金融代写|金融模型代写Modelling in finance代考|FI307

Doug I. Jones

Doug I. Jones

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我们提供的金融模型Modelling in finance及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|FI307

金融代写|金融模型代写Modelling in finance代考|Variation on a Theme

Up to now, we have worked with generic forward curves $F_X^{\mathrm{CPN}, j}(t, u, v)$ and we have not discussed their implementation. If there were infinitely many market instruments, one for each starting date $u$ – or even better, one for each starting time – the general curve would be a good enough description of the full economy. The market would provide the forward rate for each possible date and the curve description would merely be a data storage and not a data modelling tool.

In practice, there are a lot less market instruments on which we can build the forward curves; there are a lot less than one instrument for each time $u$. Often one has monthly or quarterly information on the short part of the curve, provided by FRA, futures or short term swaps, and annual information at best on the long part of the curve, provided by swaps or swap futures. One needs to resort to a modelling mechanism of some sort to find the intermediary value, often combined with interpolation. Chapter 4 discusses the subject of interpolation.

In the next two sections, we outline two ways to describe the curves; the two methods lead to different implementations. Each of them can be combined with different interpolation schemes. This dual approach was initially described in Henrard (2012c). A third approach is proposed in Fries (2013) based on coupons present values.

The first implementation uses pseudo-discount factors. This is certainly the most commonly used approach in practice; in a lot of literature and software packages it is even the definition of multi-curve framework. Starting from the description of the multi-curve framework we proposed in the previous chapter, it may seems strange to use such a convoluted approach to describe the curves. This is mainly due to the evolution of the framework from the one-curve framework. It allows us to have the same description of the curves for the discounting curve and the forward curves. As we will see in the next chapter, a lot of the curve interpolation literature focuses on the impact of interpolation in that specific framework.

The second implementation, which in many ways should really be the first, is to describe the forward curve directly by modelling the forward rates themselves. As we will see later in this chapter and in the next one, many of the drawbacks of simple interpolation, such as linear interpolation, can be reduced by using the direct curve description in terms of forward rates. Some similarities between discounting and forward curves are lost. As the theoretical descriptions of the two types of curve are very different, having different implementations is not a problem. On the positive side, the intuition of the forward curves is more direct and not obtained through a ratio of discount factors.

金融代写|金融模型代写Modelling in finance代考|Forward curves through pseudo-discount factors

The pseudo-discount factor forward curves are defined as follows.
Definition 3.1 (Coupon pseudo-discount factor curves). The forward curve $P_X^{\mathrm{CDF}, j}$ $(t, s)$ is the continuous function defined for $t \leq s$ such that $P_X^{\mathrm{CDF}, j}(t, t)=1, P_X^{\mathrm{CDF}, j}(t, s)$ is an arbitrary strictly positive function for $t \leq s<\operatorname{Spot}(t)+j$, and for $t_0 \geq t, u=$ $\operatorname{Spot}\left(t_0\right)$ and $v-u+j$ one has
F_X^{\mathrm{CPN}, j}(t, u, v)=\frac{1}{\delta}\left(\frac{P_X^{\mathrm{CDF}, j}(t, u)}{P_X^{\mathrm{CDF}, j}(t, v)}-1\right) .
The origin of the above definition can be traced back to the one-curve world, where the forward rate was written with a similar formula described by Equation (2.1). Nevertheless the substance of the two formulas is very different. The one-curve formula is a result obtained from different hypotheses and from no arbitrage condition. The above formula is merely a definition. The pseudo-discount factor should be viewed as the ‘wrong number used in the wrong formula to obtain the correct result’ type of approach. The formula is better understood through the evolution than through the foundations.

Definition 3.1, which refers to an arbitrary function, is itself arbitrary in more than one way. The definition fixes the first $j$-period as the arbitrary part; one could instead fix any other $j$ period and deduce the rest of the curve from there.

In Baviera and Cassaro (2012) the authors propose to use the arbitrariness on the second period by imposing a specific interpolation in that period and building the first period in a way similar to our definition. One could even take an arbitrary decomposition of the $j$-period time interval into subintervals and distribute those subintervals arbitrarily on the real axis in such a way that, modulo the $j$-periods, they recompose the initial $j$-period. As another arbitrary choice, one could also change the value of $P_X^{\mathrm{CDF}, j}(t, t)$ to any value different from 1 . As only the ratios between two values are used and never a value on its own, the choice of initial value has no impact on the end results. One could also impose an arbitrary value for $P_X^{\mathrm{CDF}, j}$ in $(t, \operatorname{Spot}(t))$ instead of in $(t, t)$. As the curve is used only for forward computation, it is used only with time $s>\operatorname{Spot}(t)$; the value for shorter times is irrelevant.

This arbitrariness of the pseudo-discount factors makes it very difficult to design a model for those values. All the relevant models I’m aware of model the value $F_X^{C P N}, j(t, u, v)$ and not the pseudo-discount factors $P_X^{\mathrm{CDF}, j}(t, u)$. This modelling problem with pseudo-discount factors is discussed further in Section 7.1.

金融代写|金融模型代写Modelling in finance代考|FI307



到目前为止,我们已经研究了通用的正向曲线$F_X^{\mathrm{CPN}, j}(t, u, v)$,我们还没有讨论它们的实现。如果有无限多个市场工具,每个起始日期$u$对应一个,或者更好,每个起始时间对应一个,那么总体曲线就足以很好地描述整个经济。市场将为每一个可能的日期提供远期汇率,而曲线描述将仅仅是数据存储工具,而不是数据建模工具


在接下来的两节中,我们将概述描述曲线的两种方法;这两种方法导致了不同的实现。它们中的每一个都可以与不同的插值方案相结合。Henrard (2012c)最初描述了这种双重方法。Fries(2013)提出了基于优惠券现值的第三种方法。




伪贴现因子正向曲线定义如下。3.1(息票伪贴现因子曲线)。正向曲线$P_X^{\mathrm{CDF}, j}$$(t, s)$是对$t \leq s$定义的连续函数,使得$P_X^{\mathrm{CDF}, j}(t, t)=1, P_X^{\mathrm{CDF}, j}(t, s)$对于$t \leq s<\operatorname{Spot}(t)+j$是任意严格正函数,而对于$t_0 \geq t, u=$$\operatorname{Spot}\left(t_0\right)$和$v-u+j$具有
F_X^{\mathrm{CPN}, j}(t, u, v)=\frac{1}{\delta}\left(\frac{P_X^{\mathrm{CDF}, j}(t, u)}{P_X^{\mathrm{CDF}, j}(t, v)}-1\right) .

定义3.1指的是一个任意函数,它本身在多个方面都是任意的。该定义将第一个$j$ -period固定为任意部分;相反,可以修改任何其他$j$周期,并从那里推导出曲线的其余部分

在Baviera和Cassaro(2012)中,作者建议在第二个周期中使用任意性,方法是在该周期中施加一个特定的插值,并以类似于我们的定义的方式构建第一个周期。甚至可以将$j$ -period时间间隔任意分解为子区间,并将这些子区间任意分布在实轴上,以这样一种方式,对$j$ -period取模,重新组合初始的$j$ -period。作为另一种任意选择,还可以将$P_X^{\mathrm{CDF}, j}(t, t)$的值更改为与1不同的任何值。由于只使用两个值之间的比率,而不使用单独的值,所以初始值的选择对最终结果没有影响。还可以在$(t, \operatorname{Spot}(t))$中为$P_X^{\mathrm{CDF}, j}$强加任意值,而不是在$(t, t)$中。由于该曲线仅用于前向计算,因此仅与时间$s>\operatorname{Spot}(t)$一起使用; . .

伪贴现因子的这种任意性使得为这些值设计一个模型非常困难。我所知道的所有相关模型都模拟了价值$F_X^{C P N}, j(t, u, v)$,而不是伪折扣因子$P_X^{\mathrm{CDF}, j}(t, u)$。这个带有伪贴现因子的建模问题将在7.1节中进一步讨论

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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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