## 金融代写|金融模型代写Modelling in finance代考|BUS-F541

2022年10月14日

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## 金融代写|金融模型代写Modelling in finance代考|Curve calibration

The curve calibration is the subject of Chapter 5. In this section we describe a very simple curve calibration to emphasise the difference between pseudo-discount factor curve description and the direct rate curve description of the next section.
A relatively standard way to calibrate the curves $P_X^D$ and $P_X^{C D F, j}$ is to select a set of market instruments for which the market quotes are known and an equal number of node points. An interpolation scheme is selected and the discount factors (or the associated rates) on the node points are calibrated to reproduce the market quotes. The market forward rates $F_X^{\mathrm{CPN}, j}(0, u, u+j)$ can be computed from the forward pseudo-discount factors curve using Formula (3.1). The pseudo-discount factors are calibrated using the formula and the instruments are priced from the pseudo-discount factors using the same formula. Everything is coherent and we could stop here. The big difference with a direct approach is the interpolation. Do you interpolate the pseudo-discount factors or the forward rates?

A typical forward rate curve using the pseudo-discount factors approach is displayed in black in Figure 3.1. The swap data used to calibrate the curve are those proposed in (Andersen and Piterbarg, 2010, Section 6.2) in a one-curve setting. The interpolation scheme is linear on (continuously compounded) rates. For our calibration, the swaps are fixed versus three months Ibor. We use a discounting curve with flat OIS market rates at $4 \%$. The discounting curve is mostly irrelevant for the discussion in this section.

The familiar sawtooth pattern can be seen. There are two angles in the curve for each node point. One when the fixing period end date is on a node and one when the fixing period start date is on the node. One of the reasons for this unpleasant shape is probably that we have an intuition on a market quantity – the forward rate – but model it indirectly through a ratio of discount factors where our intuition is diluted. The graph represents the forward rate, which is of interest to the trader and risk manager, but the data is stored and calibrated using a different mechanism.

## 金融代写|金融模型代写Modelling in finance代考|Direct forward curves

In this approach we do not need an intermediary function. We work directly with the curve $F_X^{C P N, j}(0, u, v)$, with $u>\operatorname{Spot}(0)$ the variable of this one-dimensional function. This approach, which seems the natural approach in the multi-curve framework, has not been documented until recently and software implementations seems to mostly ignore it.

Note that in this framework, the Ibor discounting is impossible as there is no discount factor associated to the Ibor curves. There is no longer an arbitrary part to the curve. The curve is defined unambiguously (as long as the corresponding market instruments exist) for all $u \geq \operatorname{Spot}(0)$.

Modeling the forward rate by constructing the discount or spot rates is similar to fixing a pair of eyeglasses while wearing gloves. […] Instead, if one models the forward curve directly and embeds all of the desired properties into it, then by construction the spot and discount rates will have the characteristics that the modeler requires.

The advantage of this approach is that the market rates on which we have some intuition are modelled directly. In some senses, and borrowing a well known name, it could be called the Libor Market Model of forward curve description – not of curve dynamic as its namesake.

There is no longer a requirement for an arbitrary part like in Definition $3.1$ of the pseudo-discount factor approach. The interpolation and constraints can be imposed directly on the market quantities. In Figure 3.1, the forward rate using the same data as the previous approach and the same linear interpolation scheme, but with the interpolation directly on the forward rate, is presented in dotted line.
It is to each market maker or risk manager to decide which one he prefers. With the reported data, the forward rates display less sawtooth effect with the direct forward rate approach. With some other market rates, the picture may be different. In Table $3.1$ we give the figures of the maximum variation of the three months rates three months apart. It is expected that the view of three months rates at some distant time in the future will not change too widely for relatively small differences in dates. Larger variations may indicate a problem with the way we estimate forward curves.

The three month period was chosen as this is the problematic period artificially introduced by the specific form of Definition 3.1. As can be seen from the numbers,the interpolation on pseudo-discount factors introduces a significantly larger variation. The figures using the linear interpolation on the direct forward rate are not very different from the cubic spline interpolation on the pseudo-discount factors.
In Figure 3.2, we did the same comparison as in the previous figure, this time using a smoother interpolation mechanism. The interpolation scheme is natural cubic spline.

# 金融模型代写

## 有限元方法代写

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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。