A Simplified Firm Value-Based Risky Discount Bond Pricing Model

2007 ◽  
Vol 10 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Alan T. Wang ◽  
Sheng-Yung Yang
Keyword(s):  
Firm Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Credit Spread ◽  
Bond Pricing ◽  
Computational Time ◽  
Pricing Model ◽  
Management Tools ◽  
General Parameter ◽  
Parameter Values

This paper proposes a simplified risky discount bond pricing model based on Longstaff and Schwartz (1995). The advantage of this model is that it yields a closed form solution for probability of default. Also, a practical feature with our model is that computing durations and other risk management tools become computationally less expensive, while the appealing properties in the LS model are preserved. The numerical comparisons show that the differences in credit spreads between this model and Longstaff and Schwartz are within a few basis points for fairly general parameter values. Moreover, the computational time is shown remarkably reduced by the simplified model. Sensitivity analysis of credit spread with respect to different parameter values is presented.

scholarly journals Green's functions for unsymmetric composite laminates with inclusions

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190437
Author(s):  
Chia-Wen Hsu ◽  
Chyanbin Hwu
Keyword(s):  
Composite Laminates ◽  
Green's Functions ◽  
Green’S Functions ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Displacement Discontinuity ◽  
Computational Time ◽  
Out Of Plane ◽  
Logarithmic Functions

It is known that the stretching and bending deformations will be coupled together for the unsymmetric composite laminates under in-plane force and/or out-of-plane bending moment. Although Green's functions for unsymmetric composite laminates with elliptical elastic inclusions have been obtained by using Stroh-like formalism around 10 years ago, due to the ignoring of inconsistent rigid body movements of matrix and inclusion, the existing solution may lead to displacement discontinuity across the interface between matrix and inclusion. Due to the multi-valued characteristics of complex logarithmic functions appeared in Green's functions, special attention should be made on the proper selection of branch cuts of mapped variables. To solve these problems, in this study, the existing Green's functions are corrected and a simple way to correctly evaluate the mapped complex variable logarithmic functions is suggested. Moreover, to apply the obtained solutions to boundary element method, we also derive the explicit closed-form solution for Green's function of deflection. Since the continuity conditions along the interface have been satisfied in Green's functions, no meshes are required along the interface, which will save a lot of computational time and the results are much more accurate than any other numerical methods.

Corporate bond pricing model with stochastically volatile firm value process

2016 ◽  
Vol 148 ◽  
pp. 41-44
Author(s):  
Woon Wook Jang ◽  
Young Ho Eom ◽  
Yong Joo Kang
Keyword(s):  
Firm Value ◽  
Corporate Bond ◽  
Bond Pricing ◽  
Pricing Model

scholarly journals Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

2007 ◽  
Vol 2007 ◽  
pp. 1-25
Author(s):  
M. P. Markakis
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Nonlinear Odes ◽  
Analytical Technique ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Parameter Values

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.

A Set Probability Technique for Detecting Relative Time Order Across Multiple Neurons

2006 ◽  
Vol 18 (5) ◽  
pp. 1197-1214 ◽  
Author(s):  
Anne C. Smith ◽  
Peter Smith
Keyword(s):  
Closed Form Solution ◽  
Recursive Formula ◽  
Form Solution ◽  
Upper And Lower Bounds ◽  
Firing Patterns ◽  
Large N ◽  
Cell Firing ◽  
Parameter Values ◽  
Large Groups ◽  
Multielectrode Recording

With the development of multielectrode recording techniques, it is possible to measure the cell firing patterns of multiple neurons simultaneously, generating a large quantity of data. Identification of the firing patterns within these large groups of cells is an important and a challenging problem in data analysis. Here, we consider the problem of measuring the significance of a repeat in the cell firing sequence across arbitrary numbers of cells. In particular, we consider the question, given a ranked order of cells numbered 1 to N, what is the probability that another sequence of length n contains j consecutive increasing elements? Assuming each element of the sequence is drawn with replacement from the numbers 1 through N, we derive a recursive formula for the probability of the sequence of length j or more. For n < 2j, a closed-form solution is derived. For n ≥ 2j, we obtain upper and lower bounds for these probabilities for various combinations of parameter values. These can be computed very quickly. For a typical case with small N (<10) and large n (<3000), sequences of 7 and 8 are statistically very unlikely. A potential application of this technique is in the detection of repeats in hippocampal place cell order during sleep. Unlike most previous articles on increasing runs in random lists, we use a probability approach based on sets of overlapping sequences.

Frequency domain analysis method of nonstationary random vibration based on evolutionary spectral representation

2018 ◽  
Vol 35 (2) ◽  
pp. 1098-1127
Author(s):  
Yan Zhao ◽  
L.T. Si ◽  
H. Ouyang
Keyword(s):  
Frequency Domain ◽  
Random Vibration ◽  
Closed Form Solution ◽  
Frequency Domain Analysis ◽  
Random Excitation ◽  
Form Solution ◽  
Computational Time ◽  
Content Type ◽  
Damped System ◽  
Evolutionary Spectrum

Purpose A novel frequency domain approach, which combines the pseudo excitation method modified by the authors and multi-domain Fourier transform (PEM-FT), is proposed for analyzing nonstationary random vibration in this paper. Design/methodology/approach For a structure subjected to a nonstationary random excitation, the closed-form solution of evolutionary power spectral density of the response is derived in frequency domain. Findings The deterministic process and random process in an evolutionary spectrum are separated effectively using this method during the analysis of nonstationary random vibration of a linear damped system, only modulation function of the system needs to be estimated, which brings about a large saving in computational time. Originality/value The method is general and highly flexible as it can deal with various damping types and nonstationary random excitations with different modulation functions.

scholarly journals Large Deflection of Prismatic Cantilever Beam Exposed to Combination of End Inclined Force and Tip Moment

2017 ◽  
Vol 12 (1) ◽  
pp. 98 ◽  
Author(s):  
Ibrahim Abu-Alshaikh ◽  
Hashem S. Alkhaldi ◽  
Nabil Beithou
Keyword(s):  
Numerical Solution ◽  
Cantilever Beam ◽  
Large Deflection ◽  
Elliptic Integral ◽  
Geometrical Nonlinearity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Integration Constant ◽  
Computational Time ◽  
Special Cases

The large deflection of a prismatic Euler-Bernoulli cantilever beam under a combination of end-concentrated coplanar inclined force and tip-concentrated moment is investigated. The angle of inclination of the applied force with respect to the horizontal axis remains unchanged during deformation. The cantilever beam is assumed to be naturally straight, slender, inextensible and elastic. The large deflection of the cantilever beam induces geometrical nonlinearity; hence, the nonlinear theory of bending and the exact expression of curvature are used. Based on an elliptic integral formulation, an accurate numerical solution is obtained in terms of an integration constant that should satisfy the boundary conditions associated with the cantilever beam. For some special cases this integration constant is exactly found, which leads to closed form solution. The numerical solution obtained is quite simple, accurate and involves less computational time compared with other techniques available in literature. The details of elastica and its corresponding orientation curves are presented and analyzed for extremely large load combinations. A comparative study with pre-obtained results has been made to verify the accuracy of the presented solution; an excellent agreement has been obtained.

scholarly journals Three ways to solve for bond prices in the Vasicek model

2004 ◽  
Vol 8 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Rogemar S. Mamon
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytic Solution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Bond Pricing ◽  
Rate Process ◽  
Bond Price ◽  
Martingale Approach ◽  
Forward Measure

Three approaches in obtaining the closed-form solution of the Vasicek bond pricing problem are discussed in this exposition. A derivation based solely on the distribution of the short rate process is reviewed. Solving the bond price partial differential equation (PDE) is another method. In this paper, this PDE is derived via a martingale approach and the bond price is determined by integrating ordinary differential equations. The bond pricing problem is further considered within the Heath-Jarrow-Morton (HJM) framework in which the analytic solution follows directly from the short rate dynamics under the forward measure.

Defaultable sovereign debts with macroeconomic conditions and periodic news

2021 ◽  
Vol 08 (01) ◽  
pp. 2150012
Author(s):  
Weiping Li
Keyword(s):  
Markov Chain ◽  
Sovereign Debt ◽  
Structural Model ◽  
Closed Form Solution ◽  
Form Solution ◽  
Credit Spread ◽  
Macroeconomic Conditions ◽  
Sovereign Credit ◽  
Finite State ◽  
Information Update

This paper presents a default structural model of sovereign debt under macroeconomic conditions and periodic news. I model the macroeconomic conditions to be a finite state of Markov chain, and the periodic news to be a predictable factor in the drifting and the diffusion parts in the underlying value of the representative firm. The innovation of our model is to characterize the price of sovereign debt and the sovereign credit spread associated with macroeconomic conditions, and to model periodic news with both continuous factors and periodic factors. Both the defaultable yield-to-maturity, the sovereign credit spread and the duration are related to the finite state of Markov chain and periodic news. Furthermore, we obtain a closed-form solution for the two-state Markov chain associated to macroeconomic conditions and periodical information update.

A Piecewise-Linear Approximation of the Canonical Spring-Loaded Inverted Pendulum Model of Legged Locomotion

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Zhuohua Shen ◽  
Justin Seipel
Keyword(s):  
Closed Form ◽  
Inverted Pendulum ◽  
Piecewise Linear ◽  
Closed Form Solution ◽  
Legged Locomotion ◽  
Form Solution ◽  
Piecewise Linear Approximation ◽  
Computational Time ◽  
Slip Model ◽  
Spring Loaded Inverted Pendulum

Here, we introduce and analyze a novel approximation of the well-established and widely used spring-loaded inverted pendulum (SLIP) model of legged locomotion, which has made several validated predictions of the center-of-mass (CoM) or point-mass motions of animal and robot running. Due to nonlinear stance equations in the existing SLIP model, many linear-based systems theories, analytical tools, and corresponding control strategies cannot be readily applied. In order to provide a significant simplification in the use and analysis of the SLIP model of locomotion, here we develop a novel piecewise-linear, time-invariant approximation. We show that a piecewise-linear system, with the only nonlinearity due to the switching event between stance and flight phases, can predict all the bifurcation features of the established nonlinear SLIP model over the entire three-dimensional model parameter space. Rather than precisely fitting only one particular solution, this approximation is made to quantitatively approximate the entire solution space of the SLIP model and capture all key aspects of solution bifurcation behavior and parametric sensitivity of the original SLIP model. Further, we provide an entirely closed-form solution for the stance trajectory as well as the system states at the end of stance, in terms of common functions that are easy to code and compute. Overall, the closed-form solution is found to be significantly faster than numerical integration when implemented using both matlab and c++. We also provide a closed-form analytical stride map, which is a Poincaré return section from touchdown (TD) to next TD event. This is the simplest closed-form approximate stride mapping yet developed for the SLIP model, enabling ease of analysis and numerical coding, and reducing computational time. The approximate piecewise-linear SLIP model presented here is a significant simplification over previous SLIP-based models and could enable more rapid development of legged locomotion theory, numerical simulations, and controllers.

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