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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|金融模型代写Modelling in finance代考|Forward rate agreement

An FRA with FRA discounting settlement rule is an instrument linked to a period of length $j$, a fixing date $t_0$, an accrual factor $\delta$ and a fixed rate $K$. At the fixing date $t_0$, the Ibor rate $I_X^j\left(t_0\right)$ is recorded. The contractual payment in $u=\operatorname{Spot}\left(t_0\right)$ (the start date) received by the contract buyer is
$$\frac{\delta\left(I_X^j\left(t_0\right)-K\right)}{1+\delta I_X^j\left(t_0\right)}$$
More details about the FRA conventions are given in Appendix B.3.
The origin of the formula can be traced back to the computation of the value as the difference between the Ibor fixing and the fixed rate discounted at the Ibor rate from the end date to the settlement date. The rate is not paid at the end of its natural period but at the start, and is discounted with the fixing rate itself; we insist that the Ibor discounting used in the above formula is not a modelling choice but part of the contract term sheet. It is not directly a floating coupon or a one period swap as defined in the previous section. In that sense, the above definition of an FRA, in line with a market FRA term sheet, is different from those of Ametrano and Bianchetti (2009), Bianchetti (2010), Chibane et al. (2009) and Mercurio (2009).

We would like to value this instrument in a simple way in our economy. In a general contingent claim formula with a generic numeraire $N$ and associated expected value $\mathrm{E}^N[]$, its value in 0 is
$N_0 \mathrm{E}^N\left[N_u^{-1} \frac{\delta\left(I_X^j\left(t_0\right)-K\right)}{1+\delta I_X^j\left(t_0\right)}\right]$.
Here we cannot use the usual trick of postponing the payment to $v=u+j$ by deciding to invest in $t_0$ from $u$ to $v$ at the rate $I_X^j\left(t_0\right)$ (multiplying by $1+\delta I_X^j\left(t_0\right)$ ) and selecting $P_X^j(., v)$ as the numeraire to simplify the formula. The reason is that the investment cannot be done at the Ibor rate but only at the risk-free rate.

## 金融代写|金融模型代写Modelling in finance代考|STIR Futures

A general pricing formula for interest rate futures in the one-factor Gaussian HJM model in the one-curve framework was proposed in Henrard (2005). The formula extended a previous result proposed in Kirikos and Novak (1997). The pricing in a displaced diffusion LMM with skew is analysed in Jäckel and Kawai (2005). Piterbarg and Renedo (2004) analyse the pricing of futures in some general stochastic volatility model to study the impact of the smile; they also analyse the pricing of options on futures.

The Gaussian HJM formula was extended to the multi-curve framework in Henrard (2007). In this section, we describe the pricing of futures under the hypotheses $\mathrm{I}^{\mathrm{CPN}}$ and $\mathbf{S} 0^{\mathrm{CPN}}$ in a multi-factor Gaussian $\mathrm{HJM}$ multi-curve framework. The exact notation for the multi-factor Gaussian HJM framework used here is given in Appendix 9. The pricing of the futures in the LMM with stochastic basis is proposed in Mercurio (2010b). In Mercurio and Xie (2012), the pricing of futures for an additive stochastic basis spread is analysed. The pricing of futures in a framework with multiplicative stochastic basis is proposed in Chapter 7.
The future fixing or last trading date is denoted $t_0$. The fixing is on the Ibor rate between $u=\operatorname{Spot}\left(t_0\right)$ and $v=u+j$. The fixing accrual factor for the period $[u, v]$ is $\delta$. More details on the STIR futures conventions can be found in Appendix B.4.
The futures price is denoted $\Phi_t^j$. On the fixing date, the relation between the futures price and the fixing rate is
$$\Phi_{t_0}^j=1-I_X^j\left(t_0\right) .$$
The futures margining is done on the futures price (multiplied by the notional and the futures accrual factor).

When analysing futures prices, we will use the generic futures price process theorem (Hunt and Kennedy 2004, Theorem 12.6), which states that
$$\Phi_t^j=\mathrm{E}^{\mathbb{N}}\left[\Phi_{t_0}^j \mid \mathcal{F}_t\right]$$
where $\mathrm{E}^{\mathbb{N}}$ [] is the cash account numeraire expectation. A generalisation of this theorem is proved in Chapter 8 . The formula above can be obtained as a special case of formula (8.3) for a collateral rate $c_t=0$. The link between pricing under collateral and futures is explained in that chapter.

# 金融模型代写

## 金融代写|金融模型代写modeling in finance代考|Forward rate agreement

.

$$\frac{\delta\left(I_X^j\left(t_0\right)-K\right)}{1+\delta I_X^j\left(t_0\right)}$$

$N_0 \mathrm{E}^N\left[N_u^{-1} \frac{\delta\left(I_X^j\left(t_0\right)-K\right)}{1+\delta I_X^j\left(t_0\right)}\right]$在这里我们不能使用通常的伎俩推迟付款到 $v=u+j$ 通过决定投资 $t_0$ 从 $u$ 到 $v$ 按照这个速度 $I_X^j\left(t_0\right)$ (乘以 $1+\delta I_X^j\left(t_0\right)$ )和选择 $P_X^j(., v)$ 作为数字来简化公式。原因是投资不能按Ibor利率进行，而只能按无风险利率进行

## 金融代写|金融模型代写modeling in finance代考|STIR Futures

Henrard(2005)提出了单因素高斯HJM模型在单曲线框架下利率期货的一般定价公式。该公式扩展了Kirikos和Novak(1997)先前提出的结果。在Jäckel和Kawai(2005)中分析了带有倾斜的位移扩散LMM的定价。Piterbarg和Renedo(2004)在一些一般的随机波动率模型中分析期货定价，研究微笑的影响;他们还分析期货期权的定价

Henrard(2007)将高斯型HJM公式推广到多曲线框架。在本节中，我们描述了假设$\mathrm{I}^{\mathrm{CPN}}$和$\mathbf{S} 0^{\mathrm{CPN}}$在一个多因素高斯$\mathrm{HJM}$多曲线框架下的期货定价。这里使用的多因子高斯型HJM框架的确切符号在附录9中给出。Mercurio (2010b)提出了基于随机基的LMM期货定价方法。在Mercurio和Xie(2012)中，对加性随机基价差的期货定价进行了分析。第七章提出了乘随机基框架下的期货定价问题。

$$\Phi_{t_0}^j=1-I_X^j\left(t_0\right) .$$

$$\Phi_t^j=\mathrm{E}^{\mathbb{N}}\left[\Phi_{t_0}^j \mid \mathcal{F}_t\right]$$

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## 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。

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