Put Option Valuation
As we saw in Equation 16.3, we can use the put-call parity relationship to value put options once
we know the call option value. Sometimes, however, it is easier to work with a put option valu￾ation formula directly. The Black-Scholes formula for the value of a European put option is10
P = Xe−rT
[1 − N(d2)] − S0e−δT
1 − N(d1)
10This formula is consistent with the put-call parity relationship and in fact can be derived from it. If you want to try,
remember to take present values using continuous compounding, and use the fact that when a stock pays a continuous
flow of income in the form of a constant dividend yield, δ, the present value of that dividend flow is S0(1 − e−δT).
(Notice that e− δT approximately equals 1 − δT, so the value of the dividend flow is approximately δTS0.)
EXAMPLE 16.6
Black-Scholes Put
Option Valuation
Using data from the Black-Scholes call option in Example 16.4, we find that a European put option on
that stock with identical exercise price and time to expiration is worth
$95e−.10 × .25(1 − .5714) − $100(1 − .6664) = $6.35
Notice that this value is consistent with put-call parity:
P = C + PV(X) − S0 + PV(Div) = 13.70 + 95e−.10 × .25 − 100 + 0 = 6.35
As we noted traders can do, we might then compare this formula value to the actual put price as one
step in formulating a trading strategy.
Equation 16.4 is valid for European puts. Most listed put options are American-style,
however, and offer the opportunity of early exercise. Because an American option allows its
owner to exercise at any time before the expiration date, it must be worth at least as much as
the corresponding European option. However, while Equation 16.4 describes only the lower
bound on the true value of the American put, in many applications the approximation is very
accurate.

概括

我们可以通过看跌-看涨期权平价关系来估算看跌期权价值，但直接使用布莱克-斯科尔斯公式更为便捷。该公式为欧式看跌期权定价，其表达式考虑了执行价格、股票现价、无风险利率、股息收益率及时间因素。举例说明，利用给定数据计算得出某欧式看跌期权价值为6.35美元，与看跌-看涨平价理论结果一致。尽管该公式仅适用于欧式期权，而市场上多为美式期权（允许提前行权），但该公式仍常被用作近似估值，因其结果通常与实际价值非常接近。