第8讲：二叉树期权定价模型
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Stock Price = $18 Option Price = $0
Binomial Trees and Option Valuation
5
无风险组合法（Riskless Portfolio Approach）
• Consider a portfolio: long D shares short 1 call option (covered call writing)
is worth 4.367 • The value of the shares is
D stock value = 5.000 (i.e., 0.25´20 ) • The value of the option is $0.633.
D stock value – call value = 4.367 Call value = 5.000 – 4.367 = 0.633
• Binomial Model（二叉树模型） • Risk Neutral Valuation （风险中性定价）
• Binomial Approximation （二叉树近似）
short intervals or steps. Asset prices over a short interval follow a binomial distribution: Asset prices go up or down by a certain amount. The tree represents all the possible paths that the stock price could take during the life of the option.
Binomial Trees and Option Valuation
2
Lecture 7: Binomial Model
Outline
A method of option pricing in which
• Riskless Portfolio（无风险组合） the time to maturity is divided into many
• The value of the portfolio today (r =12% per annum) is 4.5e – 0.12´0.25 = 4.3670
Binomial Trees and Option Valuation
7
期权的价值
• The portfolio that is long 0.25 shares short 1 option
• The payoff of the portfolio in 3 months is
22Dwk.baidu.com– 1
18D • Portfolio is riskless when 22D – 1 = 18D or D = 0.25
Binomial Trees and Option Valuation
6
组合的价值
• For example: An European call option on the stock has a strike price of $21, a maturity of 3 months
Stock price = $20 Option Price=?
Stock Price = $22 Option Price = $1
• 二叉树模型假设一个简单的资产价格变动过程：在一个时期以后，价 格可能从S变动到两种可能价格： 价格上涨到uS或者价格下降到dS
Example: A stock price is currently $20
In three months it will be one of the two possible prices: $22 (+10%) or $18
1
第8讲：二叉树期权定价模型 Binomial Model
Outline • 无风险证券组合Riskless Portfolio •二叉树期权定价模型Binomial Model • 风险中型定价Risk Neutral Valuation •二叉树期权定价模型的实际应用Binomial Approximation
(-10%)
Stock Price (uS)= $22
Stock price(S) = $20
Stock Price (dS)= $18
Binomial Trees and Option Valuation
4
一欧式看跌期权：二叉树定价
• Knowing the possible prices next period allows us to price a one-period call option at its maturity.
• The riskless portfolio is: long 0.25 shares short 1 call option
• The value of the portfolio in 3 months is 22´0.25 – 1 = 4.50 (no matter if the price is up or down)
Binomial Trees and Option Valuation
8
组合复制法（Replicating Portfolio Approach）
Binomial Trees and Option Valuation
3
单期二叉树定价模型 (One-Step Binomial Model）
• Binomial model assumes a simple asset price process: Over a time period
(arbitrary length), the price can go from S to two possible prices: uS (u represents 1 + rate of price appreciation) if price takes an upward step; dS if price takes a downward step.
Binomial Trees and Option Valuation
5
无风险组合法（Riskless Portfolio Approach）
• Consider a portfolio: long D shares short 1 call option (covered call writing)
is worth 4.367 • The value of the shares is
D stock value = 5.000 (i.e., 0.25´20 ) • The value of the option is $0.633.
D stock value – call value = 4.367 Call value = 5.000 – 4.367 = 0.633
• Binomial Model（二叉树模型） • Risk Neutral Valuation （风险中性定价）
• Binomial Approximation （二叉树近似）
short intervals or steps. Asset prices over a short interval follow a binomial distribution: Asset prices go up or down by a certain amount. The tree represents all the possible paths that the stock price could take during the life of the option.
Binomial Trees and Option Valuation
2
Lecture 7: Binomial Model
Outline
A method of option pricing in which
• Riskless Portfolio（无风险组合） the time to maturity is divided into many
• The value of the portfolio today (r =12% per annum) is 4.5e – 0.12´0.25 = 4.3670
Binomial Trees and Option Valuation
7
期权的价值
• The portfolio that is long 0.25 shares short 1 option
• The payoff of the portfolio in 3 months is
22Dwk.baidu.com– 1
18D • Portfolio is riskless when 22D – 1 = 18D or D = 0.25
Binomial Trees and Option Valuation
6
组合的价值
• For example: An European call option on the stock has a strike price of $21, a maturity of 3 months
Stock price = $20 Option Price=?
Stock Price = $22 Option Price = $1
• 二叉树模型假设一个简单的资产价格变动过程：在一个时期以后，价 格可能从S变动到两种可能价格： 价格上涨到uS或者价格下降到dS
Example: A stock price is currently $20
In three months it will be one of the two possible prices: $22 (+10%) or $18
1
第8讲：二叉树期权定价模型 Binomial Model
Outline • 无风险证券组合Riskless Portfolio •二叉树期权定价模型Binomial Model • 风险中型定价Risk Neutral Valuation •二叉树期权定价模型的实际应用Binomial Approximation
(-10%)
Stock Price (uS)= $22
Stock price(S) = $20
Stock Price (dS)= $18
Binomial Trees and Option Valuation
4
一欧式看跌期权：二叉树定价
• Knowing the possible prices next period allows us to price a one-period call option at its maturity.
• The riskless portfolio is: long 0.25 shares short 1 call option
• The value of the portfolio in 3 months is 22´0.25 – 1 = 4.50 (no matter if the price is up or down)
Binomial Trees and Option Valuation
8
组合复制法（Replicating Portfolio Approach）
Binomial Trees and Option Valuation
3
单期二叉树定价模型 (One-Step Binomial Model）
• Binomial model assumes a simple asset price process: Over a time period
(arbitrary length), the price can go from S to two possible prices: uS (u represents 1 + rate of price appreciation) if price takes an upward step; dS if price takes a downward step.