BD_SM21 - Chapter 21 Option Valuation 21-1 In this case the...

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Chapter 21 Option Valuation 21-1. In this case, the stock price either rises to S u = 25×1.20 = 30 or falls to S d = 25×0.80 = 20. The option payoff is therefore either C u = 5 or C d = 0. The replicating portfolio is = ( 5 - 0 29/( 30 - 20 29 = 0.5 and B = ( 0 - 20×0.5 29/ 1.06 = - 9.43. Therefore, C = 0.5×25 - 9.43 = $3.07. 21-2. The parameters are the same as in 21-1, but the payoff of the put is 0 if the stock goes up and 5 if the stock goes down. Therefore, The replicating portfolio is = ( 0 - 5 29/( 30 - 20 29 = -0.5 and B = ( 5 - 20×(-0.5) 29/ 1.06 = 14.15 . Therefore, P = -0.5×25 + 14.15 = $1.65. 21-3. Up state at time 1: = ( 4 - 0 29/( 11 - 6.50 29 = 0.889, B = ( 0 - 6.50×0.889 29/ 1.03 = - 5.61, therefore C u = 0.889×8.50 - 5.61 = $1.95 In the down state at time 1 the option is worth nothing. The call option at time 0 is therefore equivalent to the replicating portfolio = ( 1.95 - 0 29/( 8.50 - 4 29 = 0.433, B = ( 0 - 4×0.433 29/ 1.03 = - 1.68 and so, by the Law of One Price, the initial option price is 0.433×6 - 1.68 = $0.92
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166 Berk/DeMarzo • Corporate Finance 21-4. If the stock goes up twice, the put is worth zero. If the stock ends up at $6.50, the put is worth $0.50, if the stock goes down twice, the put is worth $5. Given these final values, we can calculate the value of the put at earlier dates using the binomial model. Up state at time 1: = ( 0 -0. 50 29/( 11 – 6.50 29 = -0.111 and B = ( 0.50 -6.50 ×(-0.111) 29/ 1.03 = 1.19 . Therefore, P u = -0.111×8.50 + 1.19 = $0.25. Down state at time 1: = ( 0.50 -529/( 6.50 – 2 29 = -1 and B = ( 5 -2 ×(-1) 29/ 1.03 = 6.80 Therefore, P d = -1×4 + 6.80 = $2.80. Time 0: = ( 0.25 -2.8029/( 8.50 – 4 29 = -0.567 and B = ( 2.80 -4 ×(-0.567) 29/ 1.03 = 4.92 Therefore, P d = -0.567×6 + 4.92 = $1.52. 21-5. In example 21.1, the theoretical put price is $3.30. If it actually sells for a higher price, it is overvalued, and we can sell it and buy the replicating portfolio to earn an arbitrage profit. This means that at t = 0, you will: sell the put, short 0.3333 shares of stock, and invest $23.30 in Treasury Bills. Following this strategy, you will earn the put price less (60×(-0.3333) + 23.30) = $3.30 upfront. Then, if the stock goes up, our portfolio of the put, shares, and borrowing is worth -$0 – 0.3333×72 + 23.30×1.03 = $0. If it goes down, our portfolio is worth -$6 – 0.3333×$54 + 23.30×1.03 = $0, so that at maturity, the payoff of the option and the value of the replicating portfolio cancel out.
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