二叉树期权定价--潘登同学的期权、期货及其他衍生品学习笔记
期权定价方法 – 二叉树
Option Pricing Method - Binomial Tree
- 叉树 Binomial Tree
- 在期权期限内，股票价格变动的可能路径图 A map of the possible paths of stock price movement over the option horizon
- 假设 Assume
- 股票的价格过程是一个随机游走过程 The stock price process is a Random Walk process
- 每一步，股票价格以某种概率上涨或者下跌 At each step, the stock price rises or falls with a certain probability
- 当步长足够小，模型与Black-Scholes-Merton模型一致 When the step size is small enough, the model is consistent with the Black-Scholes-Merton model
- 意义 Meaning
- 无套利假设的特点 Features of the no-arbitrage assumption
- 衍生品中的一种重要定价方法 An Important Pricing Method in Derivatives
- 引入风险中性定价原理 Introducing the Principle of Risk-Neutral Pricing
一个期权定价的例子
- 当前股价为20美元 Current share price is \$20
- 该股票的 3 个月看涨期权的执行价格为 21 美元
The 3-month call option on the stock has a strike price of \$21
- 3 个月后，股价将变为 22 美元或 18 美元 After 3 months, the share price will be \$22 or \$18
- 如果股价变为 22 美元，期权价值为 1 美元 If the stock price changes to \$22, the option is worth \$1
- 如果股价变为 18 美元，期权价值为 0 美元 If the stock price changes to \$18, the option is worth \$0
- 建立无风险投资组合 Construct a Risk-Free Portfolio
考虑一个由 ∆ 单位的股票多头和 1 份看涨期权空头的投资组合 Consider a portfolio long ∆ units of stock and short 1 call
当股价变为 22 美元时 When the stock price goes to \$22
- 投资组合价值为$22\triangle(股票/Stock) -1(期权/Option)$
当股价变为 18 美元时 When the stock price goes to $18
- 投资组合价值为$18\triangle(股票/Stock) -0(期权/Option)$
无风险投资组合 Risk free portfolio
- 如果投资组合在以上两种情况下，价值相等 If the portfolio is equal in value in the above two cases $$ 22\triangle -1 = 18\triangle \Rightarrow \triangle = 0.25 $$
- 投资组合估值 Portfolio Valuation
- 假设无风险利率为 4 % (连续复利) Assume a risk-free rate of 4% (continuously compounded)
- 无风险投资组合为 The risk-free portfolio is:：
- 多头： 买入 0.25 股股票 Long: Buy 0.25 shares of stock
- 空头： 卖出 1 份看涨期权 Short: Sell 1 call option
- 投资组合在 3 个月后的价值为
The value of the portfolio after 3 months is
- $22 ∗ 0.25 − 1 = 4.5$
- $18 ∗ 0.25 = 4.5$
- 期初投资组合的价值是
The value of the initial portfolio is
- $4.5e^{-4\%0.25}=4.455$
- 期权定价 Option Pricing
- 当前股价为 20 美元 Current share price is \$20
- 根据投资组合 3 个月后的价值
Based on portfolio value after 3 months
- 投资组合的现值为$4.455 The present value of the portfolio is $4.455
- 根据期初的现金流，假设期权价格为 𝑓
According to the cash flow at the beginning of the period, suppose the option price is 𝑓
- 投资组合的价值 Portfolio Value $$ 20 ∗ 0.25 − f = 5 − f ⇒ 5 − f = 4.455 $$ • 因此期权价格为/So the option price is $$ f = 5 − 4.455 = 0.545 $$
- 如果𝑓 > 0.545，投资组合的收益率会高于无风险利率 If 𝑓 > 0.545, the return on the portfolio will be higher than the risk-free rate
- 如果𝑓 < 0.545，卖空该投资组合可以获得低于无风险利率的借款 If 𝑓 < 0.545, short selling the portfolio can obtain borrowing below the risk-free rate
一般化方法
- 假设股票价格为$S_0$，期权价格为𝑓，期权期限为 𝑇 Suppose the stock price is $S_0$, the option price is 𝑓, and the option period is 𝑇
有效期内 Within the validity period
- 股票价格可能上涨到$S_0u$，期权价值为$f_u$ The stock price may rise to $S_0u$, the option value is $f_u$
- 股票价格可能下跌至$S_0d$，期权价值为$f_d$ The stock price may fall to $S_0d$, the option value is $f_d$
STEP1: 考虑一个 ∆ 单位的股票多头和 1 份看涨期权空头的投资组合 Consider a portfolio long ∆ units of stock and short 1 call option
- 当股价上涨至 $S_0u$ 时
When the stock price rises to $S_0u$
- 投资组合价值为 $S_0u∆ − f_u$
- 当股价下跌至 $S_0d$ 时
When the stock price drops to $S_0d$
- 投资组合价值为 $S_0d∆ − f_d$
- 当股价上涨至 $S_0u$ 时
When the stock price rises to $S_0u$
- STEP2: 无风险投资组合 Risk free portfolio
- 如果投资组合在以上两种情况下，价值相等 If the portfolio is equal in value in the above two cases $$ S_0u∆ − f_u = S_0d∆ − f_d \ ∆ = \frac{f_u-f_d}{S_0u-S_0d} $$
STEP3:根据无套利条件
- 根据投资组合在时间 T 的价值 According to the value of the portfolio at time T
- 投资组合的现值是 The present value of the portfolio is $$ (S_0u\triangle -f_u)e^{-rT} $$
- 根据期初的现金流 According to the cash flow at the beginning of the period
- 投资组合价值是/Portfolio value is $$ S_0\triangle -f $$
- 两者相等： $$ f = S_0\triangle-(S_0u\triangle -f_u)e^{-rT} $$
- 将$∆ = \frac{f_u-f_d}{S_0u-S_0d}$代入 $$ f=e^{-rT}[pf_u+(1-p)f_d],p=\frac{e^{rT}-d}{u-d} $$
p 作为概率 p as the probability
- 我们可以将 p 和 1- p 视为股价上涨与下跌的概率 We can think of p and 1-p as the probability that the stock price will rise and fall
- 衍生品的价值就是其在以无风险利率贴现的风险中性世界中的预期收益 The value of a derivative is its expected return in a risk-neutral world discounted at the risk-free rate
风险中性定价 Risk Neutral Pricing
- 风险中性 Risk Neutral
- 当投资风险上升时，投资者不需要额外的期望收益率
- When investment risk rises, investors do not need additional expected rate of return
- 风险中性定价 Risk Neutral Pricing
- 当对衍生品进行定价时，假设投资者是风险中性的
- When pricing derivatives, assume that investors are risk-neutral
- 当上涨和下跌的概率为 p 和 1- p 时，在时间 T 的预期股票价格为 $S_0e^{rT}$
The expected stock price at time T is $S_0e^{rT}$ when the probabilities of up and down are p and 1-p
- 股票价格以无风险利率均速上涨
- Stock prices rise at an average rate of risk-free rate
- 二叉树说明了对衍生品进行估值的一般结果 A binary tree illustrating the general result of valuing derivatives
使用风险中性定价对期权进行估值
- 无风险利率为 4\%
- 期限为3个月
$$ 20e^{4\%0.25}=22p+18(1-p) \ 得 p=0.5503 \ f = e^{-4\%0.25}(0.55031+(1-0.5503)*0)=0.545 $$
- 股票预期收益的无关性
The Irrelevance of Stock Expected Returns
- 当我们根据标的资产价格对期权定价时，现实世界中上涨或者下跌的概率是不相关的 When we price options based on the price of the underlying asset, the probabilities of up or down in the real world are irrelevant
- 股票价格以及期望已然包含概率信息 Stock prices and expectations already contain probabilistic information
u 和 d 的选择 Choice of u and d
匹配波动率 Matching Volatility $$ u = e^{\sigma\sqrt{\triangle t}} \ d=u^{-1} = e^{-\sigma\sqrt{\triangle t}} \ $$
- 𝜎 是波动率 volatility
- ∆𝑡是步长时间 step time
𝜎 ∆𝑡代表时间长度为∆𝑡上收益的标准差 standard deviation of returns over a time length of ∆𝑡
$\sigma^2\triangle t$为收益的方差 variance of returns
两步二叉树 Two-Step Binary Tree
- 股价从0时刻往T时刻算
- 期权价格从T时刻往前算
美式看跌期权
- 与欧式期权一样，股价从前往后算，期权从后往前算
- 但是每一个节点需要判断是否行权，判断标准是行权价与期望价值的比较
