## 金融代写|期权定价理论代写Option Pricing Theory代考|MATH485

statistics-lab™ 为您的留学生涯保驾护航 在代写期权定价理论Option Pricing Theory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。我们的专家在代写期权定价理论Option Pricing Theory代写方面经验极为丰富，各种代写期权定价理论Option Pricing Theory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|期权定价理论代写Option Pricing Theory代考|Complete models versus incomplete models

From (1.13), the market model (1.1) is complete if there exists a unique $\mathcal{F}{t^{-}}$ adapted vector $\lambda_t \in \mathbb{R}^d$ such that $$\forall i \in \text { asset, } \quad b_i\left(t, X_t\right)-r_t X_t^i=\sum{j=1}^d \sigma_{i, j}\left(t, X_t\right) \lambda_t^j$$
The unique ELMM $\mathbb{Q}$ is then given by
$$\frac{d \mathbb{Q}}{\left.d \mathbb{P}^{h i s t}\right|{\left.\right|{\mathcal{F}T}}} \equiv Y_T(-\lambda ; 1)=\prod{j=1}^d e^{-\int_0^T \lambda_t^j d W_t^j-\frac{1}{2} \int_0^T\left(\lambda_t^j\right)^2 d t}$$
The unique arbitrage-free price is
$$\mathcal{B}t\left(F_T\right)=\mathcal{S}_t\left(F_T\right)=\mathbb{E}^{\mathbb{Q}}\left[D{t T} F_T \mid \mathcal{F}t\right]$$ An inspection of (1.14) reveals that if the market is complete, then the rank of $\sigma\left(t, X_t\right)$ is equal to $d$ a.s. (which implies that #assets $\geq d$ ). In the case where #assets ${i, j}\left(t, X_t\right)\right){i \in \text { asset }, 1 \leq j \leq d}$ is invertible, then the market is complete. So, provided the volatility matrix $\sigma{i, j}\left(t, X_t\right)$ is correctly estimated, there is a unique arbitrage-free price.

Examples of complete models that are commonly used by practitioners include Dupire’s local volatility model [95]; Libor market models with local volatilities, e.g., BGM with deterministic volatilities [67]; and Markov functional models [138] (see also [11], Chapter 2).

Common examples of incomplete models are stochastic volatility models (in short SVMs). Here #assets $<d$. An example of stochastic volatility model is the double lognormal SVM, which has attracted the attention of practitioners in equity markets $[113,137]$. The dynamics of the underlying, denoted by $X_t$, reads under a risk-neutral measure $\mathbb{Q}^0 \sim \mathbb{P}^{\text {hist }}$ as
with $\chi^2=1-\rho_{\mathrm{XV}^0}^2-\frac{\left(\rho_{\mathrm{XV}}-\rho \rho_{\mathrm{XV}}\right)^2}{1-\rho^2}$ and $W_t^1, W_t^2, W_t^3$, three uncorrelated standard $\mathbb{Q}^0$-Brownian motions. $V_t$ is the instantaneous variance, and $V_t^0$ plays the role of a moving long-term average value for $V_t$. Neither $V_t$ nor $V_l^0$ are tradable instruments.

## 金融代写|期权定价理论代写Option Pricing Theory代考|Pricing in practice

In practice, the seller’s price at time $t$ is computed by picking out a particular ELMM $\mathbb{Q}$ :
$$u_t \equiv \mathbb{E}^{\mathbb{Q}}\left[D_{t T} F_T \mid \mathcal{F}_t\right]$$
Under this measure $\mathbb{Q}$, the drift for an asset $X^i$ is fixed to $b_i\left(t, X_t\right) \equiv r_t X_t^i$ (see Remark 1.1). In an incomplete market, $\mathbb{Q}$ does not necessary achieve the supremum in Theorem 1.3, and we lose the superhedging strategy paradigm. Selling options becomes a risky business. However, it seems that the idea of a “true price” (based on a “true model”) is still vivid in the community of structurers and sales people (see the quote at the beginning of this chapter). In practice, picking a particular ELMM simplifies a lot the pricing problem: it becomes a linear problem, i.e., the price of the (European) payoff $F_T^1+F_T^2$ equals the sum of the prices of the (European) payoffs $F_T^1$ and $F_T^2$.

We will always assume that there exists a (deterministic) function $r$ such that $r_t=r\left(t, X_t\right)$. Then
$$D_{t_1 t_2}=\exp \left(-\int_{t_1}^{t_2} r\left(s, X_s\right) d s\right)$$
If there exists $g$ such that $F_T=g\left(X_T\right)$, we speak of a vanilla option. In such a case, by the Markov property of $X$,
$$u_t=\mathbb{E}^{\mathbb{Q}}\left[\exp \left(-\int_t^T r\left(s, X_s\right) d s\right) g\left(X_T\right) \mid \mathcal{F}_t\right] \equiv u\left(t, X_t\right)$$
is a function $u$ of $\left(t, X_t\right)$. Below, we recall that $u(t, x)$ is a solution to a linear second order parabolic PDE, the so-called Black-Scholes pricing PDE.

# 期权理论代写

## 金融代写|期权定价理论代写Option Pricing Theory代考|Complete models versus incomplete models

$$\forall i \in \text { asset, } \quad b_i\left(t, X_t\right)-r_t X_t^i=\sum j=1^d \sigma_{i, j}\left(t, X_t\right) \lambda_t^j$$

$$\frac{d \mathbb{Q}}{d \mathbb{P}^{p i s t}|| \mathcal{F} T} \equiv Y_T(-\lambda ; 1)=\prod j=1^d e^{-\int_0^T \lambda_t^j d W_t^j-\frac{1}{2} \int_0^T\left(\lambda_t^j\right)^2 d t}$$

$$\mathcal{B} t\left(F_T\right)=\mathcal{S}t\left(F_T\right)=\mathbb{E}^{\mathbb{Q}}\left[D t T F_T \mid \mathcal{F} t\right]$$ 对 (1.14) 的检查表明，如果市场是完备的，那么 $\sigma\left(t, X_t\right)$ 等于 $d$ 作为（这意味着#assets $\geq d$ ). 在#assets 供波动率矩阵 $\sigma i, j\left(t, X_t\right)$ 被正确估计，存在唯一的无套利价格。 从业者常用的完整模型示例包括 Dupire 的局部波动率模型 [95]；具有局部波动率的 Libor 市场模型，例 如具有确定性波动率的 BGM [67]；和马尔可夫函数模型 [138]（另见 [11]，第 2 章)。 不完整模型的常见示例是随机波动率模型 (简称 SVM) 。这里#assets ${\mathrm{XV}^0}^2-\frac{\left(\rho_{\mathrm{XV}}-\rho \rho_{\mathrm{XV}}\right)^2}{1-\rho^2}$ 和 $W_t^1, W_t^2, W_t^3$ ，三个不相关的标准 $\mathbb{Q}^0$-布朗运动。 $V_t$ 是瞬时方 差，并且 $V_t^0$ 起移动长期平均值的作用 $V_t$. 两者都不 $V_t$ 也不 $V_l^0$ 是可交易的工具。

## 金融代写|期权定价理论代写Option Pricing Theory代考|Pricing in practice

$$u_t \equiv \mathbb{E}^{\mathbb{Q}}\left[D_{t T} F_T \mid \mathcal{F}t\right]$$ 在这项措施下 $\mathbb{Q}$ ，资产的漂移 $X^i$ 固定为 $b_i\left(t, X_t\right) \equiv r_t X_t^i$ (见备注 1.1)。在不完全的市场中， $\mathbb{Q}$ 不一定 达到定理 $1.3$ 中的上确界，我们就失去了超级对冲策略范式。出售期权成为一项有风险的业务。然而，“真 实价格”（基于“真实模型”）的想法似乎在架构师和销售人员群体中仍然很生动（参见本章开头的引述）。 在实践中，选择一个特定的 ELMM 大大简化了定价问题：它变成了一个线性问题，即（欧洲）收益的价格 $F_T^1+F_T^2$ 等于 (欧洲) 收益的价格总和 $F_T^1$ 和 $F_T^2$. 我们将始终假设存在一个 (确定性的) 函数 $r$ 这样 $r_t=r\left(t, X_t\right)$. 然后 $$D{t_1 t_2}=\exp \left(-\int_{t_1}^{t_2} r\left(s, X_s\right) d s\right)$$

$$u_t=\mathbb{E}^{\mathbb{Q}}\left[\exp \left(-\int_t^T r\left(s, X_s\right) d s\right) g\left(X_T\right) \mid \mathcal{F}_t\right] \equiv u\left(t, X_t\right)$$

## 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|期权定价理论代写Option Pricing Theory代考|MATH4380

statistics-lab™ 为您的留学生涯保驾护航 在代写期权定价理论Option Pricing Theory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。我们的专家在代写期权定价理论Option Pricing Theory代写方面经验极为丰富，各种代写期权定价理论Option Pricing Theory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|期权定价理论代写Option Pricing Theory代考|Arbitrage and arbitrage-free models

Let us now introduce the notion of arbitrage. An arbitrage is a self-financing strategy that is worth zero initially and yields a positive gain without any risk.

DFFINITION $1.2$ Arhitrage A self-financing admissihle portfolio is called an arbitrage if the corresponding value process $\pi_t$ satisfies $\pi_0=0$ and
$$\pi_T \geq 0 \quad \mathbb{P}^{\text {hist }}-\text { a.s } \quad \text { and } \quad \mathbb{P}^{\text {hist }}\left(\pi_T>0\right)>0$$
Arbitrageurs are a special kind of trader. Their role is precisely to detect and take full advantage of arbitrage opportunities as soon as they appear in the market. This impacts market prices: arbitrage opportunities tend to disappear as soon as they arise. Absence of arbitrage opportunities is therefore a natural modeling assumption. The next lemma gives a sufficient condition under which we exclude arbitrage opportunities in our market model.
LEMMA 1.1 Sufficient condition excluding arbitrage
Suppose there exists a measure $\mathbb{Q}$ on $\left(\Omega, \mathcal{F}T\right)$ such that ${ }^3 \mathbb{Q} \sim \mathbb{P}^{\text {hist }}$ and such that, for all asset $X^i$, the discounted price process $\left{\tilde{X}_t^i\right}{t \in[0, T]}$ is a local mar-tingale with respect to $\mathbb{Q}^4{ }^4$ Then the market $\left{X_t\right}_{t \in[0, T]}$ has no arbitrage.
Note that the assumption of Lemma $1.1$ bears only on assets $X^i$ only, not on non-tradable components of $X$, such as instantaneous interest rates, instantaneous stochastic volatility, etc.

## 金融代写|期权定价理论代写Option Pricing Theory代考|Super-replication

Let us assume that, at time $t$, we buy and delta-hedge a European option ${ }^5$ written on $m$ assets, say $X_t^1, \ldots, X_t^m$, with maturity $T$ and payoff $F_T$, at the price $z$. In general, the payoff $F_T$ is a function of the paths $\left(X_t^i, 0 \leq t \leq T\right)$ followed by the prices of the $m$ assets between times 0 and $T$. The final value of the buyer’s portfolio, discounted at time 0 , is
\begin{aligned} \tilde{\pi}T^B & =-D{0 t} z+\sum_{i=1}^m \int_t^T \Delta_s^i d \tilde{X}s^i+D{0 T} F_T \ & =-D_{0 t} z+\int_t^T \Delta_s \cdot d \tilde{X}s+D{0 T} F_T \end{aligned}
Wè can then define the buyer’s super-réplication price at time $t$ as the greatest price $z$ such that the value of the buyer’s portfolio $\tilde{\pi}T^B$ is $\mathbb{P}^{\text {hist }}$-a.S. nonnegative. To be precise, we introduce the following: DEFINITION $1.4$ Buyer’s price $\mathcal{B}_t\left(F_T\right)=\sup \left{z \in \mathcal{F}_t \mid\right.$ there exists an admissible portfolio $\Delta$ such that $$\left.\tilde{\pi}_T^B \equiv-D{0 t} z+\int_t^T \Delta_s \cdot d \tilde{X}s+D{0 T} F_T \geq 0 \mathbb{P}^{\text {hist }}-a . s .\right}$$
The price $z$ must be $\mathcal{F}t$-measurable, denoted by $z \in \mathcal{F}_t$, i.e., we cannot look into the future. Similarly, we can define the seller’s super-replication price as: DEFINITION $1.5$ Seller’s price $\mathcal{S}_t\left(F_T\right)=\inf \left{z \in \mathcal{F}_t \mid\right.$ there exists an admissible portfolio $\Delta$ such that $$\left.\tilde{\pi}_T^S \equiv D{0 t} z+\int_t^T \Delta_s \cdot d \tilde{X}s-D{0 T} F_T \geq 0 \mathbb{P}^{\text {hist }}-a . s .\right}$$

# 期权理论代写

## 金融代写|期权定价理论代写Option Pricing Theory代考|Arbitrage and arbitrage-free models

$$\pi_T \geq 0 \quad \mathbb{P}^{\text {hist }}-\text { a.s } \text { and } \mathbb{P}^{\text {hist }}\left(\pi_T>0\right)>0$$

## 金融代写|期权定价理论代写Option Pricing Theory代考|Super-replication

$$\tilde{\pi} T^B=-D 0 t z+\sum_{i=1}^m \int_t^T \Delta_s^i d \tilde{X} s^i+D 0 T F_T \quad=-D_{0 t} z+\int_t^T \Delta_s \cdot d \tilde{X} s+D 0 T F_T$$

, i. e., wecannotlookintothe future. Similarly, wecandefinetheseller’ssuper – replicatio
thereexistsanadmissibleportfolio $\$ 三角洲suchthat

## 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|期权定价理论代写Option Pricing Theory代考|MATH424

statistics-lab™ 为您的留学生涯保驾护航 在代写期权定价理论Option Pricing Theory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。我们的专家在代写期权定价理论Option Pricing Theory代写方面经验极为丰富，各种代写期权定价理论Option Pricing Theory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|期权定价理论代写Option Pricing Theory代考|Models of financial markets

Let us consider a filtered probability space $\left(\Omega,\left(\mathcal{F}t\right){0 \leq t \leq T}, \mathbb{P}^{\text {hist }}\right)$. Here $\mathbb{P}^{\text {list }}$ is the historical or real probability measure under which we model our market. A market model is defined by an $n$-dimensional stochastic differential equation $(\mathrm{SDE})$
$$d X_t^i=b_i\left(t, X_t\right) d t+\sum_{j=1}^d \sigma_{i, j}\left(t, X_t\right) d W_t^j, \quad i \in{1, \ldots, n}$$
and by another positive stochastic process $B_t$, called the money-market account, representing the value of cash, which satisfies
$$d B_t=r_t B_t d t, \quad B_0=1$$
i.e.,
$$B_t=\exp \left(\int_0^t r_s d s\right)$$

$r_t$ is the short term interest rate. It is adapted to $\mathcal{F}t$, which is the (natural) filtration generated by the $d$-dimensional uncorrelated standard Brownian motion $\left{W_t^j\right}{1 \leq j \leq d}$. In order to ensure that SDE (1.1) admits a unique strong solution (see e.g., [13]), we assume that $b$ and $\sigma$ satisfy:

Assum(SDE): The functions $b$ and $\sigma$ are Lipschitz-continuous in $x$ uniformly in $t$, and satisfy a linear growth condition: there exists a positive constant $C$ such that for all $t \geq 0, x, y \in \mathbb{R}^n$,
\begin{aligned} |b(t, x)-b(t, y)|+|\sigma(t, x)-\sigma(t, y)| & \leq C|x-y| \ |b(t, x)|+|\sigma(t, x)| & \leq C(1+|x|) \end{aligned}
We set
$$D_{t u} \equiv B_t B_u^{-1}=\exp \left(-\int_t^u r_s d s\right)$$
which is the discount factor from date $u$ to date $t$. Throughout the book, we will denote by $\tilde{Y}t \equiv D{0 t} Y_t$ the discounted value of any price process $Y_t$. Certain market components $X^i$ may not be sold or bought in the market, such as the short term interest rate, or a stochastic volatility. Throughout this book, a market component $X^i$ that can be sold and bought in the market is called an “asset.”

## 金融代写|期权定价理论代写Option Pricing Theory代考|Self-financing portfolios

Let us assume that we have a portfolio consisting of $m$ assets, say $X_t^1, \ldots, X_t^m$, and the money-market account $B_t$. It is convenient to use the notation $X^0$ for $B$. The portfolio at a time $t$ is composed of $\Delta_t^i$ assets $X_t^i$ and $\Delta_t^0$ units of $X_t^0$ (cash). The $\Delta_t^i$ ‘s must be $\mathcal{F}t$-measurable, i.e., we cannot look into the future. The portfolio value $\pi_t$ is $$\pi_t \equiv \sum{i=0}^m \Delta_t^i X_t^i$$
As time passees, wẽ can readjust the allocations $\Delta_t^i$, but no cash is éver injéctéd or removed from the portfolio: between $t$ and $t+d t$, the variation in the portfolio value is only due to the variation of the values of the assets, i.e.,
$$d \pi_t=\sum_{i=0}^m \Delta_t^i d X_t^i$$
We then speak of a self-financing portfolio. In terms of discounted values, this rears
$$d \tilde{\pi}t=\sum{i=0}^m \Delta_t^i d \tilde{X}t^i=\sum{i=1}^m \Delta_t^i d \tilde{X}_t^i$$

because for any price process $Y_t, d \tilde{Y}t=D{0 t}\left(d Y_t-r_t Y_t d t\right)$, concluding that ${ }^2$
$$\tilde{\pi}t=\pi_0+\sum{i=1}^m \int_0^t \Delta_s^i d \tilde{X}_s^i$$
We may also write this as
$$\tilde{\pi}_t=\pi_0+\int_0^t \Delta_s \cdot d \tilde{X}_s$$
where $\cdot$ denotes the usual scalar product in $\mathbb{R}^m$. As a technical condition, we need to introduce the notion of admissible portfolio:

DEFINITION 1.1 Admissible portfolio $\left(\Delta_t, 0 \leq t \leq T\right)$ defines an admissible portfolio if $\tilde{\pi}_t$ is bounded from below for all $t \mathbb{P}^{\text {hist }}$-a.s., i.e., there exists $M \in \mathbb{R}$ such that
$$\mathbb{P}^{\text {hist }}\left(\forall t \in[0, T], \tilde{\pi}_t \geq M\right)=1$$

# 期权理论代写

## 金融代写|期权定价理论代写Option Pricing Theory代考|Models of financial markets

$$d X_t^i=b_i\left(t, X_t\right) d t+\sum_{j=1}^d \sigma_{i, j}\left(t, X_t\right) d W_t^j, \quad i \in 1, \ldots, n$$

$$d B_t=r_t B_t d t, \quad B_0=1$$
$\mathrm{IE}{\mathrm{O}}$ $$B_t=\exp \left(\int_0^t r_s d s\right)$$ $\mathrm{SDE}(1.1)$ 承认唯一的强解决方案（参见例如 [13]），我们假设 $b$ 和 $\sigma$ 满足: 假设 (SDE) : 函数 $b$ 和 $\sigma$ 是 Lipschitz 连续的 $x$ 统一在 $t$ ，并且满足线性增长条件：存在正常数 $C$ 这样对于 所有人 $t \geq 0, x, y \in \mathbb{R}^n$ ， $$|b(t, x)-b(t, y)|+|\sigma(t, x)-\sigma(t, y)| \leq C|x-y||b(t, x)|+|\sigma(t, x)| \quad \leq C(1+|x|)$$ 我们设置 $$D{t u} \equiv B_t B_u^{-1}=\exp \left(-\int_t^u r_s d s\right)$$

## 金融代写|期权定价理论代写Option Pricing Theory代考|Self-financing portfolios

$$\pi_t \equiv \sum i=0^m \Delta_t^i X_t^i$$

$$d \pi_t=\sum_{i=0}^m \Delta_t^i d X_t^i$$

$$d \tilde{\pi} t=\sum i=0^m \Delta_t^i d \tilde{X} t^i=\sum i=1^m \Delta_t^i d \tilde{X}_t^i$$

$$\tilde{\pi} t=\pi_0+\sum i=1^m \int_0^t \Delta_s^i d \tilde{X}_s^i$$

$$\tilde{\pi}_t=\pi_0+\int_0^t \Delta_s \cdot d \tilde{X}_s$$

$$\mathbb{P}^{\text {hist }}\left(\forall t \in[0, T], \tilde{\pi}_t \geq M\right)=1$$

## 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|期权定价理论代写Option Pricing Theory代考|МАTH485

statistics-lab™ 为您的留学生涯保驾护航 在代写期权定价理论Option Pricing Theory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。我们的专家在代写期权定价理论Option Pricing Theory代写方面经验极为丰富，各种代写期权定价理论Option Pricing Theory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|期权定价理论代写Option Pricing Theory代考|SD Option Pricing by Pairwise Comparisons

Recall from Chap. 1 that the fundamental property of SSD is that for every agent the utility function must be increasing and concave, implying in turn that the marginal utility function is non-increasing. This fundamental property of decision-makers identified here as investors or traders in both the underlying and the option is the basis of the $\mathrm{SD}$ approach to option pricing. In market equilibrium models, this marginal utility is known as the pricing kernel and constitutes a basic element in defining the equilibrium prices of derivative assets.

We consider a market with an underlying asset with current price $S_t$ and a riskless asset with return per period equal to $R$. There is also a European call option with strike price $K$ expiring at some future time $T$. Time is initially assumed discrete $t=0,1, \ldots, T$, with intervals of length $\Delta t$, implying that $R=e^{r \Delta t}=1+r \Delta t+o(\Delta t)$, where $r$ denotes the interest rate in continuous time. In each interval the underlying asset has returns $\frac{S_{t+\Delta t}-S_t}{S_t} \equiv z_{t+\Delta t}$, whose distribution may depend on $S_t \cdot{ }^2$

Except for the trivial case where $z_{t+\Delta t}$ takes only two values the market for the index is incomplete in a discrete time context. The valuation of an option in such a market cannot yield a unique price. Our market equilibrium is derived under the following set of assumptions that are sufficient for our results:

There exists at least one utility-maximizing risk-averse investor (the trader) in the economy who holds only the index and the riskless asset.
This particular investor is marginal in the option market.
The riskless rate is non-random. ${ }^3$

## 金融代写|期权定价理论代写Option Pricing Theory代考|The FRICTIONLESS SD BoundS IN CONTINUOUS

The SSD bounds (2.16) and (2.17) are distribution-free, recursive and applicable to any number of time partitions till option expiration. The question that arises, therefore, is their relationship with the continuous time option prices that have dominated option research in the more than 40 years since the BSM model first appeared. As it turns out, the two bounds converge to the same limit, the BSM model price, when the underlying asset returns follow diffusion asset dynamics. This was shown by Perrakis (1988) for a trinomial discretization of the continuous time distribution converging to lognormal diffusion and was generalized by Oancea and Perrakis (2014) for a general discretization converging to any type of diffusion.

We model the index return $z_{t+\Delta t}$ held by the trader in the equilibrium relations (2.1) in the following general form that guarantees convergence to diffusion as $\Delta t \rightarrow 0$
$$z_{t+\Delta t}=\mu\left(S_t, t\right) \Delta t+\sigma\left(S_t, t\right) \varepsilon \sqrt{\Delta t} .$$

In this expression $\varepsilon$ has a bounded distribution of mean zero and variance one, $\varepsilon \sim D(0,1)$ and $0<\varepsilon_{\min } \leq \varepsilon \leq \varepsilon_{\max }$, but otherwise unrestricted. In (2.26) the limit is the lognormal diffusion when the parameters $\mu$ and $\sigma$ are constant.

The discretization (2.26) can be easily shown to converge to diffusion. ${ }^{10}$ The main result of this section, however, is the convergence of the transformed return distributions that underlie the two option bounds. We use the weak convergence criterion for the two return processes. For any number $m$ of time periods to expiration, we define a sequence of stock prices $\left{S_t \mid \Delta t, m\right}$ and an associated probability measure $P^m$. The weak convergence property for such processes ${ }^{11}$ stipulates that for any continuous bounded function $f$ we must have $E^P\left[f\left(S_T^m\right)\right] \rightarrow E^P\left[f\left(S_T\right)\right]$, where the measure $P$ corresponds to diffusion limit of the process, to be defined shortly. $P_m$ is then said to converge weakly to $P$ and $S_T^m$ is said to converge in distribution to $S_T$. A necessary and sufficient condition for the convergence to a diffusion is the Lindeberg condition, which was used by Merton (1992) to develop criteria for the convergence of multinomial processes. In a general form, if $\phi_t$ denotes a discrete stochastic process in $d$-dimensional space, the Lindeberg condition states that a necessary and sufficient condition that $\phi_t$ converges weakly to a diffusion is that for any fixed $\delta>0$ we must have
$$\lim {\Delta t \rightarrow 0} \frac{1}{\Delta t} \int{b+s} Q_{\Delta v}(\phi, d \varphi)=0$$
where $Q_{\Delta t}(\phi, d \varphi)$ is the transition probability from $\phi_t=\phi$ to $\phi_{t+\Delta t}=\varphi$ during the time interval $\Delta t$. Intuitively, it requires that $\phi_t$ does not change very much when the time interval $\Delta t$ goes to zero.

# 期权理论代写

## 金融代写|期权定价理论代写Option Pricing Theory代考|The FRICTIONLESS SD BoundS IN CONTINUOUS

SSD 边界 (2.16) 和 (2.17) 是无分布的、递归的并且适用于任意数量的时间分区直到期权到期。因此，出现的问 题是它们与自 BSM 模型首次出现以来 40 多年来主导期权研究的连续时间期权价格的关系。事实证明，当标的 资产收益遵循扩散资产动态时，这两个边界会收敛到相同的限制，即 BSM 模型价格。这由 Perrakis (1988) 证 明了连续时间分布的三项式离散化收敛于对数正态扩散，并被 Oancea 和 Perrakis (2014) 概括为收敛于任何类 型扩散的一般离散化。

$$z_{t+\Delta t}=\mu\left(S_t, t\right) \Delta t+\sigma\left(S_t, t\right) \varepsilon \sqrt{\Delta t} .$$

$$\lim \Delta t \rightarrow 0 \frac{1}{\Delta t} \int b+s Q_{\Delta v}(\phi, d \varphi)=0$$

## 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|期权定价理论代写Option Pricing Theory代考|MATH4380

statistics-lab™ 为您的留学生涯保驾护航 在代写期权定价理论Option Pricing Theory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。我们的专家在代写期权定价理论Option Pricing Theory代写方面经验极为丰富，各种代写期权定价理论Option Pricing Theory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|期权定价理论代写Option Pricing Theory代考|Empirical Tests of Stochastic Dominance

The SD rules that we saw in the previous sections were all derived for entire distributions of competing prospects. Such distributions, however, are rarely available in financial markets. The empirical distributions that are used in such markets have been extracted from past data, which implies that even if there is agreement about their shape there is bound to be uncertainty about their correct parameter values. Hence, the tests that are meaningful in our context are those that examine whether inferences about dominance of the underlying populations can be derived from samples extracted from these populations.

It so happens that the most powerful and useful tests for our purposes are those developed for issues that have nothing to do with financial markets, such as measurements of income inequality and welfare. These are pairwise comparisons of countries or societies, for which the social welfare function is similar to a conventional utility of income and the two different societies’ income distributions are compared as to whether one dominates the other. ${ }^{11}$ We highlight here in some detail the Davidson and Duclos (DD 2000, 2013) and Davidson (2009) SSD comparison approaches, the only ones that have had applications in option markets. These studies’ approaches were chosen because on the one hand they apply to correlated population distributions, while they require that the samples drawn from them be serially uncorrelated; both requirements are fulfilled in our tests. Further, DD (2013) and Davidson (2009) allow the testing of the more informative null hypothesis, that if $F_1$ and $F_2$ are the two compared distributions the null is $F_1 \succ{ }_2 F_2$ against the alternative that $F_1 \succ_2 F_2$.

In our tests $F_i(z), i=1,2$ are two continuous distributions with the same support $z \in[z, \bar{z}]$ from which two samples are drawn. Consistent with the definition (1.3b) in Sect. 1.1, the test statistic that verifies the dominance $F_1 \succ_2 F_2$ if the population distributions had been available is $D_2^2(z)-D_1^2(z) \geq 0$, for every $z$ in the joint support and with strict inequality for at least one value of $z$, where $D_i^2(z)=\int_{\Xi}^z(z-x) d F_i(x), i=1,2$. In practice we have two samples of $N$ paired outcomes and all DD tests replace the theoretical test statistics with their sample counterparts.

## 金融代写|期权定价理论代写Option Pricing Theory代考|Summary and Conclusions

SD is a criterion of choice under risk, more general than expected utility since it is not tied up to a specific function but to a relatively large class of functions defined on the basis of generally accepted behavioral axioms. It has had a long history in economic thinking, mostly associated with choice among alternative investment prospects. Of the three categories of SD criteria it is the second one, SSD, that is relevant for our purposes, since as it is shown in the next chapter the first degree is too broad a category and the third degree does not produce any useful results in the case of option markets.

All SD rules involved at their initial formulations pairwise comparisons of terminal probability distributions of wealth or investment returns in a single period horizon, and in all cases the dominant distribution had a larger or equal mean. The SSD rules were expressed in terms of the areas between the points of intersection of the cumulative distributions plotted against their common support. When the terminal distributions were those of returns from alternative investment prospects the intersection points of the distributions corresponded also to the intersection points of the returns. Distributions or returns with one and two intersection points are the only ones relevant for the option market results in subsequent chapters.

When the two compared distributions under SSD have the same mean then dominance implies that the dominant one has a lower risk, thus establishing a definition of risk more general than the variance. When the two prospects are random returns of two risky investment assets whose means are both higher than the return of an available riskless asset that can be combined with either one of the prospects, then it is possible to extend the definition of risk beyond prospects with equal means. Between two prospects whose returns have equal means, the dominant one is the one that shifts more probability mass to the low end of the returns.

# 期权理论代写

## 金融代写|期权定价理论代写Option Pricing Theory代考|Summary and Conclusions

SD 是风险选择标准，比预期效用更普遍，因为它不依赖于特定函数，而是依赖于根据普遍接受的行为公理定义的相对大类的函数。它在经济思想中有着悠久的历史，主要与替代投资前景的选择有关。在 SD 标准的三个类别中，第二个 SSD 标准与我们的目的相关，因为正如下一章所示，第一级是一个太宽泛的类别，第三级不会产生任何有用的结果期权市场的情况。

## 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|期权定价理论代写Option Pricing Theory代考|МАTH424

statistics-lab™ 为您的留学生涯保驾护航 在代写期权定价理论Option Pricing Theory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。我们的专家在代写期权定价理论Option Pricing Theory代写方面经验极为丰富，各种代写期权定价理论Option Pricing Theory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|期权定价理论代写Option Pricing Theory代考|Risk and SeCOND-DEgREE STOCHASTIC DOMINANCE

We shall consider distributions in the positive real line and with a bounded support, since these will form the bulk of the cases that will become relevant in many of the theoretical and in all the empirical option pricing applications. Equation (1.3b) implies that $F(x)$ should initially lie below $G(x)$. If the two don’t cross anywhere then we have FSD of $F(x)$ over $G(x)$, so we assume that they cross at least once. The two important cases for our purposes are when they cross exactly once and exactly twice, as shown in the two figures below.

In Fig. 1.1 the SSD relation (1.3b) boils down to the area between the two curves indicated by $\mathbf{A}$ exceeding the one denoted by $\mathbf{B}$. The difference between the two is equal to the difference between the means of the two distributions, which becomes thus a sufficient condition for SSD under the single crossing property. By contrast, in the double-crossing case shown in Fig. 1.2, the inequality of the means is no longer sufficient for SSD: the difference in means is equal to $\mathbf{A}+\mathbf{C}-\mathbf{B}$, but SSD exists only if $\mathbf{A}-\mathbf{B}>0$. The location of the second crossing point $\mathrm{E}$ becomes, therefore, crucial in establishing SSD.

The relation $F \succ_2 G$, if it can be established, implies that $F(x)$ has an equal or higher mean than $G(x)$ and a lower risk. If risk is not represented by variance then we need an alternative definition of it. This was discussed at length by Rothschild and Stiglitz (1970) who included two other definitions of risk, in addition to the definition $(1.2)$ that every risk averter prefers $F(x)$ to $G(x) .{ }^5$ The first one was that $F(x)$ was the distribution of a random variable $X$, while $G(x)$ represented the variable $X+\varepsilon$, where $E[\varepsilon \mid X]=0$ for all $X$. The second one considered meanpreserving spreads, namely the cases in which $F(x)$ and $G(x)$ had the same mean but $G(x)$ was obtained from $F(x)$ by shifting probability weights from the center toward the tails, chosen so that the mean stayed the same. These are also important in our applications, even though they are not always consistent with (1.2) when transaction costs are included.
The single crossing case of SSD shown in Fig. $1.1$ is more informative if we map the distributions $F(x)$ and $G(x)$ in the domain of terminal values of a function $H(X)$ and of $X$ respectively, which obviously must intersect at a single point as shown in Fig. 1.3, corresponding to Point D in Fig. 1.l. Figure $1.4$ shows the similarly mapped $F(x)$ and $G(x)$ distributions in Fig. 1.2, and Point $\mathrm{J}$ that corresponds to Point $\mathrm{E}$ in that figure. These two figures, combined with the fact that $u^{\prime}(x)$ is non-increasing for $u(x) \in U_2$, form the basis of the entire approach to option pricing presented in this book. Thus, for the case shown in Fig. 1.3, if $X$ represents terminal wealth or return on investment and $H(X)$ an alternative portfolio possibly involving options then the latter shifts the returns from the high to the low states, thus increasing utility, provided the overall expectation inequality $E[H(X)] \geq E[X]$ holds, consistent with the definition of risk.

## 金融代写|期权定价理论代写Option Pricing Theory代考|EMPIRICAL Applications and Portfolio SeLEction UNDER SSD or TSD

For all the generality of the SSD concept, the number of empirical applications in portfolio selection, its “natural” domain, has been rather limited, and it has not been able to displace the dominant mean-variance framework in spite of the latter’s theoretical weaknesses. Early empirical work used the definitions ( $1.3 \mathrm{a}, 1.3 \mathrm{~b}, 1.3 \mathrm{c})$ or their extensions in the presence of a riskless asset in pairwise comparisons among a finite number of investment prospects, generally mutual funds. ${ }^7$ Since the “true” distributions are unobservable, the tests were carried out on sample dis tributions generated from observed past data. The statistical tests on whether inferences on the basis of the sample distributions could be extrapolated to the underlying population distributions are crucial for the empirical applications of SD that involve options and will be reviewed in the next section.

Pairwise comparisons of investment prospects, even mutual funds, are meaningful only if one assumes that there are institutional or other barriers to diversification among the assets, an assumption that is hard to justify in realistic cases. The fact that $F \succ_2 G$ does not necessarily mean that dominance is preserved for the pair of portfolios formed by each one of the assets represented by the two distributions with a third asset. Hence, when full diversification is feasible there is an infinity of pairwise comparisons even when there is only a finite set of prospects. For this reason, it has not been empirically possible to generate the SSD-efficient set under such realistic conditions. This, in turn, implies that key notions associated with capital market equilibrium under mean-variance such as the risk pricing of individual assets are not even defined in connection with SSD.

An interesting theoretical extension of SSD that, however, has had few empirical applications so far is the Marginal Conditional Stochastic Dominance (MCSD). ${ }^8 \mathrm{MCSD}$ is defined relative to a reference portfolio with a given composition and derives conditional distributions for each asset in the portfolio. It then develops conditions to determine whether this composition is “efficient”: pairwise comparisons between the assets in the portfolio may uncover an asset whose weight can be decreased and allocated to another, dominating asset, thus increasing utility for all risk-averse investors. The derivation of conditional distributions in MCSD requires knowledge of the distributions of the constituent assets and the portfolio. Apart from its computational complexity, the method has to our knowledge not been integrated with the derivation of the assets’ distributions from observed time series data and its robustness properties with respect to distributions shifting over time are unknown.

# 期权理论代写

## 金融代写|期权定价理论代写Option Pricing Theory代考|Risk and SeCOND-DEgREE STOCHASTIC DOMINANCE

SSD 的单交叉情况如图 1 所示。1.1如果我们映射分布，则信息量更大 $F(x)$ 和 $G(x)$ 在函数的终值范围内 $H(X)$ 和 $X$ 如图 $1.3$ 所示，它们显然必须相交于一个点，对应于图 1.1 中的点 D。数字1.4显示类似映射 $F(x)$ 和 $G(x)$ 图 $1.2$ 中的分布和点 $\mathrm{J}$ 对应点 $\mathrm{E}$ 在那个数字中。这两个数字，结合事实 $u^{\prime}(x)$ 是不增加的 $u(x) \in U_2$ ，构 成了本书中介绍的整个期权定价方法的基础。因此，对于图 $1.3$ 所示的情况，如果 $X$ 代表终端财富或投资回报 $H(X)$ 一个可能涉及期权的替代投资组合，然后后者将回报从高状态转移到低状态，从而增加效用，前提是总 体预期不平等 $E[H(X)] \geq E[X]$ 成立，符合风险的定义。

## 金融代写|期权定价理论代写Option Pricing Theory代考|EMPIRICAL Applications and Portfolio SeLEction UNDER SSD or TSD

SSD 的一个有趣的理论扩展是边际条件随机优势 (MCSD)，但迄今为止几乎没有实证应用。8米C小号丁相对于具有给定成分的参考投资组合定义，并为投资组合中的每项资产推导条件分布。然后，它开发条件来确定这种组合是否“有效”：投资组合中资产之间的成对比较可能会发现一种资产，其权重可以降低并分配给另一种占主导地位的资产，从而增加所有规避风险的投资者的效用。MCSD 中条件分布的推导需要了解成分资产和投资组合的分布。除了其计算复杂性之外，据我们所知，该方法还没有与从观察到的时间序列数据中推导资产分布相结合，并且其关于随时间变化的分布的稳健性是未知的。

## 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|期权理论代写Mathematical Introduction to Options代考|MATH485

statistics-lab™ 为您的留学生涯保驾护航 在代写期权理论Mathematical Introduction to Options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。我们的专家在代写期权理论Mathematical Introduction to Options代写方面经验极为丰富，各种代写期权理论Mathematical Introduction to Options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|期权理论代写Mathematical Introduction to Options代考|AMERICAN OPTIONS

In the last section it was seen that the curve of the value of a European option always lies above the asymptotic lines. What of an American option which can be exercised at any time before maturity? Some very general and important conclusions can be reached using simple arbitrage arguments.
(i) First, we establish three almost trivial looking results:

• The prices of otherwise identical European and American options must obey the relationship
Price $_{\text {American }} \geq$ Price $_{\text {European }}$
This is because an American option has all the benefits of a European option plus the right of early exercise.
• An American option will always be worth at least its payoff value: if it were worth less, we would simply buy the options and exercise them. Conversely, an American option will not be exercised if its value is greater than the payoff, as this constitutes the purposeless destruction of value.
• The price of a stock falls on an ex-dividend date by the amount of the dividend which is paid. The holder of an option does not receive the benefit of a dividend, so the potential payoff of an American call drops by the value of the dividend as the ex-dividend date is crossed. If an American call is exercised, this will therefore always occur shortly before an ex-dividend date. By the same reasoning, an American put is always exercised shortly after an ex-dividend date.
(ii) American Calls: In Section 2.2(ii) we saw that the graph of a call option against price must always lie above the line representing the value of a forward, i.e. $C_{\text {European }} \geq f_{0 T}=S_0-X \mathrm{e}^{-r T}$. The first point of the last subsection then implies that $C_{\text {American }} \geq f_{0 T}=S_0-X \mathrm{e}^{-r T}$ and if $r$ and $T$ are always positive (i.e. $\mathrm{e}^{-r T} \leq 1$ ) then we must also have
$$C_{\text {American }} \geq S_0-X$$
If this is true, then by the second point of the last subsection, it can never pay to exercise an American call before maturity; but if an American call is never exercised early, this feature has no value and the price of an American call must be the same as the price of a European call.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|PUT–CALL PARITY FOR AMERICAN OPTIONS

(i) It will be apparent to the reader that given the more complex behavior of American options, there is no slick formula for put-call parity as there is for European options. However for short-term options, fairly narrow bounds can be established on the difference between American put and call prices.
$\begin{array}{rrrrr}\text { Consider American options with maturity } T \text { which } & t=0 & t=\tau & t=T \ \text { may be exercised at a time } \tau \text {. The value of the proceeds } & \begin{array}{c}t=0 \text { ex } \ \text { now }\end{array} & \text { exerse } & \text { maturity }\end{array}$
may be exercised at a time $\tau$. The value of the proceeds of each option depends not only on the price $S_T$ at maturity, but also on whether and when it is exercised. If the option is exercised early, the strike price is paid and the time value of this cash has to be taken into account. For example, an American call option might be exercised at any time $\tau$ between now and $T$. After exercise, the stock that we buy under the option will continue to vary stochastically, achieving value $S_T$ at time $T$; but the exercise price would have been paid earlier than final maturity, so that the time $T$ value of the strike price is $X \mathrm{e}^{r(T-\tau)}$ where $0 \leq \tau \leq T$. The generalized payoff value of an American call option assessed at time $T$ may therefore be written as $S_T-X \mathrm{e}^{r(T-\tau)}$; the corresponding value for an American put option is $X \mathrm{e}^{r(T-\tau)}-S_T$.

Put-call parity relations for American options may be obtained using arbitrage arguments anảlogous to those for Europeean ōtions. In thee anälysis that follows. we make the decision ahead of time to hold any American option to maturity. Any short option position may be exercised against us at time $\tau(0 \leq \tau \leq T)$ and we then maintain the resultant stock position until maturity.
(ii) Let us now compare the following two portfolios:

• A forward contract to sell one share of stock in time $T$ for a price $X$.
• Long one put option and short one call option each on one share of stock, both with strike price $X$ and maturity $T$. Our strategy in running this portfolio is only to exercise the put options on their expiry date. Our counterparty may choose to exercise the call against us before maturity, in which case we invest the cash and hang on to the short stock position until maturity.

Initial and terminal values of these two portfolios are given in Table $2.2$. The notation ${Q, 0}$ signifies a quantity which could have value $Q$ or 0 , depending on whether our counterparty has exercised the call option or not. A few seconds reflection will convince the reader that the value of the option portfolio is always equal to or less than the proceeds of the forward share sale, whatever the value of $S_T$. In terms of the present value of the two portfolios, this may be written
$$C_0(X, T)-P_0(X, T) \leq S_0-X \mathrm{e}^{-r T}$$

# 期权理论代写

## 金融代写|期权理论代写期权数学介绍代考|AMERICAN Options

.

• 其他方面相同的欧洲和美国期权的价格必须服从关系
价格$_{\text {American }} \geq$价格$_{\text {European }}$
这是因为美国期权具有欧洲期权的所有好处，加上提前行权。美式期权的价值总是至少等于它的支付价值:如果它的价值更低，我们就会直接买入期权并行权。相反，如果美式期权的价值大于收益，它就不会被执行，因为这构成了价值的无目的破坏。在除息日，股票的价格是按已支付的股息下降的。期权的持有者没有得到股息的好处，因此美国看涨期权的潜在收益随着除息日的过去而下降。如果美式看涨期权被行使，这将总是发生在除息日前不久。(ii)美国看涨期权:在2.2(ii)节中，我们看到看涨期权与价格的曲线必须总是位于代表远期价格的线(即$C_{\text {European }} \geq f_{0 T}=S_0-X \mathrm{e}^{-r T}$)之上。最后一小节的第一点意味着$C_{\text {American }} \geq f_{0 T}=S_0-X \mathrm{e}^{-r T}$，如果$r$和$T$总是正的(即$\mathrm{e}^{-r T} \leq 1$)，那么我们也必须有
$$C_{\text {American }} \geq S_0-X$$
如果这是真的，那么到最后一小节的第二点，在到期前行使美国赎回权是永远不可能的;

. . . . . . . . . .

## 金融代写|期权理论代写期权数学介绍代考| PUT-CALL奇偶校验FOR AMERICAN Options

$\begin{array}{rrrrr}\text { Consider American options with maturity } T \text { which } & t=0 & t=\tau & t=T \ \text { may be exercised at a time } \tau \text {. The value of the proceeds } & \begin{array}{c}t=0 \text { ex } \ \text { now }\end{array} & \text { exerse } & \text { maturity }\end{array}$

• 一种及时卖出一股股票的远期合约 $T$ 付出一定的代价 $X$买入一股股票的一个看跌期权，做空一个看涨期权，均有执行价 $X$ 成熟度 $T$。我们经营这个投资组合的策略是，只在到期日执行看跌期权。我们的交易对手可以选择在到期前对我们行使看涨期权，在这种情况下，我们将现金投资并持有空头股票头寸直到到期这两个投资组合的初始值和最终值见表$2.2$。${Q, 0}$表示的数量可以是$Q$或0，这取决于我们的交易对手是否行使了看涨期权。几秒钟的思考就会使读者相信，期权投资组合的价值总是等于或小于远期股票出售的收益，无论$S_T$的价值是多少。根据两个投资组合的现值，可以写成
$$C_0(X, T)-P_0(X, T) \leq S_0-X \mathrm{e}^{-r T}$$

## 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|期权理论代写Mathematical Introduction to Options代考|MATH424

statistics-lab™ 为您的留学生涯保驾护航 在代写期权理论Mathematical Introduction to Options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。我们的专家在代写期权理论Mathematical Introduction to Options代写方面经验极为丰富，各种代写期权理论Mathematical Introduction to Options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|期权理论代写Mathematical Introduction to Options代考|PAYOFFS

(i) A call option on a commodity is a contract which gives the holder of the option the right to buy a unit of the commodity for a fixed price $X$ (the strike price). The key feature of this contract is that while it confirms the right, it does not impose an obligation. If it were a contract which both allowed and obligated the option holder to buy, we would have a forward contract rather than an option. The difference is that the option holder only exercises his right if it is profitable to do so. For example, suppose an option holder has a call option with $X=\$ 10$. If the price of the commodity in the market is$\$12$, the option can be exercised for $\$ 10$and the underlying commodity sold for$\$12$, to yield a profit of $\$ 2$; on the other hand, if the market price is$\$8$, the option will not be exercised.

The outcome of this type of option contract can be summarized mathematically as follows:
$$\text { Payoff }=\max \left[0,\left(S_T-X\right)\right] \quad \text { or } \quad\left(S_T-X\right)^{+}$$
which means that the payoff equals $S_T-X$, but only if this is positive; otherwise it is zero.
The payoff may equally be regarded as the value of the call option at exercise $C_{\text {payoff. }}$. Much of this book is dedicated to the following problem: if we know $C_{\text {payoff }}$, how can we calculate the value of the option now?

A put option gives the holder the right (hut not the ohligation) to sell a unit of a commodity for a strike price $X$. This type of option is completely analogous to the call option. The payoff (option value at exercise) can be written
$$P_{\text {payoff }}=\max \left[0,\left(X-S_T\right)\right] \quad \text { or } \quad\left(X-S_T\right)^{+}$$
(ii) The payoff of a call, a put and a forward contract are shown in Figure 2.1. These are the socalled “hockey-stick” diagrams which show the value at exercise or payoff of the instruments as a function of the price of the underlying commodity.
(iii) An option is an asset with value greater than or equal to zero. If we buy an option we own an asset; but someone out there has a corresponding liability. He is the option writer and is said to be short an option in the jargon of Section 1.1.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|OPTION PRICES BEFORE MATURITY

(i) Put-Call Parity for European Options: Consider the following two portfolios:

• A forward contract to buy one share of stock in time $T$ for a price $X$.
• Long one call option and short one put option each on one share of stock, both with strike price $X$ and maturity $T$.

The values of the portfolios now and at maturity are shown in Table 2.1. It is clear that whatever the maturity value of the underlying stock, the two portfolios have the same payoff value. Therefore, by the no-arbitrage proposition $1.2$ (ii), the two portfolios must have the same value now. This important relationship is known as put-call parity and may be expressed as
$$f_{0 T}=C_0(X, T)-P_0(X, T)$$
or equivalently
$$P_0(X, T)+S_0=C_0(X, T)+X \mathrm{e}^{-r T}$$
If dividends are taken into account, the last equation may be written
$$\begin{array}{lr} P_0+\left(S_0-d \mathrm{e}^{-r \tau}\right)=C_0+X \mathrm{e}^{-r T} & \text { discrete dividend at } \tau \ P_0+S_0 \mathrm{e}^{-q T}=C_0+X \mathrm{e}^{-r T} & \text { continuous dividend rate } q \end{array}$$
i) Consider the value of a put option prior to expiry, if the stock price is much larger than the strike price. Clearly the value of this asset cannot be less than zero since it involves no obligation; on the other hand, its value must be very small if $S_0 \rightarrow \infty$, since the chance of its being exercised is small. The same reasoning applies to a call option for which $S_0 \rightarrow 0$. These can be summarized as
$$\lim {S_0 \rightarrow \infty} P_0 \rightarrow 0 ; \quad \lim {S_0 \rightarrow 0} C_0 \rightarrow 0$$
Using both these results in the put-call parity relationship of equation (2.1) gives the following general result for European options without dividends:
$$\lim {S_0 \rightarrow \infty} C_0 \rightarrow f{0 T}=S_0-X \mathrm{e}^{-r T} ; \quad \lim {S_0 \rightarrow 0} P_0 \rightarrow-f{0 T}=X \mathrm{e}^{-r T}-S_0$$
These results are illustrated in Figure 2.3. The dotted lines and the $x$-axes provide the asymptotes for the graphs of $C_0$ and $P_0$ against $S_0$, for European options.

# 期权理论代写

## 金融代写|期权理论代写期权数学介绍代考|收益

.

$$\text { Payoff }=\max \left[0,\left(S_T-X\right)\right] \quad \text { or } \quad\left(S_T-X\right)^{+}$$
，这意味着收益等于$S_T-X$，但只有当它为正时;否则就是零。在执行$C_{\text {payoff. }}$时，收益可以同等地被视为看涨期权的价值。本书的大部分内容致力于解决以下问题:如果我们知道$C_{\text {payoff }}$，我们现在如何计算期权的价值?

$$P_{\text {payoff }}=\max \left[0,\left(X-S_T\right)\right] \quad \text { or } \quad\left(X-S_T\right)^{+}$$
(ii)看涨、看跌和远期合约的收益如图2.1所示。这就是所谓的“曲棍球杆”图，它显示了工具的执行价值或支付作为相关商品价格的函数。如果我们买了期权，我们就拥有了资产;但有些人有相应的负债。他是期权作者，用1.1节的行话来说，他被称为做空期权

## 金融代写|期权理论代写期权数学介绍代考|到期前的期权价格

(i)欧洲期权的看跌期权平价:考虑以下两个组合

• 一种及时购买一股股票的远期合约 $T$ 付出一定的代价 $X$买入一份股票的看涨期权，做空一份看跌期权，均有执行价 $X$ 成熟度 $T$.

$$f_{0 T}=C_0(X, T)-P_0(X, T)$$

$$P_0(X, T)+S_0=C_0(X, T)+X \mathrm{e}^{-r T}$$

$$\begin{array}{lr} P_0+\left(S_0-d \mathrm{e}^{-r \tau}\right)=C_0+X \mathrm{e}^{-r T} & \text { discrete dividend at } \tau \ P_0+S_0 \mathrm{e}^{-q T}=C_0+X \mathrm{e}^{-r T} & \text { continuous dividend rate } q \end{array}$$
i)考虑一个看跌期权在到期前的价值，如果股票价格远远大于执行价格。显然，这种资产的价值不可能小于零，因为它不涉及任何债务;另一方面，如果$S_0 \rightarrow \infty$，它的值一定很小，因为它被执行的机会很小。同样的道理也适用于$S_0 \rightarrow 0$。这些结果可以总结为
$$\lim {S_0 \rightarrow \infty} P_0 \rightarrow 0 ; \quad \lim {S_0 \rightarrow 0} C_0 \rightarrow 0$$

$$\lim {S_0 \rightarrow \infty} C_0 \rightarrow f{0 T}=S_0-X \mathrm{e}^{-r T} ; \quad \lim {S_0 \rightarrow 0} P_0 \rightarrow-f{0 T}=X \mathrm{e}^{-r T}-S_0$$

## 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|期权理论代写Mathematical Introduction to Options代考|MATH4380

statistics-lab™ 为您的留学生涯保驾护航 在代写期权理论Mathematical Introduction to Options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。我们的专家在代写期权理论Mathematical Introduction to Options代写方面经验极为丰富，各种代写期权理论Mathematical Introduction to Options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|期权理论代写Mathematical Introduction to Options代考|ARBITRAGE

Having stated in the last section that most examples will be taken from the world of equities, we will illustrate this key topic with a single example from the world of foreign exchange; it just fits better.

Most readers have at least a notion that arbitrage means buying something one place and selling it for a profit somewhere else, all without taking a risk. They probably also know that opportunities for arbitrage are very short-lived, as everyone piles into the opportunity and in doing so moves the market to a point where the opportunity no longer exists. When analyzing financial markets, it is therefore reasonable to assume that all prices are such that no arbitrage is possible.
Let us be a little more precise: if we have cash, we can clearly make money simply by depositing it in a bank and earning interest; this is the so-called risk-free return. Alternatively, we may make rather more money by investing in a stock; but this carries the risk of the stock price going down rather than up. What is assumed to be impossible is to borrow money from the bank and invest in some risk-free scheme which is bound to make a profit. This assumption is usually known as the no-arbitrage or no-free-lunch principle. It is instructive to state this principle in three different but mathematically equivalent ways.
(i) Equilibrium prices are such that it is impossible to make a risk-free profit. Consider the following sequence of transactions in the foreign exchange market:
(A) We borrow $\$ 100$for a year from an American bank at an interest rate$r_{\$}$. At the end of the year we have to return $\$ 100\left(1+r_{\$}\right)$ to the bank. Using the conventions of the last section, its value in one year will be $-\$ 100\left(1+r_{\$}\right)$.
(B) Take the $\$ 100$and immediately do the following three things: • Convert it to pounds sterling at the spot rate$S_{\text {now }}$to give$£ \frac{100}{S_{\text {now }}}$; • Put the sterling on deposit with a British bank for a year at an interest rate of$r_£$. In a year we will receive back$£ \frac{100}{S_{\text {now }}}\left(1+r_£\right)$; • Take out a forward contract at a rate$F_{1 \text { year to }}$toxchange$£ \frac{100}{S_{\text {now }}}\left(1+r_£\right)$for$\$\frac{100}{S_{\mathrm{now}}}\left(1+r_{\mathcal{E}}\right) F_1$ year at the end of the year.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|FORWARD CONTRACTS

(i) A forward contract is a contract to buy some security or commodity for a predetermined price, at some time in the future; the purchase price remains fixed, whatever happens to the price of the security before maturity.
Clearly, the market (or spot) price and the forward price will tend to converge (Figure 1.3) as the maturity date is approached; a one-day forward price will be pretty close to the spot price.

In the last section we used the example of a forward currency contract; this is the largest, best known forward market in the world and it was flourishing long before the word “derivative” was applied to financial markets. Yet it is the simplest non-trivial derivative and it allows us to illustrate some
0
$T$

(ii) Consider some very transitory commodity which cannot be stored – perhaps some unstorable agricultural commodity. The forward price at which we would be prepared to buy the commodity is determined by our expectation of its market price at the maturity of the contract; the higher we thought its price would be, the more we would bid for the future contract. So if we were asked to quote a two-year contract on fresh tomatoes, the best we could do is some kind of fundamental economic analysis: what were past trends, how are consumer tastes changing, what is happening to area under cultivation, what is the price of tomato fertilizer, etc.

However, all commodities considered in this book are non-perishable: securities, traded commodities, stock indexes and foreign exchange. What effect does the storable nature of a commodity have on its forward price?

Suppose we buy an equity share for a price $S_0$; in time $T$ the value of this share becomes $S_T$. If we had entered a forward contract to sell the share forward for a price $F_{0 T}$, we would have been perfectly hedged, i.e. we would have paid out $S_0$ at the beginning and received a predetermined $F_{0 T}$ at time $T$. From the no-arbitrage argument 1.2(iii), this investment must yield a return equal to the interest rate. Expressed in terms of continuous interest rates, we have
$$\frac{F_{0 T}}{S_0}=\mathrm{e}^{r T} \quad \text { or } \quad F_{0 T}=S_0 \mathrm{e}^{r T}$$
This result is well known and seems rather banal; but its ramifications are so far-reaching that it is worth pausing to elaborate. Someone who knows nothing about finance theory would be forgiven for assuming that a forward rate must somehow depend on the various characteristics of each stock: growth rate, return, etc. But the above relationship shows that there is a fixed relationship between the spot and forward prices which is the same for all financial instruments and which is imposed by the no-arbitrage conditions. The reason is of course immediately obvious. With a perishable commodity, forward prices can have no effect on current prices: if we know that the forward tomatoes price is $\$ 1$million each, there is nothing we can do about it and the current price will not be affected. But if the forward copper price is$\$1$ million, we buy all the copper we can in the spot market we can, put it in a warehouse and take out forward contracts to sell it next year.

# 期权理论代写

## 金融代写|期权理论代写期权数学介绍代考|ARBITRAGE

.

(A)我们从一家美国银行借了$\$ 100$一年，利率是$r_{\$}$。在年底，我们必须把$\$ 100\left(1+r_{\$}\right)$还给银行。使用上一节的约定，它在一年后的值将是$-\$ 100\left(1+r_{\$}\right)$ .
(B)取$\$ 100$并立即做以下三件事 • 将其按即期汇率$S_{\text {now }}$兑换成英镑，得到$£ \frac{100}{S_{\text {now }}}$; • 将英镑存入一家英国银行，利率为$r_£$，存期一年。一年后，我们将收到$£ \frac{100}{S_{\text {now }}}\left(1+r_£\right)$; • 在年底以$F_{1 \text { year to }}$toxchange$£ \frac{100}{S_{\text {now }}}\left(1+r_£\right)$的利率签订一个$\$\frac{100}{S_{\mathrm{now}}}\left(1+r_{\mathcal{E}}\right) F_1$年的远期合同
金融代写|期权理论代写期权数学介绍代考|FORWARD CONTRACTS . . . .远期合约是一种在未来某个时间以预定价格购买某种证券或商品的合约;无论到期前证券的价格如何变化，买入价保持不变。显然，随着到期日的临近，市场(或现货)价格和远期价格将趋于收敛(图1.3);1天远期价格将非常接近现货价格在上一节中，我们使用了远期货币合约的例子;这是世界上最大、最著名的远期市场，早在“衍生品”一词被应用到金融市场之前，它就已经很繁荣了。然而，它是最简单的非平凡导数，它允许我们说明一些
0
$T$考虑一些无法储存的非常短暂的商品-也许是一些无法储存的农业商品。我们准备购买该商品的远期价格是由我们对该商品在合同到期时的市场价格的预期决定的;我们认为它的价格越高，我们就会为未来的合同出价越多。因此，如果我们被要求对新鲜番茄的两年合同报价，我们所能做的最好的就是某种基本的经济分析:过去的趋势是什么，消费者的口味如何变化，种植面积发生了什么，番茄肥料的价格是多少，等等然而，本书所考虑的所有商品都是不易腐烂的:证券、交易商品、股票指数和外汇。商品的可储存性对其远期价格有什么影响?假设我们以$S_0$的价格购买股票;最终，这个份额的价值$T$变成了$S_T$。如果我们签订了一个远期合约，以$F_{0 T}$的价格出售股票，我们将完全对冲，即我们将在一开始支付$S_0$，并在当时$T$收到预定的$F_{0 T}$。根据无套利论证1.2(iii)，这项投资必须产生与利率相等的回报。用连续利率表示，我们有
$$\frac{F_{0 T}}{S_0}=\mathrm{e}^{r T} \quad \text { or } \quad F_{0 T}=S_0 \mathrm{e}^{r T}$$
这个结果众所周知，但似乎相当平庸;但它的影响如此深远，值得停下来详细阐述。对金融理论一窍不通的人假设远期利率一定取决于每只股票的各种特征:增长率、回报率等，这是可以理解的。但上述关系表明，现货和远期价格之间存在一个固定的关系，这种关系对所有金融工具都是一样的，是由无套利条件所强加的。原因当然是显而易见的。对于易腐烂的商品，远期价格可能对当前价格没有影响:如果我们知道番茄远期价格是$\$ 1$万颗，我们就无能为力，当前价格也不会受到影响。但如果铜的远期价格是$\$1$万，我们就会在现货市场尽可能地买进铜，把它存入仓库，然后签订远期合约，明年卖出

## 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。