# 有限市场中的状态价格、等价鞅测度与完备性
在同一冻结二/三状态市场中求复制、定价权重和开放无套利区间。
Entry: zh-qt18 | Node: QT18 | Language: zh | Editorial revision: 2026-09-21
## Teaching instructions
你是 QT18《有限市场中的状态价格、等价鞅测度与完备性》的数学助教。面向有微积分、线性代数与基本概率背景的高年级本科或研究生。
先实际取得并读完 required_readings 中与当前任务对应的 Williams 官方 PDF 完整单元，记录版本、页/节、条件与读取失败；文件入口或摘要不算全文。同一官方 URL 已于 2026-09-21 通过正常 TLS 校验的普通 HTTP 下载取得完整 PDF，但这不替代本次课堂运行的实际读取；若本次无法取得原件，须如实说明缺口。阅读成功后的课堂运行才填写 runtime_reading_log，本导出中保持空。
本次诊断与推演：先让读者用支付矩阵解beta与Delta，再区分P/Q/pi/m；切换P后先预测哪列变化；对三状态两个端点逐行核现金、资产支付与零初值。
先让读者独立作答，再逐步反馈，最后换一个条件做迁移。反馈标准：通过须能重算100/17复制成本、区分现金账户单位与金额，解释开放区间来自具体套利；不得把Q称作已估现实概率或另造三状态P。
严格区分已证结论、引用定理、教学模型与算术验证；有限枚举不是一般证明。只使用 supplied_inputs 中当前单元的冻结市场切片，输入时点与单位不改写；不得另造第二套 EXP-STATE-01，也不得把 QT08 的抛币实验混入市场证明。不采用原件已说明的排印错误。界面只展示当前视图，静态默认表与题解同样可完成任务。
结束时问：读者只读完这个词条，真的能学明白吗？请用其独立完成的证明或计算回答，给具体缺口，不以复述结论代替理解。
Before substantive teaching, actually retrieve every required reading unit for the selected scope. Read its complete designated section, including necessary assumptions, tables and footnotes. A working URL or an editorial access date is not a runtime reading receipt. Record the actual version, location, scope and what it supports. If unavailable, use a previously verified equivalent source; if the required unit remains unavailable, identify that gap rather than teach it from memory. Start runtime_reading_log empty. Once reading is complete, use a substantive diagnostic or follow the reader's request for direct explanation. Advance one complete reasoning task at a time; skip mastered basics. Distinguish original facts, supplied teaching assumptions and inference.
## Required readings and runtime protocol
```json
{
"export_mode": "public",
"required_readings": [
{
"source_id": "QTF-WILLIAMS3",
"access": {
"kind": "pdf_full_text",
"uri": "https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf",
"verified_access_at": "2026-09-21"
},
"required_unit": {
"locator": "§3.1 printed pp.40–43/PDF2–3",
"scope": "完整读取该采用单元，含必要条件和所用证明。",
"purpose": "一期作为有限市场的特例，单位/持仓/无摩擦"
},
"supports": "一期作为有限市场的特例，单位/持仓/无摩擦",
"title": "Finite Market Model, Chapter 3",
"authors": [
"Ruth J. Williams"
],
"version": "Chapter PDF; no reliable revision date stated"
},
{
"source_id": "QTF-WILLIAMS3",
"access": {
"kind": "pdf_full_text",
"uri": "https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf",
"verified_access_at": "2026-09-21"
},
"required_unit": {
"locator": "§3.2 Definitions3.2.1–3.2.2 and Lemma3.2.3 printed pp.44–45/PDF4",
"scope": "完整读取该采用单元，含必要条件和所用证明。",
"purpose": "等价概率、贴现价格与财富"
},
"supports": "等价概率、贴现价格与财富",
"title": "Finite Market Model, Chapter 3",
"authors": [
"Ruth J. Williams"
],
"version": "Chapter PDF; no reliable revision date stated"
},
{
"source_id": "QTF-WILLIAMS3",
"access": {
"kind": "pdf_full_text",
"uri": "https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf",
"verified_access_at": "2026-09-21"
},
"required_unit": {
"locator": "§3.3 through Theorem3.3.2 printed pp.50–54/PDF7–9",
"scope": "完整读取该采用单元，含必要条件和所用证明。",
"purpose": "可复制价值和完备性含义；一般证明在P1/P2"
},
"supports": "可复制价值和完备性含义；一般证明在P1/P2",
"title": "Finite Market Model, Chapter 3",
"authors": [
"Ruth J. Williams"
],
"version": "Chapter PDF; no reliable revision date stated"
}
],
"optional_readings": [],
"runtime_reading_log": [],
"supplied_inputs": {
"market_identity": {
"experiment_id": "EXP-STATE-01",
"version": "2026-09-21-v1",
"identity": "Finite frictionless teaching market, independently reconstructed with exact rational arithmetic; no estimated probability or live security quote",
"units": {
"monetary_unit": "USD, teaching denomination per modeled asset unit",
"time": "one abstract period; not annualized",
"cash_account_holding": "number of cash-account units; cash value at t equals beta*B_t",
"probabilities": "dimensionless; state-price vector is not a probability",
"gross_cash_return_symbol": "R=B_1/B_0=51/50; reserve real scalars for mathbb R"
},
"assumptions": {
"time_grid": [
0,
1
],
"F_0": "{empty, Omega}",
"F_T": "all subsets of the displayed terminal atoms",
"physical_probability": "strictly positive on every terminal atom",
"cash_account": "B_0=1, deterministic B_1=51/50 > 0; identical borrowing and lending accumulation",
"holdings": "unrestricted finite signed real quantities; fractional positions and short sales allowed",
"timing": "holdings for (0,1] chosen from F_0; terminal state cannot be used to choose them",
"financing": "V_0=beta*B_0+sum_j h_j*S_0^j over every traded risky asset; in the augmented market this includes the added claim at price c. No external cash flows; its terminal payoff is included once in the portfolio value.",
"frictions": "zero transaction costs, no bid/ask spread, no dividends, no margin/liquidity/short-sale constraint"
},
"formula_contract": {
"state_prices": "A^T*pi=s_0; pi_i>0; sum_i pi_i=B_0/B_1=1/R",
"martingale_probability": "Q_i=R*pi_i, E_Q[S_1]=R*S_0",
"density": "z_i=Q_i/P_i; E_P[z]=1",
"pricing_kernel": "m_i=pi_i/P_i=z_i/R; E_P[m]=1/R",
"replication": "A*theta=H; price=s_0^T*theta=sum_i pi_i*H_i",
"multiperiod_cash_completion": "beta_1=(V_0-h_1 dot S_0)/B_0; beta_(k+1)=(beta_k*B_k+h_k dot S_k-h_(k+1) dot S_k)/B_k",
"discounted_wealth": "V_k/B_k=V_0/B_0+sum_(j=1)^k h_j dot (S_j/B_j-S_(j-1)/B_(j-1))",
"attainable_space": "K=span{1}+L={a*1+ell:a in real numbers,ell in L}; R is not the scalar field"
}
},
"two_state": {
"state_order": [
"up",
"down"
],
"B_0": {
"exact": "1",
"decimal": 1.0
},
"B_1": {
"exact": "51/50",
"decimal": 1.02
},
"S_0": {
"exact": "100",
"decimal": 100.0
},
"S_1": [
{
"exact": "120",
"decimal": 120.0
},
{
"exact": "90",
"decimal": 90.0
}
],
"strike": {
"exact": "105",
"decimal": 105.0
},
"claim": [
{
"exact": "15",
"decimal": 15.0
},
{
"exact": "0",
"decimal": 0.0
}
],
"P_illustrative_not_estimated": [
{
"exact": "3/5",
"decimal": 0.6
},
{
"exact": "2/5",
"decimal": 0.4
}
],
"payoff_matrix_state_rows_asset_columns_B_S": [
[
{
"exact": "51/50",
"decimal": 1.02
},
{
"exact": "120",
"decimal": 120.0
}
],
[
{
"exact": "51/50",
"decimal": 1.02
},
{
"exact": "90",
"decimal": 90.0
}
]
],
"initial_price_vector_B_S": [
{
"exact": "1",
"decimal": 1.0
},
{
"exact": "100",
"decimal": 100.0
}
],
"Q": [
{
"exact": "2/5",
"decimal": 0.4
},
{
"exact": "3/5",
"decimal": 0.6
}
],
"state_prices": [
{
"exact": "20/51",
"decimal": 0.39215686274509803
},
{
"exact": "10/17",
"decimal": 0.5882352941176471
}
],
"dQ_dP": [
{
"exact": "2/3",
"decimal": 0.6666666666666666
},
{
"exact": "3/2",
"decimal": 1.5
}
],
"pricing_kernel_m": [
{
"exact": "100/153",
"decimal": 0.6535947712418301
},
{
"exact": "25/17",
"decimal": 1.4705882352941178
}
],
"rank": 2,
"P_expected_stock": {
"exact": "108",
"decimal": 108.0
},
"Q_expected_stock": {
"exact": "102",
"decimal": 102.0
},
"P_expected_claim": {
"exact": "9",
"decimal": 9.0
},
"Q_expected_claim": {
"exact": "6",
"decimal": 6.0
},
"P_discounted_claim_expectation_not_replication_price": {
"exact": "150/17",
"decimal": 8.823529411764707
},
"replication": {
"cash_account_units": {
"exact": "-750/17",
"decimal": -44.11764705882353
},
"stock_units": {
"exact": "1/2",
"decimal": 0.5
},
"initial_cash_value": {
"exact": "-750/17",
"decimal": -44.11764705882353
},
"initial_stock_value": {
"exact": "50",
"decimal": 50.0
},
"initial_cost": {
"exact": "100/17",
"decimal": 5.882352941176471
},
"terminal_cash_value": {
"exact": "-45",
"decimal": -45.0
},
"terminal_stock_values": [
{
"exact": "60",
"decimal": 60.0
},
{
"exact": "45",
"decimal": 45.0
}
],
"terminal_values": [
{
"exact": "15",
"decimal": 15.0
},
{
"exact": "0",
"decimal": 0.0
}
]
},
"P_change_exercise": {
"P": [
{
"exact": "9/10",
"decimal": 0.9
},
{
"exact": "1/10",
"decimal": 0.1
}
],
"Q_unchanged": [
{
"exact": "2/5",
"decimal": 0.4
},
{
"exact": "3/5",
"decimal": 0.6
}
],
"price_unchanged": {
"exact": "100/17",
"decimal": 5.882352941176471
},
"new_dQ_dP": [
{
"exact": "4/9",
"decimal": 0.4444444444444444
},
{
"exact": "6",
"decimal": 6.0
}
],
"new_m": [
{
"exact": "200/459",
"decimal": 0.4357298474945534
},
{
"exact": "100/17",
"decimal": 5.882352941176471
}
]
},
"stock_simple_returns": [
{
"exact": "1/5",
"decimal": 0.2
},
{
"exact": "-1/10",
"decimal": -0.1
}
],
"cash_simple_return": {
"exact": "1/50",
"decimal": 0.02
}
},
"three_state_incomplete": {
"state_order": [
"down",
"middle",
"up"
],
"B_0": {
"exact": "1",
"decimal": 1.0
},
"B_1": {
"exact": "51/50",
"decimal": 1.02
},
"S_0": {
"exact": "100",
"decimal": 100.0
},
"S_1": [
{
"exact": "80",
"decimal": 80.0
},
{
"exact": "100",
"decimal": 100.0
},
{
"exact": "120",
"decimal": 120.0
}
],
"call_strike": {
"exact": "100",
"decimal": 100.0
},
"call_payoff": [
{
"exact": "0",
"decimal": 0.0
},
{
"exact": "0",
"decimal": 0.0
},
{
"exact": "20",
"decimal": 20.0
}
],
"P_illustrative_not_estimated": null,
"P_requirement": "Any fixed probability positive on all three terminal atoms; no default physical probabilities assigned",
"initial_price_vector_B_S": [
{
"exact": "1",
"decimal": 1.0
},
{
"exact": "100",
"decimal": 100.0
}
],
"payoff_matrix_state_rows_asset_columns_B_S": [
[
{
"exact": "51/50",
"decimal": 1.02
},
{
"exact": "80",
"decimal": 80.0
}
],
[
{
"exact": "51/50",
"decimal": 1.02
},
{
"exact": "100",
"decimal": 100.0
}
],
[
{
"exact": "51/50",
"decimal": 1.02
},
{
"exact": "120",
"decimal": 120.0
}
]
],
"rank": 2,
"Q_family": {
"parameter": "t",
"expression": [
"t",
"9/10 - 2*t",
"1/10 + t"
],
"open_parameter_interval": {
"lower": {
"exact": "0",
"decimal": 0.0
},
"upper": {
"exact": "9/20",
"decimal": 0.45
},
"endpoints_included": false
},
"default_t": {
"exact": "1/5",
"decimal": 0.2
},
"null_direction": [
{
"exact": "1",
"decimal": 1.0
},
{
"exact": "-2",
"decimal": -2.0
},
{
"exact": "1",
"decimal": 1.0
}
],
"samples": [
{
"t": {
"exact": "1/10",
"decimal": 0.1
},
"Q": [
{
"exact": "1/10",
"decimal": 0.1
},
{
"exact": "7/10",
"decimal": 0.7
},
{
"exact": "1/5",
"decimal": 0.2
}
],
"state_prices": [
{
"exact": "5/51",
"decimal": 0.09803921568627451
},
{
"exact": "35/51",
"decimal": 0.6862745098039216
},
{
"exact": "10/51",
"decimal": 0.19607843137254902
}
],
"claim_price": {
"exact": "200/51",
"decimal": 3.9215686274509802
}
},
{
"t": {
"exact": "1/5",
"decimal": 0.2
},
"Q": [
{
"exact": "1/5",
"decimal": 0.2
},
{
"exact": "1/2",
"decimal": 0.5
},
{
"exact": "3/10",
"decimal": 0.3
}
],
"state_prices": [
{
"exact": "10/51",
"decimal": 0.19607843137254902
},
{
"exact": "25/51",
"decimal": 0.49019607843137253
},
{
"exact": "5/17",
"decimal": 0.29411764705882354
}
],
"claim_price": {
"exact": "100/17",
"decimal": 5.882352941176471
}
},
{
"t": {
"exact": "2/5",
"decimal": 0.4
},
"Q": [
{
"exact": "2/5",
"decimal": 0.4
},
{
"exact": "1/10",
"decimal": 0.1
},
{
"exact": "1/2",
"decimal": 0.5
}
],
"state_prices": [
{
"exact": "20/51",
"decimal": 0.39215686274509803
},
{
"exact": "5/51",
"decimal": 0.09803921568627451
},
{
"exact": "25/51",
"decimal": 0.49019607843137253
}
],
"claim_price": {
"exact": "500/51",
"decimal": 9.803921568627452
}
}
]
},
"claim_price_function": "c(t) = (2 + 20*t)/(51/50)",
"claim_arbitrage_free_price_interval": {
"lower": {
"exact": "100/51",
"decimal": 1.9607843137254901
},
"upper": {
"exact": "550/51",
"decimal": 10.784313725490197
},
"endpoints_included": false,
"scope": "Claim is added as a freely signed, fractionally tradable asset at t=0, paying H at t=1; prices of cash and stock fixed"
},
"endpoint_arbitrages": [
{
"endpoint": "lower",
"claim_price": {
"exact": "100/51",
"decimal": 1.9607843137254901
},
"holdings_B_S_C": [
{
"exact": "5000/51",
"decimal": 98.03921568627452
},
{
"exact": "-1",
"decimal": -1.0
},
{
"exact": "1",
"decimal": 1.0
}
],
"initial_cost": {
"exact": "0",
"decimal": 0.0
},
"terminal_payoffs": [
{
"exact": "20",
"decimal": 20.0
},
{
"exact": "0",
"decimal": 0.0
},
{
"exact": "0",
"decimal": 0.0
}
]
},
{
"endpoint": "upper",
"claim_price": {
"exact": "550/51",
"decimal": 10.784313725490197
},
"holdings_B_S_C": [
{
"exact": "-2000/51",
"decimal": -39.21568627450981
},
{
"exact": "1/2",
"decimal": 0.5
},
{
"exact": "-1",
"decimal": -1.0
}
],
"initial_cost": {
"exact": "0",
"decimal": 0.0
},
"terminal_payoffs": [
{
"exact": "0",
"decimal": 0.0
},
{
"exact": "10",
"decimal": 10.0
},
{
"exact": "0",
"decimal": 0.0
}
]
}
]
},
"augmented_summary": {
"traded_call_price": {
"exact": "6",
"decimal": 6.0
},
"Q": [
{
"exact": "103/500",
"decimal": 0.206
},
{
"exact": "61/125",
"decimal": 0.488
},
{
"exact": "153/500",
"decimal": 0.306
}
],
"state_prices": [
{
"exact": "103/510",
"decimal": 0.2019607843137255
},
{
"exact": "122/255",
"decimal": 0.47843137254901963
},
{
"exact": "3/10",
"decimal": 0.3
}
],
"rank": 3,
"determinant": {
"exact": "408",
"decimal": 408.0
}
},
"view": {
"node": "QT18",
"underlying_experiment_id": "EXP-STATE-01",
"default_mode": "two"
},
"attachments": [
{
"title": "本篇完整静态阅读",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/static/QT18.html",
"kind": "html"
},
{
"title": "EXP-STATE-01 唯一冻结市场",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/qt-f-shared-state-experiment.json",
"kind": "json",
"version": "2026-09-21-v1",
"json_pointers": [
"/units",
"/assumptions",
"/two_state",
"/three_state_incomplete",
"/three_state_augmented_complete"
],
"policy": "沿同一冻结文件读取当前单元指定部分，不另造树或三状态物理概率。"
},
{
"title": "同包默认精确计算",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"kind": "json",
"json_pointers": [
"/two",
"/three",
"/complete"
]
}
]
}
}
```
## Supplied entry
给一项未来支付“取个平均”，还没有回答今天该用多少资产把它做出来。这里我们从复制开始：只要一组持仓在每个状态下都交出同样的支付，它的初始成本就提供了一个可核的价格约束。状态价格和等价鞅测度是这组约束的两种表示，不是两套额外的市场预测。
这一单元走完两个一期市场：先在二状态中复制，再增加一个终端状态，看到哪些支付失去唯一价格。一般有限期定理的完整证明分别在 QT18-P1 和 QT18-P2；本篇的矩阵、区间和套利都在这里算完。
## 1. 先固定市场、持仓与单位
取有限状态空间 $\Omega$，$\mathcal F_0=\{\varnothing,\Omega\}$，$\mathcal F_1=2^\Omega$，研究概率 $P$ 对每个状态严格为正。现金账户每单位的价格为确定的 $B_0,B_1>0$；本例 $B_0=1$、$B_1=\frac{51}{50}$，所以现金一期增长因子为 $R=B_1/B_0=\frac{51}{50}$。这是一段抽象期间，不年化。
允许在时点 0 选择任意有限、有符号的实数持仓，借贷都使用同一现金账户；没有价差、费用、分红或额外持仓限制。持仓必须在结果揭晓前确定。以上是有限市场定理的模型条件，而不是现实市场的默认事实。[^w-model]
令 $\beta$ 为现金账户单位数、$\Delta$ 为股票单位数。初始成本与终端价值是
$$
V_0=\beta B_0+\Delta S_0,
\qquad V_1=\beta B_1+\Delta S_1.
$$
$\beta$ 与现金金额 $\beta B_t$ 必须分开。负 $\beta$ 表示借款；一项支付 $H$ 也不是扣除初始成本后的净利润。
本篇市场为 EXP-STATE-01 的教学构造，计价单位为美元。它没有特定上市合约或真实成交报价；每个支付已经按这里的一份资产单位定义，不能再任意乘一个期权合约乘数。
## 2. 二状态：先复制，再谈概率
状态顺序为上涨、下跌。给定股票初价 $S_0=100$，终值与现金账户如下；看涨式支付的行权价为 $105$。
| 时点或状态 | 现金账户每单位 | 股票每单位 | 看涨式支付 |
|---|---|---|---|
| 初始 | $1$ | $100$ | 待由复制确定价格 |
| 上涨 | $\frac{51}{50}$ | $120$ | $15$ |
| 下跌 | $\frac{51}{50}$ | $90$ | $0$ |
令 $\theta=(\beta,\Delta)^T$。把状态放在行、资产放在列：
$$
A=\begin{pmatrix}\frac{51}{50}&120\\\frac{51}{50}&90\end{pmatrix},
\qquad A\theta=\begin{pmatrix}15\\0\end{pmatrix}.
$$
两行相减消去现金，得到 $\Delta=\frac{1}{2}$；再代回下跌状态，得终端现金金额 $\beta B_1=-45$，因而 $\beta=-\frac{750}{17}$。初始成本为 $V_0=\frac{100}{17}\approx5.88235294$。[^w-replicate]
| 项目 | 时点0 | 上涨终值 | 下跌终值 |
|---|---|---|---|
| 现金金额 | $-\frac{750}{17}$ | $-45$ | $-45$ |
| 股票金额 | $50$ | $60$ | $45$ |
| 合计 | $\frac{100}{17}$ | $15$ | $0$ |
这套持仓在两种状态都支付 $H$。若同一支付可以买卖，报价 $c$ 高于复制成本，就卖出支付、买入复制组合，并把差额存入现金账户。总初始成本为零，到期两项支付相消，留下确定正的现金增长额；报价更低时反向操作。因此在本模型里，可复制支付的无套利价格不能偏离复制成本。
这不是因为我们已知道上涨概率，而是因为逐状态结果完全相同。
## 3. 概率、状态价格与定价核
称满足
$$
A^T\pi=s_0,\qquad \pi_i>0
$$
的向量 $\pi$ 为一组与当前已交易资产价格相容的正状态价格。可以把 $\pi_i$ 读作这套线性定价规则给“只在状态 $i$ 支付一美元”的数字支付赋予的初始价值；但若市场不完备，这些数字支付未必已在市场中交易，也未必可复制，因此 $\pi$ 可能不唯一，不能把某一分量冒充已观察到的唯一市场报价。这里 $s_0$ 是已交易资产的初始价格向量。
若这样的 $\pi$ 存在，对任意零成本持仓 $\theta$ 都有 $\pi^TA\theta=s_0^T\theta=0$；而一个逐状态非负、至少一处严格正的终值，其严格正加权和不可能为零。这已经证明了该一期市场没有套利。反方向由有限期第一基本定理给出。[^w-ftap]
现金账户那一列满足 $B_1\sum_i\pi_i=B_0$，所以 $\sum_i\pi_i=1/R$，不是 1。将它归一化，定义 $Q_i=R\pi_i$，得到
$$
\sum_i Q_i=1,
\qquad E_Q[S_1]=RS_0.
$$
这等价于 $E_Q[S_1/B_1]=S_0/B_0$；贴现价格在 $Q$ 下为鞅。因为 $Q_i>0$，它与全支持的 $P$ 有相同的零概率事件，故称等价鞅测度。[^w-ftap]
二状态中联立概率总和与股票定价约束，算得 $Q=\left(\frac{2}{5},\frac{3}{5}\right)$，$\pi=\left(\frac{20}{51},\frac{10}{17}\right)$。再取教学概率 $P=\left(\frac{3}{5},\frac{2}{5}\right)$，就能把这些对象放在同一张表里：
| 对象 | 上涨 | 下跌 | 归一化检查 |
|---|---|---|---|
| 研究概率 $P$ | $\frac{3}{5}$ | $\frac{2}{5}$ | 分量和为1 |
| 定价概率 $Q$ | $\frac{2}{5}$ | $\frac{3}{5}$ | 分量和为1 |
| 状态价格 $\pi$ | $\frac{20}{51}$ | $\frac{10}{17}$ | 分量和为 $\frac{50}{51}$ |
| 密度 $z=Q/P$ | $\frac{2}{3}$ | $\frac{3}{2}$ | $E_Pz=1$ |
| 定价核 $m=\pi/P$ | $\frac{100}{153}$ | $\frac{25}{17}$ | $E_Pm=1/R$ |
其中密度 $z_i=Q_i/P_i$ 满足 $E_Pz=1$；定价核 $m_i=\pi_i/P_i=z_i/R$ 满足 $E_Pm=1/R$。对可复制支付，三个价格写法完全一致：
$$
\begin{aligned}
\operatorname{price}(H)
&=\sum_i\pi_iH_i\\
&=R^{-1}E_Q[H]\;=\;E_P[mH].
\end{aligned}
$$
本例 $E_PH=9$，$E_QH=6$。直接算 $R^{-1}E_PH=\frac{150}{17}$，并不会得到复制成本 $\frac{100}{17}$。研究概率在回答支付平均有多大；定价核则按资产价格约束改变了状态权重。
把界面中的 $P$ 切到 $\left(\frac{9}{10},\frac{1}{10}\right)$，先预测哪些列会变，再计算：$Q$、$\pi$ 和复制成本不变，密度与定价核改变。这个观察依赖支付矩阵与市场价格固定，不是说现实概率的变化永远不影响真实市场价格。
## 4. 三状态：缺少一列资产，留下一个自由度
接下来保留现金账户和股票初价，将终端股票改成三个状态，顺序为下跌、中间、上涨。此分支的看涨式支付行权价是 $100$，不同于二状态分支；只要求现实概率全支持，不另指定数值。
| 状态 | 现金账户终值 | 股票终值 | 新增支付终值 |
|---|---|---|---|
| 下跌 | $\frac{51}{50}$ | $80$ | $0$ |
| 中间 | $\frac{51}{50}$ | $100$ | $0$ |
| 上涨 | $\frac{51}{50}$ | $120$ | $20$ |
现金和股票只有两列，其支付矩阵秩为 2。可复制支付一定形如 $H_i=\beta B_1+\Delta S_{1i}$，即三个点落在同一直线上。给定 $H=\left(0,0,20\right)$ 并非如此：前两个状态支付都为零迫使 $\Delta=0,\beta=0$，便不可能在第三个状态支付 $20$。
从 $q_d+q_m+q_u=1$ 与 $E_QS_1=RS_0$ 出发，令 $t=q_d$，解得
$$
Q(t)=\bigl(t,\ \frac{9}{10}-2t,\ \frac{1}{10}+t\bigr).
$$
三个分量都必须严格正，所以 $0
## 5. 以一个价格补全市场，不等于选中了“真实概率”
让新增支付的成交初价固定为 $c=6$。增广矩阵为
$$
A_+=\begin{pmatrix}
\frac{51}{50}&80&0\\
\frac{51}{50}&100&0\\
\frac{51}{50}&120&20
\end{pmatrix}.
$$
它的行列式为 $408$，秩为 3。因此每个三状态支付都能解出持仓。新资产价格同时把自由参数固定，得到
$$
Q=\left(\frac{103}{500},\frac{61}{125},\frac{153}{500}\right),\qquad \pi=\left(\frac{103}{510},\frac{122}{255},\frac{3}{10}\right).
$$
这叫完备：任意终端可测支付都可复制。它不要求未来只有一个可能结果，也不把 $Q$ 解释成已估计出的现实概率。一般有限期下“无套利当且仅当存在 EMM”见 [第一基本定理证明](https://ou-liu-red-sugar.github.io/zh/notebook/qt-finite-market-ftap-proof/#qt18p1-ftap)；在已经无套利的市场中，“完备当且仅当 EMM 唯一”见 [第二基本定理证明](https://ou-liu-red-sugar.github.io/zh/notebook/qt-completeness-unique-measure-proof/#qt18p2-theorem)。后者还会实际构造三个数字支付的持仓。[^w-complete]
## 6. 两项迁移核算
问题一。只把二状态研究概率改为 $P=\left(\frac{9}{10},\frac{1}{10}\right)$，支付和初价全部不动。重新计算 $dQ/dP$、定价核以及支付价格。为什么 $E_PH$ 会变，但复制价格不变？
解析。市场方程没有变化，所以 $Q=\left(\frac{2}{5},\frac{3}{5}\right)$、$\pi=\left(\frac{20}{51},\frac{10}{17}\right)$ 不变。逐分量相除得到
$$
\frac{dQ}{dP}=\left(\frac{4}{9},6\right),\qquad m=\left(\frac{200}{459},\frac{100}{17}\right).
$$
此时 $E_PH=\frac{27}{2}$，但 $E_P[mH]=\frac{100}{17}$。改变的是研究加权与相应的核，不是已经固定的资产复制关系。若把某个 $P_i$ 改成零，本篇的全支持条件失效，也不能继续直接做这些除法。
问题二。只交易三状态市场中的现金和股票，能否复制中间状态的一美元数字支付 $(0,1,0)$？
解析。前两个方程给出 $\Delta=1/(100-80)=1/20$，以及 $\beta B_1=-80/20$。于是上涨状态的支付为 $(120-80)/20=2$，不是要求的零。不能只拟合两行就宣布复制完成。加入第三列资产后，这个支付才可以逐行精确复制；我们将在 P2 中核对它的完整资金和初始成本。
[^w-model]: Ruth J. Williams, [Finite Market Model, Chapter 3](https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf#page=2)，§3.1，印刷 pp.40–43 / PDF pp.2–3；章节 PDF 未标可靠修订日期。数值市场是 EXP-STATE-01，2026-09-21-v1 的独立教学构造，不是讲义市场报价。
[^w-replicate]: 同章 §3.3，Theorem 3.3.1，印刷 p.51 / PDF p.7，复制价值的条件期望表示。本文一期持仓由显示的矩阵直接求解。
[^w-ftap]: 同章 Definitions 3.2.1–3.2.2、Lemma 3.2.3、Theorem 3.2.4，印刷 pp.44–50 / PDF pp.4–7；正状态价格的一期充分性由本文内积直接证明，一般必要性在 QT18-P1。
[^w-complete]: 同章 §3.3，Theorems 3.3.2–3.3.3，印刷 pp.52–55 / PDF pp.8–9；本文三状态开放区间及端点套利由显示的资产和持仓直接重建。
## Experiment inputs and static equivalents
```json
[
{
"id": "EXP-STATE-01",
"title": "有限市场中的状态价格、等价鞅测度与完备性：计算视图",
"anchor": "qt18-measures",
"description": "改变本视图参数后重算；正文保留默认表、完整推导和题解，公开附件提供精确输入与结果。",
"inputs": {
"shared_experiment_id": "EXP-STATE-01",
"view": {
"node": "QT18",
"underlying_experiment_id": "EXP-STATE-01",
"default_mode": "two"
},
"public_attachments": [
{
"title": "EXP-STATE-01 唯一冻结市场",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/qt-f-shared-state-experiment.json",
"kind": "json",
"version": "2026-09-21-v1",
"json_pointers": [
"/units",
"/assumptions",
"/two_state",
"/three_state_incomplete",
"/three_state_augmented_complete"
],
"policy": "沿同一冻结文件读取当前单元指定部分，不另造树或三状态物理概率。"
},
{
"title": "同包默认精确计算",
"uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"kind": "json",
"json_pointers": [
"/two",
"/three",
"/complete"
]
}
]
},
"outputs": {
"file": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/results.json",
"scope": "two/three/complete",
"json_pointers": [
"/two",
"/three",
"/complete"
]
},
"is_view_of_existing_experiment": true,
"static_equivalent": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/static/QT18.html"
}
]
```
## Sources
- [Finite Market Model, Chapter 3](https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf): 有限状态模型、现金补足、指标持仓鞅判据、两条基本定理及有限维分离完整证明。数值输入属于另行冻结的教学市场，不归称讲义报价。
## Content relations
```json
[
{
"from": "zh-qt18",
"relation": "part_of",
"to": "quant-processes",
"reason": "主要 topic 归属"
},
{
"from": "qt18-measures",
"relation": "illustrated_by",
"to": "EXP-STATE-01",
"reason": "同一冻结输入的当前视图；证明与算术分别呈现。"
},
{
"from": "zh-qt18",
"relation": "requires",
"to": "zh-qt07",
"reason": "本证明实际调用的局部能力。",
"required_competence": "分清决定持仓与揭晓终端结果的时点。"
},
{
"from": "qt18-complete",
"relation": "derived_from",
"to": "qt18p1-ftap",
"reason": "一般有限期定理的完整证明，主篇一期例不冒充一般证明。"
},
{
"from": "qt18-complete",
"relation": "derived_from",
"to": "qt18p2-theorem",
"reason": "一般有限期定理的完整证明，主篇一期例不冒充一般证明。"
},
{
"from": "qt18-model",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "Ruth J. Williams, [Finite Market Model, Chapter 3](https://mathweb.ucsd.edu/~williams/courses/m294notes/chap3.pdf#page=2)，§3.1，印刷 pp.40–43 / PDF pp.2–3；章节 PDF 未标可靠修订日期。数值市场是 EXP-STATE-01，2026-09-21-v1 的独立教学构造，不是讲义市场报价。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-model"
]
},
{
"from": "qt18-replication",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 §3.3，Theorem 3.3.1，印刷 p.51 / PDF p.7，复制价值的条件期望表示。本文一期持仓由显示的矩阵直接求解。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-replicate"
]
},
{
"from": "qt18-measures",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 Definitions 3.2.1–3.2.2、Lemma 3.2.3、Theorem 3.2.4，印刷 pp.44–50 / PDF pp.4–7；正状态价格的一期充分性由本文内积直接证明，一般必要性在 QT18-P1。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-ftap"
]
},
{
"from": "qt18-incomplete",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 §3.3，Theorems 3.3.2–3.3.3，印刷 pp.52–55 / PDF pp.8–9；本文三状态开放区间及端点套利由显示的资产和持仓直接重建。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-complete"
]
},
{
"from": "qt18-complete",
"relation": "supported_by",
"to": "QTF-WILLIAMS3",
"reason": "本段已著明脚注的定义、条件或证明单元；例题数字由具名教学输入提供。",
"locator": "同章 §3.3，Theorems 3.3.2–3.3.3，印刷 pp.52–55 / PDF pp.8–9；本文三状态开放区间及端点套利由显示的资产和持仓直接重建。",
"scope": "沿本段原脚注的采用范围和勘误说明，不扩充为原件全部结论。",
"citation_labels": [
"w-complete"
]
}
]
```
## Related entries