Black-Scholes-Merton模型--潘登同学的期权、期货及其他衍生品学习笔记

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股票价格的正态分布性质 The Normal Distribution Properties of Stock Prices

• 考虑一只价格为 𝑆 的股票，在 ∆𝑡 的时间内，股票的收益率呈正态分布 Consider a stock with price 𝑆 whose returns are normally distributed over time ∆𝑡

$$ \frac{\triangle S}{S} \sim \phi(\mu\triangle t,\sigma^2\triangle t) $$

• 𝜇 代表股票的预期收益率 𝜇 represents the expected return on the stock • $\sigma^2$代表股票价格的波动率 $\sigma^2$ represents the volatility of the stock price
• 股价变化服从伊藤过程 $$ \triangle S = \mu S\triangle t + \sigma S\triangle z $$
• 股价变化率服从广义维纳过程 $$ \frac{\triangle S}{S} = \mu \triangle t + \sigma \triangle z $$
• 股价的对数服从广义维纳过程 $$ \begin{cases} 𝐺 = ln S \ dS = \mu Sdt + \sigma Sdz \end{cases} \ \Rightarrow \begin{aligned} dG &= (\frac{\partial{G}}{\partial{S}}\mu S + \frac{\partial{G}}{\partial{t}}+\frac{1}{2}\frac{\partial^2{G}}{\partial{S^2}}\sigma^2 S^2)dt + \frac{\partial{G}}{\partial{S}}\sigma Sdz\ &=(\frac{1}{S}\mu S + 0 + -\frac{1}{2S^2}\sigma^2 S^2)dt + \frac{1}{S}\sigma Sdz\ &=(\mu -\frac{1}{2}\sigma^2)dt + \sigma dz \end{aligned} $$

$$ ln S_t-ln S_0 \sim \phi[(\mu -\frac{1}{2}\sigma^2)T,\sigma^2T] \ ln S_t \sim \phi[ln S_0+(\mu -\frac{1}{2}\sigma^2)T,\sigma^2T] \ $$

• 由于$S_T$的对数服从正态分布，$S_T$服从对数正态分布

【问题】

• 股票的初始价格$S_0$ = \$40，期望收益为每年16%，波动率为每年20% The initial price of the stock $S_0$ = \$40, the expected return is 16% per year, and the volatility is 20% per year
• 股票价格在6个月内的概率分布为 The probability distribution of the stock price in 6 months is $$ ln S_t \sim \phi[ln S_0+(\mu -\frac{1}{2}\sigma^2)T,\sigma^2T] \ ln S_t \sim \phi[ln 40+(0.26-\frac{0.2^2}{2})0.5,0.2^20.5]\sim \phi(3.759,0.02) $$
• 95%置信区间/95% Confidence Interval $$ ln S_t \in (3.759\pm 1.96*\sqrt{0.02}) \ S_T \in (32.55,56.56) $$

对数正态分布 The Lognormal Distribution

如果一个随机变量X服从对数分布，即$log X\sim \Phi(\mu,\sigma^2)$,那么有 $$ E(X) = e^{\mu+\frac{1}{2}\sigma^2} $$ 而$ln S_t-ln S_0$服从对数正态分布，则有 $$ E(\frac{S_T}{S_0}) = e^{\mu} \Rightarrow E(S_T)=S_0e^{\mu} \ Var(S_T) = S_0^2e^{2\mu T}(e^{\sigma^2 T}-1) $$

+某股票的当前价格为$𝑆_0$ = \$20，期望收益为每年20%，波动率为每年40% The current price of a stock is $𝑆_0$ = \$20, the expected return is 20% per year, and the volatility is 40% per year

• 那么，该股票在一年内的价格期望和方差有以下推断 Then, the price expectation and variance of the stock within one year have the following inferences

$$ E(S_T)=S_0e^{\mu} = 20*e^{0.2} = 24.43 \ Var(S_T)= S_0^2e^{2\mu T}(e^{\sigma^2 T}-1)=103.54 \ SD(S_T)=\sqrt{103.54} = 10.18 $$

连续复利收益率 Continuously Compounded Return

• 假设 𝑥 是被实现的连续复利 Assume 𝑥 is realized continuous compounding

$$ S_T = S_0e^{rT} \Rightarrow x=\frac{1}{T}ln\frac{S_T}{S_0} \ ln S_T - ln S_0 = ln \frac{S_T}{S_0}\sim \phi((\mu-\frac{\sigma^2}{2})T,\sigma^2 T)\ X=\frac{1}{T}ln\frac{S_T}{S_0}\sim \phi((\mu-\frac{\sigma^2}{2}),\frac{\sigma^2}{T})\ $$

• 某股票的期望收益为每年17%，波动率为每年20% A stock has an expected return of 17% per annum and a volatility of 20% per annum
• 3年内实现的平均收益（以连续复利计算）服从正态分布 (95%置信区间) The average return (calculated with continuous compound interest) realized within 3 years follows a normal distribution (95% Confidence Interval)

$$ ln \frac{S_T}{S_0}\sim \phi((\mu-\frac{\sigma^2}{2})T,\sigma^2 T)\ X=\frac{1}{T}ln\frac{S_T}{S_0}\sim \phi((\mu-\frac{\sigma^2}{2}),\frac{\sigma^2}{T})\sim \phi(0.15,0.1155)\ X\in (15\pm1.96*0.1155)\ X\in (-7.6\%,37.6\%)\ $$

预期收益 The Expected Return

• 股票价格的期望值是$S_0e^{\mu T}$ The expected value of the stock price is$S_0e^{\mu T}$
• 股票的预期收益是$\mu-\frac{\sigma^2}{2}$，而不是 𝜇 The expected return on the stock is $\mu-\frac{\sigma^2}{2}$rather than 𝜇

• 因为以下二者并不相等 Because the following two variables are not the same

  • $ln[E(\frac{S_T}{S_0})]$
  • $E[ln(\frac{S_T}{S_0})]$

• 𝜇 是在每个时间长度为 $\triangle t$ 的时间窗口的平均收益率 𝜇 is the expected return in a very short time, $\triangle t$ , expressed with a compounding frequency of ∆𝑡

• $\mu-\frac{\sigma^2}{2}$是用连续复利表示的长期预期回报（或者，为了一个好的近似值，复利频率为 $\triangle t$） $\mu-\frac{\sigma^2}{2}$ is the expected return in a long period of time expressed with continuous compounding (or, to a good approximation, with a compounding frequency of $\triangle t$)
  • 算术平均值类似于 𝜇 The arithmetic mean is analogous to 𝜇
  • 几何平均值类似于 $\mu-\frac{\sigma^2}{2}$ The geometric mean is analogous to $\mu-\frac{\sigma^2}{2}$

利用历史数据估计波动率 Estimating Volatility from Historical Data

• 通过观察 $𝑆_0, 𝑆_1, … , 𝑆_𝑛$ 以 𝜏 年为间隔（例：对于周度数据 $\tau = \frac{1}{52}$） Take observations $𝑆_0, 𝑆_1, … , 𝑆_𝑛$ at intervals of t years (e.g. for weekly data t = 1/52)
• 每个区间内的连续复合收益为： Calculate the continuously compounded return in each interval as: $$ u_i = ln(\frac{S_i}{S_{i-1}})\ x=\frac{1}{T}ln\frac{S_T}{S_0}\sim\phi(\mu-\frac{\sigma^2}{2},\frac{\sigma^2}{T})\Rightarrow u_i = \ln \frac{s_i}{s_{i-1}}\sim \phi(\tau(\mu-\frac{\sigma^2}{2}),\tau\sigma^2)\ E(u_i) = \tau(\mu-\frac{\sigma^2}{2}),Var(u_i) = \tau\sigma^2,SD(u_i)=\sqrt{\tau}\sigma $$
• $𝑢_𝑖$ 标偏差的估计值为 The estimate of $𝑢_𝑖$′𝑠 standard deviation $$ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n(u_i-\bar{u})^2} $$
• 历史波动率估计为 The historical volatility estimate is： $$ \hat{\sigma} = \frac{s}{\sqrt{\tau}} $$

波动性的性质 Nature of Volatility

• 市场开盘（即资产正在交易）时的波动性通常比收市时高 Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed
• 出于这个原因，时间通常以“交易日”来衡量，而不是期权估值的日历日 For this reason time is usually measured in “trading days” not calendar days when options are valued
• 假设大多数资产一年有 252 个交易日 It is assumed that there are 252 trading days in one year for most assets

Black-Scholes-Merton 微分方程

• 股票价格服从以下过程 The stock price obeys the following process $$ \triangle S = \mu S\triangle t + \sigma S\triangle t $$

• 𝑓是关于𝑆的看涨期权价格，并且是𝑆和𝑡的函数 𝑓 is the call option price on 𝑆 and is a function of 𝑆 and t $$ \triangle f = (\frac{\partial f}{\partial S}\mu S + \frac{\partial f}{\partial t} + \frac{1}{2}\frac{\partial^2 f}{\partial{S^2}}\sigma^2 S^2)\triangle t + \frac{\partial f}{\partial S}\sigma S\triangle z $$

• 我们构造以下投资组合 We construct the following portfolio

  • 卖出1单位看涨期权 Sell 1 call option
  • 买入$\frac{\partial f}{\partial S}$单位股票 Buy $\frac{\partial f}{\partial S}$ unit stock

• $\Pi$为投资组合的价值 $\Pi$ is the value of the investment portfolio $$ \Pi = -f + \frac{\partial f}{\partial S} S $$

• 投资组合在∆𝑡时间区间内的变化服从 Portfolio changes within the ∆𝑡 time interval obey $$ \triangle \Pi = -\triangle f + \frac{\partial f}{\partial S} \triangle S \ \triangle \Pi = -((\frac{\partial f}{\partial S}\mu S + \frac{\partial f}{\partial t} + \frac{1}{2}\frac{\partial^2 f}{\partial{S^2}}\sigma^2 S^2)\triangle t + \frac{\partial f}{\partial S}\sigma S\triangle z) + \frac{\partial f}{\partial S} (\mu S\triangle t + \sigma S\triangle t) \ \triangle \Pi = (-\frac{\partial f}{\partial t}-\frac{1}{2}\frac{\partial^2 f}{\partial{S^2}}\sigma^2 S^2)\triangle t $$

• 由于∆𝑧被互相消除，所以该证券组合在∆𝑡时间内是无风险的 Since ∆𝑧 cancel each other out, the portfolio is risk-free for ∆𝑡 time

• 根据无套利定价 $$ \triangle \Pi = r\Pi \triangle t \ (-\frac{\partial f}{\partial t}-\frac{1}{2}\frac{\partial^2 f}{\partial{S^2}}\sigma^2 S^2)\triangle t= r(-f + \frac{\partial f}{\partial S} S)\triangle t \ (-\frac{\partial f}{\partial t}-\frac{1}{2}\frac{\partial^2 f}{\partial{S^2}}\sigma^2 S^2)= -rf + rS\frac{\partial f}{\partial S} \ \Rightarrow \frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+\frac{1}{2}\frac{\partial^2 f}{\partial{S^2}}\sigma^2 S^2=rf \quad \text{BSM微分方程} $$

• 该方程的解与边界条件相关，边界条件定义了衍生品价格在𝑆和𝑡的边界上的取值范围 The solution of this equation is related to the boundary conditions, which define the value range of the derivative price on the boundary of 𝑆 and 𝑡

  • 欧式看涨期权的关键边界条件 Key Boundary Conditions for European Call Options
    • 当𝑡 = 𝑇时，𝑓 = max（𝑆 − 𝐾, 0）
  • 欧式看跌期权的关键边界条件 Key Boundary Conditions for European Put Options
    • 当𝑡 = 𝑇时，𝑓 = max（𝐾 − 𝑆, 0）

微分方程的解为看涨期权与看跌期权的定价公式 $$ c = S_0N(d_1)-Ke^{-rT}N(d_2)\ p = Ke^{-rT}N(-d_2)-S_0N(-d_1) \ d_1 = \frac{ln(\frac{S_0}{K})+(r+\frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \ d_2 = \frac{ln(\frac{S_0}{K})+(r-\frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \ $$

• 𝑁(x)是标准正态分布的累积概率分布函数 𝑁(x)is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x

Black-Scholes-Merton 公式的性质 Properties of Black-Scholes-Merton Formula

• 当$𝑆_0$非常大时 As $𝑆_0$ becomes very large
  • $d_1,d_2 \to \infty$
  • $N(d_1),N(d_2) \to 1$
  • $c\to S_0 - Ke^{-rT}$
  • $N(-d_1),N(-d_2) \to 0$
  • $p\to 0$
• 当$𝑆_0$非常小时 As $𝑆_0$ becomes very small
  • $d_1,d_2 \to -\infty$
  • $N(d_1),N(d_2) \to 0$
  • $c\to 0$
  • $N(-d_1),N(-d_2) \to 1$
  • $p\to Ke^{-rT}-S_0$
• 当𝜎非常大/小时会发生什么？ What happens as 𝜎 becomes very large/small
  • 𝜎非常大时，$d_1\to \infty ,d_2\to -\infty$，$c\to S_0 $，$p\to Ke^{-rT}$
  • 𝜎非常小时，股票价格几乎无风险。$c\to max(S_0e^{-rT}-K,0),p\to max(K-S_0e^{-rT},0)$

$$ c = S_0N(d_1)-Ke^{-rT}N(d_2)=e^{-rT}N(d_2)[S_0e^{rT}\frac{N(d_1)}{N(d_2)}-K] $$

• $e^{-rT}$ :贴现因子/Present value factor
• $N(d_2)$ :行权概率/Probability of exercise
• $S_0e^{rT}\frac{N(d_1)}{N(d_2)}$ :如果期权被行使，在风险中性世界的预期股票价格/Expected stock price in a risk-neutral world if option is exercised
• $K$ :期权被行使时，相应的执行价格/Strike price paid if option is exercised

风险中性估值 Risk-Neutral Valuation

• 变量𝜇没有出现在 Black-Scholes-Merton 微分方程中(因为$\mu$(股票的期望收益)反映投资者的风险厌恶程度，厌恶程度越高，$\mu$值越高) The variable 𝜇 does not appear in the Black-Scholes-Merton differential equation
• 该方程独立于受风险偏好影响的所有变量(因为没有$\mu$) The equation is independent of all variables affected by risk preference
• 因此，在无风险世界中，微分方程的解与在现实世界中的解是相同的(所以可以使用风险中性投资者进行求解) The solution to the differential equation is therefore the same in a risk-free world as it is in the real world
• 这就引出了风险中性估值原则 This leads to the principle of risk-neutral valuation

应用风险中性估值 Applying Risk-Neutral Valuation

1. 假设股票价格的预期收益是无风险利率 Assume that the expected return from the stock price is the risk-free rate
2. 计算期权的预期收益 Calculate the expected payoff from the option
3. 以无风险利率折现 Discount at the risk-free rate

隐含波动率 Implied Volatility

• 期权的隐含波动率是 Black-Scholes-Merton 价格等于市场价格的波动率(根据市场价格代入B-S公式，反求波动率) The implied volatility of an option is the volatility for which the Black-Scholes-Merton price equals the market price
• 价格和隐含波动率之间存在一一对应关系 There is a one-to-one correspondence between prices and implied volatilities
• 交易员和经纪人经常引用隐含波动率而不是价格 Traders and brokers often quote implied volatilities rather than prices