### 金融代写|资产定价代写Asset Pricing代考|FN2190

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• 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|波动率模型代写Market Volatility Modelling代考|Expected Utility

Standard microeconomics represents preferences using ordinal utility functions. An ordinal utility function $\Upsilon($.$) tells you that an agent is indifferent between x$ and $y$ if $\Upsilon(x)=\Upsilon(y)$ and prefers $x$ to $y$ if $\Upsilon(x)>\Upsilon(y)$. Any strictly increasing function of $\Upsilon($. will have the same properties, so the preferences expressed by $\Upsilon($.$) are the same as those$ expressed by $\Theta(\Upsilon()$.$) for any strictly increasing \Theta$. In other words, ordinal utility is invariant to monotonically increasing transformations. It defines indifference curves, but there is no way to label the curves so that they have meaningful values.

A cardinal utility function $\Psi($.$) is invariant to positive affine (increasing linear) trans-$ formations but not to nonlinear transformations. The preferences expressed by $\Psi($.$) are$ the same as those expressed by $a+b \Psi($.$) for any b>0$. In other words, cardinal utility has no natural units, but given a choice of units, the rate at which cardinal utility increases is meaningful.

Asset pricing theory relies heavily on von Neumann-Morgenstern utility theory, which says that choice over lotteries, satisfying certain axioms, implies maximization of the expectation of a cardinal utility function, defined over outcomes.

The content of von Neumann-Morgenstern utility theory is easiest to understand in a discrete-state example. Define states $s=1 \ldots S$, each of which is associated with an outcome $x_s$ in a set $X$. Probabilities $p_s$ of the different outcomes then define lotteries. When $S=3$, we can draw probabilities in two dimensions (since $p_3=1-p_1-p_2$ ). We get the so-called Machina triangle (Machina 1982), illustrated in Figure 1.1.

We define a compound lottery as one that determines which primitive lottery we are given. For example, a compound lottery $L$ might give us lottery $L^a$ with probability $\alpha$ and lottery $L^b$ with probability $(1-\alpha)$. Then $L$ has the same probabilities over the outcomes as $\alpha L^a+(1-\alpha) L^b$.

We define a preference ordering $\succeq$ over lotteries. A person is indifferent between lotteries $L^a$ and $L^b, L^a \sim L^b$, if and only if $L^a \succeq L^b$ and $L^b \succeq L^a$.
Next we apply two axioms of choice over lotteries.
Continuity axiom: For all $L^a, L^b, L^c$ s.t. $L^a \succeq L^b \succeq L^c$, there exists a scalar $\alpha \in[0,1]$ s.t.
$$L^b \sim \alpha L^a+(1-\alpha) L^c .$$
This axiom says that if three lotteries are (weakly) ranked in order of preference, it is always possible to find a compound lottery that mixes the highest-ranked and lowest-ranked lotteries in such a way that the economic agent is indifferent between this compound lottery and the middle-ranked lottery. The axiom implies the existence of a preference functional defined over lotteries, that is, an ordinal utility function for lotteries that enables us to draw indifference curves on the Machina triangle.

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk Aversion

We now assume the existence of a cardinal utility function and ask what it means to say that the agent whose preferences are represented by that utility function is risk averse. We also discuss the quantitative measurement of risk aversion.

To bring out the main ideas as simply as possible, we assume that the argument of the utility function is wealth. This is equivalent to working with a single consumption good in a static two-period model where all wealth is liquidated and consumed in the second period, after uncertainty is resolved. Later in the book we discuss richer models in which consumption takes place in many periods, and also some models with multiple consumption goods.

For simplicity we also work with weak inequalities and weak preference orderings throughout. The extension to strict inequalities and strong preference orderings is straightforward.

An important mathematical result, Jensen’s Inequality, can be used to link the concept of risk aversion to the concavity of the utility function. We start by defining concavity for a function $f$.

Definition. $f$ is concave if and only if, for all $\lambda \in[0,1]$ and values $a, b$,
$$\lambda f(a)+(1-\lambda) f(b) \leq f(\lambda a+(1-\lambda) b) .$$
If $f$ is twice differentiable, then concavity implies that $f^{\prime \prime} \leq 0$. Figure $1.2$ illustrates a concave function.

Note that because the inequality is weak in the above definition, a linear function is concave. Strict concavity uses a strong inequality and excludes linear functions, but we proceed with the weak concept of concavity.
Now consider a random variable $\tilde{z}$. Jensen’s Inequality states that
$$\mathrm{E} f(\bar{z}) \leq f(\mathrm{E} \bar{z})$$
for all possible $\tilde{z}$ if and only if $f$ is concave.

# 波动率模型代考

## 金融代写|波动率模型代写Market Volatility Modelling代考|Expected Utility

)tellsyouthatanagentisindifferentbetween $x$ 和 $y$ 如果 $\Upsilon(x)=\Upsilon(y)$ 并且喜欢 $x$ 至 $y$ 如果
$\Upsilon(x)>\Upsilon(y)$. 的任何严格增函数 $\Upsilon($. 将具有相同的属性，因此表示的偏好 $\Upsilon$ (.) arethesameasthose 表示为 $\Theta(\Upsilon()$.) foranystrictlyincreasing $\Theta$. 换句话说，序数效用对于单调递增的变换是不变的。它 定义了无差异曲线，但无法标记曲线以使它们具有有意义的值。

von Neumann-Morgenstern 效用理论的内容在一个离散状态的例子中最容易理解。定义状态 $s=1 \ldots S$ ，每一个都与一个结果相关联 $x_s$ 在一组 $X$. 概率 $p_s$ 不同的结果然后定义彩票。什么时候 $S=3$ ，我们可以绘制二维概率 (因为 $p_3=1-p_1-p_2$ ). 我们得到所谓的 Machina 三角形 (Machina 1982)，如图 $1.1$ 所示。

$$L^b \sim \alpha L^a+(1-\alpha) L^c .$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk Aversion

$$\lambda f(a)+(1-\lambda) f(b) \leq f(\lambda a+(1-\lambda) b) .$$

$$\operatorname{E} f(\bar{z}) \leq f(\mathrm{E} \bar{z})$$

## 有限元方法代写

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## MATLAB代写

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