# 金融代写|投资组合代写Investment Portfolio代考|FINC3017

## 金融代写|投资组合代写Investment Portfolio代考|COMPLETE MARKETS AND UNIQUENESS OF THE SDF

The securities market is said to be complete if, for any $\tilde{w}$, there exists a portfolio $\theta$ such that
$$\sum_{i=1}^n \theta_i \tilde{x}_i=\tilde{w} .$$
Thus, any desired distribution of wealth across states of the world can be achieved by choosing the appropriate portfolio (if cost is not a constraint).

It should be apparent that true completeness is a rare thing. For example, if there are infinitely many states of the world, then (3.18) is an infinite number of equalities, which we are supposed to satisfy by choosing a finite-dimensional vector $\theta \in \mathbb{R}^n$. This is impossible. Note that there must be infinitely many states if we want the security payoffs $\tilde{x}_i$ to be normally distributed, or to be lognormally distributed, or to have any other continuous distribution. Thus, single-period markets with finitely many continuously distributed assets are not complete.

On the other hand, if significant gains are possible by improving risk sharing, then we would expect assets to be created to enable those gains to be realized. Also, as is shown later, dynamic trading can dramatically increase the “span” of securities markets. The real impediments to achieving at least approximately complete markets are moral hazard and adverse selection. For example, there are very limited opportunities for obtaining insurance against employment risk, due to moral hazard.

## 金融代写|投资组合代写Investment Portfolio代考|RISK-NEUTRAL PROBABILITIES

Another way to represent prices is via a risk-neutral probability. To distinguish between a risk-neutral probability and the probability under which we have been taking expectations, it is common to call the latter (whether objective or subjective) the physical probability. ${ }^5$ A risk-neutral probability is defined from the physical probability and a strictly positive SDF. Assume there is a strictly positive $\operatorname{SDF} \tilde{m}$, and, for each event $A$, let $1_A$ denote the indicator function of $A$, that is, for each state of the world $\omega, 1_A(\omega)=1$ if $\omega \in A$ and $1_A(\omega)=0$ if $\omega \notin A$.
Suppose first that there is a risk-free asset. For each event $A$, define
$$\mathbb{Q}(A)=R_f \mathrm{E}\left[\tilde{m} 1_A\right] .$$
Then, $\mathbb{Q}$ is the risk-neutral probability associated to $\tilde{m}$. In a finite-state world, (3.21) means that the risk-neutral probability of each state is the product of $R_f$, the physical probability of the state, and the value of $\tilde{m}$ in the state (it is also the product of $R_f$ with the state price). In general, $\mathbb{Q}$ is a probability: $\mathbb{Q}(A) \geq 0, \mathbb{Q}(\Omega)=1$, where $\Omega$ is the set of all states of the world, and if $A_1, A_2, \ldots$ is a sequence of disjoint events, then $\mathbb{Q}\left(\cup A_i\right)=\sum \mathbb{Q}\left(A_i\right)$. As with any probability, there is an expectation operator associated with $\mathbb{Q}$. Denote it by $E^$. The definition of $\mathbb{Q}$ can be restated as $$\mathrm{E}^\left[1_A\right]=R_f \mathrm{E}\left[\tilde{m} 1_A\right],$$
because the expectation of an indicator function is the probability of the event. More generally, the definition of $\mathbb{Q}$ implies that
$$\mathrm{E}^[\tilde{x}]=R_f \mathrm{E}[\tilde{m} \tilde{x}]$$ for every $\tilde{x}$ for which the expectation $\mathrm{E}[\tilde{m} \tilde{x}]$ exists. Because the price of any payoff $\tilde{x}$ is $\mathrm{E}[\tilde{m} \tilde{x}]$, equation (3.22) implies that the price of any payoff $\tilde{x}$ is $$\frac{1}{R_f} E^[\tilde{x}]$$

## 金融代写|投资组合代写Investment Portfolio代考|COMPLETE MARKETS AND UNIQUENESS OF THE SDF

$$\sum_{i=1}^n \theta_i \tilde{x}_i=\tilde{w} .$$

## 金融代写|投资组合代写Investment Portfolio代考|RISK-NEUTRAL PROBABILITIES

$$\mathbb{Q}(A)=R_f \mathrm{E}\left[\tilde{m} 1_A\right] .$$

$$\mathrm{E}^{\left[1_A\right]}=R_f \mathrm{E}\left[\tilde{m} 1_A\right],$$

$$\left.\mathrm{E}^{[} \tilde{x}\right]=R_f \mathrm{E}[\tilde{m} \tilde{x}]$$

$$\left.\frac{1}{R_f} E^{[} \tilde{x}\right]$$

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