### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Price–Yield Relationship

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## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Price–Yield Relationship

Following the intensity specification in (4.7) an individual security can at any time be affected by a jump-lo-default event attributed by the arrival of new information that will drive the security into a default state. The relationship between the incremental spread behavior and the security price is as follows. Let the value of the default process $d \Psi_{\tau}^{i}=1$ at a random time $\tau$. Then, given a recovery value $\beta^{i}$, we have

$$v_{\tau, T_{i}}^{i}=\sum_{\tau<m \leq T_{i}} c_{m}^{i} e^{-\beta_{i}^{i}} e^{-\left(r_{z}+\pi_{t}^{k_{i}}\right)(m-\tau)}$$
At the default announcement, the bond price will reflect the values of all payments over the bond residual life discounted at a flat risky discount rate including the risk-free interest rate and the class $k_{i}$ credit spread after default. Widening credit spreads across all the rating classes will push default rates upward without discriminating between corporate bonds within each class. On the other hand, once default is triggered at $\tau$ then the marginal spread upward movement will be absorbed by a bond price drop which is consistent with the postulated recovery rate.

Following $(4.5)$ the price movement will depend on the yield jump induced by $\eta_{t}^{i}$ This is assumed to be consistent in the mean with rating agencies recovery estimates (Moody’s Investors Service 2009 ) and carries a user-specified uncertainty reflected in a lognormal distribution with stochastic jump size $\beta^{i} \sim \ln N\left(\mu_{\beta^{i}}, \sigma_{\beta^{i}}\right)$. Then $\ln \left(\beta^{i}\right) \sim N\left(\mu_{\beta^{i}}, \sigma_{\beta^{i}}\right)$ so that $\beta_{t}^{i} \in(0,+\infty)$ and both $e^{-\beta^{i}}$ (the recovery rate) and $1-e^{-\beta^{i}}$ (the loss rate) belong to the interval $(0,1)$.

In summary the modeling framework has the following characteristic features:

• default arrivals and loss upon default are defined taking into account ratingspecific estimates by Moody’s $(2009)$ adjusted to account for a borrower’s sector of activity and market-default premium;
• over a l-year investment horizon default intensities are assumed constant and will determine the random arrival of default information to the market; upon default the marginal spread will suffer a positive shock immediately absorbed by a pricenegative movement reflecting the constant partial payments of the security over its residual life;
• the Poisson process and the Wiener processes drive the behavior of an independent ideosyncratic spread process which is assumed to determine the security price dynamics jointly with a short rate and a credit spread process;
• the credit spreads for each rating class and the short interest rate are correlated.
The model accommodates the assumption of correlated risk factors whose behavior will drive all securities across the different rating categories, but a specific default risk factor is included to differentiate securities behavior within each class. No contagion phenomena are considered and the exogenous spread dynamics impact specific bond returns depending on their duration and convexity structure.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|The Portfolio Value Process

We now extend the statistical model to describe the key elements affecting the portfolio value dynamics over a finite time horizon $T$. Assuming an investment universe of fixed income securities with annual coupon frequency we will have only one possible coupon payment and capital reimbursement over a l-year horizon. Securities may default and generate a credit loss only at those times. We assume

a wealth equation defined at time 0 by an investment portfolio value and a cash account $\mathcal{W}{0}=\sum{k \in K} \sum_{i \in I_{k}} X_{i 0}+C_{0}$, where $X_{i 0}$ is the time-0 market value of the position in security $i$, generated by a given nominal investment $x_{i 0}$ and by the current security price $v_{0, T_{i}}^{i}$. For $t=0, \Delta t, \ldots, T-\Delta t$, we have
$$\mathcal{W}{t+\Delta t, \omega t}=\sum{k \in K} \sum_{i \in h_{h}} x_{i t} v_{i, T_{i}}^{i}\left(1+\rho_{t+\Delta t}^{i}(\omega)\right)+C(t+\Delta t, \omega)$$
In (4.9) the price return $\rho_{t+\Delta t}^{i}$ is generated for each security according to (4.5) as $\frac{v^{i}(t+\Delta t, \omega)}{v^{i}(t)}-1=\left[-\delta_{t}^{i} d y_{t}^{i}+y_{t}^{i} d t+0.5 \gamma_{t}^{i}\left(d y_{t}^{i}\right)^{2}\right]$, where $y_{t}^{i}$ represents the yield return at time $t$ for security $i$. For $k=0$ we have the default-free class and the yield will coincide with the risk-free yield. As time evolves an evaluation of credit losses is made. Over the annual time span as before, we have
\begin{aligned} C(t&+\Delta t, \omega)=C(t) e^{r_{i}(\omega) \Delta t}+\sum_{k \in K} \sum_{i \in I_{k}} c_{i, t+\Delta t}+\ &-\sum_{k \in K} \sum_{i \in l_{k}} c_{i, t+\Delta t}\left(1-e^{\left[-\beta_{t+\Delta t}^{i}(\omega) d W_{i+\Delta t}^{i}(\omega)\right]}\right) . \end{aligned}
The cash balance $C(t)$ evolution will depend on the expected cash inflows $c_{i, t}$ and the associated credit losses for each position $i$, the third factor of (4.10). Coupon and capital payments in (4.10) are generated for given nominal investment by the security-specific coupon rates and expiry date. If a default occurs then, for a given recovery rate, over the planning horizon the investor will face both a cash loss in (4.10) and the market value loss of (4.9) reflecting an assumption of constant partial payments over the security residual time span within the default state.

Equation (4.10) considers payment defaults explicitly. However, the mathematical model introduced below considers an implicit default loss definition. This is obtained by introducing the coefficient specification
$$f_{i, t+\Delta t}(\omega)=c_{i}\left(e^{-\left[\beta_{i+\Delta t}^{i}(\omega) d \Psi_{l+\Delta i^{i}}^{i}(\omega)\right]}\right)$$
and, thus, $c_{i, t+\Delta t}=x_{i t} f_{i, s+\Delta t}$, where $c_{i}$ denotes the coupon rate on security $i$.
In this framework the corporate bond manager will seek an optimal dynamic strategy looking for a maximum price and cash return on his portfolio and at the same time trying to avnid possihle defanlt losses. The model is quite general and alternative specifications of the stochastic differential equations (4.2), (4.3), and (4.4) will lead to different market and loss scenarios accommodating a wide range of possible wealth distributions at the horizon.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Dynamic Portfolio Model

In this section we present a multistage portfolio model adapted to formulate and solve the market and credit risk control problem. It is widely accepted that stochastic programming provides a very powerful paradigm in modeling all those applications characterized by the joint presence of uncertainty and dynamics. In the area of portfolio management, stochastic programming has been successfully applied as witnessed by the large number of scientific contributions appearing in recent decades (Bertocchi et al. 2007; Dempster et al. 2003; Hochreiter and Pflug 2007; Zenios and Ziemba 2007). However, few contributions deal with the credit and market risk portfolio optimization problem within a stochastic programming framework. Here we mention (Jobst et al. 2006) (see also the references therein) where the authors present a stochastic programming index tracking model for fixed income securities.
Our model is along the same lines, however, introducing a risk-reward trade-off in the objective function to explicitly account for the downside generated by corporate defaults.

We consider a 1 -year planning horizon divided in the periods $t=0, \ldots, T$ corresponding to the trading dates. For each time $t$, we denote by $\mathcal{N} t$ the set of nodes at stage $t$. The root node is labeled with 0 and corresponds to the initial state. For $t \geq 1$ every $n \in \mathcal{N}{t}$ has a unique ancestor $a{n} \in \mathcal{N}{t-1}$, and for $t \leq T-1$ a non-empty set of child nodes $H{n} \in \mathcal{N}{t+1}$. We denote by $\mathcal{N}$ the whole set of nodes in the scenario tree. In addition, we refer by $t{n}$ the time stage of node $n$. A scenario is a path from the root to a leaf node and represents a joint realization of the random variables along the path to the planning horizon. We shall denote by $S$ the set of scenarios. Figure $4.1$ depicts an example scenario tree.

The scenario probability distribution $\mathcal{P}$ is defined on the leaf nodes of the scenario tree so that $\sum_{n \in N_{r}} p_{n}=1$ and for each non-terminal node $p_{n}=$ $\sum_{m \in H_{n}} p_{m}, \forall n \in N_{t}, t=T-1, \ldots, 0$, that is each node receives a conditional probability mass equal to the combined mass of all its descendant nodes. In our case study portfolio revisions imply a transition from the previous time $t-1$ portfolio allocation at the ancestor node to a new allocation through holding, buying, and selling decisions on individual securities, for $t=1, \ldots, T-1$. The last possible revision is at stage $T-1$ with one period to go. Consistent with the fixed-income portfolio problem, decisions are described by nominal, face-value, positions in the individual security $i \in I$ of rating class $k_{i} \in K$.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Price–Yield Relationship

• 违约到达和违约损失的定义考虑了穆迪对特定评级的估计(2009)调整以考虑借款人的活动部门和市场违约溢价；
• 在 1 年的投资期限内，违约强度假设为常数，并将决定违约信息随机进入市场；违约时，边际价差将立即受到正向冲击，价格负向变动反映了证券在其剩余期限内不断部分支付；
• Poisson 过程和 Wiener 过程驱动独立的异质价差过程的行为，该过程假设与短期利率和信用价差过程共同确定证券价格动态；
• 每个评级等级的信用利差和短期利率是相关的。
该模型包含相关风险因素的假设，其行为将推动不同评级类别的所有证券，但包含特定的违约风险因素以区分每个类别中的证券行为。没有考虑传染现象，外生的价差动态会根据债券的久期和凸度结构影响特定的债券回报。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|The Portfolio Value Process

C(吨+Δ吨,ω)=C(吨)和r一世(ω)Δ吨+∑到∈到∑一世∈一世到C一世,吨+Δ吨+ −∑到∈到∑一世∈一世到C一世,吨+Δ吨(1−和[−b吨+Δ吨一世(ω)d在一世+Δ吨一世(ω)]).

F一世,吨+Δ吨(ω)=C一世(和−[b一世+Δ吨一世(ω)dΨ一世+Δ一世一世一世(ω)])

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