t=6 months Arbitrage transactions Cash Flows at t=0 Sr<32 Sr>32 ell the put short s(32-S) 0 Sell the share short +$29 20 0 ( Lend32/110(=301)-32/10(=3051)+$32 +S32 0 No Arbitrage valution At maturity(two months)the payoff of the call option with strike price of $49 will be either $4 (if the stock price is $53)or $o (if the stock price is $48) Construct a portfolio consisting of A shares and Bo borrowing or lending. The payoff of the portfolio replicates the payoff the call option, therefore △53+Be12=4 △48+B 0 Solving the above two equations gives △=0.8,andB=-37.7654 The value of the portfolio today 0.8×50-37.7654=2235 To avoid arbitrage, the value of the call option must be $2. 235 Risk-neutral valuation 48 Down factor: d=-=0.96 Risk-neutral probability of an up movement 0.568l 1.06-0.96 The value of the put option is given by 0.5681×4+0 =$2235 C Gary Xu AcF2 14 Princip les of finance© Gary Xu AcF214 Principles of Finance 3 Arbitrage Transactions Cash Flows at t = 0 t = 6 months ST  32 ST  32 Sell the put short +$4 ( ) − $ 32 − ST 0 Sell the share short + $29 − $ST − $ST Buy the call long −$2.0 0 $( − 32) ST Lend $32 / 1.10(= 30.51) − 32 / 1.10(= 30.51) + $32 + $32 +$0.49 0 0 6. No Arbitrage Valution At maturity (two months) the payoff of the call option with strike price of $49 will be either $4 (if the stock price is $53) or $0 (if the stock price is $48). Construct a portfolio consisting of  shares and B0 borrowing or lending. The payoff of the portfolio replicates the payoff the call option, therefore 48 0 53 4 12 2 0.10 0 12 2 0.10 0  + =  + =   B e B e Solving the above two equations gives  = 0.8, and B0 = −37.7654 The value of the portfolio today is 0.850−37.7654 = 2.235 To avoid arbitrage, the value of the call option must be $2.235. Risk-neutral Valuation Up factor: 1.06 50 53 u = = Down factor: 0.96 50 48 d = = Risk-neutral probability of an up movement: 0.5681 1.06 0.96 0.96 12 2 0.10 = − − =  e  The value of the put option is given by $2.235 0.5681 4 0 12 2 0.10 =  +  e