4. No Arbitrage Framework: Incomplete Market

Jun He May 30, 2026

资产定价Asset Pricing 随机贴现因子SDF 无套利No Arbitrage 不完全市场Incomplete Market 学习笔记Study Note

Note

本章在不完全市场 (incomplete market) 下重新审视随机贴现因子 (SDF)。核心问题：当可交易资产张不满整个状态空间（没有全套 Arrow–Debreu 证券）时，是否仍存在 SDF $m$ 使得 $p=\mathbb E[m\tilde x]$？答案是肯定的，但要分清两个强弱不同的条件——一价定律 (Law of One Price)（弱）与无套利 (no arbitrage)（强）。主要结论：(i) 在"组合自由构造 + 一价定律"下，支付空间 $\mathcal X$ 内存在唯一的 SDF $x^*=\mathbf x'\mathbb E[\mathbf{xx}']^{-1}\mathbf p(\mathbf x)$；(ii) $\mathcal X$ 外可有无穷多 SDF，但它们在 $\mathcal X$ 上的投影 (projection) 都等于 $x^*$；(iii) 金融基本定理 (Fundamental Theorem of Finance)：无套利 $\Leftrightarrow$ 存在严格为正的 SDF；(iv) 唯有完全市场才能保证 SDF 唯一。

Note

This chapter revisits the stochastic discount factor (SDF) in an incomplete market. The key question: when tradable assets do not span the whole state space (no full set of Arrow–Debreu securities), does an SDF $m$ with $p=\mathbb E[m\tilde x]$ still exist? Yes — but we must distinguish two conditions of different strength: the Law of One Price (weaker) and no arbitrage (stronger). Main results: (i) under "free portfolio formation + Law of One Price", there is a unique SDF $x^*=\mathbf x'\mathbb E[\mathbf{xx}']^{-1}\mathbf p(\mathbf x)$ inside the payoff space $\mathcal X$; (ii) outside $\mathcal X$ there can be infinitely many SDFs, but their projection onto $\mathcal X$ all equals $x^*$; (iii) the Fundamental Theorem of Finance: no arbitrage $\Leftrightarrow$ existence of a strictly positive SDF; (iv) only a complete market makes the SDF unique.

4.1 Stochastic Discount Factor and Incomplete Market

无论是消费基础模型 (1.6) 还是完全市场 (3.3)，定价都化为同一个基本表示：

Whether from the consumption-based model (1.6) or the complete market (3.3), pricing reduces to the same basic representation:

$$ \cssId{na1}{p} = \mathbb E\big[\,\cssId{na2}{m}\,\cssId{na3}{\tilde x}\,\big] $$

这被称为资产定价基本方程 (fundamental equation of asset pricing)。在完全市场中 (3.3) 给出唯一的 SDF $m=\pi(s)$，即状态价格密度。

现实中市场往往不完全：可交易资产张不满整个状态空间，即我们没有全套 Arrow–Debreu 状态证券。关键问题是：还能否找到某个 SDF $m$ 为这些资产定价？ 答案是可以，但需要附加条件。

Tip

"$m$ 能定价"的含义。 指任意证券价格 $p$ 都能写成 $m$ 与该证券支付 $\tilde x$ 的函数 $p=\mathbb E[m\tilde x]$。What "$m$ prices" means. Any security price $p$ can be written as a function of $m$ and that security's payoff $\tilde x$: $p=\mathbb E[m\tilde x]$.

This is the fundamental equation of asset pricing. In a complete market, (3.3) gives the unique SDF $m=\pi(s)$, the state-price density.

In reality the market is often incomplete: tradable assets cannot span the entire state space, i.e. we do not have the full set of Arrow–Debreu state securities. The crucial question is: can we still find an SDF $m$ that prices these assets? Yes, under some conditions.

Tip

4.2 Setup

基础支付 (basis payoff)： 用随机向量 $\mathbf x=(x_1,x_2,\dots,x_N)'$ 记 $N$ 个基础支付。"基础"指任何可交易组合的支付都能写成这 $N$ 个基础支付的线性组合。
支付空间 (payoff space)： 设 $\mathcal X$ 为有限维支付空间，包含所有可交易证券支付及其线性组合：

Basis payoff: denote the $N$ basis payoffs by a random vector $\mathbf x=(x_1,x_2,\dots,x_N)'$. "Basis" means the payoff of any tradable combination can be written as a linear combination of these $N$ basis payoffs.
Payoff space: let $\mathcal X$ be the finite-dimensional payoff space containing all tradable payoffs and their linear combinations:

$$\mathcal X=\Big\{x:\,x=\sum_{i=1}^N c_i x_i\ \text{ for }\forall c_1,\dots,c_N\in\mathbb R\Big\}=\big\{\mathbf c'\mathbf x:\ \forall\mathbf c\in\mathbb R^N\big\}.$$

有限维来自有限的 $N$。下文主要在有限维下讨论，但当 $N\to\infty$ 时结论可推广到无穷维。关键：不完全市场下 $\mathcal X$ 一般张不满整个状态空间——它包含的支付不足以复制所有状态或有索取权。

Tip

非线性支付怎么办？ 如期权这类指示函数支付，可以要么扩展到无穷维支付空间，要么把该特定非线性支付作为额外一维直接加入基础向量，仍停留在有限维。What about non-linear payoffs? For indicator-type payoffs such as options, either extend to an infinite-dimensional payoff space, or simply add that specific non-linear payoff as an extra dimension of the basis vector and stay finite-dimensional.

Finite-dimensionality comes from finite $N$. The discussion below is mainly finite-dimensional, but extends to infinite dimensions as $N\to\infty$. Crucially, under an incomplete market $\mathcal X$ generally cannot span the whole state space — it contains fewer payoffs than needed to replicate all state-contingent claims.

Tip

4.3 Existence of Stochastic Discount Factor

4.3.1 Portfolio Formation and Law of One Price

不完全市场下 SDF 的存在性需要两条假设。

假设 1（组合构造 Portfolio Formation）。 对 $\forall y_1,y_2\in\mathcal X$ 和 $\forall a,b\in\mathbb R$，有 $ay_1+by_2\in\mathcal X$。即投资者可自由地把证券线性组合成组合，且组合仍在支付空间内（排除卖空限制、买卖价差、杠杆限制）。
假设 2（一价定律 Law of One Price）。 定价函数 $p:\mathcal X\to\mathbb R$ 是线性的：对 $\forall y_1,y_2\in\mathcal X$、$\forall a,b\in\mathbb R$，

Existence of an SDF in an incomplete market requires two assumptions.

Assumption 1 (Portfolio Formation). For $\forall y_1,y_2\in\mathcal X$ and $\forall a,b\in\mathbb R$, $ay_1+by_2\in\mathcal X$. Investors can freely form portfolios as linear combinations, and the portfolio stays in the payoff space (rules out short-sale constraints, bid–ask spreads, leverage limits).
Assumption 2 (Law of One Price). The pricing function $p:\mathcal X\to\mathbb R$ is linear: for $\forall y_1,y_2\in\mathcal X$, $\forall a,b\in\mathbb R$,

$$p(ay_1+by_2)=a\,p(y_1)+b\,p(y_2).$$

一价定律意味着"一篮子的价格等于篮子内各部分价格之和"，否则重构组合就有瞬时利润，不可能是均衡。

The Law of One Price means "the price of a bundle equals the sum of the prices of its parts"; otherwise restructuring a portfolio yields instantaneous profit, which cannot be an equilibrium.

4.3.2 Unique Stochastic Discount Factor within Payoff Space

SDF 的存在与一价定律是等价的——任一方都蕴含另一方。

定理 4.1。 SDF $m$ 的存在蕴含一价定律。

Existence of an SDF and the Law of One Price are equivalent — each implies the other.

Theorem 4.1. The existence of an SDF $m$ implies the Law of One Price.

证明 / Proof：SDF 存在 $\Rightarrow$ 一价定律

设 $y_3=ay_1+by_2\in\mathcal X$，由 SDF 定义 $p(y)=\mathbb E[my]$：

Let $y_3=ay_1+by_2\in\mathcal X$. By the SDF definition $p(y)=\mathbb E[my]$:

$$p(y_3)=\mathbb E[my_3]=\mathbb E[m(ay_1+by_2)]=a\,\mathbb E[my_1]+b\,\mathbb E[my_2]=a\,p(y_1)+b\,p(y_2).\quad\blacksquare$$

反方向更有意思。

定理 4.2。 组合构造与一价定律蕴含存在唯一的 SDF $x^*\in\mathcal X$——它本身就是支付空间里的一个随机支付。

构造：设 $x^*=\mathbf c^{*\prime}\mathbf x$。只需找到唯一的 $\mathbf c^*$ 使 $x^*$ 为基础向量定价。记基础向量价格 $\mathbf p(\mathbf x)=(p(x_1),\dots,p(x_N))'$，则

The converse is more interesting.

Theorem 4.2. Portfolio Formation and the Law of One Price imply the existence of a unique SDF $x^*\in\mathcal X$ — itself a random payoff inside the payoff space.

Construction: let $x^*=\mathbf c^{*\prime}\mathbf x$. It suffices to find the unique $\mathbf c^*$ such that $x^*$ prices the basis vector. With the basis price vector $\mathbf p(\mathbf x)=(p(x_1),\dots,p(x_N))'$,

$$p(y)=\mathbb E[x^*y]=\mathbb E[\mathbf c^{*\prime}\mathbf x\,y], \tag{4.1}$$

证明 / Proof：唯一 SDF $x^*=\mathbf x'\mathbb E[\mathbf{xx}']^{-1}\mathbf p(\mathbf x)$

要 $x^*$ 为基础向量定价，需 $\mathbf p(\mathbf x)=\mathbb E\big[(\mathbf c^{*\prime}\mathbf x)\,\mathbf x\big]=\mathbb E\big[\mathbf x(\mathbf x'\mathbf c^*)\big]=\mathbb E[\mathbf{xx}']\,\mathbf c^*$，于是

Requiring $x^*$ to price the basis vector: $\mathbf p(\mathbf x)=\mathbb E\big[(\mathbf c^{*\prime}\mathbf x)\,\mathbf x\big]=\mathbb E\big[\mathbf x(\mathbf x'\mathbf c^*)\big]=\mathbb E[\mathbf{xx}']\,\mathbf c^*$, hence

$$\mathbf c^*=\mathbb E[\mathbf{xx}']^{-1}\,\mathbf p(\mathbf x), \tag{4.2}$$

$$x^*=\mathbf c^{*\prime}\mathbf x=\mathbf x'\mathbf c^*=\mathbf x'\,\mathbb E[\mathbf{xx}']^{-1}\,\mathbf p(\mathbf x). \tag{4.3}$$

最后一步用了基础向量无冗余（$\mathbf x$ 无完全共线），等价于 $\mathbb E[\mathbf{xx}']$ 可逆。再验证它为任意 $y=\mathbf c_y'\mathbf x$ 定价：

The last step uses no redundancy in the basis (no perfect collinearity in $\mathbf x$), equivalent to invertibility of $\mathbb E[\mathbf{xx}']$. Verify it prices any $y=\mathbf c_y'\mathbf x$:

$$p(y)=\mathbb E[x^*y]=\mathbb E\big[y\,\mathbf x'\mathbb E[\mathbf{xx}']^{-1}\mathbf p(\mathbf x)\big]=\mathbf c_y'\,\mathbb E[\mathbf{xx}']\,\mathbb E[\mathbf{xx}']^{-1}\mathbf p(\mathbf x)=\mathbf c_y'\,\mathbf p(\mathbf x).\quad\blacksquare$$

Tip

$x^*$ 是"模仿组合"。 它是用现有资产线性组合出来、并能复制定价的那个 SDF。注意 $x^*$ 的存在仍依赖于给定的基础价格 $\mathbf p(\mathbf x)$：$\mathbf p(\mathbf x)$ 不是被 $x^*$ 定价，而是定义了 $x^*$（注 4.1）。给定 $\mathbf p(\mathbf x)$，$x^*$ 唯一（注 4.2）。$x^*$ is a "mimicking payoff". It is the SDF built as a linear combination of existing assets that reproduces pricing. Its existence is conditional on the given basis prices $\mathbf p(\mathbf x)$: $\mathbf p(\mathbf x)$ is not priced by $x^*$ but defines it (Remark 4.1). Given $\mathbf p(\mathbf x)$, $x^*$ is unique (Remark 4.2).

4.3.3 Projection of Stochastic Discount Factor

(4.3) 给出 $\mathcal X$ 之内的唯一 SDF $x^*$。但 $\mathcal X$ 之外可以有许多 SDF：任意随机变量 $m$ 只要对 $\forall y\in\mathcal X$ 满足

(4.3) gives the unique SDF $x^*$ inside $\mathcal X$. But outside $\mathcal X$ there can be many SDFs: any random variable $m$ is an SDF as long as for $\forall y\in\mathcal X$

$$p(y)=\mathbb E[my]. \tag{4.4}$$

记 $m=x^*+\varepsilon$。由于 $x^*$ 为 $\forall y\in\mathcal X$ 定价：

Write $m=x^*+\varepsilon$. Since $x^*$ prices $\forall y\in\mathcal X$:

$$p(y)=\mathbb E[x^*y]=\mathbb E[(m-\varepsilon)y]=\mathbb E[my]-\mathbb E[(m-x^*)y]. \tag{4.5}$$

由 (4.5)，(4.4) 成立 $\iff \mathbb E[(m-x^*)y]=0$ 对 $\forall y\in\mathcal X$，即 $x^*$ 是 $m$ 在 $\mathcal X$ 上的投影。

定义 4.1。 $x^*\in\mathcal X$ 是 $m$ 在空间 $\mathcal X$ 上的投影，若 $x^*$ 求解 $\displaystyle\min_{x\in\mathcal X}\mathbb E[(m-x)^2]$。

命题 4.1。 $x^*\in\mathcal X$ 是 $m$ 的投影 $\iff \mathbb E[(m-x^*)y]=0$ 对 $\forall y\in\mathcal X$。

By (4.5), (4.4) holds $\iff \mathbb E[(m-x^*)y]=0$ for $\forall y\in\mathcal X$, i.e. $x^*$ is the projection of $m$ onto $\mathcal X$.

Definition 4.1. $x^*\in\mathcal X$ is $m$'s projection onto $\mathcal X$ if $x^*$ solves $\displaystyle\min_{x\in\mathcal X}\mathbb E[(m-x)^2]$.

Proposition 4.1. $x^*\in\mathcal X$ is $m$'s projection $\iff \mathbb E[(m-x^*)y]=0$ for $\forall y\in\mathcal X$.

Tip

几点直觉（注 4.3–4.6）。 ① $m$ 可在 $\mathcal X$ 之外，且可有无穷多个，只要其投影是 $x^*$，它就是 SDF——不完全市场下 SDF 不唯一。② 差 $\varepsilon=m-x^*$ 必须与 $\mathcal X$ 正交；之所以存在这种正交随机性，正是因为不完全市场下 $\mathcal X$ 张不满状态空间。③ 给 SDF 添加与 $\mathcal X$ 正交的信息，不改变它对 $\mathcal X$ 内支付的定价能力。④ 完全市场下 (3.3) 给出唯一的 $\pi(s)$，没有正交于支付空间的随机性，故 SDF 唯一。Intuitions (Remarks 4.3–4.6). (1) $m$ may lie outside $\mathcal X$ and there can be infinitely many; as long as its projection is $x^*$ it is an SDF — the SDF is not unique under incomplete markets. (2) The gap $\varepsilon=m-x^*$ must be orthogonal to $\mathcal X$; such orthogonal randomness exists precisely because $\mathcal X$ does not span the state space. (3) Adding $\mathcal X$-orthogonal information to the SDF does not change its ability to price payoffs in $\mathcal X$. (4) A complete market gives the unique $\pi(s)$ from (3.3) with no orthogonal randomness, so the SDF is unique.

4.3.4 Absence of Arbitrage

(4.3) 的 $x^*$（或一般的 $m$）不保证为正：若基础支付服从正态分布，则当 $\mathbf x$ 取负值时 $x^*$ 也可能为负。我们关心 SDF 的正性，是因为它对应无套利条件。

定义 4.2（无套利 Absence of arbitrage）。 支付空间 $\mathcal X$ 与定价函数 $p(\cdot)$ 满足无套利，若：

对 $\forall y\in\mathcal X$ 且 $y\ge 0$ 几乎必然，有 $p(y)\ge 0$；
对 $\forall y\in\mathcal X$ 且 $y\ge 0$ 几乎必然、并以正概率 $y>0$，有 $p(y)>0$。

命题 4.2。 无套利蕴含一价定律（无套利更强）。

The $x^*$ in (4.3) (or a general $m$) is not guaranteed positive: with normally distributed basis payoffs, $x^*$ can be negative when $\mathbf x$ takes negative values. We care about positivity of the SDF because it corresponds to no arbitrage.

Definition 4.2 (Absence of arbitrage). Payoff space $\mathcal X$ and pricing function $p(\cdot)$ have absence of arbitrage if:

for $\forall y\in\mathcal X$ with $y\ge 0$ almost surely, $p(y)\ge 0$;
for $\forall y\in\mathcal X$ with $y\ge 0$ almost surely and $y>0$ with positive probability, $p(y)>0$.

Proposition 4.2. Absence of arbitrage implies the Law of One Price (no arbitrage is stronger).

证明 / Proof：无套利 $\Rightarrow$ 一价定律（反证）

反设满足无套利但不满足一价定律。则 $\exists y_1,y_2\in\mathcal X$、$\exists a,b\in\mathbb R$ 使 $y_3=ay_1+by_2\in\mathcal X$ 但 $p(y_3)\ne a\,p(y_1)+b\,p(y_2)$。不失一般性设

Suppose absence of arbitrage holds but the Law of One Price fails. Then $\exists y_1,y_2\in\mathcal X$, $\exists a,b\in\mathbb R$ with $y_3=ay_1+by_2\in\mathcal X$ but $p(y_3)\ne a\,p(y_1)+b\,p(y_2)$. WLOG

$$p(y_3)>a\,p(y_1)+b\,p(y_2). \tag{4.6}$$

构造 $y_4=ay_1+by_2-y_3\in\mathcal X$（买 $a$ 份 $y_1$、$b$ 份 $y_2$，卖 $1$ 份 $y_3$）。由 (4.6) 该头寸价格为负，但其支付 $y_4=0\ge 0$，无套利要求其价格非负——矛盾。$\blacksquare$

Construct $y_4=ay_1+by_2-y_3\in\mathcal X$ (buy $a$ of $y_1$, $b$ of $y_2$, sell $1$ of $y_3$). By (4.6) this position has a negative price, yet its payoff $y_4=0\ge 0$; absence of arbitrage requires a non-negative price — a contradiction. $\blacksquare$

反过来，一价定律不能推出无套利。

例 4.1（一价定律 $\nRightarrow$ 无套利）。 设 $\mathcal X$ 只有一个基础支付 $x=1$（确定性），定价 $p(x)=-1$。则一价定律可被满足（$\forall t\in\mathbb R$，$tx\in\mathcal X$ 且 $p(tx)=-t$），但这样的 $\mathcal X,p$ 永远无法满足无套利（$x=1\ge0$ 却 $p=-1<0$）。

Conversely, the Law of One Price does not imply absence of arbitrage.

Example 4.1 (LOOP $\nRightarrow$ no arbitrage). Let $\mathcal X$ have a single basis payoff $x=1$ (with certainty) and price $p(x)=-1$. The Law of One Price can hold ($\forall t\in\mathbb R$, $tx\in\mathcal X$ and $p(tx)=-t$), yet this $\mathcal X,p$ can never satisfy absence of arbitrage ($x=1\ge0$ but $p=-1<0$).

4.3.5 Fundamental Theorem of Finance

定理 4.3（金融基本定理）。 对支付空间 $\mathcal X$ 与定价函数 $p:\mathcal X\to\mathbb R$，以下两条等价：

（无套利） $\mathcal X$ 与 $p(\cdot)$ 满足无套利；
（正 SDF） 存在 $m>0$ 几乎必然，使 $\forall y\in\mathcal X$ 有 $p(y)=\mathbb E[my]$。

Theorem 4.3 (Fundamental Theorem of Finance). For payoff space $\mathcal X$ and pricing function $p:\mathcal X\to\mathbb R$, the following are equivalent:

(No arbitrage) $\mathcal X$ and $p(\cdot)$ satisfy absence of arbitrage;
(Positive SDF) there exists $m>0$ almost surely such that $\forall y\in\mathcal X$, $p(y)=\mathbb E[my]$.

证明 / Proof：正 SDF $\Rightarrow$ 无套利（易）

设正 SDF。对 $\forall y\in\mathcal X$、$y\ge0$ 几乎必然，因 $m>0$、$y\ge0$，故 $p(y)=\mathbb E[my]\ge0$。再设 $y\ge0$ 且以正概率 $y>0$，则

Assume a positive SDF. For $\forall y\in\mathcal X$ with $y\ge0$ a.s., since $m>0$ and $y\ge0$, $p(y)=\mathbb E[my]\ge0$. If moreover $y>0$ with positive probability,

$$p(y)=\mathbb E[my]=\mathbb E[my\mid y=0]\,\mathbf P\{y=0\}+\underbrace{\mathbb E[my\mid y>0]}_{>0}\,\underbrace{\mathbf P\{y>0\}}_{>0}>0,$$

满足无套利。$\blacksquare$

which satisfies absence of arbitrage. $\blacksquare$

反方向（无套利 $\Rightarrow$ 存在正 SDF）需要分离超平面定理。

定理 4.4（分离超平面定理）。 设 $\mathcal X\subset\mathbb R^N$ 为凸集，$\mathbf w\notin\mathcal X$。则 $\exists\mathbf y\in\mathbb R^N$ 使 $\forall\mathbf x\in\mathcal X$ 有 $\mathbf y'(\mathbf w-\mathbf x)\ge0$，且至少有一个 $\mathbf x^*\in\mathcal X$ 使 $\mathbf y'(\mathbf w-\mathbf x^*)>0$。

The converse (no arbitrage $\Rightarrow$ positive SDF) needs the separating hyperplane theorem.

Theorem 4.4 (Separating hyperplane theorem). Let $\mathcal X\subset\mathbb R^N$ be convex and $\mathbf w\notin\mathcal X$. Then $\exists\mathbf y\in\mathbb R^N$ such that $\forall\mathbf x\in\mathcal X$, $\mathbf y'(\mathbf w-\mathbf x)\ge0$, and at least one $\mathbf x^*\in\mathcal X$ has $\mathbf y'(\mathbf w-\mathbf x^*)>0$.

证明 / Proof：无套利 $\Rightarrow$ 存在正 SDF（分离超平面）

定义正 SDF 的凸集与其对应的价格集：

Define the convex set of positive SDFs and the corresponding set of prices:

$$\mathcal M=\big\{m>0\text{ a.s.},\ \mathbb E[|m|\,|x|]<\infty\ \forall x\in\mathcal X\big\},\qquad \mathcal P=\big\{\mathbb E[m\mathbf x]:\forall m\in\mathcal M\big\}\subset\mathbb R^N.$$

$\mathcal P$ 凸（因 $\mathcal M$ 凸）。记基础价格 $\mathbf p_0=p(\mathbf x)$，只需证 $\mathbf p_0\in\mathcal P$。反设 $\mathbf p_0\notin\mathcal P$，由分离超平面定理 4.4，$\exists\mathbf h\in\mathbb R^N$：(1) $\mathbf h'(\mathbf p-\mathbf p_0)\ge0\ \forall\mathbf p\in\mathcal P$；(2) $\mathbf h'(\mathbf p^*-\mathbf p_0)>0$ 对某 $\mathbf p^*$。令 $\hat x=\mathbf h'\mathbf x$、$\hat p=\mathbf h'\mathbf p_0$，则改写为 (1') $\mathbb E[m\hat x]\ge\hat p\ \forall m\in\mathcal M$；(2') $\mathbb E[m^*\hat x]>\hat p$。

(1') 对任意 $m$ 成立，特别对 $\varepsilon m_0$（$\forall\varepsilon>0$）成立，取 $\varepsilon\downarrow0$ 得

$\mathcal P$ is convex (since $\mathcal M$ is). Let the basis price be $\mathbf p_0=p(\mathbf x)$; it suffices to show $\mathbf p_0\in\mathcal P$. Suppose not. By the separating hyperplane theorem 4.4, $\exists\mathbf h\in\mathbb R^N$: (1) $\mathbf h'(\mathbf p-\mathbf p_0)\ge0\ \forall\mathbf p\in\mathcal P$; (2) $\mathbf h'(\mathbf p^*-\mathbf p_0)>0$ for some $\mathbf p^*$. Set $\hat x=\mathbf h'\mathbf x$, $\hat p=\mathbf h'\mathbf p_0$; these become (1') $\mathbb E[m\hat x]\ge\hat p\ \forall m\in\mathcal M$; (2') $\mathbb E[m^*\hat x]>\hat p$.

(1') holds for any $m$, in particular for $\varepsilon m_0$ ($\forall\varepsilon>0$); letting $\varepsilon\downarrow0$,

$$\lim_{\varepsilon\downarrow0}\mathbb E[\varepsilon m_0\hat x]\ge\hat p\;\Longleftrightarrow\;\hat p\le0. \tag{4.7}$$

再取另一序列 $m_\varepsilon=\big(\tfrac1\varepsilon\mathbf 1\{\hat x<0\}+1\big)m_0\in\mathcal M$，代入 (1') 并令 $\varepsilon\downarrow0$，逐状态分析（$\mathbf 1\{\hat x<0\}m_0\hat x\le0$）可得

Take another sequence $m_\varepsilon=\big(\tfrac1\varepsilon\mathbf 1\{\hat x<0\}+1\big)m_0\in\mathcal M$; substituting into (1') and letting $\varepsilon\downarrow0$, a state-by-state analysis ($\mathbf 1\{\hat x<0\}m_0\hat x\le0$) gives

$$\mathbf P\{\hat x\ge0\}=1. \tag{4.8}$$

故 $\hat x\ge0$ 几乎必然，由无套利 $\hat p\ge0$；结合 (4.7) 的 $\hat p\le0$ 得 $\hat p=0$。但 (2') 给出 $\mathbb E[m^*\hat x]>\hat p=0$，故

So $\hat x\ge0$ a.s., and by absence of arbitrage $\hat p\ge0$; together with $\hat p\le0$ from (4.7), $\hat p=0$. But (2') gives $\mathbb E[m^*\hat x]>\hat p=0$, so

$$\mathbf P\{\hat x>0\}>0. \tag{4.9}$$

(4.8)+(4.9) 经无套利得 $\hat p>0$，与 $\hat p=0$ 矛盾。故 $\mathbf p_0\in\mathcal P$：存在正 SDF $m$ 为基础向量定价，从而为 $\mathcal X$ 内任意支付定价。$\blacksquare$

(4.8)+(4.9) with absence of arbitrage give $\hat p>0$, contradicting $\hat p=0$. Hence $\mathbf p_0\in\mathcal P$: a positive SDF $m$ prices the basis vector, and therefore any payoff in $\mathcal X$. $\blacksquare$

注 4.7–4.8。 上述证明针对有限维基础向量，可推广到更一般设定；且只证明了正 SDF 的存在，未保证唯一。要由无套利得到唯一 SDF，需要完全市场。

定理 4.5。 若 $\mathcal X$ 张成完全市场（如 $\mathcal X=\mathcal L^2$，见 4.3.6），则无套利蕴含存在唯一且以概率 1 为正的 SDF $m$。

Remarks 4.7–4.8. The proof is for a finite-dimensional basis, extendable to more general settings; it shows only existence of a positive SDF, not uniqueness. Getting a unique SDF from no arbitrage requires a complete market.

Theorem 4.5. If $\mathcal X$ spans the complete market (e.g. $\mathcal X=\mathcal L^2$, see §4.3.6), then absence of arbitrage implies a unique SDF $m$ that is positive with probability 1.

证明 / Proof：完全市场下 SDF 唯一（反证）

存在性已由定理 4.3 给出，只需证唯一。反设有两个都几乎必然为正的不同 SDF $m_1,m_2$，则存在状态 $\omega$ 使 $m_1(\omega)\ne m_2(\omega)$，不妨 $m_1(\omega)>m_2(\omega)$。完全市场下非线性支付 $\mathbf 1\{m_1>m_2\}$ 也在 $\mathcal X$ 内，$m_1,m_2$ 都为它定价：

Existence is from Theorem 4.3; only uniqueness remains. Suppose two distinct, a.s.-positive SDFs $m_1,m_2$. Then some state $\omega$ has $m_1(\omega)\ne m_2(\omega)$, WLOG $m_1(\omega)>m_2(\omega)$. Under a complete market the non-linear payoff $\mathbf 1\{m_1>m_2\}$ is also in $\mathcal X$, and both $m_1,m_2$ price it:

$$\mathbb E\big[(m_1-m_2)\,\mathbf 1\{m_1>m_2\}\big]=0\;\Longrightarrow\;\underbrace{(m_1-m_2)}_{>0}\,\underbrace{\mathbf P\{m_1>m_2\}}_{>0}=0,$$

矛盾。故 SDF 唯一。$\blacksquare$

a contradiction. Hence the SDF is unique. $\blacksquare$

4.3.6 General Setting

去掉有限维限制的一般设定。定义 $\mathcal L^2=\{x:\mathbb E[|x|^2]<\infty\}$，并赋予

内积：$\forall x_1,x_2\in\mathcal L^2$，$\langle x_1\mid x_2\rangle=\mathbb E[x_1 x_2]$；
范数：$\forall x\in\mathcal L^2$，$|x|=\langle x\mid x\rangle^{1/2}$；
收敛：$\{x_j\},x_0\in\mathcal L^2$，$\{x_j\}\to x_0\iff |x_j-x_0|\to0$ 几乎必然。

设支付空间 $\mathcal X\subset\mathcal L^2$ 为完备线性空间（对极限封闭——此"完备"与"完全市场"无关）；定价 $p:\mathcal X\to\mathbb R$ 为线性且连续。则基本结论仍成立：存在唯一的 $x^*\in\mathcal X$ 使 $\forall x\in\mathcal X$，

The general setting without finite-dimensionality. Define $\mathcal L^2=\{x:\mathbb E[|x|^2]<\infty\}$ with

inner product: $\forall x_1,x_2\in\mathcal L^2$, $\langle x_1\mid x_2\rangle=\mathbb E[x_1 x_2]$;
norm: $\forall x\in\mathcal L^2$, $|x|=\langle x\mid x\rangle^{1/2}$;
convergence: for $\{x_j\},x_0\in\mathcal L^2$, $\{x_j\}\to x_0\iff |x_j-x_0|\to0$ a.s.

Let the payoff space $\mathcal X\subset\mathcal L^2$ be a complete linear space (closed under limits — this "complete" is unrelated to "complete market"); let $p:\mathcal X\to\mathbb R$ be linear and continuous. Then the basic conclusion still holds: there exists a unique $x^*\in\mathcal X$ such that $\forall x\in\mathcal X$,

$$p(x)=\mathbb E[x^*x].$$

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