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1、金融工程 第三次习题1. A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-year European call option with a strike price of $100?2. The cu
2、rrent price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The risk-free rate is 4%. For a three-month time step:(a) What is the percentage up movement?(b) What is the percentage down movement?(c) What is the probability of an up movement in a risk-neutral world?(d) What is
3、 the probability of a down movement in a risk-neutral world?Use a two-step tree to value a six-month European call option and a six-month European put option. In both cases the strike price is $150.3. A companys cash position, measured in millions of dollars, follows a generalized Wiener process wit
4、h a drift rate of 0.1 per month and a variance rate of 0.16 per month. The initial cash position is 2.0.(a) What are the probability distributions of the cash position after 1 month, 6 months, and 1 year?(b) What are the probabilities of a negative cash position at the end of 6 months and 1 year?(c)
5、 At what time in the future is the probability of a negative cash position greatest?4. Iffollows the geometric Brownian motion process in equation (14.6), what is the process followed by(a)(b)(c)(d)In each case express the coefficients of dt and dz in terms of y rather than S.5. A stock price is cur
6、rently $50. Assume that the expected return from the stock is 18% and its volatility is 30%. What is the probability distribution for the stock price in 2 years? Calculate the mean and standard deviation of the distribution. Determine the 95% confidence interval.6. Consider an option on a non-divide
7、nd-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5%, the volatility is 25% per annum, and the time to maturity is 4 months.(a) What is the price of the option if it is a European call?(b) What is the price of the option if it is an American call?
8、(c) What is the price of the option if it is a European put?(d) Verify that put-call parity holds.7. A financial institution has the following portfolio of over-the-counter options on sterling:TypePositionDelta of optionGamma of optionVega of optionCall-1,0000.502.21.8Call-5000.800.60.2Put-2,000- 0.
9、401.30.7Call-5000.701.81.4A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.(a) What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral?(b) What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral?