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BD_SM_c21 - Chapter 21 Option Valuation 21-1 Construct the...

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Chapter 21 Option Valuation 21-1. Construct the call out of the stroke and bond 30s 100b 5 20s 100b 0 + = + = Substituting these two equations leads to: 10s 5 1 s 2 = = So, 1 1 20 100b 0 and b 2 10 + = = − So the value of the option today is: 1 100 1 25 $3.066 2 1.06 10 =
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166 Berk/DeMarzo • Corporate Finance 21-2. Construct the put out of the stock and bond: 30 1.06B 0 20 1.06B 5 ∆ + = ∆ + = Substituting these into the equations: 10 5 1/ 2 ∆ = ∆ = − So, 20( 1/ 2) 1.06B 5 B 14.1509 + = = So the price of the option is 25(–1/2) + 14.1509=$1.65. 21-3. Up state at time 1: 11s 100b 4 6.5s 100b 0 + = + = So, 4.5s 4 4 s 0.8889 4.5 6.50 0.8889 b 0.05778 100 = = = × = − = So the value of the option in this date is: 100 8.50s b $1.946 1.03 + =
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Chapter 21 Option Valuation 167 In the down state at time 1 the option is worth nothing so at time 0 we have the following tree: 8.50s 100b 1.946 4s 100b 0 + = + = ( ) 4.50s 1.946 s 0.4325 4 0.4325 b 0.017298 100 = = = − = − So the price of the call is: ( ) ( ) 100 0.017298 6 0.4325 1.03 = $0.9153 21-4. Up state at time 1: 11 1.03B 0 6.50 1.03B 0.50 So, 4.50 0.50 0.5 0.1111 4.5 0.5 6.5 0.1111 B 1.1866 1.03 So the price is 8.5 0.1111 1.1866 $0.2422 ∆ + = ∆ + = ∆ = ∆ = = − × = = × − + = Down state at time 1: 6.5 1.03B 0.5 2 1.03B 5 So, 4.50 0.5 5 4.5 1 4.5 5 2 1 B 6.7961 1.03 So the price is 4 1 6.7961 $2.7961 ∆ + = ∆ + = ∆ + = ∆ = = − × − = = × − + = Time 0:
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168 Berk/DeMarzo • Corporate Finance 8.5 1.03B 0.2422 4 1.03B 2.7961 4.50 0.2422 2.7961 0.5675 2.7961 4 0.5675 B 4.9187 1.03 So the price of the call is 6 0.5675 4.9187 $1.51 ∆ + = ∆ + = ∆ + = ∆ = − × − = = × − + = 21-5. In example 2.1.1, the theoretical put price is $7.78. If it actually sells for $8, it is overvalued sell it and buy the replicating portfolio. This means that at t = 0, you will: sell $8 put, short 0.6 units of stock, and lend $40.7767. Following this strategy, you will end up with $0.22. At maturity, the payoff of the option and the value of the replicating portfolio cancel out. 21-6. In example 2.1.1, the theoretical put price is $4.85. If it actually sells for $5, it is overpriced sell it and buy the replicating portfolio. This means that at t = 0, you will: sell $5 put, short 0.5322 units of stock, and lend $31.0114.
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