Chapter 10 · Parity & Synthetics

平价关系与合成头寸：期权世界的守恒律Parity and Synthetics: The Conservation Law of Options

Put-Call Parity 不依赖正态分布或 Black-Scholes，但它有清楚的适用条件：同一标的、同一行权价与到期日的欧式 Call 和 Put，在无套利且融资、做空、股息与其他 carry 被正确计入时，组成相同到期付款的组合必须同价。在这些条件下它是到期无套利恒等式；换成美式行权、漏掉股息或遇到融资与借券限制，就需要调整等式或改用边界。理解这条边界，才会重新排列你对前九章的认识。 Put-call parity does not require lognormal returns or Black-Scholes, but it has explicit conditions: European calls and puts on the same underlying, strike, and expiry must price identical expiry payments equally when no-arbitrage holds and financing, shorting, dividends, and other carry are handled correctly. Under those conditions it is an expiry no-arbitrage identity. With American exercise, omitted dividends, or financing and borrow constraints, the equation needs adjustment or becomes a bound. Its power comes from understanding that boundary.

10.1 一个不需要模型的等式10.1 An Equation That Needs No Model

考虑两个组合。组合 A：买入一张 Call，同时卖出一张同 Strike、同到期日的 Put。组合 B：直接买入 100 股正股，并借入 K 的现值。 Consider two portfolios. Portfolio A: buy one call and sell one put at the same strike and expiry. Portfolio B: buy 100 shares outright and borrow the present value of K.

到期时，无论股价在哪里，这两个组合的价值完全相同。股价高于 K：A 的 Call 行权、Put 作废，你以 K 买入股票；B 你已经持股、还掉 K。股价低于 K：A 的 Put 被行权，你被迫以 K 买入股票；B 一样。两个组合在所有情形下都给出同一个结果，所以它们今天必须同价——否则任何人都能买低卖高、白拿差价。 At expiry, wherever the stock lands, the two portfolios are worth exactly the same. Above K: A's call is exercised and the put expires, so you buy at K; B already holds the shares and repays K. Below K: A's put is assigned and you are forced to buy at K; B, the same. Two portfolios that pay identically in every state must cost the same today — otherwise anyone could buy one, sell the other, and pocket the difference for free.

C − P = S − K · e^−rT

这就是 Put-Call Parity（看涨看跌平价关系）。请注意它的右边没有波动率——没有 σ、没有 IV、没有任何关于"股价会怎么走"的假设。它只包含股价、行权价、利率和时间，全部是可观察的量。Black-Scholes 可能给出错误的价格，平价关系不会；前者是一个关于世界如何运转的猜测，后者只是一条会计恒等式。 This is put-call parity. Note what is absent from the right-hand side: no volatility, no IV, no assumption whatsoever about how the stock will move. It contains only spot, strike, rate and time — all observable. Black-Scholes can give you a wrong price; parity cannot. The former is a conjecture about how the world works; the latter is an accounting identity.

组合 A 的盈亏图值得亲眼看一次。买 Call + 卖同 Strike 的 Put，画出来是一条斜率为 1 的直线——和持有正股一模一样。期权的所有弯折都消失了。 Portfolio A's payoff is worth seeing once with your own eyes. Long call plus short put at the same strike draws a straight line of slope 1 — identical to holding the stock. Every kink an option has vanishes.

SYNTHETIC LONG STOCK Call(+) + Put(−) 同 Strikelong call + short put, same strike

新图例：那条橙色虚线New to the charts: the amber dashed line

从这一章起，交互图上会多出一条橙色虚线。前九章的实线画的是到期那一刻的盈亏——所有时间价值都已归零；而虚线画的是此刻的盈亏，也就是把第 3 章讲的时间价值算进去之后，你的账户现在真实显示的数字。两条线之间的垂直距离，就是尚未流逝的时间价值。随着到期临近，虚线会一天天塌向实线——这正是 Theta，只不过现在你能亲眼看着它发生。灰色的竖点线标记开仓时的标的价格。 From this chapter on, the interactive charts carry an extra amber dashed line. The solid line, as in the first nine chapters, is P/L at expiry, when all time value has gone to zero. The dashed line is P/L right now — what your account actually shows once Chapter 3's time value is included. The vertical gap between them is the time value not yet surrendered. As expiry approaches, the dashed line collapses onto the solid one — that is theta, except now you can watch it happen. The faint vertical dotted line marks the underlying's price at entry.

这条等式的实际用途What the identity is actually for

你不会靠它套利——真实市场里这个价差早已被专业机构压到手续费以内。它的价值在于换算：任何一个头寸都可以用等价的另一组腿表达，而不同表达方式的手续费、保证金占用、税务处理和被指派风险完全不同。知道 A ≡ B，你就能在两者中挑成本更低的那个。本章接下来的一切都是这句话的展开。 You will not arbitrage with it — in real markets that spread is compressed inside transaction costs by professionals. Its value is translation: any position can be expressed as an equivalent set of legs, and different expressions carry different commissions, margin requirements, tax treatment and assignment risk. Knowing A ≡ B lets you pick the cheaper one. Everything that follows in this chapter unpacks that sentence.

10.2 合成头寸：同一个形状的六个名字10.2 Synthetics: Six Names for the Same Shape

把平价等式移项，就得到一张合成头寸对照表。每一行的左边和右边，到期盈亏图形状完全相同： Rearranging the identity yields a table of synthetic positions. On every row, the left and right sides have identical payoff shapes:

合成多头Synthetic long: +C −P
合成空头Synthetic short: −C +P
合成 CallSynthetic call: +股 +P+stock +P
合成 PutSynthetic put: −股 +C−stock +C
Covered Call: ≡ Short Put≡ short put

最后一行值得停下来。第 2 章用了两节分别讲 Covered Call 和 Cash-Secured Put，把它们描述成"持股者的预设卖单"和"想买者的预设买单"——两种心态、两种场景、两个名字。而平价关系说：在相同 Strike 和到期日下，它们是同一个头寸。 That last row deserves a pause. Chapter 2 spent two sections on covered calls and cash-secured puts, describing them as "a shareholder's pre-committed sell order" and "a buyer's pre-committed buy order" — two mindsets, two scenarios, two names. Parity says that at the same strike and expiry, they are one position.

COVERED CALL 持股 100 + 卖出 110 Callstock at 100 + short 110 call

SHORT PUT 卖出 110 Put — 请对比上图的形状short 110 put — compare the shape above

两张图的形状严丝合缝：同样的水平上限、同样的斜率、同样的拐点。用引擎在 $70 到 $150 之间逐点相减，两者的差值恒等于 $1.09——一个不随股价变化的常数，而这个常数恰好等于 K·(1 − e^−rT)，也就是持有 $110 现金三个月的利息。差别不在风险形态上，只在"你的钱停在哪里"——把 CSP 那 $110 担保现金按无风险利率计息，这 $1.09 恰好被补回，两者在经济上完全相等。 The two shapes fit exactly: same flat ceiling, same slope, same kink. Subtract them point by point from $70 to $150 with the engine and the difference is a constant $1.09 — invariant in the stock price, and precisely equal to K·(1 − e^−rT), i.e. three months' interest on $110 of cash. The difference isn't in the risk profile at all; it's in where your money is parked — credit the CSP's $110 of collateral cash with the risk-free rate and the $1.09 is restored exactly, making the two economically identical.

这对第 2 章和第 7 章意味着什么What this does to Chapters 2 and 7

如果 Covered Call ≡ Short Put，那么第 7 章的 Wheel 就不是"两个策略的闭环"，而是同一个头寸在两种抵押形式之间来回切换：现金担保时叫 CSP，股票担保时叫 Covered Call。轮子转的不是策略，是抵押品。 If a covered call is a short put, then Chapter 7's Wheel isn't a loop between two strategies — it is one position alternating between two forms of collateral: cash-secured it's called a CSP, share-secured it's called a covered call. What the wheel turns is not the strategy; it's the collateral.

这不会让第 7 章的结论失效——恰恰相反，它解释了那个结论为什么成立。第 7 章说"Wheel 的盈亏在第 ① 步就决定了"，现在你知道原因：整个闭环从头到尾只有一个风险敞口，就是"在 Strike 附近持有这只股票"。转轮子换不掉它。而实操上真正的区别只有三个：资金占用、被指派后的税务事件、以及股票是否付股息。 This doesn't invalidate Chapter 7 — it explains why Chapter 7 was right. That chapter said the Wheel's P/L is settled at step ①; now you know the reason: the entire loop carries exactly one exposure, "own this stock around the strike," and turning the wheel never changes it. The genuine practical differences are only three: capital tied up, the taxable event on assignment, and whether the stock pays a dividend.

思考Think 既然 Covered Call 和 CSP 是同一个头寸，为什么绝大多数人觉得前者"稳健"、后者"激进"？如果两个风险完全相同的东西给人截然不同的感受，那么这种感受是在衡量风险，还是在衡量别的什么？把这个问题带回第 9 章的仓位规则：你的仓位上限，应该由头寸的形状决定，还是由它的名字决定？ If a covered call and a CSP are the same position, why do most people feel the first is "conservative" and the second "aggressive"? If two identically risky things produce opposite feelings, is that feeling measuring risk — or measuring something else? Carry the question back to Chapter 9's sizing rules: should your size limit be set by a position's shape, or by its name?

10.3 Box Spread：把平价关系做成一笔借贷10.3 The Box Spread: Parity Turned Into a Loan

把合成多头和合成空头放在两个不同的 Strike 上，你就得到一个 Box Spread（箱体价差）：在 K₁ 合成买入，在 K₂ 合成卖出。到期时，你以 K₁ 买、以 K₂ 卖，无论股价在哪里，收益恒等于 K₂ − K₁。 Put a synthetic long and a synthetic short at two different strikes and you have a box spread: synthetically buy at K₁, synthetically sell at K₂. At expiry you buy at K₁ and sell at K₂, so the payoff equals K₂ − K₁ regardless of where the stock is.

BOX SPREAD 100/110 四条腿，两条水平线four legs, two flat lines

这张图和本书其他任何一张都不一样：两条线都是水平的。股价从 $80 走到 $130，盈亏一动不动——四条腿把方向暴露完全抵消掉了。虚线（此刻）贴在零附近，实线（到期）停在约 +$0.10，两者之间那道窄缝就是三个月的利息。拖动"已过天数"滑块，你会看到虚线像债券计息一样，一天天匀速爬向实线——没有弯折，没有加速，因为这里根本没有时间价值可言，只有利息在累积。 This chart is unlike any other in the book: both lines are flat. The stock travels from $80 to $130 and the P/L does not move — four legs cancel the directional exposure completely. The dashed line (today) sits at zero, the solid one (at expiry) rests near +$0.10, and the narrow gap between them is three months of interest. Drag the days-elapsed slider and the dashed line climbs toward the solid one at a constant rate, exactly like a bond accruing — no kink, no acceleration, because there is no time value here at all, only interest adding up.

一个到期时必然值 $10、且与股价完全无关的东西——那不是期权头寸，那是一张零息债券。所以它今天的价格必然是 $10 的现值：10 · e^−rT。按 4% 利率、三个月计，约 $9.90。 Something guaranteed to be worth $10 at expiry and utterly independent of the stock is not an options position — it is a zero-coupon bond. So today it must cost the present value of $10: 10 · e^−rT, which at 4% over three months is about $9.90.

于是 Box Spread 有了一个非常实际的用途：买入 box = 放贷，卖出 box = 借款。机构用它在期权市场里以接近国债的利率融资，绕开券商的融资利率。卖出一个 box，你收到 $9.90，到期还 $10——这就是一笔利率 4% 的借款，只是穿着四条期权腿的外衣。 This gives the box a very practical use: buying one is lending, selling one is borrowing. Institutions use them to finance at near-Treasury rates inside the options market, bypassing their broker's margin rate. Sell a box, receive $9.90, repay $10 at expiry — that is a 4% loan wearing four option legs as a costume.

ANONYMOUS FORUM SELF-REPORT · 2019-01

"它字面意义上不可能爆仓""It Literally Cannot Go Tits Up"

2019 年 1 月，Reddit 论坛 r/wallstreetbets 上一位化名 1R0NYMAN 的用户发帖，声称自己在 Robinhood 上卖出 Box Spread 来“无风险”融资。以下经过与金额来自匿名论坛自述及后续转述，不是经审计账户记录或监管调查结论。它们能证明“这个故事是如此被当事人讲述的”，不能独立证明每个数字。 In January 2019, a user posting as 1R0NYMAN on Reddit's r/wallstreetbets claimed to be selling box spreads on Robinhood for "risk-free" financing. The sequence and dollar amounts below come from anonymous forum self-report and later retellings, not audited account records or a regulatory investigation. They establish how the participant told the story, not independent verification of every number.

论坛叙事描述的是 UVXY 期权。若该产品与持仓描述属实，关键机制就在行权条款：欧式 box 的到期锁定不适用于可提前行权的美式腿。一手产品文件可以证明 UVXY 可交易期权、标准 ETP 期权采用美式行权；它们不能证明该匿名账户具体持有什么。 The forum account describes options on UVXY. If that product and position description is accurate, the key mechanism is exercise style: a European box's expiry lock does not apply unchanged to American legs that can be exercised early. Primary product documents establish that options are available on UVXY and that standard ETP options use American exercise; they do not establish what the anonymous account actually held.

按匿名帖子及转述的说法，UVXY 当时在 $50 附近，其中一条空头腿是 $15 Call，随后被提前行权。这组价格、持仓与事件顺序没有独立审计。它所演示的机制则成立：若一条深度价内空头腿先被行权，四腿到期锁定会被拆开，剩余头寸可能暴露方向风险。 According to the anonymous post and retellings, UVXY was near $50, one short leg was a $15 call, and that leg was exercised early. Those prices, positions, and sequence have not been independently audited. The mechanism they illustrate is valid: if a deep-ITM short leg is exercised first, the four-leg expiry lock is broken and the remainder may carry directional exposure.

论坛自述本金Forum-reported capital: $5,000
论坛自述授信Forum-reported credit: 约 $250,000（约 50 倍）≈ $250,000 (≈ 50×)
论坛自述负余额Forum-reported deficit: 约 $58,000≈ $58,000
证据状态Evidence status: 匿名、未审计anonymous, unaudited

匿名叙事还称发帖者此前提走 $10,000，强平后账户为负；这些结果同样未审计。能负责任得出的结论只限于机制：提前行权是欧式 box 到期恒等式没有覆盖的一条真实路径风险，不能据此完成对匿名账户的单因果归因。 The anonymous account also says the poster had withdrawn $10,000 and the account was negative after liquidation; those outcomes are likewise unaudited. The defensible conclusion is limited to mechanism: early exercise is a real path risk outside the European box's expiry identity, not a complete causal account of an anonymous portfolio.

这仍然说明一条重要方法：到期恒等式不能替代路径与合约审查。第 1 章的欧式条款、第 8 章的加速赎回和这里的美式提前行权，都会改变“等到到期”的权利与义务。匿名故事的金额不能当事实，但一手合约文件足以提醒你在建模前先核对行权、结算和提前终止条款。 The case still supports an important method: an expiry identity cannot replace path and contract review. Chapter 1's European exercise, Chapter 8's acceleration clause, and American early exercise here all change the rights and obligations involved in “waiting until expiry.” The anonymous story's amounts are not established facts, but primary contract documents are enough to require checking exercise, settlement, and early-termination terms before modelling.

证据边界：$5,000 自有资金、约 $250,000 授信、约 $58,000 亏损、提走 $10,000 与经过先后，都只是论坛自述口径，本书无法独立核实。约 50 倍只是对这组自述数字做的除法，不是券商授信记录的审计结论。可由一手文件独立验证的只是机制：ProShares 确认 UVXY 可交易期权，Cboe 确认标准 ETP 期权采用美式行权，因此深度价内空头腿可被提前行权，并打破四腿同时存在时的到期锁定。本案例不重建具体持仓或真实损益。来源与方法 Evidence boundary: the $5,000 capital, approximately $250,000 credit, approximately $58,000 loss, $10,000 withdrawal, and sequence are forum self-report that this book cannot independently verify. The roughly 50× figure is only division on those self-reported inputs, not an audit of broker credit records. Primary documents verify only the mechanism: ProShares confirms options are available on UVXY, and Cboe confirms standard ETP options use American exercise, so a deep-ITM short leg can be exercised early and break the expiry lock that exists while all four legs remain. This case does not reconstruct the actual position or P/L. Sources and methodology

机制结论：欧式 box 的到期 width 锁定不等于美式腿的路径无风险；论坛所述金额与实际损益未经审计，不作为事实结论Mechanism conclusion: a European box's expiry-width lock does not make American legs path-risk-free; the forum's amounts and actual P/L are unaudited and are not factual findings

10.4 平价关系管不到的地方10.4 Where Parity Has No Jurisdiction

平价关系强大，但它的管辖范围有明确边界，而理解这条边界正是第 3 章 Skew 的理论基础。 Parity is powerful, but its jurisdiction has a precise boundary — and understanding that boundary is the theoretical basis for Chapter 3's skew.

它管得到的：同一个 Strike 上的 Call 和 Put。由等式可以直接推出一个漂亮的结论——同一 Strike、同一到期日的 Call 和 Put，隐含波动率必然相等。因为它们的价格被恒等式锁死，而 Black-Scholes 是价格到 IV 的单调映射：价格锁死，IV 就锁死。如果你的行情软件显示同一 Strike 的 Call 和 Put 有明显不同的 IV，那不是市场机会——多半是数据源用了不同的插值或时间戳。（严格地说，这条推论只对欧式期权精确成立，SPX 这类指数期权上几乎分毫不差；美式个股期权会因提前行权权、分红与借券成本出现小而系统的偏差——而那个偏差本身是有信息的：它是市场对分红和融券成本的报价，不是白捡的套利。） Within jurisdiction: the call and put at one strike. A neat corollary follows straight from the identity — a call and a put at the same strike and expiry must have identical implied volatility. Their prices are pinned by the identity, and Black-Scholes maps price to IV monotonically: pin the price and you pin the IV. If your platform shows clearly different IVs for a call and put at the same strike, that isn't an opportunity — most likely a data feed using different interpolation or timestamps. (Strictly, the corollary is exact only for European options — on index products like SPX it holds almost to the penny. American single-stock options show small, systematic deviations from early-exercise rights, dividends and borrow costs — and that deviation is itself informative: it is the market's quote for dividends and stock-borrow, not free money.)

它管不到的：不同 Strike 之间。平价关系对 $90 的 Put 和 $110 的 Call 之间的相对价格一句话都没有说。市场因此完全自由地给它们定不同的 IV——而它确实这么做了，方向还非常固定：价外 Put 的 IV 系统性地高于价外 Call。第 8 章告诉过你这个偏斜从哪一天开始：1987 年 10 月 19 日。平价关系解释了为什么偏斜可以存在——因为没有任何恒等式禁止它。 Outside jurisdiction: across strikes. Parity says nothing whatsoever about the relative price of a $90 put and a $110 call. The market is therefore entirely free to assign them different IVs — and it does, in a very consistent direction: OTM puts carry systematically higher IV than OTM calls. Chapter 8 told you which day that began: October 19, 1987. Parity explains why skew is permitted to exist — because no identity forbids it.

一条可以直接用的推论A corollary you can use directly

既然同 Strike 的 Call 与 Put 由恒等式绑定，那么"买 Call" 和 "买股票 + 买 Put" 在风险上完全等价（第 2 章的 Protective Put 其实一直是一张合成 Call）。于是选择哪一个，就不再是风险判断，而是成本比较：哪个占用保证金少、哪个手续费低、哪个不会因为持股而在除息日被提前指派。把"我该用哪个策略"换成"同一个形状，哪种腿最便宜"，你的问题就从玄学变成了算术。 Since a call and put at one strike are bound by the identity, "buy a call" and "buy stock plus a put" are risk-equivalent — Chapter 2's protective put was a synthetic call all along. Choosing between them therefore isn't a risk judgment but a cost comparison: which ties up less margin, which costs less in fees, which avoids early assignment around an ex-dividend date. Replace "which strategy should I use" with "same shape — which set of legs is cheapest," and the question stops being metaphysics and becomes arithmetic.

10.5 Collar：为偏斜付账10.5 The Collar: Paying the Skew Bill

Collar（领口）在术语表里躺了九章，现在终于到了它的位置：持股 + 买入价外 Put + 卖出价外 Call，用卖 Call 的收入补贴买 Put 的保费。它常被宣传为“零成本对冲”，但净成本取决于行权价、期限、carry 与整条 IV 曲面。下一案例固定其他模型输入，只隔离偏斜对可卖 Call 距离的影响。 The collar has sat in the glossary for nine chapters; here is its place. Hold stock, buy an OTM put, sell an OTM call, and let the call premium subsidize the put. It is marketed as a “zero-cost hedge,” but net cost depends on strikes, tenor, carry, and the full IV surface. The next case holds other model inputs fixed and isolates skew's effect on how far away the short call can be placed.

ISSUER METHOD + 本书示意情景BOOK SCENARIO

摩根大通公开的领口方法：用自洽情景看保护如何定价J.P. Morgan's Published Collar Method: A Self-Contained Pricing Scenario

摩根大通在官方材料中说明，Hedged Equity 系列持有美国大盘股组合，并使用每三个月重置的 Put-spread Collar：买入价外 Put、卖出更深的 Put，再卖 Call 补贴差额。发行人材料支持这个结构与重置周期，但不支持媒体所述某季度的精确持仓、合约数或展仓影响。本案例因此只以官方方法为原型，不重建 JHEQX 的实际仓位。 J.P. Morgan's official materials say the Hedged Equity series holds a U.S. large-cap portfolio and uses a put-spread collar reset every three months: buy an OTM put, sell a deeper put, then sell a call to subsidize the difference. Issuer materials support that structure and reset cycle, but not media claims about a particular quarter's exact holdings, contract count, or market impact. This case therefore uses the published method as its prototype and does not reconstruct JHEQX's actual book.

为了让结构可交互复算，本书设定一个完全示意的指数水平 6,475 与三条腿。下面的 Strike、IV、合约数与损益都不是任何基金的披露或实际交易： To make the structure interactively reproducible, the book sets a wholly illustrative index level of 6,475 and three legs. The strikes, IVs, contract count, and P/L below are not a fund disclosure or actual trade:

买入 PutLong put: 6180 （−4.6%）(−4.6%)
卖出 PutShort put: 5210 （−19.5%）(−19.5%)
卖出 CallShort call: 6865 （+6.0%）(+6.0%)
净成本Net cost: 目标接近于零targeted near zero

PUT-SPREAD COLLAR 示意情景 · 虚线为当期价值，实线为到期illustrative scenario · dashed = value today, solid = at expiry

请注意它为什么需要第三条腿。如果只买 −4.6% 的 Put、只卖 +6.0% 的 Call，两者并不能互相抵消——按符合典型 SPX 偏斜形态的示意 IV 计算，这两条腿净支出约 $44/单位指数。必须再卖出一个 −19.5% 的深度价外 Put，才能把成本压下来。图中 readout 显示的约 $32 净支出，就是这组示意 IV 下仍未补平的残差——它和下一段“零成本解在 +4.0%、实际 Call 卖在 +6.0%”是同一件事：Call 卖得越远，收回的钱越少。保护成本不只由距离与期限决定；在本章固定行权价、期限和利率的对照中，下跌偏斜还额外抬高了 Put 相对 Call 的价格。 Note why it needs a third leg. Buying the −4.6% put and selling the +6.0% call does not net out: at illustrative IVs shaped like a typical SPX skew, those two legs cost roughly $44 per unit of index. Only by also selling a −19.5% deep OTM put does the cost come down. The ≈$32 net debit in the chart's readout is the residual these illustrative IVs still leave uncovered — the same fact as the next paragraph's “the zero-cost solve sits at +4.0% while the actual call is sold at +6.0%”: the further out you sell the call, the less it brings in. Protection cost depends on more than distance and tenor; in this chapter's comparison holding strikes, tenor, and rates fixed, downside skew adds to the put's price relative to the call.

把偏斜的代价量化出来最能说明问题。用同一组示意 IV 反解“能让整个结构零成本的 Call 行权价”：在这个偏斜情景里只能卖出 +4.0% 的 Call；在假想的无偏斜情景（所有 Strike 同一 IV）里可以卖出 +7.2% 的 Call。对同一组模型输入，偏斜的代价是 3.2 个百分点的示意上涨空间。这不是对 JHEQX 实际展仓或当时 SPX 偏斜的估计。 Quantifying the skew bill makes the point best. Using the same illustrative IVs, solve for the call strike that makes the structure zero-cost: in this skewed scenario the call can be sold at only +4.0%; in a hypothetical flat-IV scenario it can be sold at +7.2%. On this same set of model inputs, skew costs 3.2 percentage points of illustrative upside. This is not an estimate of JHEQX's actual roll or the SPX skew at that time.

保护的边界也要看清楚。在这组示意行权价下，指数跌到 −19.5% 之前，组合的损失被压在约 −4.6%；跌破 −19.5% 之后，卖出的那条 Put 开始反噬，保护失效。指数 −25% 时，示意组合约亏 −10.0%。这是一个为“大跌”设计、而非为“崩盘”设计的损益形状——第 4 章 ARKK 案例的同一个结构边界。 The boundary of the protection matters too. At these illustrative strikes, down to −19.5% the portfolio's loss is pinned near −4.6%; below −19.5% the short put bites back and the protection lapses. At −25% the illustrative collar loses about −10.0%. This payoff shape is built for a bad decline, not for a crash — the same structural boundary as Chapter 4's ARKK case.

方法声明：发行人一手材料只用于支持“大盘股组合 + 每三个月重置的 Put-spread Collar”。6,475、6180 / 5210 / 6865、各腿 IV（18% / 26% / 12%）及所有权利金都是本书示意输入或 Black-Scholes 输出，非实际报价、基金披露或持仓推算。保护区间按到期内在价值计算，忽略净成本残差。来源与方法 Method statement: primary issuer material supports only "large-cap portfolio plus a put-spread collar reset every three months." The 6,475 level, 6180 / 5210 / 6865 strikes, per-leg IVs (18% / 26% / 12%), and all premiums are book-supplied illustrative inputs or Black-Scholes outputs, not quotes, fund disclosures, or inferred holdings. Protection percentages use expiry intrinsic value and ignore the residual net cost. Sources and methodology

亏损减少：指数下跌 19.5% 时，裸持仓 −19.5% → 带领口约 −4.6%；但跌破 −19.5% 保护失效，−25% 时约 −10.0%Loss reduced: at −19.5% on the index, unhedged −19.5% → collared about −4.6%; below −19.5% the protection lapses, and at −25% the collar is down about −10.0%

思考Think 这个领口把组合的季度结果框定在大约 −4.6% 到 +6.0% 之间。如果你持有它十年，会发生什么？标普 500 单季涨幅超过 6% 的频率并不低。一个反复削掉右尾、却在极端下跌时失效的结构，长期的复合收益会是什么形状？把这个问题和第 9 章的"回撤算术"放在一起想。 This collar boxes each quarter's outcome between roughly −4.6% and +6.0%. What happens if you hold it for a decade? The S&P 500 gains more than 6% in a quarter reasonably often. What shape does the compounded return take for a structure that repeatedly shaves the right tail while lapsing in the extreme left one? Think about it alongside Chapter 9's arithmetic of ruin.

10.6 本章测验10.6 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.

Put-Call Parity 与 Black-Scholes 最根本的区别是什么？What is the fundamental difference between put-call parity and Black-Scholes?

C。等式右边 S − K·e^−rT 里没有 σ——它不猜测未来，只要求"付款完全相同的两个组合必须同价"。D 正好说反了。B 有一点道理但不是"根本区别"：严格形式确实要求欧式（美式的提前行权会让等式变成不等式），而这正是 10.3 案例的要害。A 混淆了范畴：Black-Scholes 是模型，可能错；平价关系是恒等式，只要能借贷和做空就不可能错。 C. The right-hand side S − K·e^−rT contains no σ — it forecasts nothing and requires only that two portfolios with identical payoffs cost the same. D says the opposite of the truth. B has a grain of truth but isn't the fundamental difference: the strict form does require European exercise (American early exercise turns the equation into an inequality), which is precisely the point of §10.3's case. A confuses categories: Black-Scholes is a model and can be wrong; parity is an identity and cannot be, given borrowing and shorting.

同 Strike、同到期日的 Covered Call 和 Short Put，风险有何不同？How does the risk of a covered call differ from a short put at the same strike and expiry?

B。平价关系直接给出这个等价：本章用引擎在 $70–150 间逐点相减，差值恒为 $1.09，正好等于 K(1 − e^−rT)。A 是最普遍的错觉——"有股票抵押"改变的是抵押品形式和资金占用，不改变风险形态（股票跌，你照样亏）。C 错在股价不会低于零（无限亏损属于 Short Call）。这道题真正要问的是：你的风险感受，是来自头寸的形状，还是来自它的名字？ B. Parity gives the equivalence directly: subtracting the two point by point from $70 to $150 with the engine yields a constant $1.09, exactly K(1 − e^−rT). A is the commonest illusion — being "backed by shares" changes the form of collateral and the capital tied up, not the risk profile (the stock falls and you lose either way). C is wrong because a share price has a floor at zero (unlimited loss belongs to short calls). The real question: does your sense of risk come from the position's shape, or from its name?

一个 100/110 的 Box Spread，三个月后到期。它今天大约值多少（利率 4%）？A 100/110 box spread expiring in three months. Roughly what is it worth today (rate 4%)?

D。box 到期必然值宽度 $10 且与股价无关，所以它就是一张零息债券，今天的价格是现值 10·e^{−0.04×0.25} ≈ $9.90。A、B 描述的是普通期权头寸——box 的全部意义就在于这两个因素被四条腿彻底抵消掉了。C 忽略了货币的时间价值，而那个被忽略的一毛钱正是整个结构的用途：它是一笔 4% 的借贷。 D. The box is worth its $10 width at expiry regardless of the stock, so it is a zero-coupon bond and today costs the present value, 10·e^{−0.04×0.25} ≈ $9.90. A and B describe ordinary option positions — the whole point of a box is that four legs cancel both factors completely. C ignores the time value of money, and the ten cents it ignores is the entire purpose of the structure: it is a 4% loan.

按匿名论坛叙事所描述的机制，为什么那组 box spread 不是路径无风险的？Under the mechanism described by the anonymous forum account, why was that box spread not path-risk-free?

A。欧式 box 的到期数学成立，但美式短腿可能在到期前被行权，从而拆开四腿锁定并暴露剩余头寸。ProShares 与 Cboe 一手文件支持 UVXY 可交易期权及标准 ETP 期权的美式行权机制；约 $50、$15 Call、账户金额和事件顺序只来自匿名论坛自述，未经独立审计。C 把到期恒等式与路径风险混为一谈，D 则忽略了真正需要核对的是合约条款。 A. The European box's expiry arithmetic holds, but an American short leg may be exercised before expiry, breaking the four-leg lock and exposing the remainder. Primary ProShares and Cboe documents support options availability on UVXY and American exercise for standard ETP options; the roughly $50 price, $15 call, account amounts, and sequence come only from anonymous forum self-report and are unaudited. C confuses an expiry identity with path risk, while D misses the need to inspect contract terms.

在本章的示意 Put-spread Collar 中，为什么除了卖 Call，还要卖出一个很深的价外 Put？In the chapter's illustrative put-spread collar, why sell a deep OTM put in addition to the call?

B。在本章固定行权价、期限和利率的示意偏斜输入下，“买 −4.6% Put + 卖 +6.0% Call”两条腿净支出约 $44/单位指数，远非零成本；必须再卖出 −19.5% 的 Put 才能继续压低成本。这只证明该模型对照中的偏斜账单，不代表距离对保费没有影响。A 恰恰说反了——卖出 Put 是在放弃保护：跌破 −19.5% 后它开始反噬，指数 −25% 时组合约亏 −10.0%。这条腿换来的是权利金，不是更多保护。 B. Under this chapter's illustrative skew inputs with strikes, tenor, and rates fixed, “buy the −4.6% put, sell the +6.0% call” costs roughly $44 per unit of index — nowhere near zero; selling the −19.5% put reduces that cost further. This identifies the skew bill in this model comparison; it does not say moneyness is irrelevant to premium. A is exactly backwards: selling a put gives protection away — below −19.5% it bites back, and at −25% the portfolio is down about −10.0%. That leg raises premium, not protection.

本章要点Key Takeaways

Put-Call Parity 是这本书里最坚固的到期恒等式之一：C − P = S − K·e^−rT 里没有波动率、没有分布假设，只有“付款相同的两个组合必须同价”。它的推论会重排你对前九章的理解——Covered Call 与 Short Put 具有相同到期损益形状（引擎逐点验证，差值恒为一个等于现金利息的常数 $1.09），于是第 7 章的 Wheel 是同一种风险形状在现金抵押与股票抵押之间切换；Protective Put 对应一张合成 Call，于是“用哪个策略”变成了“同一个形状，哪种腿的成本、条款和执行更合适”。四条欧式腿组成的 Box 在到期上等价于零息债券，但美式提前行权会引入路径风险。匿名 1R0NYMAN 故事只能作为未经审计的论坛自述；一手文件支持的是机制，不支持其金额、持仓和因果顺序。至于平价关系管不到的地方——不同 Strike 之间——市场可以开出偏斜；本章的示意输入表明，同样的下跌保护在偏斜情景中会少 3.2 个百分点的上涨空间。这是模型对比，不是 JHEQX 实际展仓的估计。 Put-call parity is one of the book's strongest expiry identities: C − P = S − K·e^−rT contains no volatility or distributional assumption, only the requirement that portfolios with identical payments have identical value. Its corollaries rearrange the first nine chapters. A covered call and a short put have the same expiry payoff shape — the engine verifies point by point that they differ by a constant $1.09 equal to interest on cash — so Chapter 7's Wheel moves one risk shape between cash and share collateral. A protective put corresponds to a synthetic call, turning “which strategy?” into “which implementation has the better costs, terms, and execution?” A box of four European legs is bond-like at expiry, but American early exercise introduces path risk. The anonymous 1R0NYMAN story is unaudited forum self-report; primary documents support the mechanism, not its amounts, positions, or causal sequence. Where parity has no jurisdiction — across strikes — the market can charge skew. With this chapter's illustrative inputs, identical downside protection gives up 3.2 more percentage points of upside in the skewed scenario. That is a model comparison, not an estimate of JHEQX's actual roll.