• 阐明随机游走假设的核心性质：
  • 正态分布与对数正态分布
  • 预期收益率、方差与标准差
  • 标准正态累积分布函数 $N(x)$
• 掌握 Black-Scholes 模型的基本假设
• 运用 Black-Scholes 定价公式计算无红利股票欧式看涨期权与看跌期权的价格
• 波动率相关分析：
  • 利用历史资产价格数据估算波动率
  • 阐明隐含波动率的定义并探讨其重要意义
• 模型拓展与应用：
  • 调整 Black-Scholes 公式以对支付红利股票的期权进行定价
  • 运用布莱克近似法（Black's approximation）估算支付红利股票的美式看涨期权价格

• Highlight the properties of the random walk assumption:
  • Normal distribution and lognormal distribution
  • Expected return, variance and standard deviation
  • Cumulative normal distribution function $N(x)$
• Assumptions underlying the Black-Scholes model
• Calculate the prices of European calls and puts on non-dividend-paying stocks using the Black-Scholes formula
• Volatility:
  • Estimate volatility from historical prices
  • Define implied volatility and discuss its importance
• Extensions:
  • Adjust the Black-Scholes formula to price options on a dividend-paying stock
  • Estimate the prices of American calls on dividend-paying stocks using Black's approximation

• 令 $\varphi [m,s]$ 表示均值为 $m$、标准差为 $s$ 的正态分布。
• 考虑一只当前价格为 $S_0$、年化预期收益率为 $\mu$、价格波动率为 $\sigma$ 的股票。
• 在随机游走假设下，在极短时间间隔 $\mathrm{\Delta }t$ 内：
  • 股票价格的变动量 $\mathrm{\Delta }S$ 服从正态分布：
    $$\mathrm{\Delta }S\sim \varphi [\mu S_0\mathrm{\Delta }t,\sigma S_0\sqrt{\mathrm{\Delta }t}]$$
  • 换言之，股票的收益率（相对价格变动比例）$\frac{\mathrm{\Delta }S}{S_0}$ 服从正态分布：
    $$\frac{\mathrm{\Delta }S}{S_0}\sim \varphi [\mu \mathrm{\Delta }t,\sigma \sqrt{\mathrm{\Delta }t}]$$

• Let $\varphi [m,s]$ denote normal distribution with mean $m$ and standard deviation $s$.
• Consider a stock with price $S_0$, expected return $\mu$ and price volatility $\sigma$.
• Under the random walk assumption, in a short period $\mathrm{\Delta }t$:
  • the change in the stock price, $\mathrm{\Delta }S$, is normally distributed:
    $$\mathrm{\Delta }S\sim \varphi [\mu S_0\mathrm{\Delta }t,\sigma S_0\sqrt{\mathrm{\Delta }t}]$$
  • In other words, the proportional stock return, $\frac{\mathrm{\Delta }S}{S_0}$, is normally distributed:
    $$\frac{\mathrm{\Delta }S}{S_0}\sim \varphi [\mu \mathrm{\Delta }t,\sigma \sqrt{\mathrm{\Delta }t}]$$

• 由此可推导得出，股票的对数价格 $\ln S_T$ 服从正态分布：
  $$\ln S_T\sim \varphi [\ln S_0+(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$$ $$\ln \frac{S_T}{S_0}\sim \varphi [(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$$
• 根据定义，若一个随机变量的自然对数服从正态分布，则该随机变量服从对数正态分布。
• 因此，若 $\ln S_T$ 与 $\ln \frac{S_T}{S_0}$ 服从正态分布，则 $S_T$ 与 $\frac{S_T}{S_0}$ 服从对数正态分布：
  
  • 股票价格的期望值（均值）为：$E[S_T]=S_0{e}^{\mu T}$
  • 其方差为：$\text{var}[S_T]=S_0^2{e}^{2\mu T}(e^{{\sigma }^{2}T}-1)$

• It follows that the stock log-return, $\ln S_T$, is normally distributed:
  $$\ln S_T\sim \varphi [\ln S_0+(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$$ $$\ln \frac{S_T}{S_0}\sim \varphi [(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$$
• A random variable whose natural logarithm is normally distributed, by definition, is log-normally distributed.
• Thus if $\ln S_T$ and $\ln \frac{S_T}{S_0}$ are normally distributed, then $S_T$ and $\frac{S_T}{S_0}$ are log-normally distributed:
  
  • The mean stock price is: $E[S_T]=S_0{e}^{\mu T}$
  • Its variance is: $\text{var}[S_T]=S_0^2{e}^{2\mu T}(e^{{\sigma }^{2}T}-1)$

• $\mu$ 为股票的年化预期收益率：
  • 股票价格的期望值为 $E(S_T)=S_0{e}^{\mu T}$
• 股票实际实现的连续复利收益率 $R$ 满足以下数学关系：
  $$S_T=S_0{e}^{RT}⟹R=\frac{1}{T}\ln \frac{S_T}{S_0}$$
• 由于 $\ln \frac{S_T}{S_0}\sim \varphi [(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$，因此：
  $$R\sim \varphi [\mu -\frac{{\sigma }^2}{2},\frac{\sigma }{\sqrt{T}}]⟹E[R]=\mu -\frac{{\sigma }^2}{2}(\ne \mu )$$
• 这是因为时间间隔 $\mathrm{\Delta }t$ 内股票收益率的算术平均值 $\mu$ 不同于这些收益率的几何平均值 $\mu -\frac{{\sigma }^2}{2}$。

• $\mu$ is the annual expected return on stock:
  • The expected value of the stock price is $E(S_T)=S_0{e}^{\mu T}$
• The actual continuously compounded return on the stock $R$ satisfies the relationship:
  $$S_T=S_0{e}^{RT}⟹R=\frac{1}{T}\ln \frac{S_T}{S_0}$$
• Since $\ln \frac{S_T}{S_0}\sim \varphi [(\mu -\frac{{\sigma }^2}{2})T,\sigma \sqrt{T}]$, then:
  $$R\sim \varphi [\mu -\frac{{\sigma }^2}{2},\frac{\sigma }{\sqrt{T}}]⟹E[R]=\mu -\frac{{\sigma }^2}{2}(\ne \mu )$$
• This is due to the fact that the arithmetic mean, $\mu$, of the stock returns over a period $\mathrm{\Delta }t$ is different to the geometric mean, $\mu -\frac{{\sigma }^2}{2}$, of these returns.

• 股票价格的波动率 $\sigma$ 是其年化连续复利收益率的标准差。
• 股票在时间段 $\mathrm{\Delta }t$ 内收益率的标准差为：
  $$\sigma \sqrt{\mathrm{\Delta }t}$$
• 算例： 若股票当前价格为 $100，其波动率为 20%，则其日收益率的标准差为多少？
  $$\frac{20\mathrm{\%}}{\sqrt{252}}=\frac{20\mathrm{\%}}{15.87}=1.26\mathrm{\%}\text{ / 交易日}$$
  股票每个交易日的预期价格波动绝对幅度（1个标准差）为：
  $$\mathrm{\$}100\times 0.0126=\mathrm{\$}1.26$$

• The volatility $\sigma$ of the stock price is the standard deviation of its annual continuously compounded return.
• The standard deviation of the return over a period $\mathrm{\Delta }t$ is:
  $$\sigma \sqrt{\mathrm{\Delta }t}$$
• Example: If the stock price is $100 and its volatility is 20%, what is the standard deviation of its daily return?
  $$\frac{20\mathrm{\%}}{\sqrt{252}}=\frac{20\mathrm{\%}}{15.87}=1.26\mathrm{\%}\text{ per trading day}$$
  The expected absolute daily price move (1 standard deviation) is:
  $$\mathrm{\$}100\times 0.0126=\mathrm{\$}1.26$$

• 期权价格与股票价格依赖于同一个底层不确定性来源（布朗运动随机扰动项）。
• 无套利论证（No-arbitrage arguments）： 我们可以构建一个由股票与期权组成的投资组合，从而完全消除这一不确定性来源：
  • 该投资组合在瞬时是无风险的，因此在瞬时必须获取无风险利率 $r$。
• Black-Scholes 模型可视为对冲组合进行无限次频繁再平衡的二叉树极限模型：
  • 当时间步长 $\mathrm{\Delta }t\to 0$ 时，二叉树期权价格的极限即为 Black-Scholes 价格。
• Black-Scholes 模型的核心假设：
  • 股票价格服从对数正态分布，且预期收益率 $\mu$ 与波动率 $\sigma$ 为常数。
  • 在期权存续期内，股票不支付红利。
  • 投资者可以按相同的恒定无风险利率 $r$ 无限制借入或贷出资金。
  • 不存在套利机会；无交易成本与税收；证券交易连续进行。

• The option price and the stock price depend on the same underlying source of uncertainty.
• No-arbitrage arguments: We can form a portfolio consisting of the stock and the option, which eliminates this source of uncertainty:
  • The portfolio is instantaneously riskless and therefore instantaneously earns the risk-free rate $r$.
• The Black-Scholes model is a binomial model where the hedge portfolio is rebalanced infinitely often:
  • The Black-Scholes price is the limit of the binomial tree price as the length of the time step $\mathrm{\Delta }t\to 0$.
• Assumptions of the Black-Scholes Model:
  • The stock price has a lognormal distribution with constant $\mu$ and $\sigma$.
  • The stock pays no dividends during the life of the option.
  • Investors can borrow or lend at the same constant risk-free rate $r$.
  • There are no arbitrage opportunities; no transaction costs or taxes; continuous security trading.

考虑执行价格为 $K$、到期期限为 $T$ 的欧式看涨期权与欧式看跌期权：

• 欧式看涨期权的价格为：
  $$c=S_{0}N(d_1)-K{e}^{-rT}N(d_2)(1)$$
• 欧式看跌期权的价格为：
  $$p=K{e}^{-rT}N(-d_2)-S_{0}N(-d_1)(2)$$
  其中：
  $$\begin{array}{rl}{d}_1& =\frac{\ln (S_0/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}\\ d_2& =\frac{\ln (S_0/K)+(r-{\sigma }^2/2)T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T}\end{array}$$
• $N(x)$ 为标准正态随机变量 $X\sim \varphi (0,1)$ 的累积分布函数（CDF）。

We consider a European call and European put with strike $K$ and maturity $T$:

• The price of the call is:
  $$c=S_{0}N(d_1)-K{e}^{-rT}N(d_2)(1)$$
• The price of the put is:
  $$p=K{e}^{-rT}N(-d_2)-S_{0}N(-d_1)(2)$$
  with:
  $$\begin{array}{rl}{d}_1& =\frac{\ln (S_0/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}\\ d_2& =\frac{\ln (S_0/K)+(r-{\sigma }^2/2)T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T}\end{array}$$
• $N(x)$ is the cumulative distribution function of a standard normal random variable $X\sim \varphi (0,1)$.

• $N(x)$ 表示均值为 0、标准差为 1 的标准正态分布随机变量小于 $x$ 的概率： $$N(d)=\text{Prob}\{z<d\},z\sim \varphi (0,1)$$
• $N(x)$ 的数值可以通过以下方式获取：
  • 使用 Excel 内置函数 NORMSDIST（或 NORM.S.DIST）
  • 查阅标准正态分布概率数值表

• 算例 – 线性插值法： 求 $N(1.645)$ 的值？
  查表可得：$N(1.64)=0.9495$ 以及 $N(1.65)=0.9505$，而 1.645 恰好位于两者中点：
  $$N(1.645)\approx 0.9495+0.5(0.9505-0.9495)=0.9500$$
  即标准正态变量小于 1.645 的概率约为 95%（构造置信区间时常用的分位数）。

• $N(x)$ is the probability that a normally distributed variable with mean 0 and standard deviation 1 is less than $x$: $$N(d)=\text{Prob}\{z<d\},z\sim \varphi (0,1)$$
• Values of $N(x)$ can be obtained by:
  • Using the Excel built-in function NORMSDIST (or NORM.S.DIST)
  • Looking up a standard normal probability table

• Example – Linear Interpolation: What is $N(1.645)$?
  From the table: $N(1.64)=0.9495$ and $N(1.65)=0.9505$, and 1.645 lies exactly halfway between them:
  $$N(1.645)\approx 0.9495+0.5(0.9505-0.9495)=0.9500$$
  So the probability that a standard normal variable is less than 1.645 is about 95% (a quantile commonly used to construct confidence intervals).

参数设定： $S_0=\mathrm{\$}100,K=\mathrm{\$}100,r=5\mathrm{\%},\sigma =20\mathrm{\%},T=1$ 年。股票不支付红利。

本例为简化算例；课本原题完整解答见《Hull 习题与解答》第 8 讲。

Setup: $S_0=\mathrm{\$}100,K=\mathrm{\$}100,r=5\mathrm{\%},\sigma =20\mathrm{\%},T=1$ year. The stock pays no dividends.

This is a simplified example. Original Hull problem with full solution: see Hull Exercises & Solutions, Lecture 8.

• 变量 $\mu$（股票预期收益率）并未出现在 Black-Scholes 公式中：
  • 投资者的风险偏好完全不影响期权理论价格！
• Black-Scholes 公式独立于风险偏好的这一性质，使其与风险中性定价原理完全契合：
  • 在“风险中性”世界中，资产的初始价值等于其在到期日预期收益按无风险利率 $r$ 进行折现的现值。
• 当前资产价格：$S_0=e^{-rT}E[S_T]⟹E[S_T]=S_0{e}^{rT}$
• 远期合约（多头）当前价值（$f_T$：合约在到期日 $T$ 的价值）：
  $$\begin{array}{rl}{f}_0& =e^{-rT}E[f_T]=e^{-rT}E[F_T-F_0]\\ & =e^{-rT}E[S_T]-F_0{e}^{-rT}\\ & =e^{-rT}\cdot \underset{E(S_T)}{\underbrace{S_0{e}^{rT}}}-F_0{e}^{-rT}\\ & =S_0-F_0{e}^{-rT}=(F_T-F_0)e^{-rT}\end{array}$$
• 看涨期权当前价值（$c_T=max(S_T-K,0)$：期权在到期日 $T$ 的回报）：
  $$c_0=e^{-rT}E[c_T]=e^{-rT}E[max[S_T-K,0]]=\dots \text{Black-Scholes 公式}$$

• The variable $\mu$ does not feature in the Black-Scholes analysis:
  • The risk preferences of investors do not affect option prices!
• The preference-independent nature of the Black-Scholes formula makes it consistent with the risk-neutral valuation principle:
  • In a “risk-neutral” world, the initial value of an asset is equal to its discounted expected payoff at maturity, discounted at $r$.
• Current asset price: $S_0=e^{-rT}E[S_T]⟹E[S_T]=S_0{e}^{rT}$
• Current value of a (long) forward contract ($f_T$: the contract's value at maturity $T$):
  $$\begin{array}{rl}{f}_0& =e^{-rT}E[f_T]=e^{-rT}E[F_T-F_0]\\ & =e^{-rT}E[S_T]-F_0{e}^{-rT}\\ & =e^{-rT}\cdot \underset{E(S_T)}{\underbrace{S_0{e}^{rT}}}-F_0{e}^{-rT}\\ & =S_0-F_0{e}^{-rT}=(F_T-F_0)e^{-rT}\end{array}$$
• Current value of a call option ($c_T=max(S_T-K,0)$: the option payoff at maturity $T$):
  $$c_0=e^{-rT}E[c_T]=e^{-rT}E[max[S_T-K,0]]=\dots \text{Black-Scholes formula}$$

• 波动率产生的根源：
  • 在有效市场假说（EMH）下，波动率反映了新信息持续流入市场的过程。
  • 交易活动本身也会产生波动（市场微观结构效应、流动性冲击）。
• 交易日 vs 日历日：
  • 市场开市交易时的资产波动率通常显著高于闭市期间（周末、节假日）。
  • 因此，在估算波动率和期权定价时，时间期限 $T$ 通常以交易日（通常按每年 252 个交易日计算）而非日历日（365 天）进行折算计量。

• Causes of volatility:
  • Under the Efficient Market Hypothesis (EMH), volatility represents new information arriving in the market.
  • Trading itself causes volatility (market microstructure, liquidity shocks).
• Trading Days vs Calendar Days:
  • Volatility is usually much greater when the market is open (i.e. the asset is trading), than when it is closed (weekends, holidays).
  • For this reason, time $T$ is usually measured in trading days (typically 252 days per year), not calendar days (365 days), for the purposes of estimating volatility and pricing options.

• 对支付红利股票的欧式期权进行估值时，只需将 Black-Scholes 公式中的现货股票价格 $S_0$ 替换为扣除红利现值 $PV(D)$（$D$：红利金额）后的调整后股票价格：
  $$S_0\to S_0^{\ast }=S_0-PV(D)$$
  注意：此处仅需扣除在期权有效期内发生除息的红利现值。
• 因此，支付红利股票的欧式看涨与看跌期权价格分别为：
  $$\begin{array}{rl}c& =S_0^{\ast }N(d_1)-K{e}^{-rT}N(d_2)(3)\\ p& =K{e}^{-rT}N(-d_2)-S_0^{\ast }N(-d_1)(4)\end{array}$$
  其中：
  $$\begin{array}{rl}{d}_1& =\frac{\ln (S_0^{\ast }/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}\\ d_2& =\frac{\ln (S_0^{\ast }/K)+(r-{\sigma }^2/2)T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T}\end{array}$$

• European options on dividend-paying stocks are valued by substituting into the Black-Scholes formula, the stock price $S_0$ with the difference between the stock price and the present value $PV(D)$ of the dividends ($D$: the dividend amount):
  $$S_0\to S_0^{\ast }=S_0-PV(D)$$
  Only dividends with ex-dividend dates during the life of the option should be included.
• Thus the prices of a call and a put on a dividend-paying stock are:
  $$\begin{array}{rl}c& =S_0^{\ast }N(d_1)-K{e}^{-rT}N(d_2)(3)\\ p& =K{e}^{-rT}N(-d_2)-S_0^{\ast }N(-d_1)(4)\end{array}$$
  where:
  $$\begin{array}{rl}{d}_1& =\frac{\ln (S_0^{\ast }/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}}\\ d_2& =\frac{\ln (S_0^{\ast }/K)+(r-{\sigma }^2/2)T}{\sigma \sqrt{T}}=d_1-\sigma \sqrt{T}\end{array}$$

• 对于不支付红利的股票，美式看涨期权绝不应提前行权（$C=c$）。
• 对于支付红利的股票，美式看涨期权仅在除息日即刻之前提前行权才可能最优。
• 美式看跌期权在满足特定条件时（深度实值、利率较高），即使股票不支付红利，通常也应提前行权。
• 美式看涨期权的价格可以通过两个欧式看涨期权价格的最大值进行近似估算（布莱克近似法 / Black's approximation）：
  • 期权 1： 到期日与美式期权相同、股价扣除红利现值后的欧式看涨期权（即使用 $S_0^{\ast }=S_0-PV(D)$，其中 $D$ 为红利，$PV(D)$ 为红利现值）。
  • 期权 2： 在最后一个除息日即刻前到期的欧式看涨期权（使用未经扣减的当前股价 $S_0$）。

• An American call on a non-dividend-paying stock should never be exercised early ($C=c$).
• An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date.
• American puts are typically exercised early under the right conditions, even when the stock pays no dividends.
• The price of the American call is estimated as the maximum of two European option prices (Black's approximation):
  • Option 1: The European call maturing at the same date as the American option (with $S_0^{\ast }=S_0-PV(D)$, where $D$ is the dividend and $PV(D)$ its present value).
  • Option 2: The European call maturing just before the final ex-dividend date (with unadjusted $S_0$).

参数设定： $S_0=\mathrm{\$}100,\sigma =20\mathrm{\%},r=5\mathrm{\%}$（与上面的算例相同）。预计 6 个月后（$t_1=0.5$ 年）支付红利 $D=\mathrm{\$}5$。看涨期权执行价格 $K=\mathrm{\$}90$（深度实值，提前行权才可能划算），期限 $T=1$ 年。

Setup: $S_0=\mathrm{\$}100,\sigma =20\mathrm{\%},r=5\mathrm{\%}$ (same as the worked example above). A dividend $D=\mathrm{\$}5$ is expected in 6 months ($t_1=0.5$ year). Call strike $K=\mathrm{\$}90$ (deep in the money, so that early exercise can pay), maturity $T=1$ year.