## 数学代写|金融衍生品代写Financial derivatives代考|AEM4210

2022年10月6日

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## 数学代写|金融衍生品代写Financial derivatives代考|Forward Rate Agreement

The forward rate agreement $(F R A)$ is a loan/deposit engagement with a pre-agreed fixed rate $(K)$ which will start at a time in the future $T_1$ and will terminate at time $T_2$. There are 3 dates in an FRA contract $\left(T_0<T_1<T_2\right)$ :
$T_0$ : inception date;
$T_1$ : expiry (or effective date) of the forward contract;
$T_2$ : termination date.
The notation $T_1 \times T_2$ is used to put emphasis on the effective date and the termination date for an FRA contract.
Example $2.2$ The following is a $3 \times 6$ FRA trade.

The settlement to the long position of an FRA may be either of the below amounts:
$\begin{cases}\text { Notional Amount } \times \frac{(r-K) \delta}{1+r \delta}, & \text { for settlement at } T_1, \text { which is the market } \ & \text { practice for FRA } \ \text { Notional Amount } \times(r-K) \delta, & \text { for settlement at } T_2,\end{cases}$
where
$r:$ the reference rate (e.g. Libor) fixed at $T_1$ for the period $T_1 \rightarrow T_2$;
$\delta$ : day count fraction calculated with the applicable convention;
$K$ : fixed rate in the FRA contract.
Proposition 2.1 For the market FRA with zero $P V$ at inception, the fixed rate $K$ is
$$K=\frac{1}{\delta}\left(\frac{P\left(T_1\right)}{P\left(T_2\right)}-1\right)$$
Proof The proof is done by replication (i.e. hedging) with zero coupon bonds of maturities $T_2$ and $T_1$. The short position of FRA for 1 dollar notional amount can be hedged with the below self-financing transactions.

## 数学代写|金融衍生品代写Financial derivatives代考|Treasury Bond Futures

The underlying of a Treasury Bond Futures contract is a virtual (or “notional”) security with standardized features for the coupon rate, the maturity and the size. The futures contract applies physical delivery at the expiry. There is a set of deliverable bonds associated with each contract which is determined by the exchange.

For example, the most active futures in Europe is of $10 \mathrm{Y}$ maturity with EUR 100,000 size and $6 \%$ coupon. ${ }^6$ The set of deliverable bonds is published by EUREX with remaining maturities of $8.5 \mathrm{Y} \Rightarrow 10.5 \mathrm{Y}$. In US, there are $2 \mathrm{Y}, 3 \mathrm{Y}, 5 \mathrm{Y}, 10 \mathrm{Y}$ Treasury Note Futures, Treasury Bond Futures (deliverable bonds of $15 \mathrm{Y} \rightarrow 25 \mathrm{Y}$ ) and Ultra T-Bond Futures (deliverable bonds of $>25 \mathrm{Y}$ ).

Treasury bond futures are related to notional bonds but delivered with real bonds from a pool of eligible bonds and notes. For each deliverable bond, a conversion factor $(C F)^7$ is applied to convert the futures contract price into the “price for the bond”:
Price for the Bond $=C F \times$ Futures Price $+$ Accrued Coupon.
At settlement, for each deliverable bond, the short party will

• receive the cash amount based on “price for the bond”
• deliver the bond from the pool of eligible ones.
The market price of the bond to be delivered and the “price for the bond” will be normally different. The bond showing the lowest value of “market price” minus “price for the bond” is called the Cheapest-To-Deliver (CTD) bond. This phenomenon is due to the fact that the market yield curve is not $6 \%$ flat in general. Moreover, the bonds have different sensitivities to the yield curve change due to the different characteristics including coupon and time to maturity.

# 金融衍生品代写

## 数学代写|金融衍生品代写金融衍生品代考|远期利率协议

$T_0$:起始日期;
$T_1$:远期合同期满(或生效日期);
$T_2$:终止日期。
$T_1 \times T_2$符号用于强调FRA合同的生效日期和终止日期。

FRA的多仓位结算可能是以下金额之一:
$\begin{cases}\text { Notional Amount } \times \frac{(r-K) \delta}{1+r \delta}, & \text { for settlement at } T_1, \text { which is the market } \ & \text { practice for FRA } \ \text { Notional Amount } \times(r-K) \delta, & \text { for settlement at } T_2,\end{cases}$

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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。