### 金融代写|利率建模代写Interest Rate Modeling代考|Covariance VaR Models

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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

## 金融代写|利率建模代写Interest Rate Modeling代考|One-factor model

The covariance VaR model assumes that the portfolio value is linear to the risk factor and normally distributed. Then, the key parameters are the mean and variance of the $\mathrm{P} \& \mathrm{~L}$ distribution, which are measured from historical data.
Let us consider a portfolio with a single risk factor $x$, the value of which is denoted by $p(x)$. From the linearity assumption, we can model $p(x)$ as $p(x)=a+b x$ for some positive constants $a$ and $b$. Let $\mu$ and $\sigma^{2}$ be, respectively, the mean and variance of $x$. For convenience, we assume that $p(\mu)$ is equal to the current value of the portfolio. It is known from the normality of the distribution that the $95 \%$ confidence level of $x$ is $\mu-1.645 \sigma$, and the $99 \%$ confidence level is $\mu-2.326 \sigma$. From this, we can see that the $95 \% \mathrm{VaR}$ is given by
\begin{aligned} 95 \% V a R &={a+\mu b}-{a+b(\mu-1.645 \sigma)} \ &=1.645 \sigma b . \end{aligned}
Fig. $1.7$ may help in understanding this scheme. Similarly, we have
$$99 \% V a R=2.326 \sigma b .$$
When $p(x)$ is not linear but almost linear, that is, when $p(x) \approx a+b x$, we can approximate the VaR from (1.42) and (1.43).

## 金融代写|利率建模代写Interest Rate Modeling代考|Historical Simulation Models

In historical simulation models, the empirical distribution of the P\&L is directly constructed from historical data.

For simplicity, we consider a portfolio consisting of two assets, $A$ and $B$. For a holding period $\Delta T$, let $t_{1}, \cdots, t_{J+1}$ be a sequence of past days with $t_{i}-t_{i+1}=\Delta T$ and $t_{1}=0$. We let $p_{A}(i)$ and $p_{B}(i)$ denote the prices of $A$ and $B$, respectively, at $t_{i}$. Let $r_{A}(i)$ and $r_{B}(i)$ be the rates of return for $A$ and $B$, respectively. These are defined by
\begin{aligned} &r_{A}(i)=\frac{p_{A}(i)-p_{A}(i+1)}{p_{A}(i+1)} \ &r_{B}(i)=\frac{p_{B}(i)-p_{B}(i+1)}{p_{B}(i+1)} \end{aligned}
for $i=1, \cdots, J$.
For the present values $p_{A}(0)$ and $p_{B}(0)$, the $\mathrm{P} \& \mathrm{~L}$ distribution of the portfolio is built from the set
$$\left{p_{A}(0) r_{A}(i)+p_{B}(0) r_{B}(i)\right}_{i=1, \cdots, J}$$
as shown in Fig. 1.6. From this distribution, we obtain the VaR, with the process working according to the idea shown in Fig. 1.6. Even for a very large portfolio, we can measure the VaR in the same way.

It is a point in favor of this model that it does not require assumptions about neither the distribution of prices nor the linearity of assets. However, it is difficult to collect the sufficient historical data for this method. Even when there are sufficiently many data points from a long period of observation, some data might be too old to reasonably reflect future risk, which is needed for measuring VaR.

## 金融代写|利率建模代写Interest Rate Modeling代考|Monte Carlo Simulation Models

A Monte Carlo method is a simulation technique that uses randomly generated numbers for simulation. Here, distribution functions of risk factors are created by using a sequence of random numbers.

Let $T$ be a holding period. For simplicity, we assume a single risk factor and let $p(x)$ and $x$ denote, respectively, the value of the portfolio and the value of the risk factor at $T$. We assume that $x$ is normally distributed with mean $\mu$ and variance $\sigma^{2}$, and that the current value of the portfolio is equal to $p(\mu)$.
Let $z_{i}, i=1, \cdots, J$ be a sequence of numbers generated randomly according to a standard normal distribution, where $J$ is the number of simulation runs. Then, we can assign $x_{i}=\mu+\sigma z_{i}$. By regarding generated sequences $x_{i}, i=$ $1, \cdots, J$ as scenarios for the risk factor $x$, we can obtain a distribution $p\left(x_{i}\right)$, where the probability of each scenario is $1 / J$. Next we rearrange $\left{p\left(x_{i}\right)\right}_{i=1, \cdots, J}$ to $\left{q_{k}\right}_{k=1, \cdots, J}$ such that
$$\left{p\left(x_{i}\right) ; i=1, \cdots, J\right}=\left{q_{k} ; k=1, \cdots, J\right}$$
with $q_{k} \leq q_{k+1}$ for all $k$.
To obtain a confidence level $1-\alpha$, we set $k_{\alpha}=\alpha J$. Then, $(1-\alpha) 100 \%$ VaR is given by
$$V a R_{\alpha}=\frac{-1}{2}\left(q_{k_{\alpha}}+q_{k_{\alpha}+1}\right) .$$
It is advantageous that Monte Carlo simulation models are applicable to nonlinear assets and path-dependent assets. This makes it possible to consider a great many scenarios. Moreover, we can employ this technique to the arbitrage-free model, which we will introduce in Chapter 3 . The disadvantages of Monte Carlo simulation are high computational time and increased model risk.

For a more advanced treatment of Monte Carlo simulation in financial engineering, interested readers are recommended to consult Glasserman (2004).

## 金融代写|利率建模代写Interest Rate Modeling代考|One-factor model

95%在一个R=一个+μb−一个+b(μ−1.645σ) =1.645σb.

99%在一个R=2.326σb.

## 金融代写|利率建模代写Interest Rate Modeling代考|Historical Simulation Models

r一个(一世)=p一个(一世)−p一个(一世+1)p一个(一世+1) r乙(一世)=p乙(一世)−p乙(一世+1)p乙(一世+1)

\left{p_{A}(0) r_{A}(i)+p_{B}(0) r_{B}(i)\right}_{i=1, \cdots, J}\left{p_{A}(0) r_{A}(i)+p_{B}(0) r_{B}(i)\right}_{i=1, \cdots, J}

## 金融代写|利率建模代写Interest Rate Modeling代考|Monte Carlo Simulation Models

\left{p\left(x_{i}\right) ; i=1, \cdots, J\right}=\left{q_{k} ; k=1, \cdots, J\right}\left{p\left(x_{i}\right) ; i=1, \cdots, J\right}=\left{q_{k} ; k=1, \cdots, J\right}

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## MATLAB代写

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