# 金融代写|投资组合代写Investment Portfolio代考|MBA6510

## 金融代写|投资组合代写Investment Portfolio代考|ARBITRAGE, THE LAW OF ONE PRICE, AND EXISTENCE OF SDFs

The remainder of this chapter analyzes the existence and structure of SDFs without reference to the first-order condition (3.7). This section describes two important concepts regarding securities markets and the relations of the concepts to the existence of SDFs. A summary of the results is as follows:
Existence of Strictly Positive SDF $\Leftrightarrow$ No Arbitrage Opportunities
$\Rightarrow \quad$ Law of One Price
$\Leftrightarrow \quad$ Existence of SDF
Let $n$ denote the number of assets, including the risk-free asset if it exists. Set $p=\left(p_1 \cdots p_n\right)^{\prime}$, and interpret a portfolio $\theta$ as a column vector. A random variable $\tilde{x}$ is said to be a marketed payoff if it is the payoff of a portfolio, meaning that $\tilde{x}=\sum_{i=1}^n \theta_i \tilde{x}i$ for some $\theta \in \mathbb{R}^n$. The “law of one price” is said to hold if each marketed payoff has a unique cost. This means that if there are two portfolios producing the same payoff, then they have the same cost. Mathematically, $$\left(\forall \theta, \hat{\theta} \in \mathbb{R}^n\right) \quad \sum{i=1}^n \theta_i \tilde{x}i=\sum{i=1}^n \hat{\theta}_i \tilde{x}_i \Rightarrow p^{\prime} \theta=p^{\prime} \hat{\theta} .$$
When we write equality of two random variables (as for the portfolio payoffs here), we always mean that they are equal with probability 1.

An arbitrage opportunity is defined to be a portfolio $\theta$ satisfying
(i) $p^{\prime} \theta \leq 0$,
(ii) $\sum_{i-1}^n \theta_i \tilde{x}i \geq 0$ with probability 1 , and (iii) either $p^{\prime} \theta<0$ or $\sum{i=1}^n \theta_i \tilde{x}_i>0$ with positive probability (or both).

## 金融代写|投资组合代写Investment Portfolio代考|The Law of One Price and SDFs

If the law of one price holds, then there is an SDF. Thus, the price $p^{\prime} \theta$ of any payoff $\sum_{i=1}^n \theta_i \tilde{x}i$ can be computed as $\mathrm{E}\left[\tilde{m} \sum{i=1}^n \theta_i \tilde{x}_i\right]$ for some $\tilde{m}$. In fact, the law of one price is equivalent to the existence of an SDF. This is true with infinitely many states of the world (limiting attention to payoffs with finite variances), but it is easier to see when there are only finitely many states.

For the remainder of this section, suppose there are $k$ possible states of the world. We make no assumptions regarding the number of states versus the number of assets, so $k$ can be smaller than, equal to, or larger than $n$. Denote the payoff of asset $i$ in state $j$ as $x_{i j}$. A state-price vector is defined to be a vector $\left(q_1 \cdots q_k\right)$ satisfying
$$\left(\begin{array}{ccc} x_{11} & \cdots & x_{1 k} \ \vdots & \vdots & \vdots \ x_{n 1} & \cdots & x_{n k} \end{array}\right)\left(\begin{array}{c} q_1 \ \vdots \ q_k \end{array}\right)=\left(\begin{array}{c} p_1 \ \vdots \ p_n \end{array}\right) .$$
This equation means that the price $p_i$ of each asset $i$ equals the sum of the payoffs of asset $i$ in the various states of the world multiplied by the state price of each state. As discussed in the introduction to this chapter, a state price is the price of the Arrow security that pays 1 unit of the consumption good in that state and 0 in all other states. Denote the matrix in (3.15) by $X$ and denote the vectors by $q$ and $p$, so we can write $(3.15)$ as
$$X q=p$$

## 金融代写|投资组合代写Investment Portfolio代考|ARBITRAGE, THE LAW OF ONE PRICE, AND EXISTENCE OF SDFs

$\Rightarrow$ 一价定律
$\Leftrightarrow$ 存在 SDF

$$\left(\forall \theta, \hat{\theta} \in \mathbb{R}^n\right) \quad \sum i=1^n \theta_i \tilde{x} i=\sum i=1^n \hat{\theta}i \tilde{x}_i \Rightarrow p^{\prime} \theta=p^{\prime} \hat{\theta}$$ 当我们写出两个随机变量的相等性 (至于这里的投资组合收益) 时，我们总是意味着它们以概率 1 相 等。 農利机会被定义为投赕组合 $\theta$ 满足 $$\text { (-) } p^{\prime} \theta \leq 0 \text {, }$$ (ii) $\sum{i-1}^n \theta_i \tilde{x} i \geq 0$ 概率为 1 ，并且 (iii) 要么 $p^{\prime} \theta<0$ 或者 $\sum i=1^n \theta_i \tilde{x}_i>0$ 具有正概率 (或两者)。

## 金融代写|投资组合代写Investment Portfolio代考|The Law of One Price and SDFs

$$X q=p$$

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