课程名称︰财务工程
课程性质︰选修
课程教师︰黄以达
开课学院：社会科学院
开课系所︰经济系所
考试日期（年月日）︰2014 年 04 月 18 日
考试时限（分钟）：14:20 ~ 18:20 左右
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
Part I. (24%) (单选题)
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
(从小于X，曲线上升到大于X，再曲线下降最后等于X) (山形)
(B) S(0) < X, S(t) < X, S(T) = X
(从小于X，曲线上升到最后等于X) (上升山坡形)
(C) S(0) = X, S(t) < X, S(T) < X
(从等于X，直线下降到小于X) (直线形)
(D) S(0) < X, S(t) < X, S(T) = X
(从小于X，先曲线下降到某低点，再曲线上升最后等于X) (谷形)
(这题其实是有图的，只是弄不上来QQ)
6. 当下面两种状况发生时，对提早执行美式卖权的动机分别有什么影响?
状况1. 市场无风险利率上升 状况2. 标的物价格的波动度增加
(A) 动机增加、动机增加 (B) 动机增加、动机减少
(C) 动机减少、动机增加 (D) 动机减少、动机减少
7. 考虑一欧式卖权，建构一期二项树模型，其中 u > 0，d > 0，r > 0 (连续复利下的无
风险利率)，δ = 0，且满足无套利假设。今你发现 u 状态以及 d 状态皆落入价内(即
P > 0 且 P > 0，关于下列叙述:
u d
1) 利用复制投资组合的技巧所计算出来的股票部位φ必定等于-1。
2) 若 r 变大，则卖权价格变贵。
3) 若连续股利δ不为零，则卖权价格变贵。
(A) 只有1是对的。 (B) 只有3是错的。 (C) 只有2是错的。 (D) 只有3是对的。
(E) 以上选项叙述都不正确。
8. 霸菱银行倒闭是由李森卖出跨式 (short straddle)，遇到神户大地震，导致日经指数
大跌，遭受到 2 亿多美元的损失。请问李森是卖出以下哪个部位?
(A) payoff (B) payoff (C) payoff (D) payoff
\ / \ /
\____ / \ / ___
\ / \ / \
\ \ / \ / \
_________\__ __\__/____ ____\/_____ _____\_____
S(T) S(T) S(T) S(T)
Part II. (24%) (多重选择题) (全对八分，错一项得四分，其他不得分)
9. 在单期二项树模型中，其中 u > 0，d > 0，r > 0 (连续复利下的无风险利率)，δ =
0，且满足无套利假设，请问下列叙述有哪些是正确的?
(A) q < 0.5 是不可能发生的。
u
(B) 假设 ud = 1，则必能推论 q >= 0.5。
u
(C) 假设 u + d = 2，则必能推论 q >= 0.5。
rT u
(D) 假设 e 越接近 d，则表示股市相对债券市场好，投资股市是利大于弊，所以买权
较贵。
10. 考虑一美式卖权，建构二期二项树模型，其中 u > 0，d > 0，r > 0 (连续复利下的
无风险利率)，且满足无套利假设。今你发现 ud 状态以及 dd 状态皆落入价内 (即
P > 0 且 P > 0)，请问下列叙述有哪些是正确的?
ud dd
(A) 在不发放股利的情况下，d 状态必定是提早执行。
(B) 若有连续股利δ = 2%，且 r = 4% 的情况下，d 状态必定是提早执行。
(C) 若有连续股利δ = 4%，且 r = 2% 的情况下，d 状态必定是提早执行。
(D) 若有连续股利δ = 3%，且 r = 3% 的情况下，d 状态必定是提早执行。
11. 振荃参加财工期中考，其中有一题是利用以 forward price 为基础的一期二项树来计
算一个欧式卖权的价格，题目有波动度，无风险利率，以及契约期间长度。根据公式
他先算出 u, d，然后计算出 P > 0，P > 0，以后再带卖权公式，得出无套利价格
u d
为 P。最后出考场才发现，他竟然漏看了连续股利的资讯，考卷上其实有δ = 2%的条
件，最后只好大喊洗洗睡了。请问下列关于他算出来的数据与真实的数据之间的叙述
，正确的有哪些? (假设他其它资讯都没看错且都计算无误)
(A) 他算出的 q (风险中立上涨机率) 必定跟真正的 q 其实一样大。
u u
(B) 他算出来的 u, d 都必定分别比真实的 u, d 来的大。
(C) 他用复制投资组合所算出来债券部位的 b 必定比真正的 b 来的大。
(D) 他算出来的 P 必定比真实的 P 来得大。
Part III. (10%) (证明题)
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%) (计算题)
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%) 有一个价平发行的亚式买权，其 Payoff 为如下：
max{Average( S ) - K, 0}
0<=t<=T t
其中Average( S )为所有历史股价(含到期日)的算数平均数，K为
0<=t<=T t
执行价。请利用二期的二项树模型，搭配下面的条件求出此选择权价格。
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%) (投资策略问题)
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