### 金融代写|金融数学作业代写Financial Mathematics代考|RISK NEUTRAL PRICING

statistics-lab™ 为您的留学生涯保驾护航 在代写金融数学Financial Mathematics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数学Financial Mathematics代写方面经验极为丰富，各种代写金融数学Financial Mathematics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 金融代写|金融数学作业代写Financial Mathematics代考|Default Status Tree

Nowadays, financial products are valued as the discounted value the expected cash flows under a risk neutral probability measure. We can discount the payments at the risk free rate using risk neutral valuation in the lines of Jarrow and Turnbull (1995). This implies that the default probabilities in the risk neutral world will be relevant.

We discretize the remaining time to maturity of the CDS, i.e., the time interval $[0 \mathrm{~T}]$, into $n$ intervals with equal distance $\Delta t$ between the grid points. For $n=3$, Figure $1.6$ pictures the evolution of the default status of the reference bond.

At $t_{0}$ the company is alive (nondefault, $N D$ ) with probability 1 . At $t_{1}$, the company can default $(D)$ or survive $(N D)$. If it defaults, this state remains the same for the future. If the firm did not default, the tree will further branch out with two possible states ( $D$ and $N D$ at $t_{2}$ ). Default at any $t_{i}$ acts as an “absorbing state” for the rest of the tree.

Let us refer back to our example in Table 1.1, where the cash flows for the buyer of a 3 year CDS with annual payments were summarized. Time was discretized in $n=6$ steps of $0.5$ years. Bankruptcy can occur on every node (column). In case of $N D$ at time $T$, the last payment occurs at $t_{4}$ or at $T$ depending on the contract specification.

In case of $D$, the payment of the premium payments stops but a last accrued premium may contractually be due at the time of default. This accrued fee is netted with the payment the protection seller has to pay in case of $D$. For the buyer, the $t_{3}$-value of the settlement of the CDS is $F(1-R)$. Depending on the contract specifications the par value can be augmented with the accrued interest from $t_{2}$ to $t_{3}$. In most general terms, the payment becomes $(F+A I)(1-R)-A F$.

## 金融代写|金融数学作业代写Financial Mathematics代考|Default Probabilities

The default status tree (Figure $1.6$ ) neglected the probabilities. The definition of the various default and survival probabilities makes the literature not very transparent. Deciphering notation is most of the time the hardest part of understanding the model.

First let us define the probability of default at each node in the tree, $p_{(i)}$, as the probability to default during period $i$. The firm still was alive at $t_{i-1}$, it defaults at $t_{i}$. Whenever the default probabilities are kept constant over time, such as in Figure 1.7, we will drop the subscript. In Figure $1.7 p=5 \%$.

These default probabilities define a number of conditional probabilities characterizing the various states of the default process.
For the first period the marginal probabilities apply:
$$\begin{gathered} P\left(D_{1}\right)=p_{(1)}=5 \% \ P\left(N D_{1}\right)=\left(1-p_{(1)}\right)=95 \% \end{gathered}$$
For the second period,
$$\begin{gathered} P\left(D_{2} \mid N D_{1}\right)=\left(1-p_{(1)}\right) p_{(2)}=4.75 \% \ P\left(N D_{2} \mid N D_{1}\right)=\left(1-p_{(1)}\right)\left(1-p_{(2)}\right)=90.25 \% \end{gathered}$$
For the third period, we get
$$\begin{gathered} P\left(D_{3} \mid N D_{2}\right)=\left(1-p_{(1)}\right)\left(1-p_{(2)}\right) p_{(3)}=4.51 \% \ P\left(N D_{3} \mid N D_{2}\right)=\left(1-p_{(1)}\right)\left(1-p_{(2)}\right)\left(1-p_{(3)}\right)=85.74 \% \end{gathered}$$
Two series of probabilities emerge:

• On the one hand, we find the probabilities of survival until time $t_{i}$, which we will denote $\pi_{i}: 95 \%, 90.25 \%$, and $85.74 \%$.
• On the other hand, we obtain the probability of default at time $t_{i}$, conditional upon survival until the previous period: $5 \%, 4.75 \%$, and $4.51 \%$. We will denote these probabilities by $p_{i}$.

In general, we can express these conditional probabilities for $t_{i}$ as
$$\left{\begin{array}{l} p_{1}=p_{(1)} \ p_{i}=P\left(D_{i} \mid N D_{i-1}\right)=p_{(i)} \prod_{j=2}^{i}\left(1-p_{(j-1)}\right) \quad \text { for } i=2,3, \ldots, n \ \pi_{i}=P\left(N D_{i} \mid N D_{i-1}\right)=\prod_{j=1}^{i}\left(1-p_{(j)}\right) \quad \text { for } i=1,2, \ldots, n \end{array}\right.$$
If the default probability is constant over the tree, these formulas for $i=1,2, \ldots, n$ collapse to
\begin{aligned} &p_{i}=P\left(D_{i} \mid N D_{i-1}\right)=p(1-p)^{i-1} \ &\pi_{i}=P\left(N D_{i} \mid N D_{i-1}\right)=(1-p)^{i} \end{aligned}
Note that the following relationships hold:

• Probability of survival until time $t_{i}$ is equal to 1 minus the sum of the conditional probabilities of default in the previous periods: $\pi_{i}=1-\sum_{j=1}^{i} p_{j}$.
• Probability of survival until time $t_{i}$ is equal to the probability of survival until the previous time minus the conditional probability to default in period $i: \pi_{i}=\pi_{i-1}-p_{i}$.
• Probability to survival until time $t_{i}$ is equal to the probability of survival until the previous period multiplied by the factor 1 minus the default probability (given that the default probability is constant): $\pi_{i}=\pi_{i-1}(1-p)$ or even more general $\pi_{i}=\pi_{i-j}$ $(1-p)^{j}$ with $j<i$ and $j \in N$.

## 金融代写|金融数学作业代写Financial Mathematics代考|Risk Neutral Pricing of the CDS

The periodic premium payments are generally labeled in the literature as the floating leg. The contingent terminal payment is known as the fixed leg of the CDS.
1.5.5.1 Under the Discrete Default Process
1.5.5.1.1 Floating Leg At the payment dates $t_{g} t_{2 g}, \ldots, t_{n}$ the protection buyer pays the premium $P=s F(g \Delta t$ ), until (and not including) default.

The expected payment for time $t_{i}$, seen today, is $P\left(N D_{i} \mid N D_{i-1}\right) P+P\left(D_{i} \mid N D_{i-1}\right) 0=\pi_{i} P$. Consequently, the present value of the expected payments is $\sum_{i=1}^{n / g} d f_{i g} \pi_{i g} P$ where $d f_{i}$ denotes the discount factor for the time $t_{\text {ig }}$ cash flow.

1. Note that the discount function can be expressed in various ways:
• Cheng (2001) uses the general zero bond price formulation: $B\left(t_{0}, t_{n}\right)$ i.e., the time $t_{0}$ value of a default free zero bond maturing at $t_{n}$.
• Brooks and Yan (1998) use simple one period forward rates: $d f_{i}=\frac{d f_{i-1}}{1+r_{w-1} \times W_{i}}$. NAD is the number of accrued days and NTD the total number of days, defining the applicable day count convention.
• Continuous interest rates would yield: $\mathrm{e}^{-r_{i} t_{i}}$.
• Scott (1998), Aonuma and Nakagawa (1998), Houweling and Vorst (2005), Longstaff, Mithal and Neis (2005) use the stochastic interest formulation: $\mathrm{e}^{-\int_{0}^{t} r(u) \mathrm{d} u}$.Also with the probabilities we can juggle a bit. Note that, as Figure $1.8$ illustrates, the survival probability, $\pi_{b}, i<n$ equals the sum of the survival probability at the end of the contract, $\pi_{n}$, plus the sum of the probabilities of defaulting between $t_{i}$ and $t_{n}$.
• $$• \begin{gathered} • \pi_{i}=\pi_{n}+\sum_{j=i}^{n} p_{j} \ • \sum_{i=1}^{n / g} d f_{i g} \pi_{i g} P \text { then becomes } \sum_{i=1}^{n / g} d f_{i g}\left(\pi_{n / g}+\sum_{j=i+1}^{n / g} p_{j}\right) P • \end{gathered} •$$
• which is a formulation Brooks and Yan (1998) use.

## 金融代写|金融数学作业代写Financial Mathematics代考|Default Probabilities

• 一方面，我们发现直到时间的生存概率吨一世, 我们将表示圆周率一世:95%,90.25%， 和85.74%.
• 另一方面，我们获得了违约概率吨一世, 以生存到前一时期为条件：5%,4.75%， 和4.51%. 我们将这些概率表示为p一世.

$$\left{p1=p(1) p一世=磷(D一世∣ñD一世−1)=p(一世)∏j=2一世(1−p(j−1)) 为了 一世=2,3,…,n 圆周率一世=磷(ñD一世∣ñD一世−1)=∏j=1一世(1−p(j)) 为了 一世=1,2,…,n\对。 一世F吨H和d和F一种在l吨pr这b一种b一世l一世吨是一世sC这ns吨一种n吨这在和r吨H和吨r和和,吨H和s和F这r米在l一种sF这r一世=1,2,…,nC这ll一种ps和吨这 p一世=磷(D一世∣ñD一世−1)=p(1−p)一世−1 圆周率一世=磷(ñD一世∣ñD一世−1)=(1−p)一世$$

• 直到时间的生存概率吨一世等于 1 减去前期违约条件概率的总和：圆周率一世=1−∑j=1一世pj.
• 直到时间的生存概率吨一世等于直到前一次的生存概率减去该期间违约的条件概率一世:圆周率一世=圆周率一世−1−p一世.
• 生存到时间的概率吨一世等于直到前一时期的生存概率乘以因子 1 减去违约概率（假设违约概率是常数）：圆周率一世=圆周率一世−1(1−p)甚至更一般圆周率一世=圆周率一世−j (1−p)j和j<一世和j∈ñ.

## 金融代写|金融数学作业代写Financial Mathematics代考|Risk Neutral Pricing of the CDS

1.5.5.1 在离散违约过程下
1.5.5.1.1 浮动腿 在付款日吨G吨2G,…,吨n保护买方支付保费磷=sF(GΔ吨)，直到（不包括）默认值。

1. 请注意，折扣函数可以用多种方式表示：
• Cheng (2001) 使用一般的零债券价格公式：乙(吨0,吨n)即时间吨0到期的违约免费零债券的价值吨n.
• Brooks and Yan (1998) 使用简单的一期远期汇率：dF一世=dF一世−11+r在−1×在一世. NAD 是累计天数，NTD 是总天数，定义了适用的天数计算惯例。
• 连续利率将产生：和−r一世吨一世.
• Scott (1998)、Aonuma 和 Nakagawa (1998)、Houweling 和 Vorst (2005)、Longstaff、Mithal 和 Neis (2005) 使用随机利率公式：和−∫0吨r(在)d在.另外，我们可以稍微调整一下概率。请注意，如图1.8说明，生存概率，圆周率b,一世<n等于合约结束时的生存概率之和，圆周率n，加上之间违约概率的总和吨一世和吨n.
• $$• \开始{聚集} • \pi_{i}=\pi_{n}+\sum_{j=i}^{n} p_{j} \ • \sum_{i=1}^{n / g} d f_{ig} \pi_{ig} P \text { 然后变成 } \sum_{i=1}^{n / g} d f_{ig}\left (\pi_{n / g}+\sum_{j=i+1}^{n / g} p_{j}\right) P • \结束{聚集} •$$
• 这是 Brooks 和 Yan (1998) 使用的公式。

## 有限元方法代写

tatistics-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。