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The Payoff to the Arbitrage Position

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Added on  2020-02-24

The Payoff to the Arbitrage Position

   Added on 2020-02-24

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c0=valueofaEuropeancallattime0p0=valueofEuropeanputattime0s0=valueofstockattime0x=excercisepricer=riskfreeinterestrateT=durationoftheoptionD=dividendProblem 1a)ct≥max⁡(0,st−D−Xe−rt)r=8%,t=4months,x=$70,st=$75,D=$1.50Now st−D−Xe−rt=75−1.5−(70∗e−0.08∗412)thisgives $ 3.6864thereforect=max⁡(0,3.6864)whichis$3.6864b)Call selling for 43Lower bound $ 3.6864To gain arbitrage profit buy the call option at c=$3. Sell the stock short at $75. Afterwards invest the proceeds 975−3¿=$72 at the rate of r=8%.At the expiration the arbitrage trader must close the short stock position if st>70,the trader should buy the stock through her call option.The payoff to the arbitrage position should be given by72(1.08)+(st−70)=7.76Problem 2a)p=Europeanputoptionprice,maturitydateis6monthsp=c+xe−rt+D−S¿5+50e−0.05+1−505+47.5614+1−50¿$3.5614b)p+s=3.5614+52=55.5614c+xe−rt=5+47.56145=52.5614an arbitrage opportunity exists with a risk-free profit of $3
The Payoff to the Arbitrage Position_1
Problem 3a)ST=30,T=6months,p=1.08,down=0.9,r=5%,X=3234.992DB 32.4E 29.16A3027CF 24.3The up with probability P and the down by probability (1−p)The f_uu=0F_ud=2.84F_dd=7.7Now p=ert−du−d=e0.05∗0.25−0.90.18=0.6254f=e−2∗rt[p2Fuu+p(1−p)Fud+(1−p)2Fdd]¿e−0.025[(0.3911∗0)+(0.4685∗2.84)+(0.1403∗7.7)]¿e−0.025∗2.41085=2.3513b)Since the portfolios are the same, the terminal payoffs i.e. Fuu,Fud,Fdd are the same tothat of European put option.DBEACFFuu=0,Fud=2.84,Fdd=7.7Now at node B we compute F_u with p=0.6254Fu=e−rt[pFuu+(1−p)Fud]¿e−0.05∗0.25[(0.6254∗0)+(0.3746∗2.84)]¿1.0506In contrast to the European option we have the right to exercise the option here St>Xmeaning its not favourable to exercise the option hence Fu=fu
The Payoff to the Arbitrage Position_2

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