization are based on the spin-then-block mechanism in which spinning–waiting threads probably waste computing resources and relinquish cores to other co-running applications after they are blocked. This paper attempts to find a new and intuitive way to improve the efficiency of barrier in competitive environments, and answer the question: Can we leverage the timeslices of waiting threads to accelerate barrier operations?Targeting this question, we propose a novel barrier synchronization mechanism named Tidon (Time Donating Barrier). The basic idea of Tidon is to donate the timeslices of waiting threads to their preempted, laggard siblings in order to accelerate barrier operations, different from traditional static spinning and blocking. We implement Tidon based on the GNU OpenMP runtime library (libgomp) and Linux kernel with new, lightweight system calls. Our evaluation with various sets of co-running applications demonstrates that the advantages of Tidon include (1) alleviating the performance degradation of barrier-intensive applications (e.g. improving the performance by up to a factor of 17.9 and 2.3 compared to the default barrier implementation of OpenMP in Completely Fair Scheduler and Balance Scheduling, respectively) while not hurting or even improving the performance of non-barrier-intensive applications, and (2) maintaining good fairness among co-running applications (e.g. improving the fairness by up to a factor of 19.8 and 1.7 compared to the default barrier implementation of OpenMP in Completely Fair Scheduler and Balance Scheduling, respectively).

Modelling default risk with occupation times

This paper develops a semi-analytic pricing formula, easily implemented via quadrature, for a structural model based on occupation times that contains both the Merton and Black–Cox models as limiting cases. In the model liquidation is triggered as soon as the firm’s asset value has spent a prespecified amount of time below the default barrier. Surprisingly, we find that the value of the firm’s debt (i) need not be monotone in the length of the grace period and (ii) need not lie between the limiting Merton and Black–Cox values.

Closed-form interpolation-based formulas for European call options written on defaultable assets

In this article, we derive closed-form, interpolation-based expressions for European call options written on defaultable assets. Our results are based on the work of Henderson et al, who derive formulas that incorporate standard static no-arbitrage restrictions, and Orosi who establishes an improved lower bound for European call options written on defaultable assets. Although, in general, the models are incapable of representing the entire call option surface because of the low number of parameters, we demonstrate their applicability to extract important quantities from quoted options. In particular, the probability of default and the size of a default barrier can be inferred from the model.

Dynamic optimal capital structure with regime switching

We investigate the optimal capital structure of a corporate when the dynamics of the assets (both growth rate and volatility) change following different states of the economy. Two structural models are examined in the paper. The first considers the case when the firm is not facing tax benefit and bankruptcy costs with a regime switching dynamics. This model extends the Black and Cox (J Financ 31:351–367, 1976) model to allow for regime switching risk. The second model incorporates both tax benefit and bankruptcy costs with a regime switching dynamics. This is is more realistic, and is an extension of the Leland (J Financ 49(4):1213–1252, 1994) model with regime switching risk. We obtain closed-form analytic solutions for the optimal capital structure and default barrier for both models.

Financial Methods: A Quantitative Approach

The study analyses finite difference methods and stochastic volatility for option pricing model till Asian and Barrier options. Simulated result is presented with VBA code. The interest rate models is analyzed in arbitrage setting e simulated environment by using an affine term structure and the drift condition in combination with inflation model by measuring the liquidity and risk premium by presenting an efficient Monte Carlo simulator. Structural Model is presented in single time maturity and default barrier in first passage model. The intensity model is also faced by analyzing the liquidity and risk premium with copula approaches as well

Credit market frictions and capital structure dynamics

We study the implications of credit market frictions for the dynamics of corporate capital structure and the risk of default of corporations. To do so, we develop a dynamic capital structure model in which firms face uncertainty regarding their ability to raise funds in credit markets and have to search for investors when seeking to adjust their capital structure. We provide a general analysis of shareholders' dynamic financing and default decisions, show when Markov perfect equilibria in financing and default barrier strategies exist, and when uniqueness can be achieved. We then use the model to generate a number of novel testable implications relating credit market frictions to target leverage, the pace and size of capital structure changes, creditor turnover, and the likelihood of default.

Higher order binaries with time dependent coefficients and two factors - model for defaultable bond with discrete default information

In this article, we consider a 2 factors-model for pricing defaultable bonds with discrete default intensity and barrier where the 2 factors are a stochastic risk free short rate process and firm value process. We assume that the default event occurs in an expected manner when the firm value reaches a given default barrier at predetermined discrete announcing dates or in an unexpected manner at the first jump time of a Poisson process with given default intensity given by a step function of time variable. Then our pricing model isgiven by a solving problem of several linear PDEs with variable coefficients and terminal value of binary type in every subinterval between the two adjacent announcing dates. Our main approach is to use higher order binaries. We first provide the pricing formulae of higher order binaries with time dependent coefficients and consider their integrals on the last expiry date variable. Then using the pricing formulae of higher binary options and their integrals, we give the pricing formulae of defaultable bonds in both cases of exogenous andendogenous default recoveries and perform credit spread analysis.

Default barrier intensity model for credit risk evaluation

We establish the Default Barrier Intensity (DBI) model, based on the conditional survival probability (also called hazard function barrier), which allows the pricing of credit derivatives with stochastic parameters. Moreover, the DBI is an analytic model which combines the structural and the reduced form approaches. It deals with the impact of the default barrier intensity on the processes around the firm. Using this model we prove the Doob–Meyer decomposition of the default process associated with the random barrier. In this framework, we present the default barrier process as the sum of its compensator (which is a predictable process) and a martingale related to the smallest filtration making the random barrier a stopping time. Furthermore, the DBI as well as the Shifted Square Root Diffusion (SSRD) Alfonsi’s model emphasizes on the dependence between the stochastic default intensity and the interest rate. This model can be useful since it can be easily generalized to all the credit derivatives products such as Collateralized Debt Obligations (CDO) and Credit Default Swaps (CDS).

Estimating Structural Models of Corporate Bond Prices in Indonesian Corporations

This  paper  applies  the  maximum  likelihood  (ML)  approaches  to  implementing  the structural  model  of  corporate  bond,  as  suggested  by  Li  and  Wong  (2008),  in  Indonesian corporations.  Two  structural  models,  extended  Merton  and  Longstaff  &  Schwartz  (LS) models,  are  used  in  determining  these  prices,  yields,  yield  spreads  and  probabilities  of default. ML estimation is used to determine the volatility of irm value. Since irm value is unobserved variable, Duan (1994) suggested that the irst step of ML estimation is to derive the likelihood function for equity as the option on the irm value. The second step is to ind parameters such as the drift and volatility of irm value, that maximizing this function. The irm value itself is extracted by equating the pricing formula to the observed equity prices. Equity,  total  liabilities,  bond  prices  data  and  the  irm's  parameters  (irm  value,  volatility of irm value, and default barrier) are substituted to extended Merton and LS bond pricing formula in order to valuate the corporate bond.These models are implemented to a sample of 24 bond prices in Indonesian corporation during  period  of  2001-2005,  based  on  criteria  of  Eom,  Helwege  and  Huang  (2004).  The equity  and  bond  prices  data  were  obtained  from  Indonesia  Stock  Exchange  for  irms  that issued equity and provided regular inancial statement within this period. The result shows that both models, in average, underestimate the bond prices and overestimate the yields and yield spread. ");} // --> activate javascript

On pricing of corporate securities in the case of jump-diffusion

Structural models of credit risk are known to present vanishing spreads at very short maturities. This shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering discontinuities of the jump type in their evolution over time. In this paper, we extend the pricing model for corporate bond and determine the default probability in jump-diffusion model to address this issue. To make the problem clearly, we first investigate the case that the firm value follows a geometric Brownian motion under similar assumptions to those in Black and Scholes (1973), Briys and de Varenne (1997), i.e, the default barrier is KD (t, T) and the recovery rate is (1 − ω), where D (t, T) is the price of zero coupon default free bond and ω is a constant (0 < ω ≤ 1). By changing the numeraire, we obtain the closed-form solution for both the price of bond and default probability. Further, we consider the case of jump-diffusion and suppose that a firm will go bankruptcy if its value V t ≤ KD (t, T) and at the same time, the bondholder will receive $(1 - \omega )\tfrac{{V_t }} {K} $ . By introducing the Green function of PDE with absorbing boundary and converting the problem to an II-type Volterra integral equation, we get the closed-form expressions in series form for bond price and corresponding default probability. Numerical results are presented to show the impact of different parameters to credit spread of bond.

Valuing Credit Default Swap under a double exponential jump diffusion model

This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geometric Brownian motion, and the default barrier follows a continuous stochastic process. Using the Gaver-Stehfest algorithm and the non-arbitrage asset pricing theory, we give the default probability of the first passage time, and more, derive the price of the Credit Default Swap.

Analytical pricing of defaultable discrete coupon bonds in unified two-factor model of structural and reduced form models

Pricing formulae for defaultable corporate bonds with discrete coupons (under consideration of the government taxes) in the united two-factor model of structural and reduced form models are provided. The aim of this paper is to generalize the two-factor structural model for defaultable corporate discrete coupon bonds (considered in [1]) into the unified model of structural and reduced form models. In our model the bond holders receive the stochastic coupon (which is the discounted value of a predetermined value at the maturity) at predetermined coupon dates and the face value (debt) and the coupon at the maturity as well as the effect of government taxes which are paid on the proceeds of an investment in bonds is considered. The expected default event occurs when the equity value is not sufficient to pay coupon or debt at the coupon dates or maturity and the unexpected default event can occur at the first jump time of a Poisson process with the given default intensity provided by a step function of time variable. We provide the model and pricing formula for equity value and using it calculate expected default barrier. Then we provide pricing model and formula for defaultable corporate bonds with discrete coupons and consider its duration.