課程名稱︰財務工程
課程性質︰選修
課程教師︰李志偉
開課學院：
開課系所︰
考試日期（年月日）︰
考試時限（分鐘）：
試題 :
Financial Engineering I Final
範圍ch10-15,ch17,ch19,ch22,選擇權作業樹作業
參考公式
https://i.imgur.com/H6tosYj.jpg
10分8題，另有2題上機考
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1. An investor sells a European call on a share for $4. The stock price is $47
and the strike price is $50. Under what circumstances does the investor make
a profit? Under what circumstances
will the option be exercised? Draw a diagram showing the variation of the inve
stor's profit withthe stock price at the maturity of the option.
2.
a) The price of a European call that expires in six months and has a strike pr
ice of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 i
s expected in two months and again in
five months. Interest rates (all maturities) are 10%. What is the price of a E
uropean put optionthat expires in six months and has a strike price of $30?
b) Explain the arbitrage opportunities in a) if the European put price is $3.
(given exp(-10% x 2/12)=0.9834, exp(-10% x 5/12) = 0.9592, exp(-10% x 6/12) =
0.9512)
3. Call options on a stock are available with strike prices of $15, $17.5, and
$20 and expirationdates in three months. Their prices are $4, $2, and $0.5 re
spectively. Explain how the options
can be used to create a butterfly spread. Construct a table showing how profit
varies with stockprice for the butterfly spread.
4. Suppose that a stock price, S, follows geometric Brownian motion with expec
ted return 符號mu and volatility 符號sigma：
dS = mu*S dt + sigma* S dz
What is the process followed by the variable S^n ? Show that S^n also follows
geometric Brownian motion.
5. A forward contract on a non-dividend-paying stock is a derivative dependent
on the stock. As such, it should satisfy equation (15.16). From equation (5.5
), prove that the value of the forward contract, f, satisfies equation (15.16)
. （證明式子5.5 符合式子15.16的要求）
6. Suppose that a portfolio is worth $60 million and the S&P 500 is at 1200. S
uppose that the portfolio has a beta of 2.0, the risk-free interest rate is 5%
per annum, and the dividend yield on
both the portfolio and the index is 3% per annum. What options should be purch
ased to provide protection against the value of the portfolio falling below $5
4 million in one year's time?
7. Consider a portfolio that is delta neutral, with a gamma of -5,000 and a ve
ga of -8,000. The options shown in the table below can be traded. What positio
n in the traded option would make the portfolio both gamma neutral and vega ne
utral?
圖表如下
https://i.imgur.com/Bc8lTWx.jpg
8. Consider a position consisting of a $100,000 investment in asset A and a $1
00,000 investment in asset B. Assume that the daily volatilities of both asset
s are 1% and that the coefficient of
correlation between their returns is 0.3. What is the 5-day 99% VaR for the po
rtfolio? Given ( N^-1 ) * (0.01) = 2.326.