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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern