4、 基于同一股票的看跌期权有相同的到期日.执行价格为$70、$65和$60，市场价格分为$5、$3和$2. 如何构造蝶式差价期权.请用一个表格说明这种策略带来的盈利性.股票价格在什么范围时，蝶式差价期权将导致损失？ 5、 基于同一股票的有相同的到期日敲定价为 $70的期权市场价格为 $4. 敲定价$65 的看跌期权的市场价格为 $6。解释如何构造底部宽跨式期权.请用一个表格说明这种策略带来的盈利性.股票价格在什么范围时，宽跨式期权将导致损失？ 答案： buy a put with the strike prices $65 and buy a call with the strike prices $70, this portfolio would need initial cost $10. The pattern of profits from the strangle is the following:Stock Price RangePayoff from Long PutPayoff from Long CallTotal PayoffTotal ProfitsST 6565- ST065- ST55 - ST65 ST 70000-10ST 700ST-70ST-70ST-80当50ST w1 = 400, w2 = 6,000, w3 =-3240. =The portfolio can be made gamma,vega and delta neutral by including long: (1) 400 of the first traded option (2) 6,000 of the second traded option. And short 3240 underlying asset. 八 1）证明在风险中性环境下，到期的欧式看涨期权被执行的概率为 ,2） 使用风险中性定价原理，假设股票1的价格和股票2的价格分别服从几何布朗运动，且独立，给到期损益为如下形式的欧式衍生品定价： Solution: Since Since Where , 九、Use two-step tree to value an American 2-year put option on a non-dividend-paying stock, current stock price is 50, the strike price is $52, and the volatility of stock price is 30% per annum, the risk-free interest rate is 5% per annum. （保留2位小数）In this case, S=50, X = 52, = 0.3, t =1, r=0.05 , the parameters necessary to construct the tree are , 91.11067.40.93502507.4337.0414.96 27.4424.56 十 If a stock price, S, follows geometric Brownian motion1) What is the process followed by the variable ? Show that also follows geometric Brownian motion.2)The expected value of ST is . What is the expected value of ?3) The varaince of ST is .What is the variance of ?4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is1)We now use Itos lemma to derive the process followed by ,Define , So that also follows geometric Brownian motion. 2) . , 3) Since and varaince of ST is .Similarly, by We get the varaince of is 十一、 In a risk-neutral world, suppose stock prices follow geometric Brownian motion1) What is t