## 金融代写|金融模型代写Modelling in finance代考|BFW3540

statistics-lab™ 为您的留学生涯保驾护航 在代写金融模型Modelling in finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融模型Modelling in finance代写方面经验极为丰富，各种代写金融模型Modelling in finance相关的作业也就用不着说。

• 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|Discounting curve

The multi-curve framework starts with the description of the fundamental curve, which is the first of the multiple curves. The starting point is the discounting of known cash-flows. This is the first hypothesis on which the multi-curve framework is based. The cash-flows considered are credit risk-free cash-flows. The impact of collateral on the framework is discussed in Chapter 8.

D: The instrument paying one unit of currency $X$ in $u$ is an asset for each $u$ and each currency $X$. Its value in $t$ is denoted $P_X^D(t, u)$. The value is continuous in $t$.

With these curves we are able to value fixed cash-flows. The discounting curves are functions $P_X^D:[0, T] \rightarrow(0,+\infty)$. To be able to model curves with the usual tools of quantitative finance one needs to fix an upper bound on the time frame on which the modelling is done. This is described in, for example, (Hunt and Kennedy, 2004, Section 7.4.4). This is why we impose an upper bound $T$. If the bound $T$ is taken sufficiently large, it will have no impact in practice.

We do not restrict discount factors to be below 1 as we want to allow negative rates. Similarly we do not impose that $P_X^D$ is monotonically decreasing in the second variable. In this sense the treatment presented here is more general than the one presented in (Andersen and Piterbarg, 2010, Section 6.1.1). We restrict only the discount factors to be positive. We want to work in an arbitrage-free world. As the payment at maturity $u$ is $1>0$, its present value today $t$, represented by the discount factor $P_X^D(t, u)$, should be strictly positive.

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

Our goal is to price Ibor- and overnight-related derivatives, in particular IRSs and OISs. We need an hypothesis to say that these instruments exist in the framework we are describing.
In this section, we adopt the misuse of language of calling instruments Ibor coupons even if the underlying index is not strictly speaking an Ibor index, but an index playing the same role or an overnight index.
As the period addition, $t+$ period $j$, is used often we adopt the notation $t+j$ for that date, without clarifying in which unit the $j$ is; it is usually clear from the context.
Our existence hypothesis for the Ibor coupons reads as
$\mathbf{I}^{\text {CPN }}$ : The value of a $j$-Ibor floating coupon is an asset for each tenor $j$, each fixing date and each currency. Its value is a continuous function of time.

The subscript CPN stands for CouPoN. We will use the same notation convention for the different quantities related to the Ibor coupons.
This hypothesis is implicit in most of the literature. It is important to state it explicitly as this is not a consequence of the existence of the discounting curve. This hypothesis is one of the foundations described in the introduction.
Once we have assumed that the instrument is an asset, we can give its value a name. We do this indirectly through the curves $F_X^j$ below. These curves are called forward curves, projections curves or estimation curves in the literature. We will use the term forward curves most of the time. At this stage we insist that the curves $F_X^j$ are pure definition and the use of the word curve should be understood in the mathematical sense, not in the financial sense.

Definition 2.1 (Forward coupon rate). The forward curve $F_X^{\mathrm{CPN}, j}$ is the continuous function such that,
$$P_X^D(t, v) \delta F_X^{\mathrm{CPN}, j}(t, u, v)$$
is the price in $t$ of the j-Ibor coupon with fixing date $t_0$, start date $u$ and maturity date $v\left(t \leq t_0 \leq u=\operatorname{Spot}\left(t_0\right)<v\right)$.

The reason for this definition is to keep the usual formulas involving forward rate computation ${ }^1$. The standard terms of discounting curve and forward rate will still be used. But at this stage we insist that they are only definitions and names. One should not attach too strong a financial intuition to those mathematical objects even if we use standard financial terms to designate them.
La mathématique est l’art de donner le même nom à des choses différentes.
Henri Poincaré.
Personal translation: Mathematics is the art of giving the same name to different items.

# 金融模型代写

## 金融代写|金融模型代写在金融建模代考|折现曲线

D:在$u$中支付一单位货币$X$的工具对于每一种货币$u$和每一种货币$X$是一种资产。它在$t$中的值表示为$P_X^D(t, u)$。$t$中连续的值

## 金融代写|金融模型代写金融建模代考|正向曲线

$$P_X^D(t, v) \delta F_X^{\mathrm{CPN}, j}(t, u, v)$$

La mathématique est l’art de donner le même nom à des choses différentes。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写金融模型Modelling in finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融模型Modelling in finance代写方面经验极为丰富，各种代写金融模型Modelling in finance相关的作业也就用不着说。

• 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|Early multi-curve framework literature

To my knowledge, the first article to propose a coherent valuation framework where the discounting is explicitly differentiated from the index forward estimation was The irony in the derivatives discounting (Henrard 2007). It was published in July 2007, just one month before the crisis started. To my knowledge there is no link between the article publication and the crisis.

The article focuses on discounting in interest rate derivatives. The starting observation is that different curves are used to value different instruments (OIS, IRS, cross-currency swaps), which is clearly incoherent and creates portfolio level arbitrages.

The approach proposed described only one Ibor curve; it is not an important feature of the approach but a description of the then reality. The approach can be used with several Ibor related curves without changes. The spread between the curves is described in a multiplicative way and is equivalent to the $\beta$ we define in Chapter 2; our notation is adapted from that used in the paper. Only the case of a constant spread is discussed in the original paper; again, it is not an important feature of the approach but more a description of the then reality.

The first instrument for which the price is discussed is the FRA and the paper provides a pricing formula, including all the hypotheses required to obtain the result. The valuation of swaps in the framework and the impact on the value and risk for not-at-par instruments is discussed. The approach is extended in a coherent way to (Short Term Interest Rate) STIR futures, caps/floors and swaptions. The Ibor fixings are clearly dissociated from the discounting rate through the introduction of an exogenous spread. The spread can be obtained from the market value of different instruments; this is equivalent to the curve building process we propose in Chapter 5.
For practical examples, the article uses an OIS-based discounting curve (called Libid in the paper). The article was probably overly simplistic, as it used only a constant spread between the curves and did not use explicitly different curves for the different Ibor tenors. Nevertheless it was an early proposal leading to today’s most commonly used approach.
The importance of the multi-curve framework, with each curve having a specific purpose, is attested by the numerous related literature that appeared in the following years. Throughout the book we have tried to cite as many sources as possible in the relevant sections. We want to give readers the opportunity to go back to the original sources. In this section we comment on a couple of the early papers which describe some particular aspects of the framework. In the literature the framework has received many names. Some of those names are: two curves, multi-curve framework, derivative tenor curves, funding-Ibor, discounting-estimation, discounting-forecast, discounting-forward, and multi-curve market. Through the book, we will use the name used in the title: multi-curve framework.

## 金融代写|金融模型代写Modelling in finance代考|Collateral and funding

The description in the first chapters of the multi-curve framework is in a credit riskfree world without the market reality of the collateral and funding issues. Even if the central theme of the book is the multi-curve framework and not the collateral and funding issues, a book on the curve framework resulting from the crisis would be missing an important part of the reality if the crisis impact on collateral and funding was not discussed at all. Chapter 8 describes some of the issues related to those features. It provides results that were initially presented in Kijima et al. (2009), Macey (2011), Pallavicini et al. (2012) and Piterbarg (2012). It also presents extensions of the above results. In particular we generalise the definition of collateral and propose results about collateral with assets themselves collateralised – called collateral square. Those results are original and were first published in Henrard (2013b). Our approach to proofs borrows heavily on Macey (2011) and Piterbarg (2012) even if we add the missing self-financing property to the original sketches.

The easiest way to read this book is to start at Chapter 1 and go to Chapter 8 , with a stop in the Appendices when required – you are allowed to eat and sleep between chapters. This is certainly not the only way to read it. In the graph below I have represented the dependency graph between the different chapters of the book. The dotted lines for the Appendices indicate where they fit best, but some readers may want to skip them entirely. The links with bidirectional arrows indicate that in some way the two chapters depend on each other and are best read in parallel if the reader has that capacity.

From the graph, the reader can see that, in the mind of the author, the most important chapters are Chapter 2 (Foundations), Chapter 5 (Curve calibration), and Chapter 8 (Collateral).

I tried to write this book in the way I would like to read it. The different axioms are clearly evidenced and for each result I have tried to be clear on which hypotheses are required. I have added as many references as possible. The reader should have the opportunity to go to the original literature and compare different approaches. I have also added a lot of cross references between sections, to allow the reader to move between sections with related subjects.

# 金融模型代写

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写金融模型Modelling in finance方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融模型Modelling in finance代写方面经验极为丰富，各种代写金融模型Modelling in finance相关的作业也就用不着说。

• 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|Foundations, evolution, and implementation

This book’s subtitle is Foundations, Evolution, and Implementation. The subtitle acts as a quick summary of the book content.

The foundations are paramount. As the framework is different from the previous one-curve approach in fundamental aspects for index-linked financial instruments, one can not rely on a ‘copy and paste’ style approach. Any claim, even the most basic one, should be backed by clear definitions and clear proofs. Before building the skyscraper that is the pricing of an exotic derivative, one has not only to be sure that the foundations are sound but also indicate explicitly where they are. In some cases, mathematically sophisticated developments are made, only to notice later that their domain of applicability is empty. Their starting hypotheses, which seem acceptable in a one-curve world, lead to contradictions for the multi-curve framework. This is why the book systematically uses an ‘axiomatic approach’. The fundamental hypotheses are displayed in the text as quotes with a bold letter in front reminding us of the content of the hypothesis – such as $\mathbf{D}$ for the discounting hypothesis at the beginning of Section 2.2. These axioms or fundamental hypotheses are the foundations of the framework. No development in interest rate modelling can be done without referring to them or to their equivalent in a different framework.

The evolution is important to understand how this new framework started and where it is coming from. Several choices made in the new framework are recycling previous approaches with appropriate and justified twists. Why and where those twists were made is an important piece of information. Looking at the final product, it may seem to have been produced by black magic. It is not any more when seen through the eyes of Darwin, as the result of evolution.
In chronological order, the evolution came before the foundations. The theory is more an explanation of the practice obtained by tinkering with different approaches than a new, ground-breaking theory that started to be used from scratch. We refer the reader to the book Antifragile by Taleb (2012) for more on how practice often precedes theory. In this book, we reverse the chronological order and start with the foundations.

Si, à l’égard de plusieurs questions traitées dans cette étude, j’ai comparé les résultats de l’observation à ceux de la théorie, ce n’est pas pour vérifier des formules établies par des méthodes mathématiques, mais pour montrer seulement que le marché, à son insu, obéit à une loi qui le domine : la loi de la probabilité.
Louis Bachelier, Théorie de la spéculation, 1900 Personal translation: If, regarding several questions analysed in this study, I compared the observed results to those of the theory, it is not to verify the formulas obtained by mathematical methods, but only to show that the market, unwittingly, complies to a law that dominates it: the law of probability.

## 金融代写|金融模型代写Modelling in finance代考|Standard textbook framework

The book by Hull (2006), one of the most popular introductory textbooks to derivative pricing, is used as an example of the way earlier literature treats the curves question. In the section on the type of rates in the Interest rates chapter (p. 76) the existence of both Libor and Libid is acknowledged. The latter is described in the above book as the rate at which a cash rich investor can invest in the interbank market. In 2006, those rates were relatively close to the OIS (Overnight Indexed Swap) rates.

When it comes to valuing the first derivatives (Section 4.7: Forward Rate Agreements) the explanation is ‘the assumption underlying the contract is that the borrowing or lending would normally be done at Libor’. This is first a misleading statement on the instrument itself. The reality is that the contract settlement amount is computed by discounting with a Libor fixing rate between the end of the accrual period and its start. There is no actual borrowing or lending and there is no assumption in the FRA ${ }^4$ contract, only a clear (contingent) settlement formula. For the valuation of the instrument before its fixing date, the approach described in the above book is to use the same Libor rate to discount the resulting quantity to the valuation date. But there is no justification in the text for choosing that particular rate from the different ones described in the previous sections. The choice of Libor for that purpose is a modelling choice and not a contractual obligation. This vagueness is certainly a witness of the consensus at that time: if the instrument is related to an Ibor rate, use similar rates for everything, even if there is no modelling or legal reason to do so. The hidden explanation is that it is easier and everybody is doing it.

Another standard textbook on swaps and curve construction is Sadr (2009). In the description of the pricing of a swap, the terms used to describe the rates linked to the Ibor leg are ‘hypothetical loan’, ‘deposits’, ‘funding’ and ‘risk-free rates’. The text indicates that to obtain the standard one-curve formula, those different rates have to be the same. Using the equivalence between those rates the book proceeds through standard arbitrage-free arguments to obtain the standard swap pricing formula.

# 金融模型代写

## 金融代写|金融模型代写在金融建模代考|基础，发展，和实现

Si, à l’égard de plusieurs questions traitées dans cette étude, j’ai comparé les résultats de l’observation à ceux de la théorie, ce n’est pas pour vérifier des formules établies par des méthodes mathématiques, mais pour montrer seulement que le marché, à son insu, obéit à une loi qui le domine : la loi de la probabilité.
Louis Bachelier, Théorie de la spéculation, 1900 Personal translation: If, regarding several questions analysed in this study, I compared the observed results to those of the theory, it is not to verify the formulas obtained by mathematical methods, but only to show that the market, unwittingly, complies to a law that dominates it: the law of probability.

## 金融代写|金融模型代写金融建模代考|标准教科书框架

.

Hull(2006)的书是衍生品定价最受欢迎的入门教材之一，它被用作早期文献处理曲线问题的一个例子。在“利率”一章的“利率类型”一节(第76页)中，承认了Libor和Libid的存在。后者在上书中被描述为现金充裕的投资者在银行间市场投资的利率。在2006年，这些利率相对接近隔夜指数掉期利率(OIS)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。