problems16

Exercise 1Page 1Arbitrage:Buy April Option Sell Feb. Option NetToday -6 6.375 0.375The strategy: If the February option is exercised, exercise the Apriloption to cover; the future cash flow will thus at worst be zero. The initial cash flow, as shown above, will be positive.The most likely explanation for the newspaper quotes is that the prices quoted are not from simultaneous trades. In order for a true arbitrage to be occur, it must be possible to simultaneouslytrade in both options at once.Exercise 1 中文Page 2套利:购买4月期权 出售2月份期权 净值今天 -6 6.375 0.375策略: 如果二月份期权执行， 通过执行四月份期权来弥补; 未来现金流最差为零.初始现金流将会是正的，正如上面显示的那样.对于报纸报价最合理的解释为报价与交易不是同时进行的.为了使套利真实发生, 期权之间必须能够立刻交易。S 60X 45Call price 2Arbitrage: Buy the option, exercise immediatelyCost -2Exercise profit 15 ST = 441 p u t + 1 c a ll b o u g h t8 07 57 06 56 05 55 04 54 0-2 0-1 5-1 0-5051 01 52 02 5S to c k p ric e a t tim e T , S TProfit1 p u t b o u g h t+ 1 c a ll b o u g h t1 c a ll b o u g h t1 p u t b o u g h t2 P u ts b o u g h t + 1 C a ll b o u g h t8 07 57 06 56 05 55 04 54 0-2 5-2 0-1 5-1 0-5051 01 52 02 5S to c k p ric e a t tim e T , S TProfit2 p u ts b o u g h t22 p u t b o u g h t1 c a ll b o u g h t + 2 p u t b o u g h t1 c a ll b o u g h tP ro fit fo r 3 P u ts + 1 C a ll8 07 06 05 04 03 0-3 5-3 0-2 5-2 0-1 5-1 0-5051 01 52 0Profit3 p u ts b o u g h t3 p u ts b o u g h t + 1 c a ll b o u g h t1 c a ll b o u g h tA ll 3 s tra te g ie s -n o te c o m m o n c ro s s in g p o in t1 0 09 08 07 06 05 04 0-4 0-3 0-2 0-1 001 02 0S TProfit 1 P u t2 P u ts3 P u tsExercise 4 中文Page 10看跌期权 X=50 P=6看涨期权 X=60 P=104.a.购买1个看跌期权和1个看涨期权 策略总结ST 看涨期权 看跌期权 利润 ST 1个看跌期权 2个看跌期权 3个看跌期权40 -10 4 -6 40 -6 -2 242 -10 2 -8 42 -8 -6 -444 -10 0 -10 44 -10 -10 -1046 -10 -2 -12 46 -12 -14 -1648 -10 -4 -14 48 -14 -18 -2250 -10 -6 -16 50 -16 -22 -2852 -10 -6 -16 52 -16 -22 -2854 -10 -6 -16 54 -16 -22 -2856 -10 -6 -16 56 -16 -22 -2858 -10 -6 -16 58 -16 -22 -2860 -10 -6 -16 60 -16 -22 -2862 -8 -6 -14 62 -14 -20 -2664 -6 -6 -12 64 -12 -18 -2466 -4 -6 -10 66 -10 -16 -2268 -2 -6 -8 68 -8 -14 -2070 0 -6 -6 70 -6 -12 -1872 2 -6 -4 72 -4 -10 -1674 4 -6 -2 74 -2 -8 -1476 6 -6 0 76 0 -6 -1278 8 -6 2 78 2 -4 -1080 10 -6 4 80 4 -2 -882 12 -6 6 82 6 0 -684 14 -6 8 84 8 2 -486 16 -6 10 86 10 4 -288 18 -6 12 88 12 6 090 20 -6 14 90 14 8 24.b.购买2个看跌期权和1个看涨期权ST 看涨期权 看跌期权 利润40 -10 8 -242 -10 4 -644 -10 0 -1046 -10 -4 -1448 -10 -8 -1850 -10 -12 -2252 -10 -12 -2254 -10 -12 -2256 -10 -12 -2258 -10 -12 -2260 -10 -12 -2262 -8 -12 -2064 -6 -12 -1866 -4 -12 -1668 -2 -12 -1470 0 -12 -1272 2 -12 -1074 4 -12 -876 6 -12 -678 8 -12 -480 10 -12 -282 12 -12 084 14 -12 286 16 -12 488 18 -12 690 20 -12 84.c.购买3个看跌期权和1个看涨期权ST 看涨期权 看跌期权 利润40 -10 12 242 -10 6 -444 -10 0 -1046 -10 -6 -1648 -10 -12 -2250 -10 -18 -2852 -10 -18 -2854 -10 -18 -2856 -10 -18 -2858 -10 -18 -2860 -10 -18 -2862 -8 -18 -2664 -6 -18 -2466 -4 -18 -2268 -2 -18 -2070 0 -18 -1872 2 -18 -1674 4 -18 -1476 6 -18 -1278 8 -18 -1080 10 -18 -882 12 -18 -684 14 -18 -486 16 -18 -288 18 -18 090 20 -18 292 22 -18 494 24 -18 64.d.通过解方程我们得到:Max(0,ST - 60) -10+Max(0, 50-ST)-6 = Max(0,ST-60)-10+2(Max(0,50-ST)-6)= ST = 44购 买 1个 看 跌 期 权 +1个 看 涨 期 权8 07 57 06 56 05 55 04 54 0-2 0-1 5-1 0-5051 01 52 02 5时 间 T股 票 价 格 , ST利润购 买 1 个 看 涨 该 期 权 + 1 个 看 跌 期 权1 个 看 涨 期 权1 个 看 跌 期 权购 买 2个 看 跌 期 权 + 1个 看 涨 期 权8 07 57 06 56 05 55 04 54 0-2 5-2 0-1 5-1 0-5051 01 52 02 5时 间 T股 票 价 格 , ST利润购 买 两 个 看 跌 期 权购 买 2 个 看 跌 期 权 + 1 个 看 涨 期 权购 买 1 个 看 涨 期 权3个 看 跌 期 权 + 1个 看 涨 期 权 的 利 润8 07 06 05 04 03 0-3 5-3 0-2 5-2 0-1 5-1 0-5051 01 52 0利润购 买 3 个 看 跌 期 权3 个 看 跌 期 权 + 1 个 看 涨 期 权购 买 1 个 看 涨 期 权所 有 3个 策 略 -注 意 共 同 交 点1 0 09 08 07 06 05 04 0-4 0-3 0-2 0-1 001 02 0S T利润1个 看 跌 期 权2个 看 跌 期 权3个 看 跌 期 权Exercise 5Page 11Write Call with X=90Buy Call with X=100Today Time = TST -10 -10The PV of 10 at time T = 10e-rT = 9.048374Since the spread today is 10, the maximum present value of the futureis 9.048374. Therefore, the difference between the two call pricesis too large and a riskless arbitrage exists.Another way to do this is to invest the proceeds in a riskless security:Today Time = TST - 10+11.05 0 -10+11.050Exercise 5 中文Page 12出售看涨期权X=90购买看涨期权X=100今天 时间 = T 时间=TST -10 -1010在时间按T = 10e-rT时的现值 = 9.048374由于今天的差价为10,未来的最大现值是9.048374.因此, 两个看涨期权价格的差值太大,存在无风险套利还有一种办法是将收益投资到无风险证券今天 时间 = TST - 10+11.05 0 -10+11.050Exercise 6Page 136abuy 1 share STbuy 1 call Max(0,ST-90)-8ST Profit80 -2382 -2184 -1986 -1788 -1590 -1392 -994 -596 -198 3100 76bbuy 1 share STbuy 2 call 2*(Max(0,ST-90)-8)ST Profit80 -3182 -2984 -2786 -2588 -2390 -2192 -1594 -996 -398 3100 96cFor 1 share + 1 CallST Profit80 -2382 -2184 -1986 -1788 -1590 -1392 -994 -596 -198 3100 7For 1 share + 2 CallsST Profit80 -3182 -2984 -2786 -2588 -2390 -2192 -1594 -996 -398 3100 9For 1 share + 3 CallsST Profit80 -39 -23 -3182 -37 -21 -2984 -35 -19 -2786 -33 -17 -2588 -31 -15 -2390 -29 -13 -2192 -21 -9 -1594 -13 -5 -996 -5 -1 -398 3 3 3100 11 7 9As shown on the graph, the profit lines cross at 98.P ro fit fo r 1 S h a re + 1 C a ll1 2 01 0 08 0- 2 5- 2 0- 1 5- 1 0- 5051 0S TProfitP ro fit fo r 1 S h a re + 2 C a lls1 2 01 0 08 0- 4 0- 3 0- 2 0- 1 001 02 0S to c k p ric e a t tim e T , S TProfitP ro fit fo r 1 S h a re + N C a lls1 2 01 0 08 0- 5 0- 4 0- 3 0- 2 0- 1 001 02 0S to c k p ric e a t tim e T , S TProfitExercise 6中文Page 146a购买1股 ST购买1个看涨期权 Max(0,ST-90)-8ST 利润80 -2382 -2184 -1986 -1788 -1590 -1392 -994 -596 -198 3100 76b购买1股 ST购买2个看涨期权 2*(Max(0,ST-90)-8)ST 利润80 -3182 -2984 -2786 -2588 -2390 -2192 -1594 -996 -398 3100 96c对于1股+1个看涨期权ST 利润80 -2382 -2184 -1986 -1788 -1590 -1392 -994 -596 -198 3100 7对于1股+2个看涨期权ST 利润80 -3182 -2984 -2786 -2588 -2390 -2192 -1594 -996 -398 3100 9对于1股+3个看涨期权ST 利润80 -39 -23 -3182 -37 -21 -2984 -35 -19 -2786 -33 -17 -2588 -31 -15 -2390 -29 -13 -2192 -21 -9 -1594 -13 -5 -996 -5 -1 -398 3 3 3100 11 7 9正如图上所示, 利润线在98处相交.1 股 + 1 个 看 涨 期 权 的 利 润1 2 01 0 08 0- 2 5- 2 0- 1 5- 1 0- 5051 0S T利润1 股 + 2 个 看 涨 期 权 利 润1 2 01 0 08 0- 4 0- 3 0- 2 0- 1 001 02 0时 间 T时 股 票 价 格 , ST利润1股 + N 个 看 涨 期 权 的 利 润1 2 01 0 08 0- 5 0- 4 0- 3 0- 2 0- 1 001 02 0时 间 T时 股 票 价 格 , ST利润Exercise 7Page 15By Put-Call ParityC+Xe-rT=P+S0C 12Bond 80 Xe-rT 76.09835Put 5S0 85r 10%T 0.5C+Xe-r 88.09835P+S0 90Therefore, we write put, short stock, buy call and buy bond.ArbitrageAt Time TToday ST S0Write Put 5 -(80-ST) 0Short Stock 85 -ST -STBuy Call -12 0 ST-80Buy Bond -76.0984 80 801.901646 0 0Exercise 7中文Page 16由看跌-看涨期权平价定理C+Xe-rT=P+S0C 12债券 80 Xe-rT 76.09835看跌期权 5S0 85r 10%T 0.5C+Xe-r 88.09835P+S0 90因此，我们出售看跌期权，卖空股票，购买看涨期权和债券。套利在时间T今天 ST S0出售看跌期权 5 -(80-ST) 0卖空股票 85 -ST -ST购买看涨期权 -12 0 ST-80购买债券 -76.0984 80 801.901646 0 0Exercise 8Page 17The prices violate the convexity arbitrage proposition, which says that 2*P(50) XBuy stock -38.50 Stock +ST StockBuy put -2.00 Put payoff X-ST Put payoffSell call 3.00 Call payoff 0 Call payoffBorrow PV(X) 38.24 Pay back loan -X Pay back loanTotal cash flow 0.74 0Thus this strategy produces positive cash flows today and guaranteed zero cash flows in the future: Arbitrage!Part c. If the price of a put is $4, then S + P C + PV(X)This means you should sell S+P and buy C+PV(X).Well leave the details to you.+ST0-(ST - X)-X0Time T, if ST XThus this strategy produces positive cash flows today and guaranteed zero cash flows in the future: Arbitrage!Formila股票价格 38.5期权执行价格, X 40期权执行时间, T 1 X购买股票 -38.50股票 +ST 股票购买看跌期权 -2.00看跌期权结算 X-ST 看跌期权结算出售看涨期权 3.00看涨期权结算 0 看涨期权结算借用现值(X) 38.24偿还贷款 -X 偿还贷款总的现金流量 0.74 0该策略今天产生正的现金流量和保证未来的现金流量为零：套利!c部分. 如果看跌期权价格为$4,那么S + P C + PV(X)着意味着你应当出售S+P 和购买 C+PV(X).我们将把细节留给你.+ST0-(ST - X)-X0时间T, 如果ST X