# Greeks：把局部导数变成带单位的风险 把delta、gamma、vega、theta写成带单位的局部导数，用50点和500点完整重定价量化误差。 Entry: zh-m19 | Node: M19 | Language: zh | Editorial revision: 2026-09-21 ## Teaching instructions 你正在教M19《Greeks：把局部导数变成带单位的风险》，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：给定单位及固定条件，计算一阶/部分二阶近似，并用完整定价指出小变动和大变动误差。 专属诊断与正向讲解：用50点后用500点，要求先判断vega是每一个百分点还是sigma=1，再将点数乘100。最后解释delta .556803与N(d2) .510441不同。 反馈尺度：输入多个变量变化时不声称只加spot gamma已经包含所有二阶项；期权点、美元和概率不混。 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。 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": "MEFG-OIC-DELTA", "version": "Undated; retrieved 2026-09-21", "access": { "kind": "html_full_text", "uri": "https://www.optionseducation.org/advancedconcepts/delta", "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": "带单位的局部敏感度与适用条件" }, "supports": "局部价格敏感度和对冲单位；概率近似不能升级成P下概率恒等。", "id": "M19-READ-01", "title": "Delta", "authors": [ "Options Industry Council" ], "retrieved_at": "2026-09-21" }, { "source_id": "MEFG-OIC-GAMMA", "version": "Undated; retrieved 2026-09-21", "access": { "kind": "html_full_text", "uri": "https://www.optionseducation.org/advancedconcepts/gamma", "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": "带单位的局部敏感度与适用条件" }, "supports": "delta的局部变化；全组合或做市商净gamma须另有持仓资料。", "id": "M19-READ-02", "title": "Gamma", "authors": [ "Options Industry Council" ], "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": "带单位的局部敏感度与适用条件" }, "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": "M19-READ-03", "title": "Vega", "authors": [ "Options Industry Council" ], "retrieved_at": "2026-09-21" }, { "source_id": "MEFG-OIC-THETA", "version": "Undated; retrieved 2026-09-21", "access": { "kind": "html_full_text", "uri": "https://www.optionseducation.org/advancedconcepts/theta", "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": "带单位的局部敏感度与适用条件" }, "supports": "固定其余变量的日历时间局部敏感度；非所有欧式期权必然价值递减。", "id": "M19-READ-04", "title": "Theta", "authors": [ "Options Industry Council" ], "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": "BSM模型假设/公式，解析与差分校验" }, "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": "M19-READ-05", "title": "15.450: Stochastic Calculus and Option Pricing", "authors": [ "Leonid Kogan", "MIT OpenCourseWare" ], "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-M19-GREEKS" }, "entry_id": "zh-m19", "node_id": "M19", "content_version": "2026-09-21-MEFG-review-v3", "experiment_ids": [ "EXP-MEFG-M19-GREEKS" ], "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/M19.html": "https://ou-liu-red-sugar.github.io/notebook/labs/m-efg/static/M19.html" } } ``` ## Supplied entry ## 一、先问“对哪个变量求导” 上一篇用价格模型把报价变成IV。本篇暂时固定模型，问一个更局部的问题：如果一个输入稍微改变，当前价格大约怎样改变？Greeks就是把这个问题按不同变量拆开。它们不是关于下一步涨跌的独立预测，也不是在任何变动幅度下都精确的赔付规则。 令期权价格为 $V(S,\sigma,\tau,r,q_{\rm div})$，其中 $\tau$ 是剩余年数。固定其他变量， $$ \Delta=\frac{\partial V}{\partial S},\qquad \Gamma=\frac{\partial^2V}{\partial S^2},\qquad \text{Vega}_{\rm abs}=\frac{\partial V}{\partial\sigma}. $$ 若用日历时间前进一日来定义theta，固定到期日时剩余年数减少 $1/365$，本篇的日theta是 $-\frac1{365}\frac{\partial V}{\partial\tau}$。有些资料按年、交易日或其他单位报告，数值不能不经换算就拼在一张表上。[^MEFG-OIC-DELTA][^MEFG-OIC-GAMMA][^MEFG-OIC-VEGA][^MEFG-OIC-THETA] ## 二、固定同一张合成call，给每个数标单位 使用与价格一篇完全相同的合成切片：$S=K=5,000$，$\tau=90/365$，$r=4\%$，$q_{\rm div}=0$，$\sigma=23.5\%$，欧式call，乘数100美元／期权点。我们不是从某个真实期权链取得这些Greeks，而是从同一BSM函数计算。[^MEFG-MIT-KOGAN] | 对象 | 默认值 | 单位及解释 | |---|---:|---| | 价格 | 256.855895 | 期权报价点 | | delta | 0.556803 | 每变动1指数点，期权点数的局部变化 | | gamma | 0.000676807 | 每变动1指数点，delta的局部变化 | | vega | 9.804437 | 波动率提高**1个百分点**时，期权点数的局部变化 | | theta | −1.556972 | 日历前进1日时，固定其余输入的局部变化 | | 合约乘数 | 100 | 每期权报价点对应美元 | 一份合约的delta金额敏感度约为55.68美元／指数点；vega约为980.44美元／波动率百分点。将 $\sigma$ 从0.235变成0.245是提高1个百分点，而不是把 $\sigma$ 加1。若使用数学导数 $\partial V/\partial\sigma$，其数值应比每百分点vega大100倍。 SPX指数也不是可直接买入的“一股股票”。0.556803可以是模型对冲比率的起点，但实际用何种ETF、期货或组合实施，需要再做单位、基差和可成交性转换；这张表只给模型对指数水平的导数。[^MEFG-SPX] ## 三、一次移动50点与500点，近似误差会发生什么 只改变 $S$ 时，泰勒展开给出 $$ \Delta V\approx\Delta\,\Delta S +\frac12\Gamma(\Delta S)^2. $$ 保留第一项是delta近似，加入第二项是delta加gamma近似。这里的 $\Delta,\Gamma$ 都在**起点**计算，不是在看到终点以后重新选择。 | 标的变动 | delta一阶变化 | delta＋gamma二阶变化 | 完整BSM重算变化 | |---:|---:|---:|---:| | ＋50指数点 | 27.840133 | 28.686142 | 28.679412 | | ＋500指数点 | 278.401326 | 363.002261 | 353.164687 | 对50点变动，一阶少算约0.839279点；二阶误差仅约0.006730点。对500点变动，二阶仍然比一阶更近，却高出约9.837574点。现金误差还要乘100。**加入gamma不是给近似加一个“精确保证”，而是多保留一阶曲率信息。** 曲率为正也可以从图上读出来：call价格随 $S$ 上升时斜率逐渐变大，起点切线会低于完整凸曲线。但移动大时，gamma自身也变；沿全区间使用起点gamma，仍是近似。OIC的局部解释和MIT的价格模型在这里各承担一层：一个教你导数意义，一个提供可重新计算的函数。[^MEFG-OIC-GAMMA][^MEFG-MIT-KOGAN] ## 四、波动率和时间的单位同样会放大错误 把 $S$ 固定，改 $\sigma$。本例的每百分点vega给出的比较为： | 波动率变化 | vega局部预测 | 完整重算 | |---:|---:|---:| | 23.5% → 24.5% | ＋9.804437点 | ＋9.805146点 | | 23.5% → 33.5% | ＋98.044371点 | ＋98.067826点 | 这一次大幅变化下误差仍小，是本切片的具体结果，不是vega在所有行权价、期限上都线性的证据。换到临近到期或远离平值的合约，敏感度形状会不同。 日theta为−1.556972，意思是保持现价、模型波动率、利率和股息输入不变，日历时间前进一日的局部效应。真实市场到明天会同时改变许多输入，所以不能把theta直接当作明天必然少掉的价格。欧式put还可能因收取 $K$ 的时点接近而出现正的日历时间效应；上一篇的深度价内反例正说明“持有期权每天必亏时间价值”不是无条件定理。[^MEFG-OIC-THETA] 若同时改变 $S,\sigma$ 和日历时间，可以先加起点delta、gamma、vega、theta项作局部估算，但跨变量项以及各导数的变化并不会消失。本实验同时展示完整重算，避免把这类估算结果误当成真实账户路径。 ## 五、delta不是“上涨概率” 在本无股息BSM call中，delta等于 $N(d_1)$；同模型定价概率下的到期价内概率为 $N(d_2)$。本例两者分别为 $$ N(d_1)=0.556803,\qquad N(d_2)=0.510441. $$ 即使在这个非常规则的模型里，它们已经不同；更不能把任意产品的delta直接解释成现实概率 $P$ 下的胜率。[^MEFG-MIT-KOGAN] 把一项价格导数叫成概率，常常是因为0到1之间的数看起来很像概率。但是它们的定义不同：delta问“当前价格对输入怎样变”，价内概率问“在某个概率模型下哪些终值状态发生”；还要再区分定价权重 $Q$ 和现实／研究概率 $P$。后面复制一篇会从交易资产价格解释这种权重区别。
实验第一视角画完整价格曲线、起点切线和起点二阶曲线；第二视角比较两种近似相对于完整重算的误差。默认可直接选50点和500点，或者改波动率百分点、时间天数。切到美元显示时只做100倍单位换算，不改变模型。 ## 六、把“近似”写进答案 **题一。** 一份本例call，指数上升50点，delta近似的美元变化是多少？二阶与完整重算分别是多少？ **解析。** 期权点变化依次为27.840133、28.686142和28.679412，乘100得到约2,784.01、2,868.61和2,867.94美元。不要再乘一次5,000，也不要把delta当作要买入0.556803股“指数股票”的实盘指令。 **题二。** 某人把每百分点vega 9.804437乘以0.01，计算波动率从23.5%变24.5%的变化。错在哪里？ **解析。** 9.804437本来已经是每一个百分点的报价量，应乘1。若要乘0.01，应使用按一整个绝对 $\sigma$ 单位报告的导数980.443707。两种单位正确换算后结果相同。 **题三。** 本例500点移动中二阶近似高于完整重算，是否否定gamma为正？ **解析。** 没有。gamma为正是价格函数在起点的曲率信息；大区间移动时曲率本身改变，还存在更高阶项。二阶多项式不是整条价格函数。 **题四。** delta为0.556803，能否宣称“我有55.68%的真实概率盈利”？ **解析。** 不能。首先delta是导数，本例连 $Q$ 下的价内概率都只有0.510441；其次价内不等于覆盖权利金后盈利；最后 $Q$ 不等于现实 $P$。这三个区别要依次保留。 [^MEFG-OIC-DELTA]: Options Industry Council，*Delta*，Undated; retrieved 2026-09-21。[原文](https://www.optionseducation.org/advancedconcepts/delta)。本篇定位：Full main concept body。 [^MEFG-OIC-GAMMA]: Options Industry Council，*Gamma*，Undated; retrieved 2026-09-21。[原文](https://www.optionseducation.org/advancedconcepts/gamma)。本篇定位：Full main concept body。 [^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-OIC-THETA]: Options Industry Council，*Theta*，Undated; retrieved 2026-09-21。[原文](https://www.optionseducation.org/advancedconcepts/theta)。本篇定位：Full main concept body。 [^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-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
初始敏感度数值单位
delta0.556802653点/标的点
gamma0.000676807476delta/标的点
vega9.804437069点/1波动率百分点
theta-1.556972481点/日
N(d2)0.510441217Q模型概率，不是P
S变化一阶加spot二阶全模型
5027.84013328.68614228.679412
500278.401326363.002261353.164687

每合约点数乘100美元。以上只改S，其余不变；同时改波动率与时间时，一阶加vega/theta，未计全部交叉二阶项。

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-M19-GREEKS", "title": "局部近似与完整重新定价", "anchor": "m19-probability", "description": "小变动和大变动各算一次；点数、美元和波动率百分点分别显示。", "data_identity": "同一90天、K=5,000、σ=23.5%的合成切片；不是实时Greeks。", "shared_dataset_ids": [ "EXP-MEFG-IV-01" ], "inputs": { "canonical_common": "shared_inputs.json", "state_contract": null, "controls": [ { "name": "ds", "label": "标的变化（指数点）", "type": "number", "default": 50, "domain_or_choices": { "min": -4000, "max": 5000, "step": 50 } }, { "name": "dvPpt", "label": "波动率变化（百分点）", "type": "number", "default": 0, "domain_or_choices": { "min": -20, "max": 100, "step": 1 } }, { "name": "elapsedDays", "label": "经过的日历天数", "type": "number", "default": 0, "domain_or_choices": { "min": 0, "max": 89, "step": 1 } }, { "name": "unit", "label": "结果单位", "type": "select", "default": "points", "domain_or_choices": [ [ "points", "期权报价点" ], [ "usd", "每合约美元（×100）" ] ] } ] }, "algorithm": "Compute analytic delta,gamma,vega_per_ppt=vega*.01,theta_per_day=theta_year/365 at shared base. Linear ΔC=delta*ΔS+vega_ppt*Δsigma_ppt+theta_day*elapsed; partial second-order adds .5*gamma*ΔS². Exact BSM at new S,sigma,T; errors report approximate-exact; dollar scale=100.", "outputs": { "default": { "base": { "price": 256.85589455779245, "delta": 0.5568026529513653, "gamma": 0.0006768074761722486, "vegaPerPpt": 9.804437069207575, "thetaPerDay": -1.556972481378402, "qITM": 0.5104412165596846, "d1": 0.14286766759947472, "d2": 0.026175237306540852 }, "ds": 50, "dvPpt": 0, "elapsedDays": 0, "linear": 27.840132647568268, "quadratic": 28.68614199278358, "exact": 28.67941201992653, "errorLinear": -0.8392793723582628, "errorQuadratic": 0.006729972857048239, "exactCash": 2867.941201992653, "multiplier": 100, "limitation": "delta/gamma curvature only in S; cross/time/vol second derivatives omitted" }, "static_equivalent": "https://ou-liu-red-sugar.github.io/notebook/labs/m-efg/static/M19.html", "unit_note": "同一90天、K=5,000、σ=23.5%的合成切片；不是实时Greeks。" }, "boundaries": [ "Telapsed<90且新S和sigma为正", "vega单位百分点、theta日、multiplier100", "只加入spot二阶，交叉和高阶未覆盖", "N(d2)≠P概率，delta≠N(d2)" ], "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-M19-GREEKS", "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/M19.html" } ] ``` ## Sources - [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。 - [Delta](https://www.optionseducation.org/advancedconcepts/delta): 局部价格敏感度和对冲单位；概率近似不能升级成P下概率恒等。 - [Gamma](https://www.optionseducation.org/advancedconcepts/gamma): delta的局部变化；全组合或做市商净gamma须另有持仓资料。 - [Theta](https://www.optionseducation.org/advancedconcepts/theta): 固定其余变量的日历时间局部敏感度；非所有欧式期权必然价值递减。 - [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。 - [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。 ## Content relations ```json [ { "from": "zh-m19", "relation": "part_of", "to": "markets-derivatives", "reason": "主要 topic 归属" }, { "from": "zh-m19", "relation": "requires", "to": "zh-m18", "reason": "仅需要这项先备能力，不由推荐次序自动生成", "required_competence": "能说明模型输入、IV及价格函数" }, { "from": "m19-variables", "relation": "supported_by", "to": "MEFG-OIC-DELTA", "reason": "局部价格敏感度和对冲单位；概率近似不能升级成P下概率恒等。", "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": "m19-variables" }, { "from": "m19-variables", "relation": "supported_by", "to": "MEFG-OIC-GAMMA", "reason": "delta的局部变化；全组合或做市商净gamma须另有持仓资料。", "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": "m19-variables" }, { "from": "m19-variables", "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": "m19-variables" }, { "from": "m19-variables", "relation": "supported_by", "to": "MEFG-OIC-THETA", "reason": "固定其余变量的日历时间局部敏感度；非所有欧式期权必然价值递减。", "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": "m19-variables" }, { "from": "m19-default", "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": "m19-default" }, { "from": "m19-default", "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": "m19-default" }, { "from": "m19-curvature", "relation": "supported_by", "to": "MEFG-OIC-GAMMA", "reason": "delta的局部变化；全组合或做市商净gamma须另有持仓资料。", "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": "m19-curvature" }, { "from": "m19-curvature", "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": "m19-curvature" }, { "from": "m19-vol-time", "relation": "supported_by", "to": "MEFG-OIC-THETA", "reason": "固定其余变量的日历时间局部敏感度；非所有欧式期权必然价值递减。", "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": "m19-vol-time" }, { "from": "m19-probability", "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": "m19-probability" }, { "from": "m19-probability", "relation": "illustrated_by", "to": "EXP-MEFG-M19-GREEKS", "reason": "小变动和大变动各算一次；点数、美元和波动率百分点分别显示。", "at_section": "m19-probability" } ] ``` ## Related entries