• 深入理解并熟练计算各项希腊字母（Greek Letters）：
  • Delta ($\mathrm{\Delta }$) • Gamma ($\mathrm{\Gamma }$)
  • Vega ($\mathcal{V}$) • Rho ($\rho$) • Theta ($\mathrm{\Theta }$)
• 构建多维度风险中性的期权投资组合：
  • Delta 中性 • Gamma 中性 • Vega 中性
• 探讨期权头寸的各类对冲模式：
  • 裸空头头寸（Naked positions）• 备兑头寸（Covered positions）
  • 止损对冲策略（Stop-loss strategies）• 动态 Delta 对冲（Dynamic delta hedging）
• 探讨合成期权的构建原理以及静态对冲与动态对冲的对比分析

• Understand and calculate the Greek Letters:
  • Delta ($\mathrm{\Delta }$) • Gamma ($\mathrm{\Gamma }$)
  • Vega ($\mathcal{V}$) • Rho ($\rho$) • Theta ($\mathrm{\Theta }$)
• Make a portfolio of options:
  • Delta neutral • Gamma neutral • Vega neutral
• Discuss the hedging of option positions:
  • Naked positions • Covered positions
  • Stop-loss strategies • Dynamic delta hedging
• Discuss the creation of synthetic options & static vs dynamic hedges

• 定义： Delta 是期权价格 $f$ 相对于基础资产价格 $S$ 的变化率（一阶偏导数）： $\mathrm{\Delta }=\frac{\partial f}{\partial S}$。
• 无红利资产的欧式看涨与看跌期权：
  $${\mathrm{\Delta }}_{\text{call}}=N(d_1)>0,{\mathrm{\Delta }}_{\text{put}}=N(d_1)-1<0$$
• Delta 取值范围：
  • 看涨期权 Delta 介于 $0$ 至 $+1$ 之间（平值 $\text{ATM}\approx 0.5$，深度实值 ITM $\to 1$，深度虚值 OTM $\to 0$）。
  • 看跌期权 Delta 介于 $-1$ 至 $0$ 之间（平值 $\text{ATM}\approx -0.5$，深度实值 ITM $\to -1$，深度虚值 OTM $\to 0$）。

• Definition: Delta is the rate of change of the option price $f$ with respect to the underlying asset price $S$: $\mathrm{\Delta }=\frac{\partial f}{\partial S}$.
• European Calls and Puts on Non-Dividend Paying Assets:
  $${\mathrm{\Delta }}_{\text{call}}=N(d_1)>0,{\mathrm{\Delta }}_{\text{put}}=N(d_1)-1<0$$
• Delta Ranges:
  • Call Delta lies between $0$ and $+1$ ($\text{ATM}\approx 0.5$, deep ITM $\to 1$, deep OTM $\to 0$).
  • Put Delta lies between $-1$ and $0$ ($\text{ATM}\approx -0.5$, deep ITM $\to -1$, deep OTM $\to 0$).

• 斜率的几何含义： Delta 是期权价格关于基础资产价格 $S$ 曲线上的切线斜率，其中 $\mathrm{\Delta }S$、$\mathrm{\Delta }f$ 分别为基础资产价格和期权价格 $f$ 的变化量（符号前的 $\mathrm{\Delta }$ 表示“变化量”，单独的 $\mathrm{\Delta }$ 则指期权的 Delta）：
  $$\text{切线斜率}=\mathrm{\Delta }=\underset{\mathrm{\Delta }S\to 0}{lim}\frac{\mathrm{\Delta }f}{\mathrm{\Delta }S}$$
  • 请与上一页的 Delta 曲线对照理解。
• Delta 与到期时间的关系：
  • 当到期时间 $T\to 0$ 时，Delta 曲线在执行价格 $K$ 附近显著变陡：
    • 对于实值看涨期权（$S>K$）：$\mathrm{\Delta }\to 1.0$
    • 对于虚值看涨期权（$S<K$）：$\mathrm{\Delta }\to 0.0$
    • 对于平值看涨期权（$S=K$）：$\mathrm{\Delta }\approx 0.5$，呈现阶跃函数的形态过渡。

• Slope Interpretation: Delta is the slope of the curve plotting option price against the underlying asset price $S$, where $\mathrm{\Delta }S$ and $\mathrm{\Delta }f$ are the changes in the asset price and in the option price $f$ (here $\mathrm{\Delta }$ in front of a symbol means "change in", while $\mathrm{\Delta }$ alone is the option's delta):
  $$\text{Slope of tangent line}=\mathrm{\Delta }=\underset{\mathrm{\Delta }S\to 0}{lim}\frac{\mathrm{\Delta }f}{\mathrm{\Delta }S}$$
  • Relate this to the previous graphs.
• Relation between Delta and Time to Maturity:
  • As time to maturity $T\to 0$, the delta curve steepens around the strike $K$:
    • For an ITM call ($S>K$): $\mathrm{\Delta }\to 1.0$
    • For an OTM call ($S<K$): $\mathrm{\Delta }\to 0.0$
    • For an ATM call ($S=K$): $\mathrm{\Delta }\approx 0.5$, transitioning like a step function.

• Delta 衡量期权价值对基础资产价格微小变动的敏感度：
  • 若资产价格 $S$ 变动 $\mathrm{\Delta }S$，则：
    • 看涨期权价值变动约为 ${\mathrm{\Delta }}_c\times \mathrm{\Delta }S$
    • 看跌期权价值变动约为 ${\mathrm{\Delta }}_p\times \mathrm{\Delta }S=-|{\mathrm{\Delta }}_p|\times \mathrm{\Delta }S$
• 数值算例：
  股票价格从 $100 上涨至 $102（$\mathrm{\Delta }S=+\mathrm{\$}2$）。
  该股票看涨期权的 Delta 为 0.5。
  预计该看涨期权价值将上涨： $$\mathrm{\Delta }\times \mathrm{\Delta }S=0.5\times \mathrm{\$}2=\mathrm{\$}1$$
• 对于深度实值期权：${\mathrm{\Delta }}_{\text{call}}\approx 1,{\mathrm{\Delta }}_{\text{put}}\approx -1$。
• 对于深度虚值期权：${\mathrm{\Delta }}_{\text{call}}\approx 0,{\mathrm{\Delta }}_{\text{put}}\approx 0$。

• Delta measures the sensitivity of option value to small changes in the underlying asset price:
  • If the underlying price $S$ changes by $\mathrm{\Delta }S$, then:
    • Call value changes by approximately ${\mathrm{\Delta }}_c\times \mathrm{\Delta }S$
    • Put value changes by approximately ${\mathrm{\Delta }}_p\times \mathrm{\Delta }S=-|{\mathrm{\Delta }}_p|\times \mathrm{\Delta }S$
• Numerical Example:
  The stock price rises from $100 to $102 ($\mathrm{\Delta }S=+\mathrm{\$}2$).
  A call option on this stock has a delta of 0.5.
  The call value is expected to increase by: $$\mathrm{\Delta }\times \mathrm{\Delta }S=0.5\times \mathrm{\$}2=\mathrm{\$}1$$
• For deep ITM options: ${\mathrm{\Delta }}_{\text{call}}\approx 1,{\mathrm{\Delta }}_{\text{put}}\approx -1$.
• For deep OTM options: ${\mathrm{\Delta }}_{\text{call}}\approx 0,{\mathrm{\Delta }}_{\text{put}}\approx 0$.

• 现货资产本身的 Delta 恒为 $1$。
• 支付连续红利率 $q$、期限为 $T$ 的资产远期合约的 Delta：
  $${\mathrm{\Delta }}_{\text{forward}}=e^{-qT}(\text{若 }q=0⟹{\mathrm{\Delta }}_{\text{forward}}=1)$$
• 支付连续红利率 $q$、期限为 $T$ 的资产期货合约的 Delta：
  $${\mathrm{\Delta }}_{\text{futures}}=e^{(r-q)T}$$
• 典型期货实例：
  • 股指期货：$\mathrm{\Delta }=e^{(r-q)T}$
  • 外汇期货：$\mathrm{\Delta }=e^{(r-r_f)T}$（$r_f$ 为外国无风险利率）
  • 商品期货（黄金/白银）：$\mathrm{\Delta }=e^{rT}$

• The delta of the spot contract (underlying asset itself) is $1$.
• The delta of a forward contract with maturity $T$ on an asset paying continuous dividend yield $q$:
  $${\mathrm{\Delta }}_{\text{forward}}=e^{-qT}(\text{if }q=0⟹{\mathrm{\Delta }}_{\text{forward}}=1)$$
• The delta of a futures contract with maturity $T$ on an asset paying continuous dividend yield $q$:
  $${\mathrm{\Delta }}_{\text{futures}}=e^{(r-q)T}$$
• Examples:
  • Index Futures: $\mathrm{\Delta }=e^{(r-q)T}$
  • Currency Futures: $\mathrm{\Delta }=e^{(r-r_f)T}$ (where $r_f$ is foreign risk-free rate)
  • Commodity (Gold/Silver) Futures: $\mathrm{\Delta }=e^{rT}$

• 基础资产支付连续红利率 $q$、期限为 $T$ 的期权，其 Delta：
  $${\mathrm{\Delta }}_{\text{call}}=e^{-qT}N(d_1),{\mathrm{\Delta }}_{\text{put}}=e^{-qT}[N(d_1)-1]$$
• 各类衍生品期权具体形式：
  • 股指期权： ${\mathrm{\Delta }}_c=e^{-qT}N(d_1),{\mathrm{\Delta }}_p=e^{-qT}[N(d_1)-1]$
  • 外汇期权： ${\mathrm{\Delta }}_c=e^{-r_{f}T}N(d_1),{\mathrm{\Delta }}_p=e^{-r_{f}T}[N(d_1)-1]$
  • 期货期权： ${\mathrm{\Delta }}_c=e^{-rT}N(d_1),{\mathrm{\Delta }}_p=e^{-rT}[N(d_1)-1]$
• 其中代入相应参数计算 $d_1$： $$d_1=\frac{\ln (S_0/K)+(r-q+\frac{1}{2}{\sigma }^2)T}{\sigma \sqrt{T}}$$

• The delta of an option with maturity $T$ on an asset paying continuous dividend yield $q$:
  $${\mathrm{\Delta }}_{\text{call}}=e^{-qT}N(d_1),{\mathrm{\Delta }}_{\text{put}}=e^{-qT}[N(d_1)-1]$$
• Examples:
  • Index Options: ${\mathrm{\Delta }}_c=e^{-qT}N(d_1),{\mathrm{\Delta }}_p=e^{-qT}[N(d_1)-1]$
  • Currency Options: ${\mathrm{\Delta }}_c=e^{-r_{f}T}N(d_1),{\mathrm{\Delta }}_p=e^{-r_{f}T}[N(d_1)-1]$
  • Futures Options: ${\mathrm{\Delta }}_c=e^{-rT}N(d_1),{\mathrm{\Delta }}_p=e^{-rT}[N(d_1)-1]$
• where by choice of $q$ (or $r_f$ or $r$), the appropriate $d_1$ is used: $$d_1=\frac{\ln (S_0/K)+(r-q+\frac{1}{2}{\sigma }^2)T}{\sigma \sqrt{T}}$$

• 定义： Theta 是期权价格 $f$ 相对于时间流逝 $t$ 的变化率：$\mathrm{\Theta }=\frac{\partial f}{\partial t}$。
• 欧式看涨期权的 Theta：
  $${\mathrm{\Theta }}_c=-\frac{S_0{N}^{\prime }(d_1)\sigma }{2\sqrt{T}}-rK{e}^{-rT}N(d_2)$$
  或者，对支付连续红利率 $q$ 的资产：
  $${\mathrm{\Theta }}_c=-\frac{S_0{N}^{\prime }(d_1)\sigma e^{-qT}}{2\sqrt{T}}-rK{e}^{-rT}N(d_2)+q{S}_{0}N(d_1)e^{-qT}$$
  其中 $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$。
• 期权多头的 Theta 通常为负值 – 随着时间流逝，期权不断损失时间价值。
• 算例： $S_0=100,K=100,r=0,\sigma =0.20,T=1$（平值看涨，$d_1=0.1$，$N^{\prime }(0.1)\approx 0.397$）： $$\mathrm{\Theta }=-\frac{100\times 0.397\times 0.20}{2}=-3.97\text{ / 年}$$ 折合 $-3.97/252=-0.0158$ / 交易日 – 每个交易日约损失 $0.016 的价值。

• Definition: Theta is the rate of change of the option price $f$ with respect to the passage of time $t$: $\mathrm{\Theta }=\frac{\partial f}{\partial t}$.
• Theta of a European call:
  $${\mathrm{\Theta }}_c=-\frac{S_0{N}^{\prime }(d_1)\sigma }{2\sqrt{T}}-rK{e}^{-rT}N(d_2)$$
  or, for an asset paying a continuous yield $q$:
  $${\mathrm{\Theta }}_c=-\frac{S_0{N}^{\prime }(d_1)\sigma e^{-qT}}{2\sqrt{T}}-rK{e}^{-rT}N(d_2)+q{S}_{0}N(d_1)e^{-qT}$$
  where $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$.
• Theta of a long option position is usually negative – as time passes, the option continually loses time value.
• Example: $S_0=100,K=100,r=0,\sigma =0.20,T=1$ (ATM call, $d_1=0.1$, $N^{\prime }(0.1)\approx 0.397$): $$\mathrm{\Theta }=-\frac{100\times 0.397\times 0.20}{2}=-3.97\text{ per year}$$ Equivalently $-3.97/252=-0.0158$ per trading day – about $0.016 of value lost per trading day.

• Theta 与基础资产价格的关系：
  • 参数设定：$K=100,r=0,\sigma =0.20,T=1$。
  • 当期权处于平值状态（$\text{ATM},S\approx K=100$）时，Theta 负值绝对值最大（时间价值损耗速度最快）。
  • 对于深度实值或深度虚值期权，Theta 趋近于 0（因为期权价格被内在价值主导，或者外在时间价值本身极小）。
• Theta 与到期时间的关系：
  • 对于平值期权，随着到期日临近（$T\to 0$），时间衰减急剧加速。
  • 交易员持有临近到期的平值多头期权时，必须警惕严峻的日度时间价值侵蚀。

• Relation between Theta and the underlying asset price:
  • Parameters: $K=100,r=0,\sigma =0.20,T=1$.
  • Theta is most negative (time value decays fastest) when the option is at-the-money ($\text{ATM},S\approx K=100$).
  • For deep ITM or deep OTM options, Theta approaches 0 (because the option price is dominated by intrinsic value, or the time value itself is tiny).
• Relation between Theta and time to maturity:
  • For ATM options, time decay accelerates sharply as expiration approaches ($T\to 0$).
  • Traders holding near-expiry ATM long options must watch out for severe daily time-value erosion.

• 定义： Gamma 是 Delta ($\mathrm{\Delta }$) 相对于基础资产价格 $S$ 的变化率（期权价格对基础资产价格的二阶偏导数）：
  $$\mathrm{\Gamma }=\frac{\partial \mathrm{\Delta }}{\partial S}=\frac{{\partial }^{2}f}{\partial S^2}$$
• 欧式看涨与看跌期权公式（两者完全相同！）：
  $$\mathrm{\Gamma }=\frac{N^{\prime }(d_1)}{S_0\sigma \sqrt{T}}\text{若含连续红利率 }q:\mathrm{\Gamma }=\frac{N^{\prime }(d_1)e^{-qT}}{S_0\sigma \sqrt{T}}$$
  其中 $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$。
• 若资产价格变动 $\mathrm{\Delta }S$，Delta 的变动量约为 $\mathrm{\Gamma }\times \mathrm{\Delta }S$。
• 算例： $S_0=100,K=100,r=0,\sigma =0.20,T=1$，得 $\mathrm{\Gamma }\approx 0.02$，$\mathrm{\Delta }\approx 0.54$。
  若股票价格从 $100 上涨至 $110（$\mathrm{\Delta }S=+\mathrm{\$}10$）： $${\mathrm{\Delta }}_{\text{new}}\approx 0.54+(0.02\times 10)=0.54+0.20=0.74$$

• Definition: Gamma is the rate of change of Delta ($\mathrm{\Delta }$) with respect to the underlying asset price $S$ (second derivative of the option price with respect to $S$):
  $$\mathrm{\Gamma }=\frac{\partial \mathrm{\Delta }}{\partial S}=\frac{{\partial }^{2}f}{\partial S^2}$$
• Formula for European calls and puts (identical for both!):
  $$\mathrm{\Gamma }=\frac{N^{\prime }(d_1)}{S_0\sigma \sqrt{T}}\text{with yield }q:\mathrm{\Gamma }=\frac{N^{\prime }(d_1)e^{-qT}}{S_0\sigma \sqrt{T}}$$
  where $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$.
• If the underlying price moves by $\mathrm{\Delta }S$, Delta changes by approximately $\mathrm{\Gamma }\times \mathrm{\Delta }S$.
• Example: $S_0=100,K=100,r=0,\sigma =0.20,T=1$, giving $\mathrm{\Gamma }\approx 0.02$, $\mathrm{\Delta }\approx 0.54$.
  If the stock price rises from $100 to $110 ($\mathrm{\Delta }S=+\mathrm{\$}10$): $${\mathrm{\Delta }}_{\text{new}}\approx 0.54+(0.02\times 10)=0.54+0.20=0.74$$

• 虚实值程度对 Gamma 的影响：
  • 深度虚值（Deep OTM）： Delta $\approx 0$，随 $S$ 变动极慢 → $\mathrm{\Gamma }\approx 0$。
  • 平值状态（ATM）： Delta $\approx 0.5$，对资产价格微小变动极度敏感 → $\mathrm{\Gamma }$ 达到峰值最大值！
  • 深度实值（Deep ITM）： Delta $\approx 1.0$，随 $S$ 变动极慢 → $\mathrm{\Gamma }\approx 0$。
• Gamma 与基础资产价格的关系：

  Gamma 随基础资产价格呈现尖锐的钟形曲线，在执行价格附近（略低于 $K$）达到峰值。

• 对冲者必须高度关注持有平值期权的组合，其 Gamma 风险极为集中。
• 左图：股价同样变动 $2，看涨期权的 Delta 只在执行价 $K$ 附近变化大；右图的 Gamma 就是左图 Delta 曲线的斜率。

• Moneyness Effect on Gamma:
  • Deep OTM: Delta $\approx 0$, changes very slowly with $S$ → $\mathrm{\Gamma }\approx 0$.
  • At-the-Money (ATM): Delta $\approx 0.5$, highly sensitive to small moves in $S$ → $\mathrm{\Gamma }$ reaches its maximum!
  • Deep ITM: Delta $\approx 1.0$, changes very slowly with $S$ → $\mathrm{\Gamma }\approx 0$.
• Relation between Gamma and Asset Price:

  The curve of Gamma against asset price forms a sharp bell-shaped peak close to the strike price (slightly below $K$).

• Hedgers must be especially attentive to portfolios with ATM options where Gamma risk is concentrated.
• Left chart: the same $2 move in the stock price changes the call's delta a lot only near the strike $K$; Gamma (right chart) is the slope of this delta curve.

• Gamma 与到期时间的关系：
  • 当 $T\to 0$ 时，平值期权的 Gamma 趋于无穷大，导致在 $S=K$ 附近的敏感度极端集中。
• Gamma 刻画了 Delta 对冲中的二阶曲率效应：
  • 若 $\mathrm{\Gamma }$ 较小：Delta 变动缓慢 → 较低频率的再平衡即可满足要求。
  • 若 $\mathrm{\Gamma }$ 较大：Delta 高度敏感 → 若不及时调整对冲比例，将引发严重的跟踪误差！
• 曲率增益直观示例：
  在 $S=100$ 时，平值看涨期权价格为 $7.97（$\mathrm{\Delta }=0.54$，$\mathrm{\Gamma }=0.02$）。若股票上涨至 $110（$\mathrm{\Delta }S=+10$）：
  一阶线性 Delta 预测值：$7.97+0.54\times 10=\mathrm{\$}13.37$
  加入 Gamma 修正：$13.37+\frac{1}{2}\times 0.02\times {10}^2=13.37+1.00=\mathrm{\$}14.37$
  考虑非线性曲率后的实际看涨期权价格（BSM）：$\mathrm{\$}14.29$
  约 +$1 的修正项（$\frac{1}{2}\mathrm{\Gamma }\mathrm{\Delta }{S}^2$）解释了线性预测 $13.37$ 与实际 $14.29$ 之间的绝大部分差距！

• Relation between Gamma and time to maturity:
  • As $T\to 0$, ATM Gamma tends to infinity, concentrating extreme sensitivity at $S=K$.
• Gamma accounts for curvature in delta hedging:
  • If $\mathrm{\Gamma }$ is small: Delta changes slowly → infrequent rebalancing suffices.
  • If $\mathrm{\Gamma }$ is large: Delta is highly sensitive → leaving the hedge unadjusted causes severe tracking error!
• Curvature Gain Illustration:
  At $S=100$, the ATM call is $7.97 ($\mathrm{\Delta }=0.54$, $\mathrm{\Gamma }=0.02$). Stock rises to $110 ($\mathrm{\Delta }S=+10$):
  Linear Delta prediction: $7.97+0.54\times 10=\mathrm{\$}13.37$
  With Gamma correction: $13.37+\frac{1}{2}\times 0.02\times {10}^2=13.37+1.00=\mathrm{\$}14.37$
  Actual call price with curvature (BSM): $\mathrm{\$}14.29$
  The ~+$1 correction ($\frac{1}{2}\mathrm{\Gamma }\mathrm{\Delta }{S}^2$) explains most of the gap between $13.37$ and $14.29$!

一句话：Delta 对冲只消除股价小幅变动的风险；剩下的是押注股价“大动”还是“不动”。

头寸方向	Delta	$\mathrm{\Gamma }$	$\mathrm{\Theta }$
多头看涨	+	+	−
多头看跌	−	+	−
空头看涨	−	−	+
空头看跌	+	−	+

$\mathrm{\Gamma }$ 与 $\mathrm{\Theta }$ 总是一正一负：对大幅波动的保护（$\mathrm{\Gamma }$）要用每天的时间损耗（$\mathrm{\Theta }$）来付费。

对 Delta 中性组合（组合 Delta $=0$）的价值变动做泰勒展开：

$\mathrm{\Pi }$ 是组合价值（不是希腊字母风险指标）。$\mathrm{\Delta }S$、$\mathrm{\Delta }t$、$\mathrm{\Delta }\mathrm{\Pi }$ 里的 $\mathrm{\Delta }$ 表示“变化量”：股价变动、经过的时间、组合价值变动，不是希腊字母 Delta。组合 Delta 中性，所以第一项为 0。

• 买入期权并对冲（$\mathrm{\Gamma }>0$，$\mathrm{\Theta }<0$）≈ 买保险：股价不动（$\mathrm{\Delta }S=0$）时每天付一点 $\mathrm{\Theta }$ 损耗；股价大涨或大跌时，$\mathrm{\Gamma }$ 带来的收益超过这笔损耗。
• 卖出期权并对冲（$\mathrm{\Gamma }<0$，$\mathrm{\Theta }>0$）≈ 卖保险：股价平稳时每天赚取 $\mathrm{\Theta }$；一旦股价大幅波动，组合亏损，波动越大亏得越多。

In one line: delta hedging removes only the risk of small moves; what is left is a bet on big moves versus no moves.

Position	Delta	$\mathrm{\Gamma }$	$\mathrm{\Theta }$
Long Call	+	+	−
Long Put	−	+	−
Short Call	−	−	+
Short Put	+	−	+

$\mathrm{\Gamma }$ and $\mathrm{\Theta }$ always have opposite signs: protection against big moves ($\mathrm{\Gamma }$) is paid for by daily time decay ($\mathrm{\Theta }$).

Taylor expansion for a delta-neutral portfolio (portfolio Delta $=0$):

$\mathrm{\Pi }$ is the portfolio value (not a Greek). The $\mathrm{\Delta }$ in $\mathrm{\Delta }S$, $\mathrm{\Delta }t$, $\mathrm{\Delta }\mathrm{\Pi }$ means “change in”, not the Greek Delta. The portfolio is delta-neutral, so the first term is 0.

• Buy options and hedge ($\mathrm{\Gamma }>0$, $\mathrm{\Theta }<0$) ≈ buying insurance: no move ($\mathrm{\Delta }S=0$): pay a little $\mathrm{\Theta }$ each day; a big move up or down: the $\mathrm{\Gamma }$ gain exceeds that cost.
• Sell options and hedge ($\mathrm{\Gamma }<0$, $\mathrm{\Theta }>0$) ≈ selling insurance: quiet days: earn $\mathrm{\Theta }$; a big move: a loss, larger the bigger the move.

一个交易日（$\mathrm{\Delta }t=1/252$）的盈亏，代入上一页的公式，其中 $\mathrm{1,000}\times \mathrm{\Theta }\mathrm{\Delta }t=-15.75$，$\mathrm{1,000}\times \frac{1}{2}\mathrm{\Gamma }=9.92$：

• 股价不动：买保险的一方亏 15.75（付了一个交易日的 $\mathrm{\Theta }$），卖保险的一方赚 15.75。
• 盈亏平衡：$-15.75+9.92(\mathrm{\Delta }S{)}^2=0\Rightarrow \mathrm{\Delta }S\approx \pm 1.26$，正好是一个交易日波动的一个标准差：$\sigma S\sqrt{\mathrm{\Delta }t}=20\mathrm{\%}\times 100\times \sqrt{1/252}=1.26$。

One trading day ($\mathrm{\Delta }t=1/252$), from the previous slide, where $\mathrm{1,000}\times \mathrm{\Theta }\mathrm{\Delta }t=-15.75$ and $\mathrm{1,000}\times \frac{1}{2}\mathrm{\Gamma }=9.92$:

• No move: the insurance buyer loses 15.75 (a trading day of $\mathrm{\Theta }$); the seller earns 15.75.
• Break-even: $-15.75+9.92(\mathrm{\Delta }S{)}^2=0\Rightarrow \mathrm{\Delta }S\approx \pm 1.26=\sigma S\sqrt{\mathrm{\Delta }t}$, one standard deviation of a trading day's move.

• 定义： Vega 是期权价格 $f$ 相对于基础资产波动率 $\sigma$ 的变化率： $\mathcal{V}=\frac{\partial f}{\partial \sigma }$。
• 欧式看涨与看跌期权公式（两者完全相同）：
  $$\mathcal{V}=S_0\sqrt{T}{N}^{\prime }(d_1)\text{若含连续红利率 }q:\mathcal{V}=S_0\sqrt{T}{N}^{\prime }(d_1)e^{-qT}$$
  其中 $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$。
• 无论看涨期权多头还是看跌期权多头，Vega 恒为正值。
• 经济含义： 若波动率 $\sigma$ 上升 1%（0.01），期权价格相应上涨约 $0.01\times \mathcal{V}$。
• Vega 与基础资产价格的关系（见下图）：平值期权（ATM）的 Vega 达到最高，随着深入实值或虚值逐渐下降。

• Definition: Vega is the rate of change of option price $f$ with respect to the volatility $\sigma$ of the underlying asset: $\mathcal{V}=\frac{\partial f}{\partial \sigma }$.
• Formula for European Call and Put (identical for both):
  $$\mathcal{V}=S_0\sqrt{T}{N}^{\prime }(d_1)\text{or with dividend }q:\mathcal{V}=S_0\sqrt{T}{N}^{\prime }(d_1)e^{-qT}$$
  where $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$.
• Vega is strictly positive for both long calls and long puts.
• Interpretation: If volatility $\sigma$ increases by 1% (0.01), the option price increases by approximately $0.01\times \mathcal{V}$.
• Relation between Vega and asset price (chart below): Vega is highest for at-the-money options and drops off for deep ITM/OTM options.

• 定义： Rho 是期权价格 $f$ 相对于无风险利率 $r$ 的变化率：
  $$\rho =\frac{\partial f}{\partial r}$$
• 欧式期权计算公式：
  $${\rho }_{\text{call}}=KT{e}^{-rT}N(d_2)>0,{\rho }_{\text{put}}=-KT{e}^{-rT}N(-d_2)<0$$
• 经济含义： 若利率 $r$ 上升 1%（0.01），期权价格相应变动约 $0.01\times \rho$。
• 外汇期权中的双 Rho 参数：
  • 本国利率 Rho（${\rho }_{\text{domestic}}$），衡量对本国无风险利率 $r$ 的敏感度。
  • 外国利率 Rho（${\rho }_{\text{foreign}}$），衡量对外国无风险利率 $r_f$ 的敏感度。

• Definition: Rho is the rate of change of option price $f$ with respect to the risk-free interest rate $r$:
  $$\rho =\frac{\partial f}{\partial r}$$
• Formulas for European Options:
  $${\rho }_{\text{call}}=KT{e}^{-rT}N(d_2)>0,{\rho }_{\text{put}}=-KT{e}^{-rT}N(-d_2)<0$$
• Interpretation: If interest rate $r$ increases by 1% (0.01), option price changes by approximately $0.01\times \rho$.
• Currency Options: There are two rho parameters:
  • Domestic Rho (${\rho }_{\text{domestic}}$) with respect to domestic rate $r$.
  • Foreign Rho (${\rho }_{\text{foreign}}$) with respect to foreign rate $r_f$.

• 期权投资组合的整体希腊字母是其各个成分合约希腊字母的加权和（组合含 $n$ 个期权，$w_i$ 为第 $i$ 个期权的头寸）：
  $${\mathrm{\Delta }}_P=\sum _{i=1}^n{w}_i{\mathrm{\Delta }}_i,{\mathrm{\Gamma }}_P=\sum _{i=1}^n{w}_i{\mathrm{\Gamma }}_i,{\mathcal{V}}_P=\sum _{i=1}^n{w}_i{\mathcal{V}}_i$$
• 实现 Delta 中性（${\mathrm{\Delta }}_P=0$）： 可通过持有现货资产或期货合约达成（现货与期货的 $\mathrm{\Gamma }=0,\mathcal{V}=0$）。
  • 若期货头寸为 $w_F$，现货头寸为 $w_S$：$w_F=e^{-(r-q)T^{\ast }}{w}_S$，其中 $T^{\ast }$ 为期货合约的到期日。
• 实现 Gamma 与 Vega 中性（${\mathrm{\Gamma }}_P=0,{\mathcal{V}}_P=0$）：
  • 现货与期货合约的 $\mathrm{\Gamma }$ 与 $\mathcal{V}$ 恒为零。
  • 因此，要调节组合的 Gamma 或 Vega，必须在组合中额外加入场内挂牌的期权合约！

• The Greek letters of a portfolio of options are weighted sums of the individual Greeks (the portfolio has $n$ options; $w_i$ = position in the $i$-th option):
  $${\mathrm{\Delta }}_P=\sum _{i=1}^n{w}_i{\mathrm{\Delta }}_i,{\mathrm{\Gamma }}_P=\sum _{i=1}^n{w}_i{\mathrm{\Gamma }}_i,{\mathcal{V}}_P=\sum _{i=1}^n{w}_i{\mathcal{V}}_i$$
• Delta Neutrality (${\mathrm{\Delta }}_P=0$): Can be achieved using positions in the underlying spot asset or in futures contracts (${\mathrm{\Gamma }}_{\text{spot}}=0,{\mathcal{V}}_{\text{spot}}=0$).
  • If $w_F$ is futures position and $w_S$ is spot position: $w_F=e^{-(r-q)T^{\ast }}{w}_S$, where $T^{\ast }$ is the maturity of the futures contract.
• Gamma & Vega Neutrality (${\mathrm{\Gamma }}_P=0,{\mathcal{V}}_P=0$):
  • The underlying asset and futures have $\mathrm{\Gamma }=0$ and $\mathcal{V}=0$.
  • Therefore, to alter Gamma or Vega, traded options must be added to the portfolio!

某金融机构的场外期权组合敏感度为 $\mathrm{\Delta }=-500$，$\mathrm{\Gamma }=-100$。市场上有一只可交易期权：${\mathrm{\Delta }}_T=0.5,{\mathrm{\Gamma }}_T=0.02$。需要多少份期权（$w$）和多少单位现货（$w_S$）才能同时实现 Gamma 与 Delta 中性？

• Gamma 与 Delta 中性联立条件（现货的 $\mathrm{\Gamma }=0$）：
  $$\mathrm{\Gamma }:-100+w\times 0.02+w_S\times 0=0$$ $$\mathrm{\Delta }:-500+w\times 0.5+w_S\times 1=0$$
• 先解 Gamma：$w=\frac{100}{0.02}=5,000$ 份期权；再解 Delta：$w_S=-(-500+5,000\times 0.5)=-2,000$ 单位。
• 对冲策略： 买入 5,000 份可交易期权，做空 2,000 单位现货。

An OTC option portfolio has $\mathrm{\Delta }=-500$, $\mathrm{\Gamma }=-100$. A traded option has ${\mathrm{\Delta }}_T=0.5,{\mathrm{\Gamma }}_T=0.02$. Find positions in the option ($w$) and in units of spot ($w_S$) for joint Gamma and Delta neutrality.

• Neutrality conditions (spot has $\mathrm{\Gamma }=0$):
  $$\mathrm{\Gamma }:-100+w\times 0.02+w_S\times 0=0$$ $$\mathrm{\Delta }:-500+w\times 0.5+w_S\times 1=0$$
• Gamma gives $w=\frac{100}{0.02}=5,000$ options; Delta then gives $w_S=-(-500+5,000\times 0.5)=-2,000$ units.
• Hedge: Buy 5,000 traded options; short 2,000 units of spot.

某场外期权组合的敏感度为 $\mathrm{\Delta }=-200,\mathrm{\Gamma }=-100,\mathcal{V}=-120$。市场上有两只可交易期权：

• 期权 1：${\mathrm{\Delta }}_1=0.5,{\mathrm{\Gamma }}_1=0.02,{\mathcal{V}}_1=0.02$
• 期权 2：${\mathrm{\Delta }}_2=0.1,{\mathrm{\Gamma }}_2=0.01,{\mathcal{V}}_2=0.02$

设加入期权 1 数量为 $w_1$，期权 2 为 $w_2$，现货为 $w_S$ 单位。三维度完全中性要求：

核心逻辑： 必须先通过两只可交易期权联立求解出满足 Gamma 和 Vega 中性的期权头寸，随后使用现货资产消除剩余 Delta！

An OTC option portfolio has $\mathrm{\Delta }=-200,\mathrm{\Gamma }=-100,\mathcal{V}=-120$. Two traded options are available:

• Option 1: ${\mathrm{\Delta }}_1=0.5,{\mathrm{\Gamma }}_1=0.02,{\mathcal{V}}_1=0.02$
• Option 2: ${\mathrm{\Delta }}_2=0.1,{\mathrm{\Gamma }}_2=0.01,{\mathcal{V}}_2=0.02$

Let $w_1$ be the number of option 1 added, $w_2$ of option 2, and $w_S$ the units of spot. Full neutrality in all three dimensions requires:

Key logic: first solve the two traded options simultaneously to achieve Gamma and Vega neutrality, then use the underlying spot asset to eliminate the remaining Delta!

联立线性方程组：

• 前两式相减：$0.01w_2=20$；回代第一式：$0.02w_1=100-20=80$，即 $$w_2=2,000\text{ 份期权 2},w_1=4,000\text{ 份期权 1}$$
• 代入 Delta 方程（期权贡献 $=2,200$）： $$-200+2,200+w_S=0⟹w_S=-2,000$$
• 对冲：买入 4,000 份期权 1、2,000 份期权 2，做空 2,000 单位现货。
• Delta 中性为微小价格变动提供保护，而 Gamma 中性则在对冲再平衡的时间间隔之间，为大幅价格变动提供保护。

Solve the simultaneous linear equations:

• Second minus first: $0.01w_2=20$; back-substitute: $0.02w_1=100-20=80$: $$w_2=2,000\text{ of Option 2},w_1=4,000\text{ of Option 1}$$
• Delta equation (options contribute $2,200$): $$-200+2,200+w_S=0⟹w_S=-2,000$$
• Hedge: Buy 4,000 Option 1 and 2,000 Option 2; short 2,000 units.
• Delta neutrality provides protection against small price moves, while gamma neutrality provides protection against large price moves between hedge rebalancing.

• 日常 Delta 动态对冲： 交易员通常确保其账户头寸每日至少实现一次 Delta 中性（在现代量化交易中，往往利用算法执行进行实时动态微调）。
• 相机管理 Gamma 与 Vega： 每当市场机会出现且交易成本允许时，交易员会交易场内期权以改善 Gamma 和 Vega 中性程度。
• 规模经济效应： 随着资产组合规模的扩大，对冲的相对成本显著降低，因为固定交易成本与买卖价差摩擦分摊到了更大的名义交易量上。

• Daily Delta Hedging: Traders usually ensure that their books and portfolios are delta-neutral at least once a day (often dynamically rebalanced in real time using algorithmic execution).
• Opportunistic Gamma and Vega Management: Whenever the opportunity arises and transaction costs allow, traders trade exchange-traded options to improve gamma and vega neutrality.
• Economies of Scale: As a portfolio becomes larger, hedging becomes relatively less expensive because fixed transaction costs and bid-ask overheads are spread over a much larger notional volume.

某金融机构以每份 $2.50 的价格售出了 1,000 份无红利股票的欧式看涨期权（每份对应 1 股），共收到权利金 $2,500。

• 参数设定： $S_0=100,K=100,r=0,\sigma =20\mathrm{\%},T=5\text{ 周}$。
• Black-Scholes 理论价格为每份 $2.47，合计 $2,470；因此该金融机构初始锁定了 $30 的理论利润。
• 金融机构如何对冲其空头风险？
  1. 裸空头头寸（Naked position）： 不采取任何行动。
  2. 备兑头寸（Covered position）： 立即全额买入 1,000 股股票。
  3. 止损对冲策略（Stop-loss strategy）： 股价涨过 $100 时买入股票，跌破 $100 时卖出股票。
  4. 动态 Delta 对冲（Dynamic delta hedging）： 动态调整股票持仓，实时维持 Delta 中性组合。

A financial institution sold 1,000 European calls on a non-dividend-paying stock at $2.50 each (1 share per option), receiving $2,500.

• Parameters: $S_0=100,K=100,r=0,\sigma =20\mathrm{\%},T=5\text{ weeks}$.
• Black-Scholes price: $2.47 each, $2,470 total; initial theoretical profit: $30.
• How to hedge the short exposure?
  1. Naked position: Do nothing.
  2. Covered position: Buy 1,000 shares immediately.
  3. Stop-loss strategy: Buy stock above $100; sell below $100.
  4. Dynamic delta hedging: Rebalance the stock holding to stay delta neutral.

• 采取裸空头看涨期权时，金融机构不采取任何对冲措施，直接留存初始收取的 $2,500 权利金。
• 到期损益特征：
  • 若 $S_T\le 100$：期权未被行权，金融机构全额保留权利金 $=+\mathrm{\$}2,500$。
  • 若 $S_T>100$：期权被对手方行权，金融机构面临无限的上行潜在亏损！
• 数值示例： 假设 $S_T=\mathrm{\$}110$：
  $$\text{损益}=(K-S_T)\times 1,000+\text{期权权利金}$$ $$=(\mathrm{\$}100-\mathrm{\$}110)\times 1,000+\mathrm{\$}2,500=-\mathrm{\$}7,500$$
• 结论： 裸卖空看涨期权使机构暴露于不可控的巨额上行市场风险之中。

• With a naked short call, the institution takes no hedging action and simply keeps the initial premium of $2,500.
• P&L Profile at Expiration:
  • If $S_T\le 100$: the option is not exercised and the institution keeps the entire premium $=+\mathrm{\$}2,500$.
  • If $S_T>100$: the option is exercised by the counterparty and the institution faces unlimited upside loss potential!
• Numerical Example: Suppose $S_T=\mathrm{\$}110$:
  $$\text{P/L}=(K-S_T)\times 1,000+\text{option premium}$$ $$=(\mathrm{\$}100-\mathrm{\$}110)\times 1,000+\mathrm{\$}2,500=-\mathrm{\$}7,500$$
• Conclusion: Writing naked calls exposes the institution to uncontrollable, large upside market risk.

• 备兑看涨期权策略（Covered Call，见第 7 讲）： 卖出 1,000 份看涨期权 + 以 $S_0=\mathrm{\$}100$ 买入 1,000 股股票（初始现金流出 $=\mathrm{\$}100,000$）。
• 损益特征：
  • 若 $S_T\ge 100$：股票在行权价 $K=100$ 被交割出售，损益 $=(100-100)\times 1,000+\mathrm{\$}2,500=+\mathrm{\$}2,500$。
  • 若 $S_T<100$：看涨期权虽作废，但所持有的现货股票遭受严重跌价损失！
• 数值示例： 假设 $S_T=\mathrm{\$}90$：
  $$\text{损益}=(S_T-S_0)\times 1,000+\text{期权权利金}$$ $$\text{损益}=(-\mathrm{\$}10)\times 1,000+\mathrm{\$}2,500=-\mathrm{\$}7,500$$
• 结论： 卖出备兑看涨期权仅仅是将上行风险转换成了严重的下行风险 – 根据买卖权平价定理，这在本质上等价于卖出裸看跌期权！

• Covered Call Strategy (see Lecture 7): Sell 1,000 call options + buy 1,000 shares at $S_0=\mathrm{\$}100$ (initial cash outlay $=\mathrm{\$}100,000$).
• P&L Profile:
  • If $S_T\ge 100$: the stock is delivered and sold at the strike $K=100$, P/L $=(100-100)\times 1,000+\mathrm{\$}2,500=+\mathrm{\$}2,500$.
  • If $S_T<100$: the calls expire worthless, but the stock held suffers a severe price loss!
• Numerical Example: Suppose $S_T=\mathrm{\$}90$:
  $$\text{P/L}=(S_T-S_0)\times 1,000+\text{option premium}$$ $$\text{P/L}=(-\mathrm{\$}10)\times 1,000+\mathrm{\$}2,500=-\mathrm{\$}7,500$$
• Conclusion: Writing a covered call merely converts upside risk into severe downside risk – by put-call parity, this is essentially equivalent to writing a naked put!

• 策略机制：
  • 一旦股票价格突破执行价格 $K=\mathrm{\$}100$，立刻买入 1,000 股股票。
  • 一旦股票价格跌破执行价格 $K=\mathrm{\$}100$，立刻卖出 1,000 股股票。
• 理论直觉： 期权在实值时持有股票（形成备兑），期权在虚值时持有现金（避免股票跌价）。
• 为何在实务中必然失败：
  • 反复拉锯震荡亏损（Whipsaw Losses）： 当股票价格在 $100 附近反复震荡时，交易员不断高买（$100.10）低卖（$99.90）——每来回一次每股亏 $0.20，1,000 股就亏 $200，资金迅速失血！
  • 价格跳空与滑点（Gapping & Slippage）： 市场价格跳跃离散，买入和卖出根本无法精确在 $100.00 成交。
  • 货币时间价值： 现金流发生在不同时间点，必须进行折现考量。

• Strategy Mechanics:
  • As soon as the stock price rises above the strike $K=\mathrm{\$}100$, immediately buy 1,000 shares.
  • As soon as the stock price falls below the strike $K=\mathrm{\$}100$, immediately sell 1,000 shares.
• Theoretical Intuition: Hold the stock when the option is in-the-money (a covered position), and hold cash when it is out-of-the-money (avoiding stock price declines).
• Why It Is Bound to Fail in Practice:
  • Whipsaw Losses: When the stock price oscillates around $100, the trader keeps buying high ($100.10) and selling low ($99.90) — each round trip loses $0.20 per share, i.e. $200 on 1,000 shares, rapidly bleeding cash!
  • Gapping & Slippage: Market prices jump discretely; purchases and sales cannot be executed at exactly $100.00.
  • Time Value of Money: Cash flows occur at different points in time and must be discounted.

• BSM 模型的理论基石： 由 $\mathrm{\Delta }$ 单位基础资产和 1 单位衍生品空头构成的资产组合在局部瞬时是完全无风险的：
  $$\mathrm{\Pi }=\mathrm{\Delta }S-f⟹d\mathrm{\Pi }=r\mathrm{\Pi }dt$$
• 其中 $\mathrm{\Delta }$ 为最优对冲比率：由衍生品价值的变化除以股价的变化计算得出（即期权的 Delta）。
• 金融机构通过持有 $1,000\times \mathrm{\Delta }$ 股股票来对冲售出 1,000 份看涨期权的空头风险敞口（例如 $\mathrm{\Delta }=0.5$ 时持有 500 股）。
• 亦可使用期货合约对冲（见第 16 页）：$w_F=e^{-(r-q)T^{\ast }}{w}_S$。
• 为何必须“动态”对冲？
  • 期权 Delta $\mathrm{\Delta }$ 是资产价格 $S$ 和时间 $t$ 的连续函数。
  • 随着股票价格变动和时间推移，$\mathrm{\Delta }$ 时刻处于变动之中。
  • 必须定期或连续再平衡持仓，才能维系 Delta 中性状态。

• Theoretical Cornerstone of the BSM Model: A portfolio of $\mathrm{\Delta }$ units of the underlying asset and 1 short unit of the derivative is locally, instantaneously risk-free:
  $$\mathrm{\Pi }=\mathrm{\Delta }S-f⟹d\mathrm{\Pi }=r\mathrm{\Pi }dt$$
• Here $\mathrm{\Delta }$ is the optimal hedge ratio. It is calculated as the change in value of the derivative divided by the change in stock price (the delta of the option).
• The institution hedges the short exposure from selling 1,000 calls by holding $1,000\times \mathrm{\Delta }$ shares (e.g. 500 shares when $\mathrm{\Delta }=0.5$).
• Futures contracts can also be used for delta hedging, see slide 16: $w_F=e^{-(r-q)T^{\ast }}{w}_S$.
• Why Must the Hedge Be “Dynamic”?
  • The option delta $\mathrm{\Delta }$ is a continuous function of the underlying price $S$ and time $t$.
  • As the stock price moves and time passes, $\mathrm{\Delta }$ changes continually.
  • The position must be rebalanced periodically or continuously to maintain delta neutrality.

在第 5 周，期权到期处于实值（$S_5=105>100$）。买方行权：按 $K=100$ 交付 1,000 股，收到 $\mathrm{\$}100,000$。
动态对冲净总成本 $=\mathrm{\$}102,500-\mathrm{\$}100,000=\mathrm{\$}2,500\approx \mathrm{\$}2,470$（理论 BS 价格）。

In week 5, the option expires in-the-money ($S_5=105>100$). The buyer exercises: 1,000 shares are delivered at $K=100$, and $\mathrm{\$}100,000$ is received.
Net total cost of dynamic hedging $=\mathrm{\$}102,500-\mathrm{\$}100,000=\mathrm{\$}2,500\approx \mathrm{\$}2,470$ (theoretical BS price).

在第 5 周，期权到期处于虚值作废（$S_5=96<100$）。无需交割，持仓已在到期前随 Delta 归零全部卖出。
动态对冲净总成本 $=\mathrm{\$}2,300\approx \mathrm{\$}2,470$（理论 BS 价格）。

In week 5, the option expires out-of-the-money and worthless ($S_5=96<100$). No delivery is required; the stock position had already been fully sold off before expiration as Delta went to zero.
Net total cost of dynamic hedging $=\mathrm{\$}2,300\approx \mathrm{\$}2,470$ (theoretical BS price).

• 动态对冲期权的总成本现值约等于其 Black-Scholes 理论价值（上面两次模拟：$2,500 与 $2,300，均围绕 $2,470 波动，多次模拟的平均值收敛于理论价值）。
• 动态对冲组合构建了期权的合成多头头寸，从而精确中和了期权卖方的空头敞口。
• 再平衡频率的权衡： 提高再平衡频率可以降低跟踪误差及对冲成本的波动，但会显著增加累积的交易手续费。
• 利用远期合约进行静态对冲：
  • 由于远期合约的 Delta 恒等于 1（$\mathrm{\Delta }=1$），由 1 股股票多头与 1 份远期合约空头构成的组合完全不需要任何再平衡！
  • 远期合约可以实现“对冲后即可高枕无忧”（静态对冲）。而期权因具有内在非线性曲率，本质上依赖动态 Delta 对冲。

• The present value of the total cost of dynamically hedging the option is approximately equal to its Black-Scholes theoretical value (the two simulations above gave $2,500 and $2,300, both scattered around $2,470; averaged over many simulations the cost converges to the theoretical value).
• The dynamic hedge portfolio creates a synthetic long position in the option, exactly offsetting the option writer's short exposure.
• Rebalancing Frequency Trade-off: Rebalancing more frequently reduces the tracking error and the variability of hedging cost, but significantly increases cumulative transaction costs.
• Static Hedging with Forward Contracts:
  • Since a forward contract has a constant delta of 1 ($\mathrm{\Delta }=1$), a portfolio of long 1 share of the underlying stock and short 1 forward contract requires no rebalancing at all!
  • Forward contracts allow “hedge and forget” (static hedging). Options, with their inherent nonlinear curvature, inherently rely on dynamic delta hedging.