A4_sol(3)

# The loan guarantee does not affect the shareholders

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The loan guarantee does not affect the shareholders’ payoffs however. So the equity value does not change. Total value of firm=equity value + debt value =16.36 + 4.09=20.45. The increase in the firm value is attributed to the government loan guarantee. The increase of 20.45 – 20 = .45 is exactly the value of the loan guarantee. 5. Consider an American put option written on a non-dividend paying stock. The strike price is 50. The put will expire in two years. The stock price follows a two-period binomial as in the graph. At each date, the natural probability that the stock price increases is one half. The stock is currently selling at 50. A two-year annual coupon bond with annual coupon rate of 10% is selling at par. 6

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(a) What is the price of the American put option? (b) What is the price of an American call option written on the same stock with the same strike price and expiration date? (c) Prove or disprove: the put-call parity holds for American puts and American calls. Answer (a) u=1.6 and d=0.6. R=1.1. The risk neutral probability is q=(R – d)/(u – d)=1/2. The payoffs of the American put on date 2 are P uu =0, P ud =2 and P dd =32. The payoffs at date 1 is P u =max{0, 2*0.5/1.1}=0.91 and P d =max{20, (2*0.5+32*0.5)/1.1}=20. That is, it would be optimal to early exercise at the downward node on the intermediate date. The price of the put is P=(0.5*0.91+0.5*20)/1.1=9.5. (c) The payoffs of an American call on date 2 is C uu =78, C ud =0 and C dd =0. The payoffs at date 1 is C u =0.5*78/1.1=35.45 and C d =0. The price of the call is C=0.5*35.45/1.1=16.12. (d) P+S=9.5+50=59.5 > C+k/1.1 2 = 57.4 Disprove the put-call parity. 7
• Fall '12
• Stefina
• Economics, Strike price

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