## 1. The one principle behind everything > **If two portfolios have identical payoffs in every future state of the world, they must have identical prices today.** This is the **no-arbitrage principle (无套利原则)**. Every parity relationship in options is a special case of it. ### The recipe for finding any parity relationship 1. Pick the instrument you want to understand 2. Build a portfolio out of _other_ instruments with the **same payoff at expiration** 3. Set today's prices equal 4. Solve --- ## 2. Put–call parity, derived from scratch Payoffs at expiration: - **Long call** at strike $K$: $\max(S - K,, 0)$ - **Long put** at strike $K$: $\max(K - S,, 0)$ - **Long stock**: $S$ - **Cash of $K$ at expiration**: costs $K e^{-rT}$ today Algebraic identity: $\max(S-K,0) ;=; (S-K) ;+; \max(K-S,0)$ Translate to instruments and set prices equal: $\boxed{;C - P ;=; S - K e^{-rT};}$ That's **put–call parity (看跌-看涨平价)**. --- ## 3. The basic building blocks ### 3.1 Long stock (多头股票) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="10" fill="#444" text-anchor="middle">S₀ (entry)</text> <line x1="100" y1="200" x2="300" y2="60" stroke="#1f77b4" stroke-width="2.5"/> <circle cx="200" cy="130" r="3" fill="#444"/> <text x="200" y="120" font-size="9" fill="#444" text-anchor="middle">BE</text> <text x="305" y="55" font-size="9" fill="#2ca02c">↑ max gain: unlimited</text> <text x="60" y="240" font-size="9" fill="#d62728">max loss: −S₀ (if stock → 0)</text> </svg> A 45° line up. Breakeven at the entry price; **max gain unlimited**, **max loss = entire investment**. ### 3.2 Short stock (空头股票) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="10" fill="#444" text-anchor="middle">S₀ (entry)</text> <line x1="100" y1="60" x2="300" y2="200" stroke="#d62728" stroke-width="2.5"/> <circle cx="200" cy="130" r="3" fill="#444"/> <text x="200" y="120" font-size="9" fill="#444" text-anchor="middle">BE</text> <text x="60" y="55" font-size="9" fill="#2ca02c">max gain: +S₀ (if stock → 0)</text> <text x="350" y="220" font-size="9" fill="#d62728" text-anchor="end">↓ max loss: unlimited</text> </svg> Mirror image. **Max gain capped** at the entry price; **max loss unlimited**. ### 3.3 Cash / bond (现金 / 债券) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="60" y1="115" x2="350" y2="115" stroke="#2ca02c" stroke-width="2.5"/> <text x="350" y="113" font-size="9" fill="#2ca02c" text-anchor="end">+ interest (constant)</text> </svg> Flat line. Independent of $S$. Cash is a **shifter** — adding it to any position slides the whole diagram up or down without changing its shape. ### 3.4 Long call (多头看涨, strike K) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="11" fill="#444" text-anchor="middle">K</text> <line x1="50" y1="155" x2="350" y2="155" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="168" font-size="9" fill="#d62728" text-anchor="end">max loss = −premium</text> <line x1="60" y1="155" x2="200" y2="155" stroke="#1f77b4" stroke-width="2.5"/> <line x1="200" y1="155" x2="325" y2="30" stroke="#1f77b4" stroke-width="2.5"/> <circle cx="225" cy="130" r="3" fill="#444"/> <text x="225" y="120" font-size="9" fill="#444" text-anchor="middle">BE = K + prem</text> <text x="325" y="48" font-size="9" fill="#2ca02c" text-anchor="end">↑ max gain: unlimited</text> </svg> Hockey stick pointing up-and-right. **Flat at −premium below K**, then 45° up. Breakeven at K + premium. ### 3.5 Short call (空头看涨) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="11" fill="#444" text-anchor="middle">K</text> <line x1="50" y1="105" x2="350" y2="105" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="100" font-size="9" fill="#2ca02c" text-anchor="end">max gain = +premium</text> <line x1="60" y1="105" x2="200" y2="105" stroke="#d62728" stroke-width="2.5"/> <line x1="200" y1="105" x2="325" y2="210" stroke="#d62728" stroke-width="2.5"/> <circle cx="225" cy="130" r="3" fill="#444"/> <text x="225" y="148" font-size="9" fill="#444" text-anchor="middle">BE = K + prem</text> <text x="325" y="225" font-size="9" fill="#d62728" text-anchor="end">↓ max loss: unlimited</text> </svg> Upside-down hockey stick. **Max gain capped** at premium; **max loss unlimited**. ### 3.6 Long put (多头看跌) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="220" y1="30" x2="220" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="220" y="225" font-size="11" fill="#444" text-anchor="middle">K</text> <line x1="50" y1="155" x2="350" y2="155" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="168" font-size="9" fill="#d62728" text-anchor="end">max loss = −premium</text> <line x1="95" y1="60" x2="220" y2="155" stroke="#1f77b4" stroke-width="2.5"/> <line x1="220" y1="155" x2="350" y2="155" stroke="#1f77b4" stroke-width="2.5"/> <circle cx="195" cy="130" r="3" fill="#444"/> <text x="170" y="120" font-size="9" fill="#444" text-anchor="middle">BE = K − prem</text> <text x="100" y="55" font-size="9" fill="#2ca02c">max gain ≈ K − prem (if stock → 0)</text> </svg> Hockey stick pointing up-and-left. Breakeven at K − premium. **Max gain large but bounded** (stock can only fall to zero). ### 3.7 Short put (空头看跌) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="220" y1="30" x2="220" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="220" y="225" font-size="11" fill="#444" text-anchor="middle">K</text> <line x1="50" y1="105" x2="350" y2="105" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="100" font-size="9" fill="#2ca02c" text-anchor="end">max gain = +premium</text> <line x1="95" y1="200" x2="220" y2="105" stroke="#d62728" stroke-width="2.5"/> <line x1="220" y1="105" x2="350" y2="105" stroke="#d62728" stroke-width="2.5"/> <circle cx="195" cy="130" r="3" fill="#444"/> <text x="170" y="148" font-size="9" fill="#444" text-anchor="middle">BE = K − prem</text> <text x="100" y="220" font-size="9" fill="#d62728">max loss ≈ −(K − prem)</text> </svg> Upside-down. **Max gain capped** at premium; **max loss large but bounded** (stock can fall to zero). > [!tip] The mental model Six shapes. That's it. Anything else in this note is a **superposition** of these. --- ## 4. Named structures ### 4.1 Income / yield enhancement #### Covered call (备兑看涨) = long stock + short call <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="10" fill="#444" text-anchor="middle">K = S₀</text> <line x1="50" y1="105" x2="350" y2="105" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="100" font-size="9" fill="#2ca02c" text-anchor="end">max gain = +premium</text> <line x1="120" y1="185" x2="200" y2="105" stroke="#9467bd" stroke-width="2.5"/> <line x1="200" y1="105" x2="350" y2="105" stroke="#9467bd" stroke-width="2.5"/> <circle cx="175" cy="130" r="3" fill="#444"/> <text x="150" y="120" font-size="9" fill="#444" text-anchor="middle">BE = S₀ − prem</text> <text x="60" y="220" font-size="9" fill="#d62728">max loss ≈ −(S₀ − prem) (if stock → 0)</text> </svg> Stock-like upside until $K$, then **capped at +premium**. Below $K$ you take full stock downside cushioned by the premium. View: "sideways or modestly up; happy to give up tail upside for premium." Equivalent in shape to a **short put** + cash (parity). #### Cash-secured put (现金担保看跌) = short put + cash equal to strike Same payoff _shape_ as a covered call. Different construction; chosen for capital, tax, or mandate reasons. ### 4.2 Hedging #### Protective put (保护性看跌) = long stock + long put <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="10" fill="#444" text-anchor="middle">K = S₀</text> <line x1="50" y1="155" x2="350" y2="155" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="168" font-size="9" fill="#d62728" text-anchor="end">max loss = −premium (the floor)</text> <line x1="60" y1="155" x2="200" y2="155" stroke="#9467bd" stroke-width="2.5"/> <line x1="200" y1="155" x2="325" y2="30" stroke="#9467bd" stroke-width="2.5"/> <circle cx="225" cy="130" r="3" fill="#444"/> <text x="245" y="120" font-size="9" fill="#444" text-anchor="middle">BE = S₀ + prem</text> <text x="325" y="48" font-size="9" fill="#2ca02c" text-anchor="end">↑ max gain: unlimited</text> </svg> A **floor below K**, full upside above. Synthetically identical to a **long call** (parity). #### Collar (领口) = long stock + long put (K₁) + short call (K₂) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="140" y1="30" x2="140" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="200" y1="30" x2="200" y2="210" stroke="#bbb" stroke-dasharray="2,4"/> <line x1="260" y1="30" x2="260" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="140" y="225" font-size="11" fill="#444" text-anchor="middle">K₁</text> <text x="200" y="225" font-size="9" fill="#888" text-anchor="middle">S₀</text> <text x="260" y="225" font-size="11" fill="#444" text-anchor="middle">K₂</text> <line x1="50" y1="70" x2="350" y2="70" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="65" font-size="9" fill="#2ca02c" text-anchor="end">max gain = K₂ − S₀</text> <line x1="50" y1="190" x2="350" y2="190" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="203" font-size="9" fill="#d62728" text-anchor="end">max loss = −(S₀ − K₁)</text> <line x1="60" y1="190" x2="140" y2="190" stroke="#9467bd" stroke-width="2.5"/> <line x1="140" y1="190" x2="260" y2="70" stroke="#9467bd" stroke-width="2.5"/> <line x1="260" y1="70" x2="350" y2="70" stroke="#9467bd" stroke-width="2.5"/> <circle cx="200" cy="130" r="3" fill="#444"/> <text x="200" y="120" font-size="9" fill="#444" text-anchor="middle">BE</text> </svg> Boxed between a floor and a ceiling. Often built **zero-cost** (call premium pays for put). Both tails traded away. ### 4.3 Pure volatility (directionally neutral) #### Long straddle (多头跨式) = long call + long put, **same strike** <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="11" fill="#444" text-anchor="middle">K</text> <line x1="50" y1="180" x2="350" y2="180" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="193" font-size="9" fill="#d62728" text-anchor="end">max loss = −total premium</text> <line x1="120" y1="50" x2="200" y2="180" stroke="#9467bd" stroke-width="2.5"/> <line x1="200" y1="180" x2="280" y2="50" stroke="#9467bd" stroke-width="2.5"/> <circle cx="150" cy="130" r="3" fill="#444"/> <circle cx="250" cy="130" r="3" fill="#444"/> <text x="150" y="120" font-size="9" fill="#444" text-anchor="middle">BE₁</text> <text x="250" y="120" font-size="9" fill="#444" text-anchor="middle">BE₂</text> <text x="115" y="42" font-size="9" fill="#2ca02c">↑ unlimited</text> <text x="285" y="42" font-size="9" fill="#2ca02c" text-anchor="end">↑ unlimited</text> </svg> A **V centered at K**. Two breakevens at K ± total premium. View: "something big is coming, don't know which direction." Pure long-vol bet. #### Short straddle = short call + short put, same strike <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="200" y="225" font-size="11" fill="#444" text-anchor="middle">K</text> <line x1="50" y1="80" x2="350" y2="80" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="75" font-size="9" fill="#2ca02c" text-anchor="end">max gain = +total premium</text> <line x1="120" y1="210" x2="200" y2="80" stroke="#d62728" stroke-width="2.5"/> <line x1="200" y1="80" x2="280" y2="210" stroke="#d62728" stroke-width="2.5"/> <circle cx="150" cy="130" r="3" fill="#444"/> <circle cx="250" cy="130" r="3" fill="#444"/> <text x="150" y="148" font-size="9" fill="#444" text-anchor="middle">BE₁</text> <text x="250" y="148" font-size="9" fill="#444" text-anchor="middle">BE₂</text> <text x="115" y="222" font-size="9" fill="#d62728">↓ unlimited</text> <text x="285" y="222" font-size="9" fill="#d62728" text-anchor="end">↓ unlimited</text> </svg> Upside-down V (a tent). Max profit at $S = K$. **Theoretically unlimited loss** in both directions — what blew up funds in Volmageddon 2018. #### Long strangle (多头宽跨式) = long call (K₂) + long put (K₁) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="170" y1="30" x2="170" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="230" y1="30" x2="230" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="170" y="225" font-size="11" fill="#444" text-anchor="middle">K₁</text> <text x="230" y="225" font-size="11" fill="#444" text-anchor="middle">K₂</text> <line x1="50" y1="160" x2="350" y2="160" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="173" font-size="9" fill="#d62728" text-anchor="end">max loss = −total premium</text> <line x1="110" y1="100" x2="170" y2="160" stroke="#9467bd" stroke-width="2.5"/> <line x1="170" y1="160" x2="230" y2="160" stroke="#9467bd" stroke-width="2.5"/> <line x1="230" y1="160" x2="290" y2="100" stroke="#9467bd" stroke-width="2.5"/> <circle cx="140" cy="130" r="3" fill="#444"/> <circle cx="260" cy="130" r="3" fill="#444"/> <text x="140" y="120" font-size="9" fill="#444" text-anchor="middle">BE₁</text> <text x="260" y="120" font-size="9" fill="#444" text-anchor="middle">BE₂</text> <text x="105" y="92" font-size="9" fill="#2ca02c">↑ unlimited</text> <text x="295" y="92" font-size="9" fill="#2ca02c" text-anchor="end">↑ unlimited</text> </svg> Like a straddle but with a **flat bottom** between the strikes. Cheaper (both legs OTM) but needs a bigger move to pay off. ### 4.4 Directional with capped risk/reward (vertical spreads) #### Bull call spread (多头看涨价差) = long call (K₁) + short call (K₂), K₁ < K₂ <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="150" y1="30" x2="150" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="260" y1="30" x2="260" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="150" y="225" font-size="11" fill="#444" text-anchor="middle">K₁</text> <text x="260" y="225" font-size="11" fill="#444" text-anchor="middle">K₂</text> <line x1="50" y1="80" x2="350" y2="80" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="75" font-size="9" fill="#2ca02c" text-anchor="end">max gain = (K₂−K₁) − net debit</text> <line x1="50" y1="160" x2="350" y2="160" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="173" font-size="9" fill="#d62728" text-anchor="end">max loss = −net debit</text> <line x1="60" y1="160" x2="150" y2="160" stroke="#9467bd" stroke-width="2.5"/> <line x1="150" y1="160" x2="260" y2="80" stroke="#9467bd" stroke-width="2.5"/> <line x1="260" y1="80" x2="350" y2="80" stroke="#9467bd" stroke-width="2.5"/> <circle cx="190" cy="130" r="3" fill="#444"/> <text x="190" y="120" font-size="9" fill="#444" text-anchor="middle">BE</text> </svg> Staircase going up. **Both gain and loss capped**. View: "modestly bullish, willing to cap upside in exchange for paying less premium." #### The four vertical spreads at a glance |Name|Construction|Net|View| |---|---|---|---| |Bull call spread|long call K₁ + short call K₂|debit|bullish| |Bull put spread|short put K₂ + long put K₁|credit|bullish| |Bear call spread|short call K₁ + long call K₂|credit|bearish| |Bear put spread|long put K₂ + short put K₁|debit|bearish| Same staircase shape (or its mirror); only the construction differs. Pick by parity / capital / tax considerations. ### 4.5 Pinned-outcome structures #### Long butterfly (蝶式) = long call (K₁) + 2× short call (K₂) + long call (K₃) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="120" y1="30" x2="120" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="200" y1="30" x2="200" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="280" y1="30" x2="280" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="120" y="225" font-size="11" fill="#444" text-anchor="middle">K₁</text> <text x="200" y="225" font-size="11" fill="#444" text-anchor="middle">K₂</text> <text x="280" y="225" font-size="11" fill="#444" text-anchor="middle">K₃</text> <line x1="50" y1="60" x2="350" y2="60" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="55" font-size="9" fill="#2ca02c" text-anchor="end">max gain = (K₂−K₁) − net debit</text> <line x1="50" y1="155" x2="350" y2="155" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="350" y="168" font-size="9" fill="#d62728" text-anchor="end">max loss = −net debit</text> <line x1="60" y1="155" x2="120" y2="155" stroke="#9467bd" stroke-width="2.5"/> <line x1="120" y1="155" x2="200" y2="60" stroke="#9467bd" stroke-width="2.5"/> <line x1="200" y1="60" x2="280" y2="155" stroke="#9467bd" stroke-width="2.5"/> <line x1="280" y1="155" x2="350" y2="155" stroke="#9467bd" stroke-width="2.5"/> <circle cx="140" cy="130" r="3" fill="#444"/> <circle cx="260" cy="130" r="3" fill="#444"/> <text x="140" y="120" font-size="9" fill="#444" text-anchor="middle">BE₁</text> <text x="260" y="120" font-size="9" fill="#444" text-anchor="middle">BE₂</text> </svg> A **tent peaking at K₂**. View: "stock will pin near K₂ at expiration." Small premium upfront, **bounded both directions**. #### Long condor (兀鹰式) = same idea, with a **flat top** <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="100" y1="30" x2="100" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="170" y1="30" x2="170" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="240" y1="30" x2="240" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="310" y1="30" x2="310" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="100" y="225" font-size="10" fill="#444" text-anchor="middle">K₁</text> <text x="170" y="225" font-size="10" fill="#444" text-anchor="middle">K₂</text> <text x="240" y="225" font-size="10" fill="#444" text-anchor="middle">K₃</text> <text x="310" y="225" font-size="10" fill="#444" text-anchor="middle">K₄</text> <line x1="50" y1="80" x2="350" y2="80" stroke="#2ca02c" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="55" y="75" font-size="9" fill="#2ca02c">max gain</text> <line x1="50" y1="155" x2="350" y2="155" stroke="#d62728" stroke-dasharray="2,3" stroke-width="0.8" opacity="0.7"/> <text x="55" y="168" font-size="9" fill="#d62728">max loss = −net debit</text> <line x1="60" y1="155" x2="100" y2="155" stroke="#9467bd" stroke-width="2.5"/> <line x1="100" y1="155" x2="170" y2="80" stroke="#9467bd" stroke-width="2.5"/> <line x1="170" y1="80" x2="240" y2="80" stroke="#9467bd" stroke-width="2.5"/> <line x1="240" y1="80" x2="310" y2="155" stroke="#9467bd" stroke-width="2.5"/> <line x1="310" y1="155" x2="350" y2="155" stroke="#9467bd" stroke-width="2.5"/> <circle cx="120" cy="130" r="3" fill="#444"/> <circle cx="290" cy="130" r="3" fill="#444"/> </svg> Trapezoid. Profitable anywhere in [K₂, K₃]. **Wider profit zone than a butterfly, smaller max gain.** > [!note] Iron variants **Iron butterfly** = short straddle + long strangle around it. Same tent shape, built mixing calls and puts. Net credit. **Iron condor** = short strangle + long strangle. Same trapezoid shape. Probably the most popular retail income structure. ### 4.6 Skew / asymmetric #### Risk reversal (风险逆转) = long call (K₂) + short put (K₁) <svg viewBox="0 0 380 250" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="30" x2="50" y2="210" stroke="#444" stroke-width="1.5"/> <line x1="50" y1="130" x2="355" y2="130" stroke="#444" stroke-width="1.5"/> <text x="45" y="134" font-size="10" fill="#444" text-anchor="end">0</text> <text x="42" y="22" font-size="11" fill="#444" font-weight="bold">P/L</text> <text x="362" y="134" font-size="11" fill="#444" font-weight="bold">S</text> <text x="56" y="44" font-size="9" fill="#2ca02c">+ profit</text> <text x="56" y="204" font-size="9" fill="#d62728">− loss</text> <line x1="140" y1="30" x2="140" y2="210" stroke="#888" stroke-dasharray="3,3"/> <line x1="260" y1="30" x2="260" y2="210" stroke="#888" stroke-dasharray="3,3"/> <text x="140" y="225" font-size="11" fill="#444" text-anchor="middle">K₁</text> <text x="260" y="225" font-size="11" fill="#444" text-anchor="middle">K₂</text> <line x1="80" y1="190" x2="140" y2="130" stroke="#9467bd" stroke-width="2.5"/> <line x1="140" y1="130" x2="260" y2="130" stroke="#9467bd" stroke-width="2.5"/> <line x1="260" y1="130" x2="320" y2="70" stroke="#9467bd" stroke-width="2.5"/> <text x="320" y="60" font-size="9" fill="#2ca02c" text-anchor="end">↑ unlimited</text> <text x="80" y="205" font-size="9" fill="#d62728">↓ large (down to K₁ → 0)</text> <text x="200" y="120" font-size="9" fill="#888" text-anchor="middle">flat "dead zone"</text> </svg> Looks like long stock with a **dead zone** in the middle (between K₁ and K₂, P/L = 0 if zero-cost). Often structured zero-cost. Also the standard structure for **trading skew itself** — its price is largely determined by the IV difference between OTM call and OTM put. ### 4.7 Synthetics — re-expressing the basics These don't express new views. They re-express something simple in a different form for capital / tax / mandate reasons. |Synthetic|Construction|Equals| |---|---|---| |Synthetic long stock|long call + short put (same K, T)|long stock| |Synthetic short stock|short call + long put|short stock| |Synthetic long call|long stock + long put|long call (= protective put)| |Synthetic long put|short stock + long call|long put| All four are **just put–call parity rearranged**. --- ## 5. The workflow that makes this stick > [!tip] How to actually use this note > > 1. Start with a **view** ("modestly bullish, capped risk") > 2. Sketch the **payoff shape** you want (a staircase going up) > 3. Figure out which **basic blocks** make that shape (long call K₁ + short call K₂) > 4. _Then_ look up what people **call** it (bull call spread) > > Memorize the shapes. The names will stick by themselves. --- ## 6. Why "equivalent" ≠ "the same in practice" Put–call parity tells you two structures have **the same payoff**. But in real markets a position is also: - a **margin profile** (covered call ≈ $0 extra margin; naked short put can demand 20% of strike) - a **tax event** (US: covered call vs short put have very different qualified-treatment rules) - a **regulatory category** (most IRAs / pension funds prohibit naked short puts entirely) - a **cash-flow profile** (stockholder collects dividends; put-seller doesn't) - a **line item on a report** (a book full of "short puts" reads very differently to risk than one full of "covered calls") This is the **limits-to-arbitrage** insight: parity violations can persist precisely because the "cheap" side often requires a structure most participants literally can't hold. --- ## 7. Open threads / next steps - [ ] Plot butterfly = bull spread + bear spread on the same axes — visual proof - [ ] Greeks of each named structure (the _shape at expiration_ tells you almost nothing about behavior _before_ expiration) - [ ] How risk reversal price ↔ skew (deriving the "25-delta risk reversal" as a skew quote) - [ ] Box spread = synthetic zero-coupon bond (a beautiful pure-arbitrage construction) ## See also - [[Put-Call Parity Derivation]] - [[Volatility Surface - Skew and Term Structure]] - [[Greeks - Delta Gamma Theta Vega]] - [[Limits to Arbitrage]]