学习目标

Learning Objectives

期权与基础资产的头寸组合

Positions in Options and Underlying Assets

备兑看涨期权 (Covered Call)

Covered Call

案例分析 – 备兑看涨期权

本讲全部期权价格都是 Black-Scholes 价格（$\sigma =20\mathrm{\%}$，期限 1 年，$r=0$，不派红利）取整到美元。

设定：以 $S_0=\mathrm{\$}100$ 买入股票，并卖出 $K=\mathrm{\$}100$、$c=\mathrm{\$}8$ 的看涨期权。
净投入 $S_0-c=\mathrm{\$}92=-p+K$（$-8+100$），与卖出看跌期权同形。

到期价格 $S_T$	股票多头	看涨期权空头 ($K=100$)	权利金合计	净损益总计
$S_T<100$	$S_T-100$	$0$	$+8$	$S_T-92$
$S_T\ge 100$	$S_T-100$	$100-S_T$	$+8$	+8

Example – Covered Call

All option prices in this lecture are Black-Scholes prices ($\sigma =20\mathrm{\%}$, 1 year, $r=0$, no dividends) rounded to whole dollars.

Setup: Buy stock for $S_0=\mathrm{\$}100$ and short a call with $K=\mathrm{\$}100$, $c=\mathrm{\$}8$.
Net outlay $S_0-c=\mathrm{\$}92=-p+K$ ($-8+100$): the same shape as a short put.

Terminal Price	Long Stock	Short Call ($K=100$)	Total Premium	Total P/L
$S_T<100$	$S_T-100$	$0$	$+8$	$S_T-92$
$S_T\ge 100$	$S_T-100$	$100-S_T$	$+8$	+8

备兑看涨期权：策略属性

Covered Call: Strategy Properties

保护性看跌期权 (Protective Put)

Protective Put

案例分析 – 保护性看跌期权

设定：以 $S_0=\mathrm{\$}100$ 买入资产，并买入行权价 $K=\mathrm{\$}100$、期权费 $p=\mathrm{\$}8$ 的看跌期权（$r=0$，不派红利）。
期初净投入 $S_0+p=\mathrm{\$}108$。买卖权平价：$S_0+p=c+K$（$108=8+100$），与买入看涨期权同形。

到期价格 $S_T$	股票多头	看跌期权多头 ($K=100$)	权利金合计	净损益总计
$S_T<100$	$S_T-100$	$100-S_T$	$-8$	−8
$S_T\ge 100$	$S_T-100$	$0$	$-8$	$S_T-108$

Example – Protective Put

Setup: Buy asset for $S_0=\mathrm{\$}100$ and a long put with $K=\mathrm{\$}100$ and premium $p=\mathrm{\$}8$ ($r=0$, no dividends).
Net outlay $S_0+p=\mathrm{\$}108$. Put-call parity: $S_0+p=c+K$ ($108=8+100$), the same shape as a long call.

Terminal Price	Long Stock	Long Put ($K=100$)	Total Premium	Total P/L
$S_T<100$	$S_T-100$	$100-S_T$	$-8$	−8
$S_T\ge 100$	$S_T-100$	$0$	$-8$	$S_T-108$

保护性看跌期权：策略属性

Protective Put: Strategy Properties

备兑看涨期权 vs 保护性看跌期权：不对称风险与收益特征

Covered Call vs Protective Put: Asymmetric Risk & Profit Profiles

差价策略 (Spreads)

Spreads

利用看涨期权的牛市差价策略

• 头寸构成：买入较低行权价 $K_1$ 的看涨期权，同时卖出较高行权价 $K_2$ 的看涨期权 ($K_1<K_2$)。
• 适用市场环境：温和看涨 / 高波动率（中度乐观的市场预期）。
• 策略优势：
  • 同时锁定了最大下行风险与最大上涨盈利。
  • 期初建仓成本较低；相比直接买入看涨期权更为便宜，因为卖出高行权价看涨期权所得的权利金补贴了买入期权的支出。
• 策略特性：
  • 买入的看涨期权 ($K_1$) 通常处于实值 (ITM) 或平值 (ATM)（较为昂贵）；整体看法是看涨。
  • 卖出的看涨期权 ($K_2$) 通常处于虚值 (OTM)（较为便宜）；它有助于为头寸融资，但会减少潜在利润。
  • 当两只看涨期权均为虚值时，构成最进取型的牛市差价策略。
• 面临问题：建仓需要净支付现金期权费（借方差价 Debit Spread）。

Bull Spreads using Calls

• Position: Long call with low strike $K_1$ and short call with high strike $K_2$ ($K_1<K_2$).
• Market profile: Mildly bullish / high volatility (moderately optimistic view).
• Benefits:
  • Limits both downside risk and upside reward.
  • Low setup cost strategy; cheaper than buying a call outright because the short call premium subsidizes the long call.
• Properties:
  • Long call ($K_1$) is typically ITM/ATM (expensive); the overall view is bullish.
  • Short call ($K_2$) is typically OTM (cheaper); it helps to finance the position, but it reduces the profit potential.
  • Most aggressive bull spread occurs when both calls are OTM.
• Problem: Requires an upfront net cash premium (debit spread).

利用看跌期权的牛市差价策略

• 头寸构成：买入较低行权价 $K_1$ 的看跌期权，同时卖出较高行权价 $K_2$ 的看跌期权 ($K_1<K_2$)。
• 短期收益增强策略：
  • 期初带来正向现金净流入（贷方差价 Credit Spread）。
  • 卖出较高行权价看跌期权收取的期权费，高于买入较低行权价看跌期权的支出 ($p_2>p_1$)。
• 保证金要求：
  • 由于包含较高行权价的看跌期权空头，交易所会要求缴纳保证金以确保履行履约担保义务。

Bull Spreads using Puts

• Position: Long put with low strike $K_1$ and short put with high strike $K_2$ ($K_1<K_2$).
• Short-term income strategy:
  • Provides an upfront positive cash flow (credit spread).
  • Selling the higher-strike put generates more premium than the cost of buying the lower-strike put ($p_2>p_1$).
• Margin requirements:
  • Because it involves a short put with high strike, it attracts exchange margin requirements to guarantee performance.

牛市差价：看涨借方差价 vs 看跌贷方差价

• 结构性差异：看涨期权牛市差价需要期初支付现金（无需占用保证金），而看跌期权牛市差价前期产生现金流入，但需要占用保证金以担保看跌期权空头的履约责任。
• 箱式差价联结 (Box Spread)：买入看涨牛市差价（付 $10）+ 买入看跌熊市差价（付 $14-4=\mathrm{\$}10$）构成了箱式差价 (Box spread)，其到期回报固定为 $K_2-K_1=\mathrm{\$}20$，期初成本 $10+10=\mathrm{\$}20$ 正是 $20 的现值（$r=0$，不需要折现）。若组合成本偏离这一现值，套利者将通过该无风险套利机会赚取差价。

Bull Spreads: Call Debit Spread vs Put Credit Spread

• Structural Difference: Call bull spread requires paying cash upfront (no margin), whereas put bull spread yields upfront cash but requires margin to back the short put obligation.
• Box Spread Connection: Long Call Bull Spread (pay $10) + Long Bear Put Spread (pay $14-4=\mathrm{\$}10$) forms a box spread with guaranteed payoff equal to $K_2-K_1=\mathrm{\$}20$; its cost of $10+10=\mathrm{\$}20$ is exactly the present value of $20 ($r=0$, so no discounting). If the cost differs from this present value, arbitrageurs exploit the mispricing.

利用看跌期权的熊市差价策略

• 头寸构成：买入较高行权价 $K_2$ 的看跌期权，同时卖出较低行权价 $K_1$ 的看跌期权 ($K_1<K_2$)。
• 适用市场环境：温和看跌 / 高波动率（中度悲观的市场预期）。
• 策略优势：
  • 同时锁定了最大下行收益与最大上行亏损。
  • 期初建仓成本较低；比直接买入看跌期权更便宜，因为卖出看跌期权的权利金抵消了部分买入成本。
• 策略特性：
  • 买入的看跌期权 ($K_2$) 通常处于实值 (ITM)（较为昂贵）；整体看法是看跌。
  • 卖出的看跌期权 ($K_1$) 通常处于虚值 (OTM)（较为便宜）；它有助于为头寸融资，但会减少潜在利润。
• 面临问题：需要期初净支付权利金（借方差价 Debit Spread）。

Bear Spreads using Puts

• Position: Long put with high strike $K_2$ and short put with low strike $K_1$ ($K_1<K_2$).
• Market profile: Mildly bearish / high volatility (moderately pessimistic view).
• Benefits:
  • Limits both downside and upside risk.
  • Low setup cost strategy; cheaper than buying a put outright because short put premium offsets long put expense.
• Properties:
  • Long put ($K_2$) is typically ITM (expensive); the overall view is bearish.
  • Short put ($K_1$) is typically OTM (cheaper); it helps to finance the position, but it reduces the profit potential.
• Problem: Requires an upfront net premium (debit spread).

利用看涨期权的熊市差价策略

• 头寸构成：买入较高行权价 $K_2$ 的看涨期权，同时卖出较低行权价 $K_1$ 的看涨期权 ($K_1<K_2$)。
• 短期收益增强策略：
  • 期初带来正向现金净流入（贷方差价 Credit Spread）。
  • 卖出较低行权价看涨期权 ($K_1$) 收取的期权费，高于买入较高行权价看涨期权 ($K_2$) 的支出。
• 保证金要求：
  • 交易所要求缴纳保证金，以应对看涨期权空头腿在基础资产大幅上涨时可能面临的履约义务。

Bear Spreads using Calls

• Position: Long call with high strike $K_2$ and short call with low strike $K_1$ ($K_1<K_2$).
• Short-term income strategy:
  • Provides an upfront positive cash flow (credit spread).
  • Selling the lower-strike call ($K_1$) generates more premium than the cost of buying the higher-strike call ($K_2$).
• Margin requirements:
  • Attracts exchange margin requirements to cover potential obligations on the short call leg.

利用看涨期权的蝶式差价策略

• 构建方法：买入1份较低行权价 $K_1$ 的看涨期权，买入1份较高行权价 $K_3$ 的看涨期权，同时卖出2份中间行权价 $K_2$ 的看涨期权：
  $K_2=\frac{K_1+K_3}{2}$

• 适用市场环境：平稳 / 低波动率预期；预计股票价格将在到期时维持在中间行权价 $K_2$ 附近。
• 优势与局限：建仓成本低且盈利目标明确；但需要期初支付净权利金。
• 反向蝶式差价 (Reverse Butterfly)：卖出 $K_1$、卖出 $K_3$、买入2份 $K_2$ → 适合预期市场将发生剧烈波动的情形。

Butterfly Spreads using Calls

• Construction: Long one call with low strike $K_1$, long one call with high strike $K_3$, and short two calls with middle strike $K_2$:
  $K_2=\frac{K_1+K_3}{2}$

• Market profile: Stable / low volatility outlook; expects stock to remain near $K_2$.
• Benefits & Problem: Low setup cost with modest profit; requires upfront net premium.
• Reverse Butterfly: Short $K_1$, short $K_3$, long two $K_2$ → profitable in high volatility.

案例：利用看涨期权的蝶式差价策略

Example: Butterfly Spread using Calls

案例分析 – 蝶式差价策略评估

• 盈亏平衡点：$S_T=\mathrm{\$}92$ 与 $S_T=\mathrm{\$}108$。
• 最大利润：当 $S_T=\mathrm{\$}100$（中间行权价）时取得最大利润 $+\mathrm{\$}8.00$。
• 最大亏损：当 $S_T\le 90$ 或 $S_T\ge 110$ 时，亏损锁定为期初净权利金支出 $-\mathrm{\$}2.00$。
• 盈利价格区间：当 $\mathrm{\$}92<S_T<\mathrm{\$}108$ 时该策略实现净盈利。
• 利用看跌期权的蝶式差价：买入看跌 $K_1$、卖出2份看跌 $K_2$、买入看跌 $K_3$ 会产生完全一致的损益图。

Example – Butterfly Spread Analysis

• Breakeven points: $S_T=\mathrm{\$}92$ and $S_T=\mathrm{\$}108$.
• Maximum profit: $+\mathrm{\$}8.00$ when $S_T=\mathrm{\$}100$ (the middle strike price).
• Maximum loss: Capped at $-\mathrm{\$}2.00$ (the initial net premium) when $S_T\le 90$ or $S_T\ge 110$.
• Profitable range: Strategy is profitable when $\mathrm{\$}92<S_T<\mathrm{\$}108$.
• Butterfly Spreads using Puts: Long put $K_1$, short two puts $K_2$, long put $K_3$ yields identical payoffs.

日历差价策略 (Calendar Spread)

• 核心概念：利用相同行权价 $K$ 但不同到期日的期权构建组合。
• 利用看涨期权的日历差价：
  • 卖出近期看涨期权，同时买入远期看涨期权。
  • 利用了近期期权时间价值衰减速度快于远期期权（近期期权具有更高的 $\mathrm{\Theta }$ 绝对值，$\mathrm{\Theta }$：期权价值随时间流逝而发生的变化，即时间衰减）的特性。
  • 若在近期期权到期时，股票价格接近行权价 $K$，近期期权将毫无价值地作废，而远期期权依然保留着可观的时间价值。
• 利用看跌期权的日历差价：
  • 卖出近期看跌期权，同时买入远期看跌期权。
• 反向日历差价 (Reverse Calendar Spread)：
  • 买入近期期权并卖出远期期权；当股票价格发生剧烈单向大幅突破时获利。

Calendar Spread

• Core Concept: Use options with the same strike price $K$ but different expiration dates.
• Calendar spread using Calls:
  • Sell a short-dated call option and buy a longer-dated call option.
  • Capitalizes on the fact that the short-dated option decays faster (higher $\mathrm{\Theta }$ in absolute value; $\mathrm{\Theta }$ is theta, the change in option value as time passes) than the long-dated option.
  • If the stock price remains close to strike $K$ when the short option expires, the short option expires worthless while the long-dated option retains substantial time value.
• Calendar spread using Puts:
  • Sell a short-dated put option and buy a longer-dated put option.
• Reverse Calendar Spread:
  • Buy a short-dated option and sell a longer-dated option; profitable when stock makes large directional moves.

组合策略 (Combinations)

Combinations

跨式期权 (Straddles)

• 底部跨式 / 买入跨式 (Bottom Straddle / Long Straddle)：
  • 买入1份看涨期权和1份看跌期权，两者具有相同行权价 $K$ 与相同到期日 $T$。
  • 做多波动率策略：若价格向任意方向发生大幅跳跃，即可享受无限盈利潜力。
  • 最大可能亏损严格锁定为所支付的全部权利金合计 ($c+p$)。
• 顶部跨式 / 卖出跨式 (Top Straddle / Short Straddle)：
  • 卖出1份看涨期权和1份看跌期权，两者具有相同的行权价 $K$ 与到期日 $T$。
  • 做空波动率策略：在低波动率、波澜不惊的市场中赚取期权权利金收入。
  • 风险极高：若股价向任一方向发生剧烈波动，都将面临无限亏损的敞口。

Straddles

• Bottom Straddle (Straddle Purchase / Long Straddle):
  • Buy 1 call and 1 put with the same strike $K$ and same maturity $T$.
  • Long Volatility strategy: Unlimited profit potential if a major price jump occurs in either direction.
  • Limited maximum loss capped at total premium paid ($c+p$).
• Top Straddle (Straddle Write / Short Straddle):
  • Sell 1 call and 1 put with the same strike $K$ and maturity $T$.
  • Short Volatility strategy: Earn premium income in low volatility, stagnant markets.
  • Highly risky: Unlimited loss potential from large price movements in either direction.

买入 vs 卖出跨式期权：波动率机制与双重盈亏平衡点

Long vs Short Straddles: Volatility Mechanics & Dual Breakevens

条式组合与带式组合 (Strips and Straps)

• 买入条式组合 (Long Strip)：
  • 买入1份看涨期权和2份看跌期权，均具有相同的行权价 $K$ 与到期日 $T$。
  • 适用市场预期：预期波动率剧烈增大，且带有偏空头倾向（下行斜率的陡峭程度是上行斜率的两倍）。
• 买入带式组合 (Long Strap)：
  • 买入2份看涨期权和1份看跌期权，均具有相同的行权价 $K$ 与到期日 $T$。
  • 适用市场预期：预期波动率剧烈增大，且带有偏多头倾向（上行斜率的陡峭程度是下行斜率的两倍）。

Strips and Straps

• Long Strip:
  • Buy 1 call and 2 puts with the same strike $K$ and maturity $T$.
  • Market profile: High volatility expected, with a bearish bias (downside slope is twice as steep as upside).
• Long Strap:
  • Buy 2 calls and 1 put with the same strike $K$ and maturity $T$.
  • Market profile: High volatility expected, with a bullish bias (upside slope is twice as steep as downside).

宽跨式期权 (Strangles)

• 底部垂直组合 / 买入宽跨式 (Bottom Vertical Combination / Long Strangle)：
  • 买入相同到期日但不同行权价的看涨期权与看跌期权：买入较低行权价 $K_1$ 的看跌期权，并买入较高行权价 $K_2$ 的看涨期权 ($K_1<K_2$)。
  • 即：买入高行权价看涨期权与低行权价看跌期权，或者买入低行权价看涨期权与高行权价看跌期权。二者有什么区别？
  • 做多波动率策略：建仓成本比跨式期权更便宜，因为两只期权通常均为虚值期权 (OTM)。
  • 权衡取舍：股价必须发生比跨式期权更大幅度的偏离波动，策略方能开始实现盈利。
• 顶部垂直组合 / 卖出宽跨式 (Top Vertical Combination / Short Strangle)：
  • 卖出不同行权价 ($K_1<K_2$) 且相同到期日的看涨期权与看跌期权。
  • 做空波动率策略；在 $K_1$ 与 $K_2$ 之间提供了一个更宽阔的安全无损盈利带。

Strangles

• Bottom Vertical Combination (Strangle Purchase / Long Strangle):
  • Long a call and a put with different strikes and the same maturity: Long put with low strike $K_1$ and long call with high strike $K_2$ ($K_1<K_2$).
  • That is: long call with high strike and long put with low strike, or long call with low strike and long put with high strike. What is the difference?
  • Long volatility strategy: Cheaper to buy than a straddle because both options are typically OTM.
  • Trade-off: The stock price must move further than in a straddle to generate profits.
• Top Vertical Combination (Strangle Write / Short Strangle):
  • Short a call and a put with different strikes ($K_1<K_2$) and same maturity.
  • Short volatility strategy; provides a wider safety band between $K_1$ and $K_2$.

买入宽跨式：分段损益表与双重盈亏平衡点

Long Strangle: Piecewise Profit Table & Dual Breakevens

做空波动率策略对比

Comparison of Short Volatility Strategies

做空波动率策略：四种组合叠加在同一张图上

• 跨式：在 $100 处利润见顶 (+16)，两侧亏损无限。
• 宽跨式：在 $90 与 $110 之间为扁平的最大收益 (+8)，两侧亏损无限。
• 蝶式：尖峰收益 (+8) 低于跨式，亏损封顶 (−2)；鹰式：扁平收益且最小 (+5)，亏损封顶 (−5)。

Short Volatility Strategies Overlaid on One Chart

• Straddle: profit peaks at $100 (+16), unlimited loss on both sides.
• Strangle: flat maximum return (+8) between $90 and $110, unlimited loss on both sides.
• Butterfly: peak (+8) below the straddle's, loss capped (−2); Condor: flat, smallest (+5), loss capped (−5).

做空波动率策略对比

Comparison of Short Volatility Strategies