发帖
雷达卡
1. 根据欧式(行权价和期限都相同)CALL与PUT之间关系（所谓Put–call parity)：S(0)+Put = K的现值+Call, 其中S(0)现在股价，K是行权价,如果不等就有riskless profit opportunities.
K=130, S(0)=115, T=1 (year), put=18, Call=9, r=6%（题意不清，不知是否是连续复利. 便假设是按年计息的利率，不是连续复利，不过不影响解题思路）
左边=S(0)+Put=115+18=133，右边=K/((1+r)^T)+Call=130/1.06+9=131.64, 左边-右边=1.36>0, so, Put is relatively over-valued (or Call is relatively under-valued).
So, you can do the following to make a riskless profit (assuming no transaction cost, no credit risk, and you can sell shares short, ...):
you start with nothing in your pocket, sell (short) one share for $115 and write/sell a put option at $18, you will get $133 cash, and then use part of cash (i.e. $9) to buy one call option and deposit remaining cash (133-9=$124) into a bank account (to earn interest at 6% per year). Then, at the end of year,
- if the stock price=S(1)>130 (=k), you will exercise the call option by paying $130 cash, and you get one share, and use this share to settle the short sales of one share. Then, your bank account will have 124*1.06-130=$1.44 cash remaining. The put option will not be exercised by the counterparty, because the S(1)>K.
- if the stock price=S(1)=130, you will not exercise the call, and put option will not be exercised by the counterparty, but you need to buy one share (at the then price, i.e. 130) to settle the short sales of one share, again your bank account will have 124*1.06-130=$1.44 cash remaining.
- if the stock price=S(1)<130, The put option will be exercised by the counterparty, you will pay $130 and get one share. Then, you use this share to settle the short sales of one share. Then, your bank account will also have 124*1.06-130=$1.44 cash remaining. The call option will not be exercised by you, because S(1)<K.
So, in any case, you will always have 124*1.06-130=$1.44 cash remaining. since you start with nothing, but in one year's time, you will get $1.44 no matter what happens, that $1.44 is riskless profit (which has a present value of $1.36) you make.
2. 这题有点难，水平有限，概念不是很明白（如tracking portfolio，tracking error），没把握，只能试试：
第a小题：
设tracking portfolio是 P= 股票持有a股，现金持有b元。那么，beginning of year（时刻0）tracking portfolio 的价值，P(0)=S(0)*a+b=2a+b
call option的pay-off (在时刻1, i.e. end of year) = $1.8 (if S(1)=$5.3, i.e. good economy & win lawsuit) or 0 (in other cases).
为了模仿call option的pay-off, tracking portfolio 在时刻1应该是：$1.8 (if S(1)=$5.3, i.e. good economy & win lawsuit) or 0 (in worst case)，注意，因为end of year时有4个状态，但是只有2个资产，所以2个资产组成的tracking portfolio不能完全模仿(replicate) call option在时刻1的pay-off, 取worst case是因为这样能确保tracking portfolio的payoff不会比call option的pay-off差（从风险管理角度考虑，tracking portfolio要对冲风险，不能再有未对冲的风险）。
如何求tracking portfolio 在时刻1的payoff是：$1.8 (if S(1)=$5.3, i.e. good economy & win lawsuit) or 0 (in worst case)呢：
若S(1)=5.3, S(1)*a+b=5.3*a+b*1.1=$1.8 (注×1.1是因为有利息10%，假设10%利率是年计息的利率，不是连续复利）
若S(1)=0.2（worst case）, S(1)*a+b=0.2*a+b*1.1=0
解这2个方程组成的两元一次方程组得：a=0.3529412, b=-0.0641712 （负数表明是借钱，不是持有现金）
由此得: P(0)=S(0)*a+b=2a+b=$0.641711
可以把这个tracking portfolio定为call option的价值的下限（即：如果call option的价值高于此价时，有riskless profit套利机会；如果call option的价值低于此价时，没有riskless profit套利机会）。
第b小题：
当end of year的股价=0.45或3.3时，tracking portfolio 的pay-off会大于call option的pay-off，即：
end of year的股价=3.3时，tracking portfolio 的pay-off（3.3*a+b*1.1=$1.094）会大于call option的pay-off （$0）
end of year的股价=0.45时，tracking portfolio 的pay-off（0.45*a+b*1.1=$0.088）会大于call option的pay-off （$0）
这些差异应该就是tracking error了吧。不过，不需要worry，因为，tracking portfolio 的pay-off大于call option的pay-off，从风险管理角度考虑，tracking portfolio对冲了CALL OPTION的风险，能获利的可能，没有未对冲的风险。
京公网安备 11010802022788号
