AN IMPROVED APPROACH TO EVALUATE DEFAULT PROBABILITIES AND DEFAULT CORRELATIONS WITH CONSISTENCY

We provide (i) a simplified analytic closed form formula for evaluating joint default probability, (ii) an improved method to resolve the inconsistency between the univariate process underlying firm-specific default probability and the correlated bivariate process of the first-passage-time default correlation model, (iii) illustration of risk management implications from misspecification of the default state space. Our closed form formula provides a natural extension of previous structural first-passage-time models and shows the sensitivities of default correlation numerically with respect to the underlying asset correlation, obligor credit quality and time horizon. We emphasize the disparate credit risk management implications of our result in contrast to commonly used risk measurement methods.

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Three Essays on Stopping

Risks ◽

10.3390/risks7040105 ◽

2019 ◽

Vol 7 (4) ◽

pp. 105

Author(s):

Eberhard Mayerhofer

Keyword(s):

First Passage Time ◽

Passage Time ◽

Alternative Proof ◽

Sufficient Condition ◽

Diffusion Characteristic ◽

Reflected Brownian Motion ◽

First Passage ◽

Brownian Motion With Drift ◽

Spectrally Negative Lévy Process ◽

Closed Form Formula

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ / σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).

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The Impact of Default Correlations on the Prices of Collateralized Bond Obligat

Journal of Derivatives and Quantitative Studies ◽

10.1108/jdqs-01-2002-b0005 ◽

2002 ◽

Vol 10 (1) ◽

pp. 113-142

Author(s):

In Joon Kim ◽

Suk Joon Byun ◽

Yuen Jung Park

Keyword(s):

Firm Value ◽

First Passage Time ◽

Market Price ◽

Numerical Procedure ◽

Passage Time ◽

Correlation Model ◽

Default Correlation ◽

First Passage ◽

Time Model ◽

The Impact

This paper presents a numerical procedure for pricing collateralized bond obligations (CBO) and analyze the impact of default correlations for the prices of collateralized bond obligations. Specifically, we adopt default correlation model of Zhou (2001) and first passage time model of Black and Cox (1976). The model of Black and Cox is used for estimating the value of the firm and the volatility of the firm value which are unobservable variables. We find that the impact of default correlations on the prices of collateralized bond obligations is generally quite large. This can be tested by carrying out Monte-Carlo simulations for firm value processes, assuming first no default correlations and second modeling default correlations between the processes. We also compare the model prices and recently issued CBO market price and find that no default correlation model over prices the issued CBO and default correlation model under prices the issued CBO. These results in this paper emphasize that modeling default correlations is very important in analyzing CBO and a more complicated further analysis is required.

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A note on the Volterra integral equation for the first-passage-time probability density

Journal of Applied Probability ◽

10.1017/s0021900200103092 ◽

1995 ◽

Vol 32 (03) ◽

pp. 635-648 ◽

Cited By ~ 3

Author(s):

R. Gutiérrez Jáimez ◽

P. Román Román ◽

F. Torres Ruiz

Keyword(s):

Integral Equation ◽

Closed Form ◽

Volterra Integral Equation ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Closed Form Solutions ◽

Actual Use ◽

Probability Densities

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

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On first-passage-time and transition densities for strongly symmetric diffusion processes

Nagoya Mathematical Journal ◽

10.1017/s0027763000006140 ◽

1997 ◽

Vol 145 ◽

pp. 143-161 ◽

Cited By ~ 9

Author(s):

A. Di Crescenzo ◽

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi

Keyword(s):

Diffusion Process ◽

Closed Form ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Time Dependent ◽

Closed Form Expression ◽

First Passage ◽

One Dimensional ◽

Transition Functions

One dimensional diffusion processes have been increasingly invoked to model a variety of biological, physical and engineering systems subject to random fluctuations (cf., for instance, Blake, I. F. and Lindsey, W. C. [2], Abrahams, J. [1], Giorno, V. et al [10] and references therein). However, usually the knowledge of the ‘free’ transition probability density function (pdf) is not sufficient; one is thus led to the more complicated task of determining transition functions in the presence of preassigned absorbing boundaries, or first-passage-time densities for time-dependent boundaries (see, for instance, Daniels, H. E. [6], [7], Giorno, V. et al. [10]). Such densities are known analytically only in some special instances so that numerical methods have to be implemented in general (cf., for instance, Buono-core, A. et al [3], [4], Giorno, V. et al [11]). The analytical approach becomes particularly effective when the diffusion process exhibits some special features, such as the symmetry of its transition pdf. For instance, in [10] special symmetry conditions on the transition pdf of one-dimensional time-homogeneous diffusion process with natural boundaries are investigated to derive closed form results concerning the transition pdf’s and the first-passage-time pdf for particular time-dependent boundaries. On the other hand, by using the method of images, in [6] Daniels has obtained a closed form expression for the transition pdf of the standard Wiener process in the presence of a particular time-dependent absorbing boundary. It is interesting to remark that such density cannot be obtained via the methods described in [10], even though the considered process exhibits the kind of symmetry discussed therein.

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A note on the Volterra integral equation for the first-passage-time probability density

Journal of Applied Probability ◽

10.2307/3215118 ◽

1995 ◽

Vol 32 (3) ◽

pp. 635-648 ◽

Cited By ~ 24

Author(s):

R. Gutiérrez Jáimez ◽

P. Román Román ◽

F. Torres Ruiz

Keyword(s):

Integral Equation ◽

Closed Form ◽

Volterra Integral Equation ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Closed Form Solutions ◽

Actual Use ◽

Probability Densities

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Closed-form solutions for the first-passage-time problem and neuronal modeling

Ricerche di Matematica ◽

10.1007/s11587-015-0248-6 ◽

2015 ◽

Vol 64 (2) ◽

pp. 421-439 ◽

Cited By ~ 10

Author(s):

Aniello Buonocore ◽

Luigia Caputo ◽

Giuseppe D’Onofrio ◽

Enrica Pirozzi

Keyword(s):

Closed Form ◽

First Passage Time ◽

Passage Time ◽

Neuronal Modeling ◽

First Passage ◽

Closed Form Solutions ◽

Time Problem ◽

First Passage Time Problem

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First-Passage Time Model Driven by Lévy Process for Pricing CoCos

Mathematical Problems in Engineering ◽

10.1155/2017/5171470 ◽

2017 ◽

Vol 2017 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Xiaoshan Su ◽

Manying Bai

Keyword(s):

Closed Form ◽

Lévy Process ◽

First Passage Time ◽

Passage Time ◽

Levy Process ◽

Closed Form Expression ◽

Lévy Measure ◽

First Passage ◽

Time Model ◽

Time Problem

Contingent convertible bonds (CoCos) are typical form of contingent capital that converts into equity of issuing firm or writes down if a prespecified trigger occurs. This paper proposes a general Lévy framework for pricing CoCos. The Lévy framework indicates that the difficulty in giving closed-form expression for CoCos price is the possible introduction of the Lévy process whose first-passage time problem has not been solved. According to characteristics of new Lévy measure after the measure transform, three specific Lévy models driven by drifted Brownian motion, spectrally negative Lévy process, and double exponential jump diffusion process are proposed to give the solution keeping the form of the driving process unchanged under the measure transform. These three Lévy models provide closed-form expressions for CoCos price while the latter two possess them up to Laplace transform, whose pricing results are given by combining with numerical Fourier inversion and Laplace inversion. Numerical results show that negative jumps have large influence on CoCos pricing and the Black-Scholes model would overestimate CoCos price by simply compressing jumps information into volatility while the other two models would give more accurate CoCos price by taking jump risk into consideration.

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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