Chapter 03 · The Greeks

Greeks：读懂期权的仪表盘The Greeks: Reading the Instrument Panel

第 1 章说期权价格由三个旋钮支配：距离、时间、波动率。Greeks 就是这三个旋钮的刻度表——它们回答的不是"会涨吗"，而是"每转动一格，我的仓位会动多少"。看不懂仪表盘的人，是在盲飞。 Chapter 1 said an option's price is governed by three dials: distance, time, volatility. The Greeks are those dials' gauges — they answer not "will it go up" but "how much does my position move per notch." Fly without reading them and you are flying blind.

3.1 仪表盘的意义：Greeks 是灵敏度，不是预测3.1 What the Panel Is For: Sensitivities, Not Forecasts

最常见的误解，是把 Greeks 当成信号。"Delta 高所以要买"、"Theta 为正所以稳赚"——这两句话都错得很彻底。Greeks 不预测任何事，它们只描述"如果某件事发生，你的仓位会怎样"。Delta 0.6 不代表股价会涨，它只说：如果股价涨 $1，你的期权大约涨 $0.60。 The most common misreading is to treat the Greeks as signals. "High delta, so buy" or "positive theta, so it's free money" are both thoroughly wrong. The Greeks forecast nothing; they describe what happens to your position if something happens. A 0.6 delta does not say the stock will rise — it says: if the stock rises $1, your option gains roughly $0.60.

这个区别有实际后果。仪表盘的正确用法是在开仓前做压力测试：股价跌 5% 我亏多少？IV 回落 10 个点我亏多少？放一周不动我亏多少？这三个问题分别由 Delta、Vega、Theta 回答，而且能在下单之前算出来。期权交易里绝大多数"意外"，都是本来可以在开仓前用四个数字算清楚的事。 The distinction has practical consequences. The right use of the panel is to stress-test before you enter: what do I lose if the stock drops 5%? If IV falls 10 points? If I simply hold for a week? Delta, vega, and theta answer those three, and all of them can be computed before you click buy. Nearly every "surprise" in options trading was something four numbers could have told you in advance.

Delta: 股价动 $1 → 期权动多少Per $1 of stock
Gamma: 股价动 $1 → Delta 动多少Per $1, how delta moves
Theta: 过一天 → 期权动多少Per day elapsed
Vega: IV 动 1 点 → 期权动多少Per 1 IV point

关于本章的所有曲线About every curve in this chapter

下面的 Greeks 曲线均由 Black-Scholes 模型计算，输入已固定为：Strike $100、波动率 30%、无风险利率 4%、不含分红，横轴为股价。它们是模型输出，不是市场报价——真实期权链会因为波动率偏斜、分红和流动性而偏离。但形状是对的，而形状正是这一章要教的东西。 Every Greek curve below is computed from the Black-Scholes model with fixed inputs: $100 strike, 30% volatility, 4% risk-free rate, no dividend, with spot price on the horizontal axis. These are model outputs, not market quotes — a real chain will deviate because of volatility skew, dividends and liquidity. But the shapes are right, and the shapes are what this chapter teaches.

3.2 Delta：一个数字，两重身份3.2 Delta: One Number, Two Identities

Call 范围Call range: 0 → +1
Put 范围Put range: −1 → 0
身份一Identity 1: 方向暴露（等效股数）Directional exposure
身份二Identity 2: 到期 ITM 的粗略概率Rough odds of finishing ITM
股票的 DeltaStock's delta: 1.00

Delta 的第一重身份是局部方向暴露。一张 0.60 Delta 的 Call 对应 100 股，当前这个价位附近等效于持有约 60 股正股——这句话让你可以把期权和股票放进同一个尺子里衡量。手里有 300 股 AAPL 加上三张 0.40 Delta 的 Call？此刻的一阶方向暴露约为 300 + 120 = 420 股。但这是一个会随股价、时间和 IV 变化的瞬时近似，不是整个价格路径上的固定仓位。 Delta's first identity is local directional exposure. One 0.60-delta call covers 100 shares and, near the current spot, behaves like roughly 60 shares — which lets you measure options and stock on one ruler. Hold 300 AAPL shares plus three 0.40-delta calls? Your first-order directional exposure at this moment is about 300 + 120 = 420 shares. But this is an instantaneous approximation that changes with spot, time and IV, not a fixed share count across the whole price path.

第二重身份是风险中性概率的粗略代理。交易员常把 0.30 Delta 读成“约 30% 概率到期时 ITM”，但两者在模型里不是同一个量：Call Delta 是 N(d1)，到期 ITM 的风险中性概率是 N(d2)，真实世界概率还会受风险溢价与分布偏斜影响。它可以用来快速比较 Strike，不能当成真实概率，更不能直接等同于美式期权在到期前被指派的概率。 The second identity is a rough proxy for risk-neutral probability. Traders often read a 0.30 delta as “about a 30% chance of finishing ITM,” but the two are not the same model quantity: call delta is N(d1), the risk-neutral expiry probability is N(d2), and real-world probability also reflects risk premia and skewed distributions. It is useful for quickly comparing strikes; it is not a literal forecast, and it is not the probability that an American option will be assigned before expiry.

DELTA （Call，横轴为股价，Strike $100）(call, spot on x-axis, $100 strike)

请仔细看这两条线的差别，它是本章后面一切的种子。60 天的 Delta 是一条平缓的 S 形：股价从 $90 到 $110，Delta 从 0.23 走到 0.82，一路渐变。7 天的 Delta 却几乎是一个台阶：$95 时才 0.12，$100 时 0.52，$105 时已经 0.89。同一张合约，临近到期时对股价的反应从"渐变"变成了"开关"。这个从平滑到陡峭的转变，有一个名字。 Look carefully at the difference between these two lines — it is the seed of everything that follows. The 60-day delta is a gentle S-curve: from $90 to $110 it travels from 0.23 to 0.82, gradually. The 7-day delta is very nearly a step: 0.12 at $95, 0.52 at $100, already 0.89 at $105. The same contract, near expiry, stops responding to the stock like a dimmer and starts responding like a switch. That transition from smooth to steep has a name.

3.3 Gamma：Delta 的加速度3.3 Gamma: The Acceleration of Delta

定义Definition: Delta 的变化率Rate of change of delta
最大处Peaks at: ATM，且临近到期ATM, near expiry
买方Buyer: Gamma 为正（朋友）Long gamma (friend)
卖方Seller: Gamma 为负（敌人）Short gamma (enemy)

如果 Delta 是速度，Gamma 就是加速度。它衡量"股价再动 $1，我的 Delta 会变成多少"。买方永远是 Gamma 为正的一方：股价朝你走，你的 Delta 自动变大，赚得越来越快；股价背你而去，Delta 自动缩小，亏得越来越慢。这个"自动加仓、自动减仓"的特性，是期权买方最珍贵的东西，也是那笔 Premium 真正买到的东西。 If delta is speed, gamma is acceleration: how much your delta changes per further $1 in the stock. Buyers are always long gamma: when the stock moves your way your delta grows on its own, so you make money faster; when it moves against you your delta shrinks, so you lose more slowly. This automatic scaling in and out is the most valuable property an option buyer has — and it is what the premium actually bought.

GAMMA （×100，注意 7 天曲线的尖峰）(×100 — note the 7-day spike)

这张图解释了期权世界里最危险的现象之一。60 天的 Gamma 峰值约 3.3，分布得很宽；7 天的 Gamma 在 ATM 处飙到约 9.6，几乎是三倍，但只集中在 Strike 附近很窄的一段。这就是"到期周的 ATM 期权像手榴弹"的数学解释：股价一个小小的波动，就能让 Delta 从 0.3 翻到 0.7。对买方这是天堂，对卖方——尤其是裸卖方——这是最容易把小亏变成大亏的地带。 This chart explains one of the most dangerous phenomena in options. At 60 days gamma peaks near 3.3 and is spread wide; at 7 days it spikes to about 9.6 at the money — nearly triple — but concentrated in a narrow band around the strike. This is the mathematics behind "expiry-week ATM options are hand grenades": a small wiggle in the stock can take delta from 0.3 to 0.7. For buyers that's heaven; for sellers — especially naked ones — it is the zone where small losses most easily become large ones.

CASE FILE · 2021-01

GameStop：一个流行叙事，和数据给出的反证GameStop: A Popular Narrative, and the Evidence Against It

2021 年 1 月 4 日，GME 收盘 $17.25。1 月 27 日，它收在 $347.51；次日盘中触及 $483。从 1 月 4 日到 27 日共 17 个交易日收盘点，端点涨幅 +1,915%。事后最流行的解释之一是 Gamma Squeeze；但“机制听起来合理”不等于“它在这次事件里真的发生”。 On January 4, 2021, GME closed at $17.25. On January 27 it closed at $347.51, and touched $483 intraday the next day. January 4 through 27 contains 17 trading-session closes, with a +1,915% endpoint gain. One of the most popular explanations afterward was a gamma squeeze — but a plausible mechanism is not evidence that the mechanism actually drove this episode.

GME · JAN 2021 （收盘价）(closes)

教科书里的机制是这样的：如果客户集中买入 Call、做市商因此净卖出 Call，他们通常会买入相应比例的正股来对冲 Delta。股价上涨会抬高这些 Call 的 Delta，迫使净短 Gamma 的做市商继续买股；买盘再推高股价，正反馈就可能闭合。注意开头那个“如果”：没有做市商净头寸和实际对冲流量的数据，这只能证明机制可能存在，不能证明某次上涨由它造成。 The textbook mechanism works like this: if customers buy calls in concentration and dealers become net call sellers, dealers commonly buy a matching fraction of shares to hedge delta. A rising stock lifts those calls' deltas and can force dealers who are net short gamma to buy still more shares; that buying can lift the stock again and close a positive-feedback loop. Notice the opening “if”: without data on dealers' net positions and actual hedging flow, this establishes a possible mechanism, not the cause of a particular rally.

用数字看这个反事实模型。取一张 30 天到期、Strike $60、IV 150% 的 Call：股价 $40 时 Delta 约 0.23，$60 时 0.59，$80 时 0.81。如果某做市商净卖出 1 万张且持续 Delta 中性，从 $40 涨到 $80 会把理论对冲量从 23 万股推到 81 万股，相差 58 万股。这组数字解释了 Gamma Squeeze 为什么可能猛烈；它不是对 GME 实际做市商持仓的重建。 Put numbers on that counterfactual model. Take a 30-day, $60-strike call at 150% IV: its delta is about 0.23 with the stock at $40, 0.59 at $60 and 0.81 at $80. If a dealer were net short 10,000 such contracts and continuously delta-neutral, the theoretical hedge would rise from 230,000 to 810,000 shares as the stock moved from $40 to $80 — a difference of 580,000 shares. Those numbers show why a gamma squeeze can be violent; they do not reconstruct actual GME dealer positions.

证据边界：SEC 工作人员基于 CAT 等监管数据审查后明确写道，没有发现 2021 年 1 月 GME 发生 Gamma Squeeze 的证据；客户期权成交量的增量主要来自买 Put，且数据显示做市商当时是买入而非卖出 Call，这些观察与 Gamma Squeeze 不一致。参见 SEC Staff Report（2021），第 29 页。本书的 Delta 与“1 万张”仍只是机制演示。 Evidence boundary: after reviewing regulatory data including CAT, SEC staff wrote that they did not find evidence of a gamma squeeze in GME in January 2021. The increase in customer options volume was driven mostly by put buying, and dealers were buying rather than writing calls — observations inconsistent with a gamma squeeze. See the SEC Staff Report (2021), p. 29. This book's deltas and “10,000 contracts” remain a mechanism illustration only.

归因被否：GME 在 1 月 4 日至 27 日的 17 个交易日收盘点之间上涨 1,915%，但 SEC 未发现 Gamma Squeeze 证据；“价格暴涨”不能反推出“Gamma 是原因”Attribution rejected: GME gained 1,915% across the 17 session closes from January 4 through 27, but the SEC found no evidence of a gamma squeeze; a price surge does not prove gamma caused it

Gamma 是一把双刃剑，而且反面更常见Gamma cuts both ways, and the reverse is more common

上面的反事实让人记住 Gamma 的爆发力，却也容易让人忘记证据边界。单一期权的正 Gamma 通常与负 Theta 同时出现——买方为凸性持续支付时间价值；卖方在收取时间价值的同时承担跳空与负凸性风险。但组合层面还会叠加不同到期日、IV、利率和持仓腿，不能把“正 Theta = 必然负 Gamma”当成普遍定律，更不能从一次暴涨反推谁赚了多久、又因哪个 Greek 亏了多少。 The counterfactual above makes gamma's explosive potential memorable, but it also shows why evidence boundaries matter. For a single vanilla option, positive gamma usually comes with negative theta — the buyer continually pays time value for convexity, while the seller collects time value and carries jump and negative-convexity risk. At portfolio level, however, expiries, IV, rates and multiple legs interact. “Positive theta means necessarily negative gamma” is not a universal law, and a price surge alone cannot tell us who earned how much theta or lost how much gamma.

思考Think Gamma Squeeze 需要三个条件同时成立：大量集中的 OTM Call 买盘、做市商机械地对冲、以及标的流通盘不足以吸收对冲买入。这三个条件里，哪一个是散户可以制造的，哪一个不是？如果只有一个能被制造，那么"复制 GameStop"这件事的真实成功率是多少？ A gamma squeeze needs three conditions at once: heavy concentrated OTM call buying, mechanical dealer hedging, and a float too small to absorb the hedge. Which of the three can retail actually manufacture, and which cannot? If only one can, what is the real success rate of "recreating GameStop"?

3.4 Theta：每天流逝的租金3.4 Theta: The Rent That Runs Every Day

定义Definition: 其他条件不变时，理论价值对日历时间的敏感度Sensitivity of theoretical value to calendar time, all else equal
单一期权买方Single-option buyer: 通常为负Usually negative
单一期权卖方Single-option seller: 通常为正Usually positive
衰减最明显Decay most pronounced: 临近到期的 ATM 期权Near-expiry ATM options

日历时间一定会向前，但Theta 本身不是一笔保证到账的损益。它回答的是一个条件句：“如果股价、IV、利率等其他输入都不变，过一天，模型价格大约变多少？”现实里这些输入通常同时变化，所以实际一天的 P&L 不能只归给 Theta。卖方收取的权利金是承担波动、跳空、偏斜与流动性等风险的总对价，Theta 只是把其中的时间敏感度单独标出来。 Calendar time certainly moves forward, but theta is not guaranteed P&L. It answers a conditional question: “if spot, IV, rates and the other inputs stay unchanged, about how much does the model price change after a day?” Those inputs normally move together in real markets, so a day's realized P&L cannot be assigned to theta alone. Premium compensates a seller for volatility, jumps, skew, liquidity and other risks; theta isolates only the time sensitivity.

THETA （期权价值随剩余天数衰减，股价固定 $100）(option value vs. days left, spot pinned at $100)

两件事值得从这张图里读出来。第一，衰减是加速的，不是线性的。那张 ATM 的 Call 在 90 天时值 $6.41，每天流失约 $0.038；到只剩 10 天时，每天流失约 $0.104——速度快了近三倍。所谓"最后 30 天最凶险"不是江湖传说，是曲线的形状。 Two things to read off this chart. First, decay accelerates; it is not linear. That ATM call is worth $6.41 at 90 days and bleeds about $0.038 a day; with 10 days left it bleeds roughly $0.104 a day — nearly three times faster. "The last 30 days are the cruel ones" isn't folklore, it's the shape of the curve.

第二，OTM 期权的衰减比 ATM 残酷得多。从 90 天走到 30 天，ATM 的 Call 保住了原值的 56%，而 $110 的 OTM Call 只剩 24%——在股价完全没动的前提下。这就是"买彩票"策略的真实成本结构：你不仅需要方向对，还需要它快点对。等待本身就是一项持续支出。 Second, OTM options decay far more brutally than ATM ones. Going from 90 days to 30, the ATM call retains 56% of its value while the $110 OTM call retains just 24% — with the stock not moving at all. That is the true cost structure of the lottery-ticket trade: you need to be right, and you need to be right soon. Waiting is itself an ongoing expense.

Theta 陷阱：正的 Theta 不等于正的期望The theta trap: positive theta ≠ positive expectancy

“我的组合每天收 $50 Theta”是新手最容易上瘾的一句话。但 Theta 是模型在当前输入附近给出的条件敏感度，不是每日利息。一个正 Theta 组合可能同时暴露于负 Gamma、Vega、偏斜、相关性和跳空风险；具体是哪一种，必须看完整头寸。真正的问题不是“模型显示我收多少 Theta”，而是“哪些市场变化会让这项局部优势失效，以及压力情景下会亏多少”。 “My book collects $50 of theta a day” is an addictive sentence for beginners. But theta is a conditional model sensitivity around today's inputs, not daily interest. A positive-theta portfolio may also carry negative gamma, vega, skew, correlation and jump exposures; the actual mix depends on the whole position. The real question is not “how much theta does the screen show?” but “which market moves make this local advantage disappear, and what do I lose under stress?”

3.5 Vega：你真正在交易的东西3.5 Vega: What You Are Actually Trading

定义Definition: IV 每变动 1 点的价格变化Price change per 1 IV point
买方Buyer: 做多波动率Long volatility
卖方Seller: 做空波动率Short volatility
最大处Largest at: ATM，且到期日越远越大ATM, and grows with time

这是本章最重要的一节。多数人以为自己在交易方向，实际上他们在交易波动率。当你买入一张期权，你同时下了两个赌注：股价往哪走（Delta），以及市场对"能走多远"的预期会怎么变（Vega）。第二个赌注常常比第一个大——尤其在事件前后。 This is the most important section in the chapter. Most people think they are trading direction; they are in fact trading volatility. When you buy an option you place two bets at once: where the stock goes (delta), and how the market's expectation of how far it can go will change (vega). The second bet is frequently larger than the first — especially around events.

VEGA （时间越短，Vega 越小——保护也越"纯"）(less time, less vega — protection gets "purer")

注意 Vega 与 Gamma 的相反关系：Gamma 在临近到期时飙升，Vega 却在临近到期时萎缩（ATM 处从 60 天的 0.161 降到 7 天的 0.055）。这个对比给出一条极实用的规则——想赌方向和爆发，买近月（高 Gamma、低 Vega）；想赌波动率本身，买远月（高 Vega、低 Gamma）。买错了期限，你就在赌一个自己没打算赌的东西。 Note that vega and gamma run opposite: gamma spikes into expiry while vega shrinks into it (at the money, from 0.161 at 60 days to 0.055 at 7). That contrast yields a very practical rule — to bet on direction and explosiveness, buy near-dated (high gamma, low vega); to bet on volatility itself, buy longer-dated (high vega, low gamma). Choose the wrong tenor and you are betting on something you never intended to bet on.

CASE FILE · 2024-08

日元套息平仓夜：方向对了，仓位亏了 42%The Yen Carry Unwind: Right on Direction, Down 42%

2024 年 8 月 5 日是近年最纯粹的一堂 Vega 课。日本央行 7 月 31 日加息，叠加 8 月 2 日疲弱的美国就业数据，触发了日元套息交易的集中平仓。日经 225 当日暴跌 12.4%（-4,451 点，该指数历史上最大的单日点数跌幅）；标普 500 下跌 3.0%；VIX 从 8 月 2 日收盘的 23.39 盘中一度冲到约 65，收在 38.57。 August 5, 2024 was the purest vega lesson in recent memory. A Bank of Japan hike on July 31, compounded by a weak US jobs report on August 2, triggered a concentrated unwind of the yen carry trade. The Nikkei 225 fell 12.4% that day (−4,451 points, the largest single-day point decline in its history); the S&P 500 dropped 3.0%; the VIX went from a 23.39 close on August 2 to roughly 65 intraday, settling at 38.57.

VIX · AUG 2024 （收盘，约值）(closes, approx.)

现在跟着一个在 8 月 5 日恐慌中买入保护的人走一遍。他买入一张 30 天到期的 ATM SPY Put（SPY 约 $517.5，IV 约 38.6%），成本约 $21.74。次日，8 月 6 日：SPY 只反弹了约 1.1% 到 $523，但 VIX 从 38.57 塌到 27.71。同一张 Put 的价值变成约 $12.67——一夜 -42%。 Now walk through someone who bought protection into the panic on August 5: one 30-day ATM SPY put (SPY around $517.5, IV about 38.6%), costing roughly $21.74. The next day, August 6: SPY recovered only about 1.1% to $523, but the VIX collapsed from 38.57 to 27.71. That same put was worth about $12.67 — −42% overnight.

把亏损拆开看，结论很刺眼。这张 Put 的 Vega 约 $0.59/点，IV 跌了 10.9 个点，仅 Vega 一项就吃掉 $6.42——占全部 $9.07 亏损的约 71%。剩下的才是股价反弹和一天 Theta 的贡献。这个人对"市场处于危险中"的判断并没有错（后续两周市场仍在震荡），他只是在保险最贵的那一刻买了保险。 Decompose the loss and the conclusion stings. That put's vega was about $0.59 per point; IV fell 10.9 points, so vega alone accounts for $6.42 — roughly 71% of the total $9.07 loss. The stock's bounce and one day of theta explain the rest. This person was not wrong that the market was in danger — it kept churning for another two weeks. They simply bought insurance at the exact moment insurance was most expensive.

Counterfactual：同样一份 30 天 ATM 保护，在 8 月 2 日（VIX 23.39，SPY 约 $532.9）买入，成本约 $13.16，占标的市值 2.5%；8 月 5 日买入则占 4.2%。晚了三天，同一份保护贵了约 65%。这与第 2 章 COVID 案例的结论完全一致，只是这次我们能把它拆成一个 Greek：保险要在着火前买，因为着火时你付的是 Vega。 Counterfactual: the same 30-day ATM protection bought on August 2 (VIX 23.39, SPY around $532.9) cost about $13.16, or 2.5% of the underlying's value; bought on August 5 it cost 4.2%. Three days later, the same protection cost about 65% more. This is exactly Chapter 2's COVID conclusion, except now we can name the Greek responsible: buy insurance before the fire, because during the fire what you are paying for is vega.

简化声明：VIX、日经、标普与 SPY 的价格为真实历史约值。期权价格与 Vega 由 Black-Scholes 按上述输入计算（利率 5%、无分红），是模型值而非实际成交报价；真实 SPY 期权链存在偏斜，ATM Put 的报价会与此有出入。 Stated simplification: VIX, Nikkei, S&P and SPY levels are approximate real history. The option prices and vega are Black-Scholes values computed from those inputs (5% rate, no dividend) — model figures, not actual fills; a real SPY chain carries skew, so the quoted ATM put would differ somewhat.

亏损减少：同一份 30 天保护，恐慌前买入占市值 2.5%，恐慌中买入占 4.2%——提前三天，成本低约 39%Loss reduced: the same 30-day protection cost 2.5% of spot before the panic vs. 4.2% during it — three days earlier, about 39% cheaper

思考Think 如果这个人买的不是 30 天的 Put，而是 5 天的 Put，Vega 会小得多（回看上面的 Vega 曲线），IV 塌缩造成的伤害也会小得多。那么"恐慌中买短期期权"是不是一个好办法？在你回答之前，先看一眼 Gamma 那张图——你躲开了 Vega，接住了什么？ Had this person bought a 5-day put instead of a 30-day one, vega would have been far smaller (look again at the vega curves) and the IV collapse would have hurt far less. So is "buy short-dated during a panic" a good answer? Before you say yes, glance back at the gamma chart — you dodged vega, but what did you catch?

3.6 把四个表盘一起读3.6 Reading All Four Gauges Together

单独看一个 Greek 没有意义，因为它们是被捆绑出售的。市场不允许你只买 Gamma 不付 Theta，也不允许你只收 Theta 不担 Gamma。每一个头寸都是这四个数字的一个特定组合，而你选择的策略，本质上就是在选择这个组合。 Reading one Greek in isolation is meaningless, because they are sold as a bundle. The market will not let you buy gamma without paying theta, nor collect theta without carrying gamma. Every position is one particular combination of these four numbers, and choosing a strategy is precisely choosing that combination.

拿第 2 章的 Covered Call 做例子：持有 100 股（Delta +1.00、其余 Greeks 为零）并卖出一张 0.35 Delta 的 Call。整体的仪表盘读数是： Take Chapter 2's covered call: hold 100 shares (delta +1.00, all other Greeks zero) and sell one 0.35-delta call. The combined panel reads:

Delta: +0.65 （还是多头）(still long)
Gamma: 负（怕跳空）Negative (fears gaps)
Theta: 正（每天收租）Positive (collects rent)
Vega: 负（盼 IV 下降）Negative (wants IV to fall)

COVERED CALL · Delta 就是这条线的斜率· delta is the slope of this line

这四个数字合起来是一句完整的话："我温和看多，我希望市场平静，我怕跳空，我靠时间赚钱。"——这正是 Covered Call 的全部内容，只不过用 Greeks 说了一遍。反过来，当你说不清一个头寸的四个 Greek 符号时，你其实说不清自己在赌什么。 Together those four numbers form one complete sentence: "I'm mildly long, I want calm, I fear gaps, and I get paid by time." — which is the entire content of a covered call, restated in Greeks. Conversely, when you can't name the signs of a position's four Greeks, you can't actually say what you're betting on.

最后一件实用的事：把上面那张盈亏图的斜率看一遍。$100 附近斜率接近 1（Delta 高），$115 之后曲线变平（Delta 趋近 0）。到期盈亏图的斜率就是 Delta，斜率的变化就是 Gamma。你在第 2 章和第 4 章看过的每一张图，其实都已经把这两个 Greek 画出来了——只是当时你还没有名字可以称呼它们。 One last practical thing: look at the slope of the payoff diagram above. Near $100 it's close to 1 (high delta); past $115 it flattens (delta approaching 0). The slope of an expiry payoff diagram is delta, and the change in that slope is gamma. Every chart you've seen in Chapters 2 and 4 had both Greeks drawn on it already — you simply had no names for them yet.

3.7 本章测验3.7 Chapter Quiz

点击选项即可作答，答案与解析即时显示。Click an option to answer; the correct answer and explanation appear immediately.

你持有 200 股 AAPL，另外买入 3 张 Delta 0.5 的 Call。你的方向暴露相当于多少股？You hold 200 AAPL shares plus three 0.5-delta calls. Your directional exposure is equivalent to how many shares?

B。每张 Call 对应 100 股，Delta 0.5 意味着等效 50 股；三张就是 150 股。200 + 150 = 350。A 忘了合约乘数 100（这是第 1 章的教训）；C 把 Call 当成了满仓的 300 股，那只有 Delta = 1 时才成立。 B. Each call covers 100 shares, and a 0.5 delta makes it equivalent to 50; three calls are 150. 200 + 150 = 350. A forgets the 100 multiplier (Chapter 1's lesson); C treats the calls as a full 300 shares, which would require delta = 1.

同样是 ATM 的 Call，7 天到期的和 60 天到期的相比，哪个说法正确？Comparing two ATM calls, one with 7 days to expiry and one with 60, which statement is correct?

C。这是本章最实用的一条对立关系：临近到期，Gamma 飙升而 Vega 萎缩。本章曲线里 ATM Gamma 从 60 天的约 3.3 升到 7 天的约 9.6，同时 Vega 从 0.161 降到 0.055。实践含义：赌爆发买近月，赌波动率买远月。 C. This is the chapter's most useful opposition: into expiry, gamma spikes while vega shrinks. In the curves above, ATM gamma rises from about 3.3 at 60 days to about 9.6 at 7, while vega falls from 0.161 to 0.055. Practical consequence: buy near-dated for explosiveness, longer-dated for volatility.

财报后股价没动，你的 Long Call 却亏了 45%。最主要的原因是哪个 Greek？After earnings the stock didn't move, yet your long call is down 45%. Which Greek is chiefly responsible?

D。股价没动，Delta 和 Gamma 就没有贡献（A、B 排除）。Theta 一夜的消耗远达不到 45%（回看衰减曲线：ATM 期权在最后 10 天每天也只流失约 $0.10 / 2 美元级的价值）。只剩 Vega：财报落地，事件溢价消失，IV 骤降——这就是 IV Crush，而 IV Crush 只是"Vega 乘以一个很大的负数"的通俗叫法。 D. With the stock still, delta and gamma contribute nothing (rules out A and B). One night of theta comes nowhere near 45% (revisit the decay curve: even in its last 10 days the ATM option sheds only about $0.10 a day on a ~$2 price). That leaves vega: the event premium vanishes when results land and IV collapses. That is IV crush, which is simply the popular name for "vega times a large negative number."

"我的组合每天收 $80 的 Theta，所以这是一个稳定盈利的系统。"这句话的问题在哪？"My book collects $80 of theta a day, so it's a steadily profitable system." What's wrong with that?

A。Theta 是在其他输入不变时的局部模型输出，不是账户每天保证收到的 $80，更不能单独证明策略有正期望。单一期权卖方通常正 Theta、负 Gamma，但多腿与跨期限组合不满足这个绝对等号；还必须检查 Vega、跳空、偏斜、流动性和压力损失。B 把“条件模型输出”误解成“不准确”；C 错在 Theta 并非只到期日才存在；D 只看 Delta 同样不完整。 A. Theta is a local model output conditional on the other inputs staying fixed, not $80 the account is guaranteed to receive each day and not proof of positive expectancy. A single short vanilla option is usually positive theta and negative gamma, but multi-leg and cross-expiry portfolios do not obey that absolute equivalence; vega, jumps, skew, liquidity and stress losses still matter. B confuses “conditional” with “inaccurate”; C is wrong because theta exists before expiry; D is equally incomplete if it looks only at delta.

2024 年 8 月 5 日 VIX 冲上 38，你确信市场还会跌，于是买入 30 天的 SPY Put。第二天 SPY 只涨了 1%，你却亏了 42%。这说明什么？On August 5, 2024 with the VIX at 38, you're sure the market will fall further and buy a 30-day SPY put. The next day SPY rises just 1% and you're down 42%. What does this show?

B。拆开那 $9.07 的亏损：Vega 贡献了约 $6.42（71%），股价和 Theta 只占其余。方向判断是否正确在这里几乎无关紧要（A 错）——市场此后两周确实继续震荡。价格也没有错（C 错）：$21.74 是 IV 38.6% 下的公允价，你只是在 IV 38.6% 的时候买了它。至于 D，在一个由 Vega 主导亏损的头寸上加仓，是把同一个错误做大。 B. Decompose the $9.07 loss: vega contributed about $6.42 (71%), with the stock and theta making up the rest. Whether the directional view was right barely matters here (so not A) — the market did keep churning for two more weeks. Nor was the option mispriced (not C): $21.74 is fair at 38.6% IV; you simply bought it at 38.6% IV. As for D, adding to a position whose loss is vega-driven is enlarging the same mistake.

本章要点Key Takeaways

Greeks 不是信号，是局部敏感度——它们让你在开仓之前回答“如果某个输入单独变化，我会怎样”。Delta 衡量当前价位附近的一阶方向暴露，也常被用作风险中性概率的粗略代理；Gamma 是 Delta 对股价的变化率，在临近到期的 ATM 区域最集中；Theta 是其他输入不变时的时间敏感度，不是保证到账的收益；Vega 则显示 IV 变化会怎样影响价格。2024 年 8 月 5 日那张示例 Put 的一天损益里，71% 来自 IV 假设的变化，正说明任何一个 Greek 都不能脱离其他输入单独解释实际 P&L。真正的问题从来不是“哪个 Greek 好”，而是你承担了哪一组局部风险，以及期限和 Strike 是否真的表达了你的意图。 The Greeks are not signals but local sensitivities — they let you ask, before entering, “what happens if one input changes?” Delta measures first-order directional exposure near today's spot and is often used as a rough proxy for risk-neutral probability; gamma is the rate at which delta changes with spot and is most concentrated near-the-money close to expiry; theta is time sensitivity with the other inputs fixed, not guaranteed income; and vega describes sensitivity to IV. In the August 5, 2024 example put, 71% of the one-day change came from the assumed IV move, illustrating that no Greek can explain realized P&L in isolation from the other inputs. The question is never “which Greek is good?” but which bundle of local risks you carry and whether the tenor and strike express what you intend.