21-1
第 21章期权定价
Option Valuation
21-2
期权定价
Option Valuation
21.1 期权定价简介
21,2 期权价值的限制
21.3 二项式 期权定价模型
21.4 布莱克 -斯克尔斯 期权定价
21.5 布莱克 -斯克尔斯公式的运用
21.6 经验证明
21-3
期权定价
Option Values
内在价值 – 立即执行该期权能获的收入
Intrinsic value - profit that could be made if
the option was immediately exercised
– 看涨期权,股票价格 – 执行价格
Call,stock price - exercise price
– 看跌期权,执行价格 - 股票价格
Put,exercise price - stock price
时间价值 – 期权价格与内在价值间的差异
Time value - the difference between the
option price and the intrinsic value
21-4
期权的时间价值,看涨期权
Time Value of Options,Call
期权价值
Option
value
X 股票价格 Stock Price
看涨期权价值
Value of Call
内在价值
Intrinsic Value时间价值
Time value
21-5
影响看涨期权价值的因素
Factors Influencing Option Values,Calls
因素 Factor 对 价值的作用 Effect on value
股票价格 Stock price increases
执行价格 Exercise price decreases
股票价格的波动 Volatility of stock price increases
到期 Time to expiration increases
利息率 Interest rate increases
股利 Dividend Rate decreases
21-6
二项式期权定价模型：举例
Binomial Option Pricing,Text Example
100
200
50
股票价格 Stock Price
C
75
0
看涨期权价值
Call Option Value
X = 125
21-7
二项式期权定价模型：举例
Binomial Option Pricing,Text Example
另一个组合 Alternative Portfolio
买 1股 100元的股票借 $46.30
8%的利率）净支出是 $53.70
Buy 1 share of stock at $100
Borrow $46.30 (8% Rate)
Net outlay $53.70
收入 Payoff
Value of Stock 50 200
Repay loan - 50 -50
Net Payoff 0 150
53.70
150
0
收入结构正好是看涨期权的 2倍
Payoff Structure is
exactly 2 times the Call
21-8
二项式期权定价模型：举例
Binomial Option Pricing,Text Example
53.70
150
0
C
75
0
2C = $53.70
C = $26.85
21-9
收入和期权价值的另一种观点
Another View of Replication of Payoffs and Option Values
另一个组合 – 1股股票和两个售出的看涨期权的组合恰好被套期保值
Alternative Portfolio - one share of stock and 2 calls
written (X = 125) Portfolio is perfectly hedged
股票价值 Stock Value 50 200
两个售出的看涨期权的义务
Call Obligation 0 -150
净收入 Net payoff 50 50
Hence 100 - 2C = 46.30 or C = 26.85
21-10
布莱克 -斯科尔斯期权定价模型
Black-Scholes Option Valuation
Co = Soe-dTN(d1) - Xe-rTN(d2)
d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2)
d2 = d1 - (s T1/2)
式中 where
Co = 当前看涨期权的价值 Current call option value.
So = 当前股票价格 Current stock price
N(d) = 随机的偏离标准正态分布的概率小于 d
probability that a random draw from a normal dist,
will be less than d.
21-11
布莱克 -斯科尔斯期权定价模型
Black-Scholes Option Valuation
X = 执行价格 Exercise price.
d = 标的股票的年股利收益率 Annual dividend
yield of underlying stock
e = 2.71828,自然对数函数的底数 the base of the
nat,log.
r = 无风险利率 Risk-free interest rate (annualizes
continuously compounded with the same
maturity as the option.
21-12
布莱克 -斯科尔斯期权定价模型
Black-Scholes Option Valuation
T = 期权到期前的时间（以年为单位）
time to maturity of the option in years.
ln = 自然对数 Natural log function
s = 股票连续复利年收益率的标准差 Standard
deviation of annualized cont,compounded
rate of return on the stock
21-13
标准正态曲线布莱克 -斯科尔斯看涨期权举例
Call Option Example
股票价格 So = 100 执行价格 X = 95
利率 r =,10 到期时间 T =,25 (quarter)
标准差 s =,50 股利收益率 d = 0
d1 = [ln(100/95)+(.10-0+(.5 2/2))]/(.5,251/2)
=,43
d2 =,43 - ((.5)(,251/2)
=,18
21-14
正态分布的概率
Probabilities from Normal Dist.
N (.43) =,6664
Table 17.2
d N(d)
.42,6628
.43,6664 添写 Interpolation
.44,6700
21-15
正态分布的概率
Probabilities from Normal Dist.
N (.18) =,5714
Table 17.2
d N(d)
.16,5636
.18,5714
.20,5793
21-16
看涨期权价值
Call Option Value
Co = Soe-dTN(d1) - Xe-rTN(d2)
Co = 100 X,6664 - 95 e-,10 X,25 X,5714
Co = 13.70
隐含波动性 Implied Volatility
用布莱克 -斯科尔斯公式和实际期权价格来解决波动性
Using Black-Scholes and the actual price of the
option,solve for volatility.
隐含波动性是否与股票具有一致性？
Is the implied volatility consistent with the stock?
21-17
看跌期权价值：布莱克 -斯科尔斯
Put Option Value,Black-Scholes
P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)]
用上例的数据 Using the sample data
P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664)
P = $6.35
21-18
看跌期权定价：用看涨 -看跌期权平价关系
Put Option Valuation,Using Put-Call Parity
P = C + PV (X) - So
= C + Xe-rT - So
Using the example data
C = 13.70 X = 95 S = 100
r =,10 T =,25
P = 13.70 + 95 e -.10 X,25 - 100
P = 6.35
21-19
用布莱克 -斯科尔斯公式
Using the Black-Scholes Formula
套期：套期保值率或 d系数
Hedging,Hedge ratio or delta
股票价格需要套期保值防范所持有期权的价格变化风险 The number of stocks required to hedge
against the price risk of holding one option
看涨期权的套期保值率 Call = N (d1)
看跌期权的套期保值率 Put = N (d1) - 1
21-20
用布莱克 -斯科尔斯公式
Using the Black-Scholes Formula
期权弹性 Option Elasticity
期权价值变化 1%所对应的股票价值的百分比变化
Percentage change in the option’s value given a 1%
change in the value of the underlying stock
21-21
投资组合保险 – 保护股票价值的下跌
Portfolio Insurance - Protecting Against Declines in Stock Value
买看跌期权 -价格下跌保护和没有限制的向上的潜在收益蔼 Buying Puts - results in downside protection with
unlimited upside potential
限制 Limitations
– 如果使用指数到看跌期权则会产生循迹误差
Tracking errors if indexes are used for the puts
– 看跌期权的到期时间可能很短
Maturity of puts may be too short
– 套期保值率或 d 系数在股票价值变化时随之变化
Hedge ratios or deltas change as stock values
change
21-22
Summary
期权的价值包括内在价值与时间价值，或“波动性”
价值。波动性价值是如果股票价格与预测变动方向相反则选择不执行期权的权利。因此，不论股票价格如何变动，期权拥有者的损失不会超过获得期权的成本。
Option values may be viewed as the sum of intrinsic
value plus time or,volatility” value,The volatility value
is the right to choose not to exercise if the stock price
moves against the holder,Thus the option holder
cannot lose more than the cost of the option
regardless of stock price performance.
21-23
Summary
当期权的执行价格较低，股票的价格较高，利率较高，到期时间长，风险大时，看涨期权更有价值。
Call options are more valuable when the
exercise price is lower,when the stock price is
higher,when the interest rate is higher,when
the time to maturity is greater,when the stock’s
volatility is greater,and when dividends are
lower.
21-24
Summary
看涨期权的价值应该至少等于股票价格减去执行价格与到期前支付的红利的现 值，这说明不支付红利的股票看涨期权的价格可能比立即执行所获得的收入要高。因为不支付红利的美式看涨期权的提前执行没有价值，所以欧式看涨期权与不支付红利 的股票美式看涨期权具有相同的价值。
Call options must sell for at least the stock price less the
present value of the exercise price and dividends to be paid
before maturity,This implies that a call option on a non-
dividend-paying stock may be sold for more than the proceeds
from immediate exercise,Thus European calls are worth as
much as American calls on stocks that pay no dividends,
because the right to exercise the American call early has no
value.
21-25
Summary
可以用两时期、两状态定价模型对期权进行定价。随时期数的增加，期权公式可以更近似地反映股票价格的分布。布莱克 -舒尔斯期权定价公式可以看作是当时间间隔持续地分为更小的期间时，在利率与股票的波动性保持不变的情况下，二项式期权定价公式的极限情况。
Options may be priced relative to the underlying stock
price using a simple two- period,two-state pricing model,As
the number of periods increases,the model can approximate
more realistic stock price distributions,The Black-Scholes
formula may be seen as a limiting case of the binomial option
model,as the holding period is di- vided into progressively
smaller subperiods when the interest rate and stock volatility
are constant.
21-26
Summary
布莱克 -舒尔斯期权定价公式对于不支付红利的股票期权定价是正确的，它对于支付红利的股票欧洲看涨期权的定价也是充分了。但是，对于支付红利的股票美式看涨期权的定价则需要更复杂的公式。
The Black-Scholes formula is valid for options on
stocks that pay no dividends,Dividend adjustments
may be adequate to price European calls on
dividend-paying stocks,but the proper treatment of
American calls on dividend-paying stocks requires
more complex formulas.
21-27
Summary
不管股票是否支付红利，看跌期权都可提前执行。因此，一般来讲，美式看跌期权比欧式看跌期权更有价值。
Put options may be exercised early,whether
the stock pays dividends or not,Therefore,
American puts generally are worth more than
European puts.
21-28
Summary
欧式看跌期权的价值可以从与看涨期权的平价关系中得到，但是由于美式看跌期权有提前执行的可能，欧式看跌期权的定价方法不适用于美式看跌期权。
European put values can be derived from the
call value and the put-call parity relation- ship,
This technique cannot be applied to American
puts for which early exercise is a possibility.
21-29
Summary
套期保值率是在出售期权时，为抵消期权的价格风险所需要的股票的数量，深度虚值看涨期权的套期保值率接近于 0，而深度实值的看涨期权的套期保值率接近 1。
The hedge ratio is the number of shares of
stock required to hedge the price risk involved
in writing one option,Hedge ratios are near
zero for deep out-of-the-money call options
and approach 1.0 for deep in-the-money calls.
21-30
Summary
虽然套期保值率小于 1，但看涨期权的弹性却大于 1。股票价格的波动带来的看涨期权的收益率大于 1比 1。
Although hedge ratios are less than 1.0,call
options have elasticities greater than 1.0,The
rate of return on a call (as opposed to the dollar
return) responds more than one for one with
stock price movements.
21-31
Summary
通过购买股权头寸的保护性看跌期权可以获得资产组合保险，当交易适当的跌期权时，卖出等于预期的看跌期权得尔塔值的比例的股权，换成无风险的证券，就可以实现资产组合保险的动态套期保值策略。
Portfolio insurance can be obtained by purchasing a
protective put option on an equity position,When the
appropriate put is not traded,portfolio insurance
entails a dynamic hedge strategy where a fraction of
the equity portfolio equal to the desired put option’s
delta is sold and placed in risk-free securities.
第 21章期权定价
Option Valuation
21-2
期权定价
Option Valuation
21.1 期权定价简介
21,2 期权价值的限制
21.3 二项式 期权定价模型
21.4 布莱克 -斯克尔斯 期权定价
21.5 布莱克 -斯克尔斯公式的运用
21.6 经验证明
21-3
期权定价
Option Values
内在价值 – 立即执行该期权能获的收入
Intrinsic value - profit that could be made if
the option was immediately exercised
– 看涨期权,股票价格 – 执行价格
Call,stock price - exercise price
– 看跌期权,执行价格 - 股票价格
Put,exercise price - stock price
时间价值 – 期权价格与内在价值间的差异
Time value - the difference between the
option price and the intrinsic value
21-4
期权的时间价值,看涨期权
Time Value of Options,Call
期权价值
Option
value
X 股票价格 Stock Price
看涨期权价值
Value of Call
内在价值
Intrinsic Value时间价值
Time value
21-5
影响看涨期权价值的因素
Factors Influencing Option Values,Calls
因素 Factor 对 价值的作用 Effect on value
股票价格 Stock price increases
执行价格 Exercise price decreases
股票价格的波动 Volatility of stock price increases
到期 Time to expiration increases
利息率 Interest rate increases
股利 Dividend Rate decreases
21-6
二项式期权定价模型：举例
Binomial Option Pricing,Text Example
100
200
50
股票价格 Stock Price
C
75
0
看涨期权价值
Call Option Value
X = 125
21-7
二项式期权定价模型：举例
Binomial Option Pricing,Text Example
另一个组合 Alternative Portfolio
买 1股 100元的股票借 $46.30
8%的利率）净支出是 $53.70
Buy 1 share of stock at $100
Borrow $46.30 (8% Rate)
Net outlay $53.70
收入 Payoff
Value of Stock 50 200
Repay loan - 50 -50
Net Payoff 0 150
53.70
150
0
收入结构正好是看涨期权的 2倍
Payoff Structure is
exactly 2 times the Call
21-8
二项式期权定价模型：举例
Binomial Option Pricing,Text Example
53.70
150
0
C
75
0
2C = $53.70
C = $26.85
21-9
收入和期权价值的另一种观点
Another View of Replication of Payoffs and Option Values
另一个组合 – 1股股票和两个售出的看涨期权的组合恰好被套期保值
Alternative Portfolio - one share of stock and 2 calls
written (X = 125) Portfolio is perfectly hedged
股票价值 Stock Value 50 200
两个售出的看涨期权的义务
Call Obligation 0 -150
净收入 Net payoff 50 50
Hence 100 - 2C = 46.30 or C = 26.85
21-10
布莱克 -斯科尔斯期权定价模型
Black-Scholes Option Valuation
Co = Soe-dTN(d1) - Xe-rTN(d2)
d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2)
d2 = d1 - (s T1/2)
式中 where
Co = 当前看涨期权的价值 Current call option value.
So = 当前股票价格 Current stock price
N(d) = 随机的偏离标准正态分布的概率小于 d
probability that a random draw from a normal dist,
will be less than d.
21-11
布莱克 -斯科尔斯期权定价模型
Black-Scholes Option Valuation
X = 执行价格 Exercise price.
d = 标的股票的年股利收益率 Annual dividend
yield of underlying stock
e = 2.71828,自然对数函数的底数 the base of the
nat,log.
r = 无风险利率 Risk-free interest rate (annualizes
continuously compounded with the same
maturity as the option.
21-12
布莱克 -斯科尔斯期权定价模型
Black-Scholes Option Valuation
T = 期权到期前的时间（以年为单位）
time to maturity of the option in years.
ln = 自然对数 Natural log function
s = 股票连续复利年收益率的标准差 Standard
deviation of annualized cont,compounded
rate of return on the stock
21-13
标准正态曲线布莱克 -斯科尔斯看涨期权举例
Call Option Example
股票价格 So = 100 执行价格 X = 95
利率 r =,10 到期时间 T =,25 (quarter)
标准差 s =,50 股利收益率 d = 0
d1 = [ln(100/95)+(.10-0+(.5 2/2))]/(.5,251/2)
=,43
d2 =,43 - ((.5)(,251/2)
=,18
21-14
正态分布的概率
Probabilities from Normal Dist.
N (.43) =,6664
Table 17.2
d N(d)
.42,6628
.43,6664 添写 Interpolation
.44,6700
21-15
正态分布的概率
Probabilities from Normal Dist.
N (.18) =,5714
Table 17.2
d N(d)
.16,5636
.18,5714
.20,5793
21-16
看涨期权价值
Call Option Value
Co = Soe-dTN(d1) - Xe-rTN(d2)
Co = 100 X,6664 - 95 e-,10 X,25 X,5714
Co = 13.70
隐含波动性 Implied Volatility
用布莱克 -斯科尔斯公式和实际期权价格来解决波动性
Using Black-Scholes and the actual price of the
option,solve for volatility.
隐含波动性是否与股票具有一致性？
Is the implied volatility consistent with the stock?
21-17
看跌期权价值：布莱克 -斯科尔斯
Put Option Value,Black-Scholes
P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)]
用上例的数据 Using the sample data
P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664)
P = $6.35
21-18
看跌期权定价：用看涨 -看跌期权平价关系
Put Option Valuation,Using Put-Call Parity
P = C + PV (X) - So
= C + Xe-rT - So
Using the example data
C = 13.70 X = 95 S = 100
r =,10 T =,25
P = 13.70 + 95 e -.10 X,25 - 100
P = 6.35
21-19
用布莱克 -斯科尔斯公式
Using the Black-Scholes Formula
套期：套期保值率或 d系数
Hedging,Hedge ratio or delta
股票价格需要套期保值防范所持有期权的价格变化风险 The number of stocks required to hedge
against the price risk of holding one option
看涨期权的套期保值率 Call = N (d1)
看跌期权的套期保值率 Put = N (d1) - 1
21-20
用布莱克 -斯科尔斯公式
Using the Black-Scholes Formula
期权弹性 Option Elasticity
期权价值变化 1%所对应的股票价值的百分比变化
Percentage change in the option’s value given a 1%
change in the value of the underlying stock
21-21
投资组合保险 – 保护股票价值的下跌
Portfolio Insurance - Protecting Against Declines in Stock Value
买看跌期权 -价格下跌保护和没有限制的向上的潜在收益蔼 Buying Puts - results in downside protection with
unlimited upside potential
限制 Limitations
– 如果使用指数到看跌期权则会产生循迹误差
Tracking errors if indexes are used for the puts
– 看跌期权的到期时间可能很短
Maturity of puts may be too short
– 套期保值率或 d 系数在股票价值变化时随之变化
Hedge ratios or deltas change as stock values
change
21-22
Summary
期权的价值包括内在价值与时间价值，或“波动性”
价值。波动性价值是如果股票价格与预测变动方向相反则选择不执行期权的权利。因此，不论股票价格如何变动，期权拥有者的损失不会超过获得期权的成本。
Option values may be viewed as the sum of intrinsic
value plus time or,volatility” value,The volatility value
is the right to choose not to exercise if the stock price
moves against the holder,Thus the option holder
cannot lose more than the cost of the option
regardless of stock price performance.
21-23
Summary
当期权的执行价格较低，股票的价格较高，利率较高，到期时间长，风险大时，看涨期权更有价值。
Call options are more valuable when the
exercise price is lower,when the stock price is
higher,when the interest rate is higher,when
the time to maturity is greater,when the stock’s
volatility is greater,and when dividends are
lower.
21-24
Summary
看涨期权的价值应该至少等于股票价格减去执行价格与到期前支付的红利的现 值，这说明不支付红利的股票看涨期权的价格可能比立即执行所获得的收入要高。因为不支付红利的美式看涨期权的提前执行没有价值，所以欧式看涨期权与不支付红利 的股票美式看涨期权具有相同的价值。
Call options must sell for at least the stock price less the
present value of the exercise price and dividends to be paid
before maturity,This implies that a call option on a non-
dividend-paying stock may be sold for more than the proceeds
from immediate exercise,Thus European calls are worth as
much as American calls on stocks that pay no dividends,
because the right to exercise the American call early has no
value.
21-25
Summary
可以用两时期、两状态定价模型对期权进行定价。随时期数的增加，期权公式可以更近似地反映股票价格的分布。布莱克 -舒尔斯期权定价公式可以看作是当时间间隔持续地分为更小的期间时，在利率与股票的波动性保持不变的情况下，二项式期权定价公式的极限情况。
Options may be priced relative to the underlying stock
price using a simple two- period,two-state pricing model,As
the number of periods increases,the model can approximate
more realistic stock price distributions,The Black-Scholes
formula may be seen as a limiting case of the binomial option
model,as the holding period is di- vided into progressively
smaller subperiods when the interest rate and stock volatility
are constant.
21-26
Summary
布莱克 -舒尔斯期权定价公式对于不支付红利的股票期权定价是正确的，它对于支付红利的股票欧洲看涨期权的定价也是充分了。但是，对于支付红利的股票美式看涨期权的定价则需要更复杂的公式。
The Black-Scholes formula is valid for options on
stocks that pay no dividends,Dividend adjustments
may be adequate to price European calls on
dividend-paying stocks,but the proper treatment of
American calls on dividend-paying stocks requires
more complex formulas.
21-27
Summary
不管股票是否支付红利，看跌期权都可提前执行。因此，一般来讲，美式看跌期权比欧式看跌期权更有价值。
Put options may be exercised early,whether
the stock pays dividends or not,Therefore,
American puts generally are worth more than
European puts.
21-28
Summary
欧式看跌期权的价值可以从与看涨期权的平价关系中得到，但是由于美式看跌期权有提前执行的可能，欧式看跌期权的定价方法不适用于美式看跌期权。
European put values can be derived from the
call value and the put-call parity relation- ship,
This technique cannot be applied to American
puts for which early exercise is a possibility.
21-29
Summary
套期保值率是在出售期权时，为抵消期权的价格风险所需要的股票的数量，深度虚值看涨期权的套期保值率接近于 0，而深度实值的看涨期权的套期保值率接近 1。
The hedge ratio is the number of shares of
stock required to hedge the price risk involved
in writing one option,Hedge ratios are near
zero for deep out-of-the-money call options
and approach 1.0 for deep in-the-money calls.
21-30
Summary
虽然套期保值率小于 1，但看涨期权的弹性却大于 1。股票价格的波动带来的看涨期权的收益率大于 1比 1。
Although hedge ratios are less than 1.0,call
options have elasticities greater than 1.0,The
rate of return on a call (as opposed to the dollar
return) responds more than one for one with
stock price movements.
21-31
Summary
通过购买股权头寸的保护性看跌期权可以获得资产组合保险，当交易适当的跌期权时，卖出等于预期的看跌期权得尔塔值的比例的股权，换成无风险的证券，就可以实现资产组合保险的动态套期保值策略。
Portfolio insurance can be obtained by purchasing a
protective put option on an equity position,When the
appropriate put is not traded,portfolio insurance
entails a dynamic hedge strategy where a fraction of
the equity portfolio equal to the desired put option’s
delta is sold and placed in risk-free securities.
