### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Hedging Market and Credit Risk in Corporate Bond Portfolios

statistics-lab™ 为您的留学生涯保驾护航 在代写金融中的随机方法Stochastic Methods in Finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融中的随机方法Stochastic Methods in Finance方面经验极为丰富，各种代写金融中的随机方法Stochastic Methods in Finance相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Hedging Market and Credit Risk in Corporate Bond Portfolios

The 2007-2009 financial turmoil witnessed an unprecedented market downturn for financial instruments carrying credit risk (Abaffy et al. 2007; Berndt 2004; Crouhy et al. 2000; Duffie and Singleton 1999) spanning both the secondary markets for corporate and sovereign securities and the markets for derivatives based on such instruments. The crisis propagated in US markets from mortgage-backed securities (MBS) and collateralized debt obligations (CDO) to credit instruments traded overthe-counter and thus into international portfolios. Widespread lack of liquidity in the secondary market induced first the Federal Reserve, then the European Central Bank, to adopt an expansive monetary policy through a sequence of base rate

reductions. Such policy partially limited the fall of bond prices but could do very little against instability in the corporate equity and fixed income markets.

The debate over the causes and possible remedies of such a prolonged financial crisis involved, from different perspectives, policy makers, financial intermediaries, and economists (European Central Bank 2010), all interested in analyzing the equilibrium recovery conditions, possibly within a quite different market architecture.
Financial investors, on the other hand, suffered dramatic portfolio losses within increasingly illiquid money, secondary stock and bond and credit derivative markets. In this chapter we present a stochastic model for interest rate and credit risk applied to a portfolio of corporate bonds traded in the Eurobond market with portfolio strategies tested over the 2008-2009 crisis. The portfolio management problem is formulated as a dynamic stochastic program with recourse (Consigli and Dempster 1998; Pflug and Römisch 2007 ; Zenios and Ziemba 2007). This chapter provides evidence of the potential offered by dynamic policies during a dramatic market crisis. Key to the results presented are the definitions of

• a statistical model capturing common and bond-specific credit risk factors that will determine jointly with the yield curve the defaultable bonds price behavior (Dai and Singleton 2003; Das and Tufano 1996; Duffie and Singleton 2000; Jarrow and Turnbull 2000; Kijima and Muromachi 2000; Longstaff et al. 2005) and
• a multistage strategy determined by a risk-reward objective function explicitly considering an extreme risk measure (Bertocchi et al. 2007; Consigli et al. 2010; Dempster et al. 2003; Jobst et al. 2006; Jobst and Zenios 2005; Rockafellar and Uryasev 2002). The problem considers a constant investment universe excluding credit derivatives and exogenous hedging strategies.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Corporate Bonds Risk Exposure

The recent credit crisis witnessed an unprecedented credit spread increase across all maturities in the Eurobond market while, first in the USD monetary area and then in the UK and Europe, interest rates were rapidly, though ineffectively, driven down to try to facilitate a liquidity recovery in the markets. In this work we assume a risk model incorporating both common and specific risk factors and allow the investor to determine her optimal policy by exploiting a scenario representation of the credit spreads evolution which may be affected at random times by severe market shocks. We consider a universe of Euro-denominated bonds with ratings from $A A A$ to $C C C-C$ (see (Abaffy et al. 2007)) plus one default-free government bond. Ratings are alphanumeric indicators specified by international rating agencies sueh as standard and poor, moody and fiteh, defining the eredit merit of the bond issuer and thus the likelihood of possible defaults over a given risk horizon. Rating revisions can be induced by market pressures or expert opinions and only occur infrequently. Bonds trading in the secondary market on the other hand generate a continuous information flow on expected interest rates and yield movements.

Bonds belonging to the same rating class may be distinguished according to their maturity and more importantly the activity sector to which the issuer belongs. Bond issuers within the same rating class and industry sector may finally be distinguished according to their market position and financial strength: the riskier the issuer, according to the above classification, the higher the credit spread that will be requested by investors to include a specific fixed income security in the portfolio.
Corporate prices are thus assumed to be driven by

• a common factor affecting every interest-sensitive security in the market (Cox et al. 1985), related to movements of the yield curve,
• a credit risk factor, related to movements of the credit curves (Abaffy et al. 2007; Duffie and Singleton 1999; Jobst and Zenios 2005), one for each rating class, and
• a specific bond factor, related to the issuer’s economic sector and its general financial health.

A degree of complexity may be added to this framework when trying to take into account the possible transition over the portfolio lifetime of bond issuers across different rating classes (Jarrow et al. 1997) and bond-specific liquidity features resulting in the bid-ask spread widening in the secondary market. In this work we will focus on the solution of a l-year corporate bond portfolio management problem with monthly portfolio revision, not considering either transition risk or liquidity risk explicitly.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Market and Credit Risk Model

We consider a credit-risky bond market with a set $I$ of securities. For each bond $i \in I$, we denote by $T_{i}$ its maturity. Let $K$ be the set of rating classes. Assuming

a canonical $\mathrm{S} \& \mathrm{P}$ risk partition from AAA to $\mathrm{CCC}-\mathrm{C}$ and $\mathrm{D}$ (the default state), we will denote by $k=1,2, \ldots, 8$ the credit risk indicator associated, respectively, with default-free sovereign and corporate AAA, AA, A, BBB, BB, B, CCC-C, and D ratings. A difference is thus postulated between a sovereign and a corporate AAA rating: the first one will be associated with the prevailing market yield for defaultfree securities with null credit spread over its entire market life, while the second may be associated with a positive, though negligible, credit spread. The security $i$ credit risk class is denoted by $k_{i}$, while $I_{k}$ is the set of bonds belonging to rating class $k$. The price of security $i$ at time $t$ will be denoted by $v_{t, T_{i}}^{i}$. For given security price and payment structure the yield $y_{t, T_{i}}^{i}$ can be inferred from the classical price-yield relationship:
$$v_{t, T_{i}}^{i}=\sum_{t<m \leq T_{i}} c_{m}^{i} e^{-y_{t, T_{i}}(m-t)}$$
where the cash payments over the security residual life are denoted by $c_{m}^{i}$ up to and including the maturity date. In (4.1) the yield $y_{t, T_{i}}^{i}$ will in general reflect the current term structure of risky interest rates for securities belonging to class $k_{i}$. We assume $y_{t, T_{i}}^{i}$ generated by a bond-relevant credit spread $\pi_{t, T_{i}}^{i}$ and a defaul $t$-free interest rate $r_{t, T_{i}}$, i.e. $y_{t, T_{i}}^{i}=r_{t, T_{i}}+\pi_{t, T_{i}}^{i}$

A one-factor model is considered for both the default-free interest rate curve and the security-specific credit spread. The latter is assumed to be generated by a rating-specific factor $\pi_{t}^{k}$ and an idyosincratic factor $\eta_{t}^{i}$.
The three state variables of the model are assumed to follow the s.d.e.’s:
\begin{aligned} d r_{t}(\omega) &=\mu_{r} d t+\sigma_{r}(t, r) d W_{t}^{r}(\omega) \ d \pi_{t}^{k}(\omega) &=\mu_{k} d t+\sigma_{k}\left(t, \pi^{k}\right) \sum_{l \in K} q^{k l} d W_{t}^{l}(\omega) \quad \forall k \ d \eta_{t}^{i}(\omega) &-\mu_{\eta} d t+\sigma_{i} d W_{t}^{i}(\omega)+\beta^{i}(\omega) d \Psi_{t}^{i}\left(\lambda^{i}\right) \quad \forall i \end{aligned}
where $\omega$ is used to identify a generic random variable. The first equation describes the short interest rate evolution as a diffusion process with constant drift and possibly state- and time-dependent volatility. $d W_{t}^{r} \sim N(0, d t)$ is a normal Wiener increment. The $k=1,2, \ldots, 7$ credit spread processes in $(4.3)$ are also modeled as diffusion processes with time- and state-varying volatilities. Credit spreads and the risk-free rate are correlated: the coefficients $q^{k i}$ denote the elements of the correlation matrix lower triangular Choleski factor linking the eight risk factors together, given the independent Wiener increments $d W_{t}^{k}, k=1,2, \ldots, 7$.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Hedging Market and Credit Risk in Corporate Bond Portfolios

2007-2009 年的金融风暴见证了带有信用风险的金融工具的前所未有的市场低迷（Abaffy 等人 2007；Berndt 2004；Crouhy 等人 2000；Duffie 和 Singleton 1999），横跨企业和主权证券的二级市场以及基于此类工具的衍生品市场。危机在美国市场从抵押贷款支持证券 (MBS) 和债务抵押债券 (CDO) 蔓延到场外交易的信贷工具，进而传播到国际投资组合。二级市场普遍缺乏流动性，首先是美联储，然后是欧洲央行，通过一系列基准利率采取扩张性货币政策

• 一个统计模型，捕捉共同的和特定于债券的信用风险因素，这些因素将与收益率曲线共同决定违约债券的价格行为（Dai 和 Singleton 2003；Das 和 Tufano 1996；Duffie 和 Singleton 2000；Jarrow 和 Turnbull 2000；Kijima 和 Muromachi 2000 ; Longstaff 等人，2005 年）和
• 一种多阶段策略，由明确考虑极端风险度量的风险回报目标函数确定（Bertocchi et al. 2007; Consigli et al. 2010; Dempster et al. 2003; Jobst et al. 2006; Jobst and Zenios 2005; Rockafellar and Uryasev 2002）。该问题考虑了一个不包括信用衍生品和外生对冲策略的恒定投资范围。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Corporate Bonds Risk Exposure

• 影响市场上每一种对利息敏感的证券的共同因素（Cox 等，1985），与收益率曲线的变动有关，
• 一个信用风险因素，与信用曲线的移动有关（Abaffy 等人 2007；Duffie 和 Singleton 1999；Jobst 和 Zenios 2005），每个评级类别一个，以及
• 与发行人的经济部门及其总体财务状况相关的特定债券因素。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Market and Credit Risk Model

v吨,吨一世一世=∑吨<米≤吨一世C米一世和−是吨,吨一世(米−吨)

dr吨(ω)=μrd吨+σr(吨,r)d在吨r(ω) d圆周率吨到(ω)=μ到d吨+σ到(吨,圆周率到)∑一世∈到q到一世d在吨一世(ω)∀到 d这吨一世(ω)−μ这d吨+σ一世d在吨一世(ω)+b一世(ω)dΨ吨一世(λ一世)∀一世

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。