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

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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
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• 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|Nested Simulation

There are many exotic options whose price depends on the path of an underlying asset. If a structured product involves an exotic derivative, then the price of the product is also path-dependent. Since the pricing of complex products is not trivial, recombining trees and Monte Carlo simulation are employed for pricing.

When a portfolio contains exotic products such as those described above, it is difficult to price them for a future date, which is necessary for measuring VaR. Nested simulation is a computing technique for measuring VaR of a derivatives portfolio, which we briefly introduce below.

Let $T$ be a holding period for measuring VaR. We denote by $x_{t}$ the value of the risk factor at time $t$, and by $p_{T}$ the price of a derivative asset at $T$. As illustrated in Fig. $1.8$, the distribution of the risk factor $x_{T}$ at $T$ is simulated by a Monte Carlo method in the outer step. Then, we consider $x_{T}$ to represent a scenario in which VaR is to be measured. At each scenario $x_{T}$, the derivative is priced as $p_{T} \mid x_{T}$ by another Monte Carlo simulation starting at time $T$. This is the inner step. Once we obtain the distribution of $p_{T} \mid x_{T}$, we can compute the VaR in the manner described in Section 1.6.3 of this chapter.
A problem intrinsic to nested simulation is that the number of simulation becomes enormous. There are many studies on algorithms for accelerating nested simulation. As examples, see Devineau and Loisel (2009) and Gordy and Juneja (2010), among others.

## 金融代写|利率建模代写Interest Rate Modeling代考|Validity of VaR

VaR is widely used as a risk measure in regulatory guidelines in the banking sectors. However it is known that VaR at a particular confidence level does not tell us the loss beyond that level, although the validity of VaR has been studied mathematically and financially in many papers. In this section, we make some brief remarks about the properties of $\mathrm{VaR}$ as a downside risk measure and define expected shortfall as a coherent risk measure. We address these issues basically according to the Bank for International Settlement (BIS, 2004b) for tail risk and according to Artzner et al. (1999) for coherent risk measures.
For this purpose, we denote by $X$ the value of a portfolio at time $T$, and by $\rho$ a risk measure for $X$. Then, $\rho(X)$ represents the size of the risk of $X$ measured by $\rho$. Naturally, the portfolio is allowed to sell some options, which creates the possibility that $X$ will take a negative value. To avoid confusion, we note that a large positive $\rho(X)$ represents a larger risk than a small positive or negative $\rho(X)$. We remark that up to the previous section, risk measures have been defined in terms of the profit-loss measures of a portfolio, such as sensitivity, convexity, and VaR. In this section, we work with a risk measure for $X$ that does not represent profit or loss. In this context, we define a VaR of $V_{\alpha}$ at confidence level $1-\alpha$ over a holding period $T$ as
$$V_{\alpha}(X)=-\inf {x \mid \mathbf{P}(X \leq x)>\alpha}$$
Tail risk
To explain the tail risk, we consider two portfolios, whose prices at time $T$ are denoted by $A$ for one portfolio and $B$ for the other. We use $A$ and $B$ as the names of these portfolios. We assume that $A$ and $B$ have the same VaR at confidence level $1-\alpha$, so that $V_{\alpha}(A)=V_{\alpha}(B)$. In Fig. $1.9$, the solid line shows

the distribution of future value of $A$, and the dashed line shows that of $B$. We see that the downside risk of portfolio $B$ is larger than that of $A$. This shows that $V a R_{\alpha}$ does not capture the difference in downside risk between portfolios, as exemplified by $A$ and $B$; this risk is noted in BIS (2004b) as tail risk.

Under VaR regulations, risk managers will choose portfolio $B$ in preference to A because there is a larger opportunity for profit in B. Thus, VaR might mislead risk managers who optimally control their portfolios. For further argument along these lines, see Basak and Shapiro (2001) and Yamai and Yoshiba $(2002)$, among others.

## 金融代写|利率建模代写Interest Rate Modeling代考|Probability Space

Probability space
We denote by $\Omega$ a set of all possible outcomes, called the sample space, and an element $w \in \Omega$, called a sample. The family $\mathcal{F}$ is a collection of subsets of $\Omega$ and is assumed to be a $\sigma$-algebra. A stochastic event $A$ is represented as an element in $\mathcal{F}$.

The function $\mathbf{P}$ is a probability function defined on $\mathcal{F}$ such that $\mathbf{P}(\Omega)=1$ and $\mathbf{P}(\phi)=0 ;$ this is also called a probability measure. For $A \in \mathcal{F}, \mathbf{P}(\mathbf{A})$ indicates probability of event $A$ occurring. From these, we characterize a probability space by a triplet $(\Omega, \mathcal{F}, \mathbf{P})$.
Example 2.1: Sample space
In financial engineering, a sample $w \in \Omega$ is regarded as a history of up and down moves of a stock price observed in a fixed period. This makes $\Omega$ the set of up and down sequences of the stock price. If a period consists of two days $\left{t_{1}, t_{2}\right}$, then $\Omega$ will contain four samples and be given by
$$\Omega={u u, u d, d u, d d}$$
where we represent by $u$ and $d$ the up and down movements, respectively, of the stock price. Accordingly, $\mathcal{F}$ consists of all subsets of $\Omega$,
$$\mathcal{F}={\phi, \Omega, u u, u d, d u, d d,{u u, u d},{u u, d d,}, \cdots}$$

which means that $\mathcal{F}$ has $16\left(=2^{4}\right)$ events.
Real-world measure
In financial market analysis, the probability measure derived from observation of market prices is called the market measure, with many synonyms, such as historical measure, physical measure, actual measure, and real-world measure. We regard historical data as being observed under the real-world measure.
Recall the previous example. If our observation estimates that the probability of an up move is constant at $0.5$ and the down-move probability is $0.5$, then we have
$$\mathbf{P}(u u)=\mathbf{P}(u d)=\mathbf{P}(d u)=\mathbf{P}(d d)=0.25$$
Accordingly, for an arbitrary event $A \in \mathcal{F}$, it is possible to give $\mathbf{P}(A)$. This probability measure $\mathbf{P}$ is an example of the real-world measure.

## 金融代写|利率建模代写Interest Rate Modeling代考|Validity of VaR

VaR 被广泛用作银行业监管指南中的风险度量。然而，众所周知，特定置信水平的 VaR 并不能告诉我们超出该水平的损失，尽管在许多论文中已经对 VaR 的有效性进行了数学和财务研究。在本节中，我们对属性的一些简要说明在一个R作为下行风险衡量标准，并将预期短缺定义为连贯的风险衡量标准。我们基本上根据国际清算银行 (BIS, 2004b) 的尾部风险和 Artzner 等人的方法来解决这些问题。(1999) 用于连贯的风险措施。

## 金融代写|利率建模代写Interest Rate Modeling代考|Probability Space

Ω=在在,在d,d在,dd

F=φ,Ω,在在,在d,d在,dd,在在,在d,在在,dd,,⋯

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## MATLAB代写

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