課程名稱︰財務工程
課程性質︰選修
課程教師︰黃以達
開課學院：社會科學院
開課系所︰經濟系所
考試日期（年月日）︰2014 年 04 月 18 日
考試時限（分鐘）：14:20 ~ 18:20 左右
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Part I. (24%) (單選題)
1. Suppose you own an asset. Which of the following additions to your portfolio
would represent "insurance" against the downside price risk associated with
your long underlying asset position?
(A) Long call (B) Long put (C) Short call (D) Short put
(E) The corrent answer is not given by (A), (B), (C), or (D)
2. Which of the following is the correct relationship associated with a
synthetic forward contract?
(A) Zero-coupon bond = stock + forward
(B) Stock = forward - zero-coupon bond
(C) Forward = stock + zero-coupon bond
(D) None of (A) through (C)
3. A stock has a current spot price of $90, and a nine-month forward price of
$95. The continuously compounded annual interest rate is 10%. Find the
stock's annualized continuous dividend yield which is consistent with this
forward price.
(A) 2.0% (B) 2.8% (C) 3.4% (D) 4.2% (E) 5.0%
4. You are given the following information:
Spot price of market index today = $1,500.
Forward price of nine-month forward contract on market index = $1,540.
Spot price of market index nine months from today = $1,560.
A $1,000 face value nine-month zero-coupon bond is selling for $936.39.
Find the difference, nine months from today, between the profits associated
with a long index strategy versus a long forward strategy.
(A) $0 (B) $3 (C) $10 (D) $17 (E) $20
5. Which one is the correct early exercise boundary of the American put options
on a non-dividend paying stock? (Early exercise boundary divides the space
into two parts: early exercising and holding option.) (T is the maturity of
the option; X is the strike price)
(A) S(0) < X, S(t) > X, S(T) = X
(從小於X，曲線上升到大於X，再曲線下降最後等於X) (山形)
(B) S(0) < X, S(t) < X, S(T) = X
(從小於X，曲線上升到最後等於X) (上升山坡形)
(C) S(0) = X, S(t) < X, S(T) < X
(從等於X，直線下降到小於X) (直線形)
(D) S(0) < X, S(t) < X, S(T) = X
(從小於X，先曲線下降到某低點，再曲線上升最後等於X) (谷形)
(這題其實是有圖的，只是弄不上來QQ)
6. 當下面兩種狀況發生時，對提早執行美式賣權的動機分別有甚麼影響?
狀況1. 市場無風險利率上升 狀況2. 標的物價格的波動度增加
(A) 動機增加、動機增加 (B) 動機增加、動機減少
(C) 動機減少、動機增加 (D) 動機減少、動機減少
7. 考慮一歐式賣權，建構一期二項樹模型，其中 u > 0，d > 0，r > 0 (連續複利下的無
風險利率)，δ = 0，且滿足無套利假設。今你發現 u 狀態以及 d 狀態皆落入價內(即
P > 0 且 P > 0，關於下列敘述:
u d
1) 利用複製投資組合的技巧所計算出來的股票部位φ必定等於-1。
2) 若 r 變大，則賣權價格變貴。
3) 若連續股利δ不為零，則賣權價格變貴。
(A) 只有1是對的。 (B) 只有3是錯的。 (C) 只有2是錯的。 (D) 只有3是對的。
(E) 以上選項敘述都不正確。
8. 霸菱銀行倒閉是由李森賣出跨式 (short straddle)，遇到神戶大地震，導致日經指數
大跌，遭受到 2 億多美元的損失。請問李森是賣出以下哪個部位?
(A) payoff (B) payoff (C) payoff (D) payoff
\ / \ /
\____ / \ / ___
\ / \ / \
\ \ / \ / \
_________\__ __\__/____ ____\/_____ _____\_____
S(T) S(T) S(T) S(T)
Part II. (24%) (多重選擇題) (全對八分，錯一項得四分，其他不得分)
9. 在單期二項樹模型中，其中 u > 0，d > 0，r > 0 (連續複利下的無風險利率)，δ =
0，且滿足無套利假設，請問下列敘述有哪些是正確的?
(A) q < 0.5 是不可能發生的。
u
(B) 假設 ud = 1，則必能推論 q >= 0.5。
u
(C) 假設 u + d = 2，則必能推論 q >= 0.5。
rT u
(D) 假設 e 越接近 d，則表示股市相對債券市場好，投資股市是利大於弊，所以買權
較貴。
10. 考慮一美式賣權，建構二期二項樹模型，其中 u > 0，d > 0，r > 0 (連續複利下的
無風險利率)，且滿足無套利假設。今你發現 ud 狀態以及 dd 狀態皆落入價內 (即
P > 0 且 P > 0)，請問下列敘述有哪些是正確的?
ud dd
(A) 在不發放股利的情況下，d 狀態必定是提早執行。
(B) 若有連續股利δ = 2%，且 r = 4% 的情況下，d 狀態必定是提早執行。
(C) 若有連續股利δ = 4%，且 r = 2% 的情況下，d 狀態必定是提早執行。
(D) 若有連續股利δ = 3%，且 r = 3% 的情況下，d 狀態必定是提早執行。
11. 振荃參加財工期中考，其中有一題是利用以 forward price 為基礎的一期二項樹來計
算一個歐式賣權的價格，題目有波動度，無風險利率，以及契約期間長度。根據公式
他先算出 u, d，然後計算出 P > 0，P > 0，以後再帶賣權公式，得出無套利價格
u d
為 P。最後出考場才發現，他竟然漏看了連續股利的資訊，考卷上其實有δ = 2%的條
件，最後只好大喊洗洗睡了。請問下列關於他算出來的數據與真實的數據之間的敘述
，正確的有哪些? (假設他其它資訊都沒看錯且都計算無誤)
(A) 他算出的 q (風險中立上漲機率) 必定跟真正的 q 其實一樣大。
u u
(B) 他算出來的 u, d 都必定分別比真實的 u, d 來的大。
(C) 他用複製投資組合所算出來債券部位的 b 必定比真正的 b 來的大。
(D) 他算出來的 P 必定比真實的 P 來得大。
Part III. (10%) (證明題)
Please prove the following inequalities:
(K is the strike price, S is the spot price of the stock, r is the risk-free
A 0 E
rate, P is the American put option price, P is the European put price.)
A A -rT
(1) (7%) S - K <= C - P <= S - Ke
0 0
A E -rT
(2) (3%) K >= (P - P ) / (1 - e )
Part IV. (34%) (計算題)
一、 (4%) You are given:
(i) The price of a stock is 43.00.
(ii) The continuously compounded risk-free rate is 5%.
(iii) The stock pays a dividend of 1.00 three months from now.
(iv) A 3-month European call option on the stock with strike 44.00
costs 1.90.
You with to create this stock synthetically, using a combination of
options and leanding. Determine the amount of money you should lend.
(Hint: 4_.____)
二、 (4%) A 1-year European option on a stock is modeled with a 1-period
binomial tree based on forward prices. You are given:
(i) r = 6%.
(ii) δ = 2%.
(iii) The risk-neutral probability of an increase in price is 0.45.
Determine σ. (Hint: 0.2_0_)
三、 (4%) For 2 non-dividend paying stock X and Y, the current prices are
both 100. There are three possible outcomes for their prices after 1
year:
Outcome Price of X Price of Y
1 $200 $0
2 $50 $0
3 $0 $300
Let C(X) be the price of an European call option on X, and P(Y) be the
price of an European put option on Y. Both options expire in one year
and have a strike price of $95. The continuously compounded risk-free
rate in dollar is 10%. Calculate C(X) - P(Y). (Hint: 4.___)
四、 (4%) 有一個價平發行的亞式買權，其 Payoff 為如下：
max{Average( S ) - K, 0}
0<=t<=T t
其中Average( S )為所有歷史股價(含到期日)的算數平均數，K為
0<=t<=T t
執行價。請利用二期的二項樹模型，搭配下面的條件求出此選擇權價格。
rT/2
S = 100, u = 1.2, d = 0.9, R = e = 1.05. (Hint: _.__39)
0
五、 (4%) An European put option is modeled with a 1-period binomial tree.
You are given:
(i) The stock price is 20.
(ii) The strike price is 20.
(iii) The continuously compounded risk-free rate is 3%.
(iv) The continuous dividend rate is 2%.
(v) Δ for a 6-month call option is 0.4.
Determine Δ. (Hint: -0.5___)
六、 (4%) You are given:
(i) The continuously compounded risk-free rate for dollars is 4%.
(ii) The continuously compounded risk-free rate for pounds is 6%.
(iii) A 6-month dollar-denominated European call option on pounds with
strike 1.45 costs $0.05.
(iv) A 6-month dollar-denominated European put option on pounds with
strike 1.45 costs $0.02.
Determine the 6-month forward exchange rate of dollars per pound.
(Hint: 1._8__)
七、 (4%) An investor, wishing to insure herself against a decrease in value
of her stock without incurring the total cost of buying an European put
option, make use of a collar strategy, whereby she sells an European
call option and purchase a put option. Assume following:
(i) Stock price change quarterly.
(ii) The options mature in six months.
(iii) The current stock price is 50.
(iv) The call option strike price is 60.
(v) The put option strike price is 40.
(vi) Each quarter, the stock price will either increase or decrease by
20%.
(vii) The risk-free interest rate is 5% per annum, compounded
continuously.
Determine the initial cost of the collar. (Hint: -0._2_)
八、 (6%) A 6-month euro-denominated European call option to buy dollars is
modeled with a 6-period binomial tree. You are given:
(i) The spot exchange is 1.25 $/€.
(ii) The tree is constructed using forward price.
(iii) The continuously compounded risk-free rate in euro is 3%.
(iv) The continuously compounded risk-free rate in dollars is 5%.
(v) σ = 0.05.
(vi) The strike price is 1.35 $/€.
Determine the premium, in euro, for a call option on $1,000,000.
(Hint: 50___)
Part V. (11%) (投資策略問題)
1. (1) (5%) Given the spot price of stock A is $100, risk-free rate is 8%,
and the three-month option prices of stock A are as follow: put with
exercise price $90 is $3.1, and call with exercise price $110 is $3.05.
Calculate the establishment cost of the potfolio whose payoff function is
the same as the following: $20 regardless of future stock price.
(Hint: You may need four different options)
(Hint: __._0_)
(2) (3%) Under the same information of stock and derivative market as (1)
, now we face a three-month zero coupon bond, which is 2.55% in bond
market. Is there any arbitrage opportunity? If yes, please identify the
arbitrage opportunity.
2. (3%) Given the spot price of stock A is $105, risk-free rate is 0%, and
the one-year option prices of stock A are as follow: put with exercise
price $90 is $2, put with exercise price $100 is $4, call with exercise
price $110 is $5, and call with exercie price $110 is $2. Calculate the
establishment cost of the potfolio whose payoff function is the same as
the following graph.
Payoff
10