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

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• 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|Constraints

The portfolio composition is optimized under various restrictions. The model includes the classical inventory balance constraints on the nominal amount invested in each bond, for each node of the scenario tree:
$$\begin{gathered} x_{i 0}=\bar{x}{i}+x{i 0}^{+}-x_{i 0}^{-} \quad \forall i \in I \ x_{i n}=x_{i a_{n}}+x_{i n}^{+}-x_{i n}^{-} \quad \forall i \in I, \quad \forall n \in N \end{gathered}$$
The cash balance constraint is imposed for the first and later stages:
\begin{aligned} &\sum_{i \in I} v_{i 0} x_{i 0}^{+}\left(1+\chi^{+}\right)+z_{0}-g_{0} \ &=C_{0}-\bar{g}+\sum_{i \in I} v_{i 0} x_{i 0}^{-}\left(1-\chi^{-}\right) \ &\sum_{i \in I} v_{i n} x_{i n}^{+}\left(1+x^{+}\right)+z_{n}-g_{n} \ &=\sum_{i \in I} v_{i n} x_{i n}^{-}\left(1-x^{-}\right)+z_{a_{n}} e^{r_{e_{n}}} \ &-g_{a_{n}} e^{b_{a_{n}}}+\sum_{i \in I} f_{i n} x_{i a_{n}} \quad \forall n \in \mathcal{N} \end{aligned}
P. Bcraldi et al.
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The model also includes constraints bounding the amount invested in each rating class (4.19) as well as in investment grade (4.20) and speculative grade (4.21) classes, respectively, as fractions of the current portfolio value:
$$\begin{gathered} \sum_{i \in I_{k}} v_{i n} x_{i n} \leq v_{k} \sum_{i \in I} v_{i n} x_{i n} \quad \forall k \in K, \quad \forall n \in \mathcal{N}, \ \sum_{k=0}^{4} \sum_{i \in I_{k}} v_{i n} x_{i n} \leq \phi \sum_{i \in I} v_{i n} x_{i n} \quad \forall n \in \mathcal{N}, \ \sum_{k=5}^{7} \sum_{i \in I_{k}} v_{i n} x_{i n} \leq \zeta \sum_{i \in I} v_{i n} x_{i n} \quad \forall n \in \mathcal{N} \end{gathered}$$
Finally, a limit on the debt level for each node of the scenario tree is imposed:
$$g_{n} \leq \gamma\left(C_{0}+\sum_{i \in I} v_{i 0} \bar{x}_{i}-\bar{g}\right) \quad \forall n \in \mathcal{N}$$

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

The goal of financial planning is twofold: maximize the expected wealth generated by the investment strategy while controlling the market and credit risk exposure of the portfolio. This trade-off can be mathematically represented by adopting a risk-reward objective function:
$$\max (1-\alpha) E\left[\mathcal{W}{n}\right]-\alpha \theta{c}\left[\mathcal{W}{n}\right]$$ where $\alpha$ is a user-defined parameter accounting for the risk aversion attitude and $n$ are the leaf nodes $\left(n \in \mathcal{N}{T}\right)$ of the scenario tree. The higher the $\alpha$ the more conservative, but also the less profitable, the suggested financial plan. The first term of (4.23) denotes the expected value of terminal wealth, computed as
$$E\left[W_{n}\right]=\sum_{n \in N_{T}} p_{n} W_{n}$$
where the wealth at each node $n$ is
$$\mathcal{W}{n}=\sum{i \in l} v_{i n} x_{i n}+z_{n}-g_{n}$$
The second term in (4.23) accounts for risk. In particular, we have considered the conditional value at risk (CVaR) at a given confidence level $\epsilon$ (usually $95 \%$ ). CVaR measures the expected value of losses exceeding the value at risk (VaR). It is a “coherent” risk measure, suitable for asymmetric distributions and thus able to control the downside risk exposure. In addition, it enjoys nice computational properties (Andersson et al. 2001; Artzner et al. 1999; Rockafellar and Uryasev 2002) and

admits a simple linear reformulation. In tree notation the CVaR of the portfolio terminal wealth can be defined as
$$\theta_{c}=\xi_{c}+\frac{1}{1-\epsilon} \sum_{n \in \mathcal{N}{T}} p{n}\left[L_{n}-\xi_{c}\right]{+\uparrow}$$ where $\xi{e}$ denotes the VaR at the same confidence level. Here $L_{n}$ represents the loss at node $n$, measured as the negative deviation from a given target value of the portfolio terminal wealth:
$$L_{n}=\max \left[0, \tilde{W}{n}-\mathcal{W}{n}\right]$$
where $\tilde{W}{n}$ represents a reference value, computed on the initial wealth l-year compounded value for given current (known) risk-free rate: $$\begin{gathered} \mathcal{W}{0}=\left(C_{0}+\sum_{i \in l} v_{i} \bar{x}{i}-\bar{g}\right), \ \tilde{W}{n}=\mathcal{W}{0} e^{r{0} f_{n}} \quad \forall n \neq 0 . \end{gathered}$$
The overall objective function can be linearized through a set of auxiliary variables $\left(\zeta_{n}\right)$ and the constraints as follows:
$$\begin{gathered} \theta_{c}=\xi_{c}+\frac{1}{1-\epsilon} \sum_{n \in \mathcal{N}{T}} p{n} \zeta_{n} \ \zeta_{n} \geq L_{n}-\xi_{c} \quad \forall n \in \mathcal{N}{T} \ \zeta{n} \geq 0 \quad \forall n \in \mathcal{N} T \end{gathered}$$
This model specification leads to an easily solvable large-scale linear multi-stage stochastic programming problem. Decisions at any node explicitly depend on the corresponding postulated realization for the random variables and have a direct impact on the decisions at descendant nodes. Depending on the number of nodes in the scenario tree, the model can become very large calling for the use of specialized solution methods.

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

The bond portfolio model implementation requires the definition of an extended set of random coefficients (Dupačová et al. 2001; Heitsch and Rōmisch 2003; Pflug 2001). We focus in this section on the relationship between the statistical model

actually implemented to develop the case study and the set of scenario-dependent coefficients included in the stochastic programming formulation.

The statistical model drives the bond returns over the planning horizon $0, \ldots, T$. At time 0 all coefficients in $(4.2),(4.3),(4.4),(4.5),(4.7)$, and (4.8) have been estimated and, for a given initial portfolio, Monte Carlo simulation can be used to estimate the credit risk exposure of the current portfolio. In this application a simple random sampling algorithm for a fixed, pre-defined, tree structure is adopted to define the event tree structure underlying the stochastic programming formulation.

A method of moments (Campbell et al. 1997) estimation is first performed on $(4.2)$, (4.3), and (4.4) from historical data, then Moody’s statistics (Moody’s Investors Service 2009 ) are used to estimate the natural default probability and the recovery rates, which are calibrated following $(4.7)$, to account for recent market evidence and economic activity, and (4.8), allowing a limited dispersion from the average class-specific recovery rates.

In the case study implementation the risk-free rate and the credit spread processes are modeled as correlated square root processes according to the tree specification, for $s \in S, t=1, \ldots, T, n \in \mathcal{N}{t}$, where $h{n}$ denotes the child node in the given scenario $s$. For each $k \in K$ and initial states $r_{0}$ and $\pi_{0}^{k}$, we have
$$\begin{gathered} \Delta r_{n}=\mu^{r}\left(t_{h_{n}}-t_{n}\right)+\sigma^{r} \sqrt{r_{n}} \sqrt{t_{h_{n}}-t_{n}} e_{n} \ \Delta \pi_{n}^{k}=\mu^{k}\left(t_{h_{n}}-t_{n}\right)+\sigma^{k} \sqrt{\pi_{n}^{k}} \sqrt{t_{h_{n}}-t_{n}} \sum_{l \in K} q_{n}^{k l} e_{n}^{l} \end{gathered}$$
In $(4.33), e_{n}^{l} \sim N(0,1)$, for $l=0,1, . ., 7$, independently and $q^{k l}$ denote the Choleski coefficients in the lower triangular decomposition of the correlation matrix. The nodal realizations of the risk-free rate and the credit spreads are also used to identify the investor’s borrowing rate $b_{n}=r_{n}+\pi_{n}^{k}$ in the dynamic model implementation, where $\bar{k}$ denotes the investors specific rating class.

The incremental spread $\eta_{n}^{i}$ for security $i$ has been implemented in the case study as a pure jump-to-default process with null mean and volatility. The associated idlosyncratic tree processes, all independent from each other, will in this case for all $i$ follow the dynamic
$$d \eta_{n}^{i}=\beta_{n}^{i} d \Psi^{i}\left(\lambda^{i}, n\right)$$

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

X一世0=X¯一世+X一世0+−X一世0−∀一世∈一世 X一世n=X一世一种n+X一世n+−X一世n−∀一世∈一世,∀n∈ñ

∑一世∈一世v一世0X一世0+(1+χ+)+和0−G0 =C0−G¯+∑一世∈一世v一世0X一世0−(1−χ−) ∑一世∈一世v一世nX一世n+(1+X+)+和n−Gn =∑一世∈一世v一世nX一世n−(1−X−)+和一种n和r和n −G一种n和b一种n+∑一世∈一世F一世nX一世一种n∀n∈ñ
P. Bcraldi 等人。
84

∑一世∈一世到v一世nX一世n≤v到∑一世∈一世v一世nX一世n∀到∈到,∀n∈ñ, ∑到=04∑一世∈一世到v一世nX一世n≤φ∑一世∈一世v一世nX一世n∀n∈ñ, ∑到=57∑一世∈一世到v一世nX一世n≤G∑一世∈一世v一世nX一世n∀n∈ñ

Gn≤C(C0+∑一世∈一世v一世0X¯一世−G¯)∀n∈ñ

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

(4.23) 中的第二项说明了风险。特别是，我们考虑了给定置信水平下的条件风险值 (CVaR)ε（通常95%）。CVaR 衡量损失的预期价值超过风险价值 (VaR)。它是一种“连贯”的风险度量，适用于非对称分布，因此能够控制下行风险敞口。此外，它还具有很好的计算特性（Andersson et al. 2001; Artzner et al. 1999; Rockafellar and Uryasev 2002）和

θC=XC+11−ε∑n∈ñ吨pn[大号n−XC]+↑在哪里X和表示相同置信水平下的 VaR。这里大号n表示节点的损失n，测量为与投资组合终端财富给定目标值的负偏差：

θC=XC+11−ε∑n∈ñ吨pnGn Gn≥大号n−XC∀n∈ñ吨 Gn≥0∀n∈ñ吨

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

Δrn=μr(吨Hn−吨n)+σrrn吨Hn−吨n和n Δ圆周率n到=μ到(吨Hn−吨n)+σ到圆周率n到吨Hn−吨n∑一世∈到qn到一世和n一世

d这n一世=bn一世dΨ一世(λ一世,n)

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