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 ================================ 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. --

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