基于结构化方法的含信用等级迁移的公司债券定价

考虑在债券发行方可能发生信用等级迁移的情况下的公司零息债券定价问题.假设公司资产价值变化满足几何Brownian运动,而债券的信用等级只与公司的资产有关.运用结构化方法的思想,通过给定不同的等级迁移边界条件,建立了两个具信用等级迁移可能性的债券定价模型.定价模型均可以用在迁移边界耦合的偏微分方程表示.分析了两个模型的关系,并求出第二个模型的显式解.最后作图展示了两种模型下债券价格关于各参数的变化情况,并分析了其金融意义.     

On dynamic programming equations for utility indifference pricing under delta constraints

In this paper we study the problem of utility indifference pricing in a constrained financial market, using a utility function defined over the positive real line. We present a convex risk measure −v(•:y) satisfying q(x,F)=x+v(F:u₀(x)), where u₀(x) is the maximal expected utility of a small investor with the initial wealth x, and q(x,F) is a utility indifference buy price for a European contingent claim with a discounted payoff F. We provide a dynamic programming equation associated with the risk measure (−v), and characterize v as a viscosity solution of this equation.     

Convergence of Utility Indifference Prices to the Superreplication Price

A discrete-time financial market model is considered with a sequence of investors whose preferences are described by concave strictly increasing functions defined on the positive axis. Under suitable conditions, we show that the utility indifference prices of a bounded contingent claim converge to its superreplication price when the investors’ absolute risk-aversion tends to infinity.     

Convergence of Utility Indifference Prices to the Superreplication Price: the Whole Real Line Case

A discrete-time financial market model is considered with a sequence of investors whose preferences are described by concave strictly increasing functions defined on the whole real line. Under suitable conditions we prove that, whenever their absolute risk-aversion tends to infinity, the respective utility indifference prices of a given bounded contingent claim converge to the superreplication price. We also prove that there exists an accumulation point of the optimal strategies’ sequence which is a superhedging strategy.     

Utility-based indifference pricing in regime-switching models

In this paper, we study utility-based indifference pricing and hedging of a contingent claim in a continuous-time, Markov, regime-switching model. The market in this model is incomplete, so there is more than one price kernel. We specify the parametric form of price kernels so that both market risk and economic risk are taken into account. The pricing and hedging problem is formulated as a stochastic optimal control problem and is discussed using the dynamic programming approach. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution to the problem is given. An issuer’s price kernel is obtained from a solution of a system of linear programming problems and an optimal hedged portfolio is determined.     

Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach

This paper deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modelled as an Ornstein-Uhlenbeck process. Here, only the stock price and interest rate can be observable for an investor. It is reduced to a partially observed stochastic control problem. Combining the filtering theory with the dynamic programming approach, explicit representations of the optimal value functions and corresponding optimal strategies are derived. Moreover, closed-form solutions are provided in two cases of exponential utility and logarithmic utility. In particular, logarithmic utility is considered under the restriction of short-selling and borrowing.      

On a multiple credit rating migration model with stochastic interest rate

In this paper, we explore a pricing model for corporate bond accompanied with multiple credit rating migration risk and stochastic interest rate. The bond price volatility strongly depends on potentially multiple credit rating migration and stochastic change of interest rate. A free boundary problem of partial differential equation is presented, which is the equivalent transformation of the pricing model. The existence, uniqueness, and regularity for the free boundary problem are established to guarantee the rationality of the pricing model. Due to the stochastic change of interest rate, the discontinuous coefficient in the free boundary problem depends explicitly on the time variable but is convergent as time tends to infinity. Accordingly, an auxiliary free boundary problem is constructed, whose coefficient is the convergent limit of the coefficient in the original free boundary problem. With some constraint on the risk discount rate satisfied, we prove that a unique traveling wave exists in the auxiliary free boundary problem. The inductive method is adopted to fit the multiplicity of credit rating. Then we show that the solution of the original free boundary problem converges to the traveling wave in the auxiliary free boundary problem. Returning to the pricing model with multiple credit rating migration and stochastic interest rate, we conclude that the bond price profile can be captured by a traveling wave pattern coupling with a guaranteed bond price with face value equal to one at the maturity.     

Valuation of power plants by utility indifference and numerical computation

This paper presents a real option valuation model of a power plant, which accounts for physical constraints and market incompleteness. Switching costs, minimum on-off times, ramp rates, or non-constant heat rates are important characteristics that can lead, if neglected, to overestimated values. The existence of non-hedgeable uncertainties is also a feature of energy markets that can impact assets value. We use the utility indifference approach to define the value of the physical asset. We derive the associated mixed optimal switching-control problem and provide a characterization of its solution by means of a coupled system of reflected Backward Stochastic Differential Equations (BSDE). We relate this system to a system of variational inequalities, and we provide a numerical comparative study by implementing BSDE simulation algorithms, and PDE finite differences schemes.     

用等效用原理对马氏链市场的UL生存合约定价

利用效用无差异原理,根据动态规划原则,最大化财富的期望指数效用,在马氏链驱动的市场下,导出HJB方程,给出unit-linked(UL)生存合约在简单Poisson市场下的保费方程,并给出它的数值模拟．这个结果推广了Brown运动驱动的市场下的保费方程,使得UL生存合约在联接到纯跳的市场时,可以用效用无差异原理定价．     

Portfolio selection problem with multiple risky assets under the constant elasticity of variance model

This paper focuses on the constant elasticity of variance (CEV) model for studying the utility maximization portfolio selection problem with multiple risky assets and a risk-free asset. The Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem is established. By applying a power transform and a variable change technique, we derive the explicit solution for the constant absolute risk aversion (CARA) utility function when the elasticity coefficient is −1 or 0. In order to obtain a general optimal strategy for all values of the elasticity coefficient, we propose a model with two risky assets and one risk-free asset and solve it under a given assumption. Furthermore, we analyze the properties of the optimal strategies and discuss the effects of market parameters on the optimal strategies. Finally, a numerical simulation is presented to illustrate the similarities and differences between the results of the two models proposed in this paper.     

Estimation of rating classes and default probabilities in credit risk models with dependencies

Let Y = m(X) + ε be a regression model with a dichotomous output Y and a one‐step regression function m . In the literature, estimators for the three parameters of m , that is, the breakpoint θ and the levels a and b , are proposed for independent and identically distributed (i.i.d.) observations. We show that these standard estimators also work in a non‐i.i.d. framework, that is, that they are strongly consistent under mild conditions. For that purpose, we use a linear one‐factor model for the input X and a Bernoulli mixture model for the output Y . The estimators for the split point and the risk levels are applied to a problem arising in credit rating systems. In particular, we divide the range of individuals' creditworthiness into two groups. The first group has a higher probability of default and the second group has a lower one. We also stress connections between the standard estimator for the cutoff θ and concepts prevalent in credit risk modeling, for example, receiver operating characteristic. Copyright © 2014 John Wiley & Sons, Ltd.     

Kolmogorov–Smirnov‐type testing for the partial homogeneity of Markov processes—with application to credit risk

In banking, the default behaviour of the counterpart is not only of interest for the pricing of transactions under credit risk but also for the assessment of a portfolio credit risk. We develop a test against the hypothesis that default intensities are chronologically constant within a group of similar counterparts, e.g. a rating class. The Kolmogorov–Smirnov‐type test builds up on the asymptotic normality of counting processes in event history analysis. The right censoring accommodates for Markov processes with more than one no‐absorbing state. A simulation study and two examples of rating systems demonstrate that partial homogeneity can be assumed, however occasionally, certain migrations must be modelled and estimated inhomogeneously. Copyright © 2007 John Wiley & Sons, Ltd.     

债券定价及利率风险的市场价格的重构方法

假设利率变化的模型是由随机微分方程给出,则可以用推导Black-Scholes方程的方法来推出债券价格满足的偏微分方程,得到一个抛物型的偏微分方程.但是,在债券定价的方程中隐含有一个参数λ称为利率风险的市场价格.所谓债券定价的反问题,就是由不同到期时间的债券的现在价格来得到利率风险的市场价格.对随机利率模型下债券定价的正问题先给予介绍和差分数值求解方法,并介绍了反问题,且对反问题给出了数值方法.     

带注资的二维复合泊松模型的最优分红

研究建立两类理赔关系的二维复合泊松模型的最优分红与注资问题,目标为最大化分红减注资的折现. 该问题由随机控制问题刻画, 通过解相应的哈密尔顿-雅克比-贝尔曼(HJB)方程,得到了最优分红策略,并在指数理赔时明确地解决该问题.