put-call parity

问题如下：

The price of a six-month, USD 25 strike price, European call option on a stock is USD 3. The stock price is USD 24. A dividend of USD 1 is expected in three months. The continuously compounded risk-free rate for all maturities is 5% per year. Which of the following is closest to the value of a put option on the same underlying stock with a strike price of USD 25 and a time to maturity of six months?

选项：

A.

USD 3.60

B.

USD 2.40

C.

USD 4.37

D.

USD 1.63

解释：

From the equation for put-call parity, this can be solved by the following equation:

p = c + PV (K) + PV (D) - S₀

where PV represents the present value, so that

PV (K) = K* e^-rT and PV (D) = D* e^-rt

Where:

p represents the put price, c is the call price,

K is the strike price of the put option,

D is the dividend,

S0 is the current stock price.

T is the time to maturity of the option,

and t is the time to the next dividend distribution.

Calculating PV (K), the present value of the strike price, results in a value of 25* e^{-0 05}*^0.5 or 24.38, while PV (D) is equal to 1 * e^{-0 05}*^0.25, or 0.99. Hence p = 3 + 24.38 + 0.99 - 24 = US 4.37.