The entropic measure transform

This article presents a structural model of default risk under macroeconomic conditions. The macroeconomic conditions are assumed to be a finite state of a Markov chain. The innovation of the author’s model is to characterize the firm default, the default-free pure discount bond price, the defaultable bond price, and the credit spread associated with macroeconomic conditions. By using the Wiener–Hopf factorization, the author shows that there is a risk-neutral measure under regime switching. Under macroeconomic conditions, he derives both the default-free bond price and the price of the defaultable bond governed by fundamental systems of partial differential equations. The defaultable yield to maturity, credit spread, and duration are related to the finite state of the Markov chain.

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. 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 risk-free interest rate, high payout rate and high volatility.

By exploiting the first public bond default event in China—the Chaori bond default in 2014—we examine how implicit government guarantees shape the role of media information in bond pricing. We find an insignificant association between preissuance media tones and bond issuances’ yield spreads before the event; however, the association becomes negative and significant after the event. The role of media tone in bond pricing is more pronounced for regions with greater information demand for default risk and for media outlets with stronger information provisions toward implicit government guarantees after the event. Mechanism analyses suggest increased bond investors’ risk awareness, rather than strengthened information provisions in media tones, as the force behind the more pronounced role of media tone in bond pricing. Finally, lower yield spreads driven by more positive media tone do not suggest bond overpricing. Our study reveals the role of media tone in bond pricing and displays the evolution of the media tone's role in bond pricing due to changing institutions.

In this paper we consider a stochastic model of systems (phase dynamics . -irreducible homogeneous Markov chain) that accumulate a resource (earn money in economic interpretation) and use it for functioning, reservation, etc. The essentially ubiquitous inhomogeneity of the phase space. In the phase space (general type) of such a system, a non-negative function X, a function Z with values 1,2, ..., d are given. The values >0 indicate the resource level of the system at time n, the values indicate the mode determined in the economic interpretation, for example, the number of service points, devices, service lines, the number of offices, workers, etc. .d., as well as the type of operation - repair, preventive maintenance, downtime. Let be the average increments X for a unit period for an arbitrary set of states where где ; ) is the probability of transition from state to mode j, P ( ) is the corresponding stochastic matrixand is the stationary probability distribution for it. The paper gives a mathematical proof , announced earlier by the authors: The existence of a monotone ,integrable on the interval [0,∞) function ψ(x)≥0 for which ( for large X ( )) is sufficient fact (if some practically general conditions ) for tending to 1 almost surely the relative fraction of those n up to the moment N ( N → ∞) for which > C (C - arbitrary). From the point of view of economic interpretation, the system is acceptable in the sense that it ensures the movement over time to 100% of the time of financial wealth and the growth of the level of wealth itself, despite the possibility of wealth falling to a minimum level (with 0-recurrence of the Markov chain, for sure an unlimited number of times).

Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility

A stochastic minimax optimal time-delay state feedback control strategy for uncertain quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal state feedback control problem of uncertain quasi-integrable Hamiltonian system with time delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed state feedback control forces are approximated by the control forces without time delay and the original problem is converted into a stochastic optimal state feedback control problem of an uncertain system without time delay. After that, by following a procedure based on the stochastic averaging method and stochastic differential game, the worst-case disturbances and the optimal controls are obtained from solving a Hamiltonian–Jacobi–Isaacs (HJI) equation. As an example, the stochastic minimax optimal state feedback control of a Duffing oscillator with parametric disturbances and time-delayed feedback control is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.

A four‐factor model (the extended model of Schmid and Zagst) is presented for pricing credit risk related instruments such as defaultable bonds or credit derivatives. It is an advancement of an earlier three‐factor model. In addition to a firm‐specific credit risk factor, a new systematic risk factor in the form of GDP growth rate is included. This new model is set in the context of other hybrid defaultable bond pricing models and empirically compared to specific representatives. We find that a model based only on firm‐specific variables is unable to capture changes in credit spreads completely. However, it is shown that in this model, market variables such as GDP growth rates, non‐defaultable interest rates and firm‐specific variables together significantly influence credit spread levels and changes.

The stochastic optimal nonlinear control of coupled adjacent building structures is studied based on the stochastic dynamical programming principle and the stochastic averaging method. The coupled structures with control devices under random seismic excitation are first condensed to form a reduced-order structural model for the control analysis. The stochastic averaging method is applied to the reduced model to yield stochastic differential equations for structural modal energies as controlled diffusion processes. Then a dynamical programming equation for the energy processes is established based on the stochastic dynamical programming principle, and solved to determine the optimal nonlinear control law. The seismic response mitigation of the coupled structures is achieved through the structural energy control and the dimension of the optimal control problem is reduced. The seismic excitation spectrum is taken into account according to the stochastic dynamical programming principle. Finally, the nonlinear controlled structural response is predicted by using the stochastic averaging method and compared with the uncontrolled structural response to evaluate the control efficacy. Numerical results are given to demonstrate the response mitigation capabilities of the proposed stochastic optimal control method for coupled adjacent building structures.

The problem of term structure of interest rates modelling is considered in a continuous-time framework. The emphasis is on the bond prices, forward bond prices and so-called LIBOR rates, rather than on the instantaneous continuously compounded rates as in most traditional models. Forward and spot probability measures are introduced in this general set-up. Two conditions of no-arbitrage between bonds and cash are examined. A process of savings account implied by an arbitrage-free family of bond prices is identified by means of a multiplicative decomposition of semimartingales. The uniqueness of an implied savings account is established under fairly general conditions. The notion of a family of forward processes is introduced, and the existence of an associated arbitrage-free family of bond prices is examined. A straightforward construction of a lognormal model of forward LIBOR rates, based on the backward induction, is presented.

Evidence of Personal Tax Effects on the Pricing of Corporate Bonds

Pricing nondiversifiable credit risk in the corporate Eurobond market

This paper derives a maximum principle for dynamic systems with continuous lags, i.e., systems governed by integrodifferential equations, using dynamic programming. As a result, the adjoint variables can be provided with useful economic interpretations.

A minimax stochastic optimal semi-active control strategy for stochastically excited quasi-integrable Hamiltonian systems with parametric uncertainty by using magneto-rheological (MR) dampers is proposed. Firstly, the control problem is formulated as an n-degree-of-freedom (DOF) controlled, uncertain quasi-integrable Hamiltonian system and the control forces produced by MR dampers are split into the passive part and the semi-active part. Then the passive part is incorporated into the uncontrolled system. After that, the stochastic optimal semi-active control problem is solved by applying the minimax stochastic optimal control strategy based on the stochastic averaging method and stochastic differential game. The worst-case disturbances and the optimal controls are obtained by the minimax dynamical programming equation with the constraints of disturbance bounds and MR damper dynamics. Finally, the system response and the control performance are evaluated by using Monte Carlo simulation. An example of a two-DOF system with coupling damping and parametric uncertainty under Gaussian white noise excitations is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy, which is also compared with the clipped linear-quadratic-Gaussian control strategy to show the advantages.

It is well known that a discounting function describes the price of a default free bond. In this paper, some mathematical properties of this function are studied when it is subadditive. Subadditive discounting means that the total discount is minor when the year is divided into months and implies that the more divided the interval is into subintervals, the smaller the total discount will be. First, two sufficient conditions for a discounting function being subadditive (resp. superadditive) are shown; more precisely, the increase (resp. decrease) of the instantaneous discount rate and certain integral inequality are proposed as such sufficient conditions. On the other hand, the convexity (resp. concavity) of the Napierian logarithm of the discounting function is shown as a particular case of another more general sufficient condition. Finally, under certain conditions, subadditive discounting functions are necessarily regular.

In this article, an optimized leader-following consensus control scheme is proposed for the nonlinear strict-feedback-dynamic multi-agent system by learning from the controlling idea of optimized backstepping technique, which designs the virtual and actual controls of backstepping to be the optimized solution of corresponding subsystems so that the entire backstepping control is optimized. Since this control needs to not only ensure the optimizing system performance but also synchronize the multiple system state variables, it is an interesting and challenging topic. In order to achieve this optimized control, the neural network approximation-based reinforcement learning (RL) is performed under critic-actor architecture. In most of the existing RL-based optimal controls, since both the critic and actor RL updating laws are derived from the negative gradient of square of the Hamilton-Jacobi-Bellman (HJB) equation's approximation, which contains multiple nonlinear terms, their algorithm are inevitably intricate. However, the proposed optimized control derives the RL updating laws from the negative gradient of a simple positive function, which is correlated with the HJB equation; hence, it can be significantly simple in the algorithm. Meanwhile, it can also release two general conditions, known dynamic and persistence excitation, which are required in most of the RL-based optimal controls. Therefore, the proposed optimized scheme can be a natural selection for the high-order nonlinear multi-agent control. Finally, the effectiveness is demonstrated by both theory and simulation.