7. No-arbitrage valuation At maturity(three months)the payoff of the European put option with strike price of $40 will be either 0 (if the stock price is $45 )or $5(if the stock price is $45) Construct a portfolio consisting of A shares and Bo borrowing or lending. The payoff of the portfolio replicates the payoff the put option, therefore △45+B0(1+002)=0 △35+B(1+002)=5 Solving the equations gi Bo The value of the portfolio today is (-0.5)×40+220588=$20588 Therefore, the value of the put option is $2.0588 Risk-neutral valuation Up factor,, 45 1.125 Down factor: d =35=0.875 Risk-neutral probability of an up movement 102-0.875 1.125-09750.58 The value of the put option is given by 0.58×0+042×5 1.02 =$2.0588 a)S=52,X=50,r=12%,G=0.30,=0.25 h(52/50)+(012+032)×025 030√0.25 =0.5365 d2=d1-o√r=0.5365-0.30√0.25=0.3865 N(05365)=0.7042,N(0.3865)=06504 C Gary Xu AcF2 14 Princip les of Finance© Gary Xu AcF214 Principles of Finance 4 7. No-arbitrage valuation At maturity (three months) the payoff of the European put option with strike price of $40 will be either 0 (if the stock price is $45) or $5 (if the stock price is $45). Construct a portfolio consisting of  shares and B0 borrowing or lending. The payoff of the portfolio replicates the payoff the put option, therefore ( ) 35 (1 0.02) 5 45 1 0.02 0 0 0  + + =  + + = B B Solving the equations gives  = −0.5, and B0 = 22.0588 The value of the portfolio today is (− 0.5)40 + 22.0588 = $2.0588 Therefore, the value of the put option is $2.0588. Risk-neutral Valuation Up factor: 1.125 40 45 u = = Down factor: 0.875 40 35 d = = Risk-neutral probability of an up movement: 0.58 1.125 0.975 1.02 0.875 = − −  = The value of the put option is given by $2.0588 1.02 0.58 0 0.42 5 ＝  +  8. a) S = 52, X = 50,r = 12%, = 0.30, = 0.25 ( ) ( ) 0.5365 0.30 0.25 ln 52 / 50 0.12 0.3 0.25 2 1 = + +  d = d2 = d1 −  = 0.5365 − 0.30 0.25 = 0.3865 N(0.5365) = 0.7042, N(0.3865) = 0.6504