# 期权价格、内在价值与隐含波动率 解释价格因素与欧式put的迟收货款反例；在明确BSM假设下将报价区间反解为IV区间。 Entry: zh-m18 | Node: M18 | Language: zh | Editorial revision: 2026-09-21 ## Teaching instructions 你正在教M18《期权价格、内在价值与隐含波动率》，内容版本2026-09-21-MEFG-review-v3。读者已有高年级本科至研究生的数学基础。 先确认选用本篇共同正文和哪些分支，再实际读取agent_packet列出的当前必读完整原文。PDF需读脚注、表图与符号；只拿到摘要/目录/搜索片段不算完成。记录实际版本、范围与内容对应，不沿用编辑端“已读”充当本次读取；同会话已完整取得相同版本单元可以复用。若所需单元失败，尝试机构正式等价全文；仍缺失则指出具体单元，不凭记忆补成已读教学。runtime_reading_log从空开始。 独立学习任务：在完全给定的树中分别倒推欧式和美式put；给定模型与报价侧后反解IV，界外拒绝。 专属诊断与正向讲解：先问欧式持有人今天能否收到100；写S50,K100,u1.2,d.9,R1.02，无股息和抽象期间，逐节点比较继续/行权。之后才看SPX尺寸的另一组模型报价。 反馈尺度：不能把两个不同市场写回EXP-STATE-01；解释迟收K和决策集合，而非仅报反例数字。IV不是真实sigma或P概率。 读者要求直接讲解时，按本篇连贯推导讲清，不反复问已会先修。静态例、实验、练习必须使用本包同一输入和单位；实例价、合成价、模型价、规则时点不能混换。练习要给完整解析，不仅打分。仅当选读分支被采用时，将其optional reading转入当前必读。 本篇数值与IV实验使用无股息BSM；股息只作现金/行权机制说明，不把未读的非零股息公式扩展归给Kogan来源。 说明BSM时必须列出常参数GBM动态 $dS_t=\mu S_tdt+\sigma S_tdW_t$、q=0和复制后物理漂移不进入定价公式。 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": "MA-OCC", "version": "June 2024", "access": { "kind": "pdf_full_text", "uri": "https://www.theocc.com/getmedia/a151a9ae-d784-4a15-bdeb-23a029f50b70/riskstoc.pdf", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "Premium pp11–12；Intrinsic Value and Time Value pp14–15完整小节", "scope": "完整指定单元、必要脚注与表图；不以搜索摘要代替", "purpose": "价格与内在/剩余价值的条件" }, "supports": "规则、权利金、行权/指派和同财富假设例；本课指定单元不替代实际交易者对整份ODD的阅读。无真实行情或个人券商操作保证。", "id": "M18-READ-01", "title": "Characteristics and Risks of Standardized Options", "authors": [ "The Options Clearing Corporation" ], "retrieved_at": "2026-09-21" }, { "source_id": "PFH-MIT-BSM", "version": "Fall 2010", "access": { "kind": "pdf_full_text", "uri": "https://ocw.mit.edu/courses/15-450-analytics-of-finance-fall-2010/0d1260b891a96241316d883d4f5bfaec_MIT15_450F10_lec02.pdf", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "slides16–21完整BSM市场/复制/公式单元", "scope": "完整指定单元、必要脚注与表图；不以搜索摘要代替", "purpose": "模型假设和价格函数" }, "supports": "常参数、无股息模型条件下价格/偏导与Q；不宣称完整连续时间定理证明。", "id": "M18-READ-02", "title": "15.450: Stochastic Calculus and Option Pricing", "authors": [ "Leonid Kogan", "MIT OpenCourseWare" ], "retrieved_at": "2026-09-21" }, { "source_id": "PFH-MIT-OPTIONS", "version": "Fall 2008", "access": { "kind": "pdf_full_text", "uri": "https://ocw.mit.edu/courses/15-401-finance-theory-i-fall-2008/c40ecc0cc0dce0fbf2d229bc4027c43b_MIT15_401F08_lec10.pdf", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "slides16–21完整复制/树单元", "scope": "完整指定单元、必要脚注与表图；不以搜索摘要代替", "purpose": "独立教学树的倒推关系" }, "supports": "模型内的支付和复制定价；不代替当前产品规则，讲义现金增长记号不能误写成年化净利率。", "id": "M18-READ-03", "title": "15.401 Finance Theory I, Lecture 10–11: Options", "authors": [ "Andrew W. Lo", "MIT OpenCourseWare" ], "retrieved_at": "2026-09-21" }, { "source_id": "MEFG-OIC-VEGA", "version": "Undated; retrieved 2026-09-21", "access": { "kind": "html_full_text", "uri": "https://www.optionseducation.org/advancedconcepts/vega", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "完整概念正文", "scope": "完整指定单元、必要脚注与表图；不以搜索摘要代替", "purpose": "模型价格随vol变化的局部含义" }, "supports": "波动率参数的偏导和每百分点单位；IV不是独立的P下预测。", "id": "M18-READ-04", "title": "Vega", "authors": [ "Options Industry Council" ], "retrieved_at": "2026-09-21" }, { "source_id": "PFH-EXERCISE", "version": "Undated; retrieved 2026-09-21", "access": { "kind": "html_full_text", "uri": "https://www.optionseducation.org/referencelibrary/faq/options-exercise", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "FAQ中提前行权、剩余价值和closing相关完整问答", "scope": "完整指定单元、必要脚注与表图；不以搜索摘要代替", "purpose": "欧式与美式决策集合" }, "supports": "行权vs平仓、ex-by-ex与客户指令、提前行权和分红；不提供任何个人券商统一截止。", "id": "M18-READ-05", "title": "Options Exercise FAQ", "authors": [ "Options Industry Council" ], "retrieved_at": "2026-09-21" } ], "optional_readings": [], "runtime_reading_log": [], "supplied_inputs": { "content_version": "2026-09-21-MEFG-draft-v1", "data": { "schema": "markets-mefg-adopted-inputs-v1", "content_version": "2026-09-21-MEFG-draft-v1", "scope": [ "M15", "M16", "M17", "M18", "M19", "M20", "M21", "M22", "M23" ], "state_tree": { "experiment_id": "EXP-STATE-01", "source_file": "data/qt-f-shared-state-experiment.json", "source_version": "2026-09-21-v1", "source_sha256": "8489fd3b74f469c3424d32a2b0738280e4e83709f081dfb24282d9afcfa8804b", "source_pointer": "two_state / three_state_incomplete / three_state_augmented_complete", "identity": "Exact received QT-F/M21 frozen contract; file preserved byte-for-byte; generated browser data is a derived copy", "physical_up_probability_variants": [ 0.2, 0.6, 0.8, 0.9 ], "period_unit": "one abstract period; R is gross growth, not annual net rate" }, "odd_equal_wealth": { "identity": "OCC ODD June2024 printed p61 hypothetical; exact interest versus rounded official display distinguished", "initial_wealth_usd": 5000, "spot_usd": 50, "strike_usd": 50, "premium_per_share_usd": 5, "shares_per_contract": 100, "stock_position_shares": 100, "mixed_contracts": 1, "all_option_contracts": 10, "mixed_cash_usd": 4500, "simple_annual_cash_rate": 0.0325, "horizon_years": 0.5, "terminal_spots_usd": [ 62, 58, 54, 50, 46, 42, 38 ], "cost_convention": "No fees, tax or stock dividends in base; terminal economic payoff is not a funding guarantee for physical exercise" }, "cash_examples": { "identity": "Synthetic amounts under documented product rules; not historical quotes/series/accounts", "equity_call": { "multiplier": 100, "contracts": 1, "strike": 100, "purchase_premium": 4, "later_underlying_bid": 105, "later_option_bid": 6, "later_option_ask": 6.2, "exercise_gross_cash_assumption": 10000, "exercise_model": "sequential gross payment first; no assumed netting/credit; broker policy not inferred" }, "spxw_call": { "multiplier_usd_per_point": 100, "strike_points": 5000, "purchase_premium_points": 50, "settlement_value_points": 5025, "shares_delivered": 0 }, "uncovered_equity_call_margin": { "contract_type": "OCC ordinary equity uncovered call minimum; not ETF/index or broker house rules", "multiplier": 100, "strike": 105, "initial_spot": 100, "initial_option_premium": 4, "later_spot": 120, "later_option_mark": 17, "base_fraction": 0.2, "minimum_call_fraction": 0.1, "other_positions_or_broker_addons": false, "default_external_cash_available": 2200, "model": "cash collateral; liability is mark not daily futures VM; no external deposits beyond the explicitly displayed steps" } }, "early_exercise_counterexample": { "identity": "Separate teaching market, not EXP-STATE-01", "spot": 50, "strike": 100, "up_factor": 1.2, "down_factor": 0.9, "gross_cash_growth_per_period": 1.02, "periods": [ 1, 2 ], "dividend": 0, "time_unit": "one abstract period; not a year; American exercise at every displayed node" }, "parity_quotes": { "identity": "Separate synthetic quote/funding scenario; no friction toggle on EXP-STATE-01", "strike": 105, "gross_cash_growth": 1.02, "stock": { "bid": 99.95, "ask": 100.05 }, "call": { "bid": 5.8, "ask": 6.2 }, "put": { "bid": 8.7, "ask": 9.1 }, "extra_fee_per_underlying_unit": 0, "cost_convention": "same borrowing/lending, zero dividend European same K/T; all legs synchronous and fully fillable only by explicit teaching assumption", "execution_assumptions": { "borrowing": true, "stock_short": true, "synchronous": true, "depth": true, "exercise_terms_match": true }, "default_capital_available": 1 }, "surface": { "experiment_id": "EXP-MEFG-IV-01", "identity": "Entirely synthetic SPX-sized European call/put quotes; actual multiplier/tick only; no historical chain", "quote_date": null, "spot_index_points": 5000, "annual_continuous_rate": 0.04, "annual_continuous_dividend_yield": 0, "dividend_note": "zero is teaching assumption, not S&P500 dividend forecast", "day_count": "ACT/365 teaching whole-day expiries", "strikes_points": [ 4500, 5000, 5500 ], "maturities": [ { "days": 30, "sigma_by_strike": [ 0.28, 0.24, 0.22 ] }, { "days": 90, "sigma_by_strike": [ 0.27, 0.235, 0.22 ] }, { "days": 180, "sigma_by_strike": [ 0.25, 0.23, 0.22 ] } ], "quote_half_width_points": 0.2, "contract_multiplier_usd_per_point": 100, "min_quote_tick_below3": 0.05, "min_quote_tick_at_or_above3": 0.1, "greek_example": { "strike": 5000, "days": 90, "sigma": 0.235, "spot_changes_points": [ 50, 500 ], "vol_changes_absolute": [ 0.01, 0.1 ] }, "validation_scope": "displayed grid only; not a global no-arbitrage surface or observed chain" }, "vix": { "official_sample": { "identity": "Methodology v6.0 Appendix3 explicitly hypothetical (2022-09-27 10:45:15 ET label is not observed VIX)", "minutes_near": 34484, "minutes_next": 44954, "target_minutes": 43200, "year_minutes": 525600, "near_variance": 0.019233906, "next_variance": 0.019423884, "displayed_T_near": 0.0656088, "displayed_T_next": 0.0855289, "displayed_vix": 13.93, "one_strike": { "strike": 1370, "delta_K": 5, "mid_quote": 0.2, "R_rounded": 0.000317, "T_rounded": 0.0656088, "displayed_contribution": 5.328e-07 } }, "two_term_teaching_variant": { "identity": "separate variance interpolation, not a VIX-strip reproduction", "days_near": 24, "days_next": 36, "target_days": 30, "near_vol": 0.2, "next_vol": 0.24 } }, "information_example": { "identity": "Synthetic contract volumes classified by initiating side; size-weighted KIT Eq(2), not unweighted transaction counts; not motives", "unit": "option contracts", "buyer_call_contracts": 30, "seller_call_contracts": 10, "buyer_put_contracts": 25, "seller_put_contracts": 35, "non_event_mean_TOI": 0.05, "baseline_window": "non-announcement days tau-40 through tau-10 per paper; 0.05 here a teaching value", "table_2_column_3": { "identity": "published regression coefficients with macro/non-macro separately standardized variables; not raw-return slopes/profits", "OI_coefficient": 0.0033, "OI_macro_interaction": 0.025, "column_1_OI_coefficient": 0.0078, "sample_observations": 51522 } }, "runtime_reading_log": [] }, "additional_files": [], "experiment_id": "EXP-MEFG-M18-IV" }, "entry_id": "zh-m18", "node_id": "M18", "content_version": "2026-09-21-MEFG-review-v3", "experiment_ids": [ "EXP-MEFG-M18-IV" ], "source_id_aliases": { "MEFG-ODD": "MA-OCC", "MEFG-OCC-EQUITY": "PFH-EQUITY", "MEFG-OCC-ETF": "PFH-ETF", "MEFG-OIC-T1": "PFH-T1", "MEFG-SPX": "MA-SPX", "MEFG-MIT-OPTIONS": "PFH-MIT-OPTIONS", "MEFG-MIT-KOGAN": "PFH-MIT-BSM", "MEFG-OIC-PARITY": "PFH-PARITY", "MEFG-OIC-EXERCISE": "PFH-EXERCISE", "MEFG-CHICAGOFED-2025": "PFH-CHIFED2025" }, "source_paths": { "shared_inputs.json": "https://ou-liu-red-sugar.github.io/notebook/labs/m-efg/inputs.json", "data/qt-f-shared-state-experiment.json": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-f/data/qt-f-shared-state-experiment.json", "static/M18.html": "https://ou-liu-red-sugar.github.io/notebook/labs/m-efg/static/M18.html" } } ``` ## Supplied entry ## 一、当前价格买到的不是“今天就行权” 一张期权的到期支付由合约给出，当前交易价却买到一段尚未结束的选择。看涨与看跌、行权价、剩余期限、行权方式和分配条款都属于这项选择的一部分。权利金因市场交易而形成，模型则回答：在一组明确假设下，哪些因素通过什么关系约束这个价格。[^MEFG-ODD] 先定义一个常用比较量。给定当前标的价 $S$，call的现时立即行权差额为 $(S-K)^+$，put为 $(K-S)^+$。它常被称作spot intrinsic value。但只有合约此刻允许行权，持有人才有权立即取得相应交换差额。欧式期权此刻没有这个动作，不能把“看起来已经价内”直接当成可以立即兑现的现金。 因此，权利金减去这个现时差额，可用来讨论价格中超出立即交换价值的部分；不能先把差值命名为“时间价值”，再从名称推出它永远非负。我们先把一个完整反例走完，再看价格模型。[^MEFG-ODD] ## 二、一个越晚到期反而越便宜的欧式put 下面是独立的教学树，不是SPX行情，也不是后面复制一篇的冻结市场。初始 $S_0=50$，$K=100$；每个抽象期间，股票乘以 $u=1.2$ 或 $d=0.9$，现金增长因子 $R=1.02$，无股息，允许按相同条件借贷。这里一期不是默认一年。 一期股票为60或45；两期为72、54或40.5，均低于100。定价权重由股票／现金关系得到 $$ q_{\rm tree}=\frac{R-d}{u-d}=\frac{1.02-0.9}{1.2-0.9}=0.4. $$ 因为每个终端状态的put支付都等于 $K-S_T$，只需持有到期付 $K$ 的现金、同时空一股就能复制。所以 $n$ 期欧式价格为 $$ P_E(n)=\frac{K}{R^n}-S_0. $$ | 到期期间数 | 欧式put价格 | 当前 $K-S_0$ | 价格减当前立即行权差额 | |---:|---:|---:|---:| | 1 | $2450/51\approx48.039216$ | 50 | −1.960784 | | 2 | $119950/2601\approx46.116878$ | 50 | −3.883122 | 这不是凭空“违反常识”。欧式持有人要等到到期才能取得 $K$；在此树中所有到期状态仍深度价内，选择权没有额外打开“价格高于 $K$ 就放弃卖出”的终端分支，推迟收到100的资金影响直接显现。复制关系与现金时间价值给出了价格下降，而不是一个“所有put都随期限便宜”的定理。[^MEFG-MIT-OPTIONS] 现在让同一树上的put变为每个节点都能行权的美式版本。两期末支付依次为28、46、59.5。回到第一期节点，继续持有值必须和立即行权值比较： | 美式树节点 | 股票价 | 继续持有价值 | 立即行权价值 | 节点选择 | |---|---:|---:|---:|---:| | 第一期上涨 | 60 | $(0.4\times28+0.6\times46)/1.02=38.039216$ | 40 | 40 | | 第一期下跌 | 45 | $(0.4\times46+0.6\times59.5)/1.02=53.039216$ | 55 | 55 | | 初始节点 | 50 | $(0.4\times40+0.6\times55)/1.02=48.039216$ | 50 | 50 | 一期美式初值同样为50。美式之所以不低于立即行权差额，是因为立即行权属于它的可行决策集，而非因为所有期权都天然附带一个非负“时间价值”。这也为提前行权与分红、融资之间的比较提供了正确起点。[^MEFG-OIC-EXERCISE] ## 三、价格因素各自改变哪一件事 接下来固定一种合约与模型，一次只动一个变量。提高 $S$ 让按固定 $K$ 买入更有利，call价值上升，put方向相反；提高 $K$ 则让call不利、put有利。剩余时间会同时改变可等待多久和未来货款的折现，刚才的欧式put反例说明不能只看前者。 利率和股息也不是公式里的装饰。持有股票可能收分红，持有call并不会自动获得同一现金；支付行权货款的时间不同，融资成本也不同。提前行权是否值得考虑时，这两类现金差异也会进入比较。[^MEFG-OIC-EXERCISE] 本批真正拿来算数的 Kogan 单元采用**无股息**的常参数欧式模型。在这个已经读到公式的模型里，较高利率降低到期 $K$ 的现值，对call和put的作用相反。若以后要把连续股息率作为非零模型输入，应另采用明确包含该扩展的公式来源，而不是把一般现金直觉静默塞进当前公式。[^MEFG-MIT-KOGAN] 波动率的作用要多说一句。看涨与看跌的支付是凸函数。在合适的“保持均值而增加分散”的比较下，更多尾部变化可以增加这种选择的价值；仅说“某分布方差更大”，并不能在任意两个分布间无条件得出所有凸支付价格更高。BSM模型给出了一个具体可算的比较：其余参数固定，vanilla期权对波动率的vega为正。它说明这个模型中的变化方向，不是所有市场与产品的万能因果命题。[^MEFG-OIC-VEGA] ## 四、把一个报价放进一个有条件的模型 本批使用一个完全合成的SPX尺度网格：$S=5,000$，连续复利年利率4%，**股息设为0**，期限用天数／365。零股息是教学假设，不是对S&P 500实际分红的估计。真实规格只提供每点100美元乘数与报价增量；没有取得对应某日期的历史期权链。[^MEFG-SPX] 为与实际采用的 Kogan 单元和共享输入保持完全一致，这里只写本批使用的**无股息 BSM 欧式公式**。模型进一步假定标的在连续时间服从常参数几何布朗运动 $$ dS_t=\mu S_t\,dt+\sigma S_t\,dW_t, $$ 并假定连续可交易、无摩擦、可按同一常数利率融资和对冲。复制定价消去了物理漂移 $\mu$；在风险中性表示下，标的漂移相应由 $r$ 进入价格公式。本批股息率固定为0，因此不在下面的数值公式里额外放连续股息项。[^MEFG-MIT-KOGAN] 在这些条件下： $$ \begin{aligned} d_1&=\frac{\log(S/K)+(r+\sigma^2/2)\tau}{\sigma\sqrt{\tau}},\\ d_2&=d_1-\sigma\sqrt{\tau},\\ C&=S N(d_1)-Ke^{-r\tau}N(d_2),\\ P&=Ke^{-r\tau}N(-d_2)-S N(-d_1). \end{aligned} $$ $N$ 是标准正态分布函数。这里把公式作为明确模型的计算入口；连续时间复制的完整证明不在本篇里偷压成几句话。[^MEFG-MIT-KOGAN] 对90天、$K=5,000$、$\sigma=23.5\%$，模型call价格为256.855895点。为形成教学bid／ask，先按相应增量把中价取成256.9，再给两边各0.2点价差： | 报价身份 | 期权点数 | 一份对应美元 | 同模型反解的IV | |---|---:|---:|---:| | bid | 256.7 | 25,670 | 23.4841% | | mid | 256.9 | 25,690 | 23.5045% | | ask | 257.1 | 25,710 | 23.5249% | 模型输入23.5%与中价反解23.5045%的细微差异来自报价取整，不是推断出另一条“真实波动率”。这一点正好提醒我们：IV是由一个价格和其他约定反解出的参数，不是额外独立观察。 ## 五、IV是价格的一种坐标，不是未来的保证 给定合约及本批无股息模型的 $S,K,\tau,r$，隐含波动率解决的是方程 $C_{\rm model}(\sigma)=C_{\rm quote}$。可以从较小和较大的 $\sigma$ 起步，反复二分，直到模型价格接近给定报价。买入通常面对ask，卖出面对bid；相应IV区间比只报一个中价IV更贴近交易对象。 反解前还要检查价格界。在本例无股息模型中，call的正有限波动率价格位于 $\max(S-Ke^{-r\tau},0)$ 与 $S$ 之间。若30天、$K=4,500$ 的call被输入1点，远低于模型下界，算法应拒绝给出IV，而不是挤出一个接近零的数字来掩盖不相容。
实验提供价格随 $\sigma$ 变化的曲线和提前行权树两个视角。选择网格中的bid、mid、ask可以看反解位置；切到put反例时，节点比较使用上面的完整 $u,d,R$，不会把它改成SPX的连续时间模型。 市场报价还会受库存、供求、成交条件和资金限制影响。把报价反解为IV有助于跨行权价、跨期限比较，但不等于证明它就是未来真实波动率或真实上涨概率；后面会把这种报价表达与研究中的风险补偿分开。 ## 六、从反例回到自己的判断 **题一。** 为什么一期欧式put的48.039216低于当前差额50，不构成这里的套利？ **解析。** 持有人现在没有立即以100卖出股票的权利，不能买put后今天就取得50。合约只承诺期末支付，复制需要今天存入 $100/1.02$ 并空一股，成本48.039216。将美式的立即行权操作擅自加给欧式，才制造了虚假的套利。 **题二。** 两期美式put为什么不是46.116878？请在上涨节点核一次。 **解析。** 上涨后股票60，继续持有两期末支付的折现价值38.039216，而立即行权值40；美式可取较大者40。下跌节点取55，初始继续值48.039216仍低于立即行权50，故初值50。46.116878只属于必须等待两期末的欧式版本。 **题三。** 为什么表中ask IV高于bid IV？它们是否是对未来的两个独立预测？ **解析。** 在其他BSM输入相同且vega为正时，更高期权价格需要更大的 $\sigma$ 来重现。因此两边反解出区间；它们首先表达同一交易价差，不是从两个独立样本估出的未来波动预测。 **题四。** 30天、$K=4,500$ 的call报价1点，能否输出“市场几乎确定不会波动”？ **解析。** 不能。先检查模型价格下界；该输入已与无股息、给定现价／利率的价格界不相容。可能是单位、合约、时点或输入错误，也可能模型条件不适用。在识别原因前，不应强行输出IV。 [^MEFG-ODD]: The Options Clearing Corporation，*Characteristics and Risks of Standardized Options*，June 2024。[原文](https://www.theocc.com/getmedia/a151a9ae-d784-4a15-bdeb-23a029f50b70/riskstoc.pdf)。本篇定位：Ch I printed pp3–5；Ch II vanilla terms pp6–9; Premium/Opening and Closing pp11–12; Intrinsic and Time Value pp14–15；Ch VIII pp55–57; Ch IX pp58–59；Ch X holder pp61–63; writer pp63–67（至所采用第11项）。 [^MEFG-MIT-OPTIONS]: Andrew W. Lo, MIT OpenCourseWare，*15.401 Finance Theory I, Lecture 10–11: Options*，Fall 2008。[原文](https://ocw.mit.edu/courses/15-401-finance-theory-i-fall-2008/c40ecc0cc0dce0fbf2d229bc4027c43b_MIT15_401F08_lec10.pdf)。本篇定位：Slides 3–9: option definitions and payoff；Slides 16–21: binomial replication and risk-neutral valuation。 [^MEFG-OIC-EXERCISE]: Options Industry Council，*Options Exercise FAQ*，Undated; retrieved 2026-09-21。[原文](https://www.optionseducation.org/referencelibrary/faq/options-exercise)。本篇定位：Full FAQ main body through exercise/closing distinction。 [^MEFG-MIT-KOGAN]: Leonid Kogan, MIT OpenCourseWare，*15.450: Stochastic Calculus and Option Pricing*，Fall 2010。[原文](https://ocw.mit.edu/courses/15-450-analytics-of-finance-fall-2010/0d1260b891a96241316d883d4f5bfaec_MIT15_450F10_lec02.pdf)。本篇定位：Slides 16–21: BSM market, replication/PDE/formula；Slides 52–58 and 60: Q pricing and European application。 [^MEFG-OIC-VEGA]: Options Industry Council，*Vega*，Undated; retrieved 2026-09-21。[原文](https://www.optionseducation.org/advancedconcepts/vega)。本篇定位：Full main concept body。 [^MEFG-SPX]: Cboe Global Markets，*S&P 500 Index Options (SPX) Fact Sheet*，©2026; WF-451400-KC; no precise issue date shown。[原文](https://cdn.cboe.com/resources/spx/spx-fact-sheet.pdf)。本篇定位：Full physical pp1–2; p2 specifications and relevant footnotes。 ## Additional teaching material ### 本篇默认结果与静态等价
90天/K5,000 call报价点反解IV
bid256.7023.484100%
mid256.9023.504499%
ask257.1023.524897%

BSM:S=5,000,r=4%,q_div=0,ACT/365；标的采用常参数GBM dS=μSdt+σSdW。复制定价消去物理漂移μ；价格界以同一无股息模型验证，q_div=0不是SPX实际股息。

put反例欧式美式现在行权差额
148.03921568650.0050
246.11687812450.0050
两期树节点（t,上次数）S立即行权美式继续值美式价值
0,050.0050.0048.03921650.000000
1,160.0040.0038.03921640.000000
1,045.0055.0053.03921655.000000
2,272.0028.00—28.000000
2,154.0046.00—46.000000
2,040.5059.50—59.500000

独立教学树S0=50,K=100,u=1.2,d=0.9,R=1.02，每期抽象期间，无股息；欧式今天不能行权。

完整冻结输入：https://ou-liu-red-sugar.github.io/notebook/labs/m-efg/inputs.json。来源内的包路径按 source_paths 取得。 ## Experiment inputs and static equivalents ```json [ { "id": "EXP-MEFG-M18-IV", "title": "报价反解IV，与欧式put反例", "anchor": "m18-inversion", "description": "先检查报价是否在模型价格界内，再看同一树在不同决策集合中的价值。", "data_identity": "EXP-MEFG-IV-01合成报价；提前行权树为独立教学市场，绝不改写EXP-STATE-01。", "shared_dataset_ids": [ "EXP-MEFG-IV-01" ], "inputs": { "canonical_common": "shared_inputs.json", "state_contract": null, "controls": [ { "name": "days", "label": "到期天数", "type": "select", "default": "90", "domain_or_choices": [ [ "30", "30天" ], [ "90", "90天" ], [ "180", "180天" ] ] }, { "name": "strike", "label": "执行价", "type": "select", "default": "5000", "domain_or_choices": [ [ "4500", "4,500" ], [ "5000", "5,000" ], [ "5500", "5,500" ] ] }, { "name": "kind", "label": "类型", "type": "select", "default": "call", "domain_or_choices": [ [ "call", "Call" ], [ "put", "Put" ] ] }, { "name": "side", "label": "报价侧", "type": "select", "default": "mid", "domain_or_choices": [ [ "bid", "Bid" ], [ "mid", "Mid" ], [ "ask", "Ask" ], [ "custom", "另设教学报价" ] ] }, { "name": "quote", "label": "自定义报价（点）", "type": "number", "default": 256.9, "domain_or_choices": { "step": 0.1 } }, { "name": "periods", "label": "put反例抽象期数", "type": "select", "default": "2", "domain_or_choices": [ [ "1", "一期" ], [ "2", "两期" ] ] } ] }, "algorithm": "BSM no-dividend/general continuous-yield price; strict positive finite IV if model lower0且价格严格界内才反解有限正IV", "q_div=0非真实SPX估计；σ是模型参数", "无全局无套利surface保证" ], "views": { "first": "数值/单位/现金表", "second": "定价、现金、状态或事件图；见同一实验实现" }, "implementation": { "html": "/notebook/labs/m-efg/interactions.html?experiment=EXP-MEFG-M18-IV", "engine": "/notebook/labs/m-efg/engine.js", "static_available_without_js": true }, "source_id_aliases": { "MEFG-ODD": "MA-OCC", "MEFG-OCC-EQUITY": "PFH-EQUITY", "MEFG-OCC-ETF": "PFH-ETF", "MEFG-OIC-T1": "PFH-T1", "MEFG-SPX": "MA-SPX", "MEFG-MIT-OPTIONS": "PFH-MIT-OPTIONS", "MEFG-MIT-KOGAN": "PFH-MIT-BSM", "MEFG-OIC-PARITY": "PFH-PARITY", "MEFG-OIC-EXERCISE": "PFH-EXERCISE", "MEFG-CHICAGOFED-2025": "PFH-CHIFED2025" }, "input_uri": "https://ou-liu-red-sugar.github.io/notebook/labs/m-efg/inputs.json", "static_equivalent": "https://ou-liu-red-sugar.github.io/notebook/labs/m-efg/static/M18.html" } ] ``` ## Sources - [Characteristics and Risks of Standardized Options](https://www.theocc.com/getmedia/a151a9ae-d784-4a15-bdeb-23a029f50b70/riskstoc.pdf): 本会话实际读取的指定单元，不是96页全文；支持权利、单位、权利金、费用和保证金的区分。 契约、买卖双方权利义务、权利金/担保品、行权/指派/结算与多腿独立性。不给本链报价、客户截止时点或自动处理保证。 M-E/F/G 本批采用：规则、权利金、行权/指派和同财富假设例；本课指定单元不替代实际交易者对整份ODD的阅读。无真实行情或个人券商操作保证。 - [SPX Options Fact Sheet](https://cdn.cboe.com/resources/spx/spx-fact-sheet.pdf): 现金结算、欧式行权、100美元乘数及SPX/SPXW结算差别；不采用营销/税务部分。 SPXW欧式、PM、每点100美元、现金于到期后营业日交付；传统SPX AM/SOQ与SPXW分开。无实际报价。 M-E/F/G 本批采用：SPX/SPXW欧式现金、100美元乘数、最小报价增量及AM/PM结算差别；并非某条历史上市期权链。 - [Vega](https://www.optionseducation.org/advancedconcepts/vega): 波动率参数的偏导和每百分点单位；IV不是独立的P下预测。 - [Options Exercise FAQ](https://www.optionseducation.org/referencelibrary/faq/options-exercise): 行权、平仓、成员exercise-by-exception与客户指令分开；股息和时间价值影响选择，但不保证每个客户腿自动执行。 M-E/F/G 本批采用：行权vs平仓、ex-by-ex与客户指令、提前行权和分红；不提供任何个人券商统一截止。 - [15.450 Stochastic Calculus and Option Pricing](https://ocw.mit.edu/courses/15-450-analytics-of-finance-fall-2010/0d1260b891a96241316d883d4f5bfaec_MIT15_450F10_lec02.pdf): 无分红、常利率/常波动率模型、复制、PDE与BSM公式；本例采用r=0，非一般American模型或实际概率。 M-E/F/G 本批采用：常参数、无股息模型条件下价格/偏导与Q；本批数值只采用q=0版本，不把非零股息扩展归给该读取单元；不宣称完整连续时间定理证明。 - [15.401 Finance Theory I: Lecture 10–11, Options](https://ocw.mit.edu/courses/15-401-finance-theory-i-fall-2008/c40ecc0cc0dce0fbf2d229bc4027c43b_MIT15_401F08_lec10.pdf): 逐腿相加的保护、价差和跨式支付；不承担现行结算规则。 M-E/F/G 本批采用：模型内的支付和复制定价；不代替当前产品规则，讲义现金增长记号不能误写成年化净利率。 ## Content relations ```json [ { "from": "zh-m18", "relation": "part_of", "to": "markets-derivatives", "reason": "主要 topic 归属" }, { "from": "zh-m18", "relation": "requires", "to": "zh-m15", "reason": "仅需要这项先备能力，不由推荐次序自动生成", "required_competence": "分清call/put、premium、单位和行权方式" }, { "from": "zh-m18", "relation": "uses_method", "to": "zh-m17", "reason": "本篇相应段落就地调用报价、履约、时间价值或价格敏感度，不要求重新学习整篇" }, { "from": "m18-price", "relation": "supported_by", "to": "MA-OCC", "reason": "规则、权利金、行权/指派和同财富假设例；本课指定单元不替代实际交易者对整份ODD的阅读。无真实行情或个人券商操作保证。", "locator": "Ch I printed pp3–5；Ch II vanilla terms pp6–9; Premium/Opening and Closing pp11–12; Intrinsic and Time Value pp14–15；Ch VIII pp55–57; Ch IX pp58–59；Ch X holder pp61–63; writer pp63–67（至所采用第11项）", "scope": "本段具名采用内容；教学生成数字仍按自身身份", "at_section": "m18-price" }, { "from": "m18-tree", "relation": "supported_by", "to": "PFH-MIT-OPTIONS", "reason": "模型内的支付和复制定价；不代替当前产品规则，讲义现金增长记号不能误写成年化净利率。", "locator": "Slides 3–9: option definitions and payoff；Slides 16–21: binomial replication and risk-neutral valuation", "scope": "本段具名采用内容；教学生成数字仍按自身身份", "at_section": "m18-tree" }, { "from": "m18-tree", "relation": "supported_by", "to": "PFH-EXERCISE", "reason": "行权vs平仓、ex-by-ex与客户指令、提前行权和分红；不提供任何个人券商统一截止。", "locator": "Full FAQ main body through exercise/closing distinction", "scope": "本段具名采用内容；教学生成数字仍按自身身份", "at_section": "m18-tree" }, { "from": "m18-factors", "relation": "supported_by", "to": "PFH-EXERCISE", "reason": "分红、融资与提前行权的现金比较；不把经纪商操作或特定截止写成统一规则。", "locator": "FAQ中提前行权、分红、closing与exercise相关完整问答", "scope": "解释持股现金分配与行权选择的现实边界；不作为BSM非零股息公式来源", "at_section": "m18-factors" }, { "from": "m18-factors", "relation": "supported_by", "to": "PFH-MIT-BSM", "reason": "常参数、无股息模型条件下价格/偏导与Q；不宣称完整连续时间定理证明。", "locator": "Slides 16–21: BSM market, replication/PDE/formula；Slides 52–58 and 60: Q pricing and European application", "scope": "本段具名采用内容；教学生成数字仍按自身身份", "at_section": "m18-factors" }, { "from": "m18-factors", "relation": "supported_by", "to": "MEFG-OIC-VEGA", "reason": "波动率参数的偏导和每百分点单位；IV不是独立的P下预测。", "locator": "Full main concept body", "scope": "本段具名采用内容；教学生成数字仍按自身身份", "at_section": "m18-factors" }, { "from": "m18-model", "relation": "supported_by", "to": "MA-SPX", "reason": "SPX/SPXW欧式现金、100美元乘数、最小报价增量及AM/PM结算差别；并非某条历史上市期权链。", "locator": "Full physical pp1–2; p2 specifications and relevant footnotes", "scope": "本段具名采用内容；教学生成数字仍按自身身份", "at_section": "m18-model" }, { "from": "m18-model", "relation": "supported_by", "to": "PFH-MIT-BSM", "reason": "常参数、无股息模型条件下价格/偏导与Q；不宣称完整连续时间定理证明。", "locator": "Slides 16–21: BSM market, replication/PDE/formula；Slides 52–58 and 60: Q pricing and European application", "scope": "本段具名采用内容；教学生成数字仍按自身身份", "at_section": "m18-model" }, { "from": "m18-inversion", "relation": "illustrated_by", "to": "EXP-MEFG-M18-IV", "reason": "先检查报价是否在模型价格界内，再看同一树在不同决策集合中的价值。", "at_section": "m18-inversion" } ] ``` ## Related entries ## Optional reading path 理解合约与市场机制: step 17/17 先辨欧式可行权约束和模型条件，再检查报价界并反解 IV。