四、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？五、 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.当 50<ST<80时，组合会带来损失六、远期/期货价格公式及其价值公式，B-S 公式的使用()()12()()q T t r T t c Se N d Xe N d ----=>=-()()21()()r T t q T t p Xe N d Se N d ----=---21d = 21d d =-1).What is the price of a European call option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is six months?()()()()()r T t r T t r q T t F Se S I e Se ----==-=)()(t T r t T q KeSe f -----=)()(,t T r t T r Ke I S f Ke S f ------=-=2). Suppose the current value of the index is 500, continuous dividend yields of index is 4% per annum, the risk-free interest rate is 6% per annum . if the price of three-month European index call option with exercise price 490is $20, What is the price of a three-month European index put option with exercise price 490?by put-call parity3) What is the price of a European futures put option ：current futures price is $19, the strike price is $20, the risk-free interest rate is 12% per annum, the volatility is 20% per annum, and the time to maturity is five months? （保留2位小数）Solution: In this case F=19,X=20, r=0.12, σ=0.20, T -t=0.42,210.33d ==-210.46d d =-=-(0.33)0.6293,(0.46)0.6772N N ==12()(0.33)0.6293,()(0.46)0.6772N d N N d N -==-==The price of the European put is()()210.120.420.120.42()()200.6772190.6293 1.51r T t r T t p Xe N d Fe N d e e -----⨯-⨯=---=⨯-⨯=4) A one-year-long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.(a)What are the forward price and the initial value of the forward contract?(b)Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract?The forward price, 21.44401.0)(===-e Se F t T r ， The initial value of the forward contract is zero.0=f (a)The delivery price K in the contract is $44.21. The value of the forward contract after six months is given:95.221.44455.01.0)(=-=-=⨯---e Ke S f t T rThe forward price, 31..47455.01.0)(===⨯-e Se F t T r七 Consider a portfolio that is delta neutral, with a gamma of -5,000 and a vega of -8,000. Suppose that a traded option has a gamma of 0.5, a vega of 2.0, and a delta of 0.6.Another traded option with a gamma of 0.8, a vega of 1.2, and a delta of 0.5.What position in the traded two call options and in the underlying asset would make the portfolio gamma ,vega and delta neutral ？Solution: If , w1 ,w2 , ,w3 are the amounts of the two traded options and underlying asset included in the portfolio, we require that -5,000 + 0.5w1 + 0.8 w2 = 0- 8,000 + 2.0w1 + 1.2w2 = 0w3 +0.6w1 + 0.5 w2 =0=> 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 )(2d N ,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：121,2 : 0 elseT T T K S X S X T f ⎧>>⎪=⎨⎪⎩ Solution: Since ())(),)2/(ln ~ln 22t T t T r S N S T ---+σσ（ )N(d )))(2/(ln ln ( )))(2/(ln ln (1)ln p(ln 1)ln p(ln )p(222=------=------=<-=>=>tT t T r S X N tT t T r S X N X S X S X S T T T σσσσ Since 121,22 : and p() N(d )0 elseT T T T K S X S X T f S X ⎧>>⎪=>=⎨⎪⎩121,2212122122()()2122[] P()K[P() *P() ]K[N(d )*N(d )][][N(d )*N(d )]T T T T T r T t r T t T E f K S X S X S X S X f e E f e K ----=>>=>>===Where1221d =,2222d =,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,σ = 0.3, Δt =1, r=0.05 , the parametersnecessary to construct the tree are11.35,0.74u e d u====, 0.05*1=1.10e 0.05*10.51,10.49e d p p u d -==-=-{}])1([,max .11.1,j i j i t r j i j j i f p pf e d Su X f +++∆---+-=十 If a stock price, S, follows geometric Brownian motiont SdW Sdt dS σμ+=1) What is the process followed by the variable n S ? Show that n S also follows geometric Brownian motion.2)The expected value of ST is =)(T S E )(t T Se -μ. What is the expected value of nT S ? 50 7.43 50 227.4424.5691.11 0 37.0414.9667.4 0.933) The varaince of ST is =)(T S D )1()()(222---t T t T e e S σμ.What is the variance of n T S ? 4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is:n T T T f S =1)We now use Ito's lemma to derive the process followed by n S , Define n S G =,0,)1(,2221=∂∂-=∂∂=∂∂--tG S n n S G nS S G n n t SdW Sdt dS σμ+=222)(21dS SG dS S G dt t G dG ∂∂+∂∂+∂∂= 221)()1(21)(t n t n SdW Sdt S n n SdW Sdt nS σμσμ+-++=--dt S n n dW S n dt S n n t n n 2)1(21σσμ-++= t n n n dW S n dt S n n S n σσμ+-+=])1(21[2 t n n dW S n dt S n n n dG σσμ+-+=])1(21[2 t GdW n Gdt n n n σσμ+-+=])1(21[2 n S G =ΘSo that n S also follows geometric Brownian motion. 2) t SdW Sdt dS ce σμ+=sin=)(T S E )(t T Se -μ. dG t GdW n Gdt n n n σσμ+-+=])1(21[2 =⇒)(T G E )]()1(21[2t T n n n Ge--+σμ n S G =Θ, =∴)(n T S E )]()1(21[2t T n n n n e S --+σμ3) Since t SdW Sdt dS σμ+= and varaince of ST is=)(T S D )1()()(222---t T t T e e S σμ.Similarly, by dG t GdW n Gdt n n n σσμ+-+=])1(21[2 We get the varaince of n T S is=)(n T S D =)(T G D ]1[)()]()1(2[2222----+t T n t T n n n e e G σσμ ]1[)()()]()1(2[2222-=---+t T n t T n n n n e e S σσμ十一、 In a risk-neutral world, suppose stock prices follow geometric Brownian motion,dS rSdt SdW σ=+1) What is the process followed by the variable n S by Ito’s lemma? Show that n S also follows geometric Brownian motion.2) The expected value of T S is =)(T S E ()r T t Se -. What is the expected value of n T S ?4) Using risk-neutral valuation to value the derivative, whose payoff at maturity is:n T T at T f S =十二、Consider the price of a stock, S , which is the following processt dW dt dS σμ+=where t W is a standard Brownian motion. For the first three years, 5,211==σμ; for the next three years, 4,322==σμ. If the initial value of stock price is $10, what is the expect value of the stock price at the end of year 6?The change in S during the first three years has the probability distribution21~(23,53)(6,75)S N N ∆⨯⨯=The change in S during the next three years has the probability distribution)48,9()34,33(~22N N S =⨯⨯∆The probability distribution of the change is therefore12~(15,123)S S S N ∆=∆+∆012T S S S S =+∆+∆Since the initial value of the variable is 10,01225T ES ES E S E S =+∆+∆=, 127548123T DS D S D S =∆+∆=+=~(25,123)T S Nthe expect value of the stock price at the end of year 6 is 25.。