fm3_problems17

# fm3_problems17 - TWO-DATE BINOMIAL OPTION PRICING Up Down...

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TWO-DATE BINOMIAL OPTION PRICING Up 1.40 ### Down 0.80 ### Initial stock price 25 Interest rate 1.08 Exercise price 30 Stock price Bond price 35 ### 1.08 25 1 20 ### 1.08 Call option Put option 5 ### 0 ??? ??? 0 ### 10 A 0.3333 #MACRO? B -6.1728 #MACRO? Call price 2.1605 <-- Exercise 1, using simultaneous equations Put price 4.9383 <-- Exercise 2, using state prices Check: confirm that state prices State prices actually price the stock and the bond 0.4321 ### 1.08 ### 0.4938 ### 25 ### Put-call parity--Exercise 2 Stock + put 29.9383 ### Call + PV(X) 29.9383 ### Solving for the portfolio parameters: A is the number of shares and B is the number of bonds. 55*A + 108*B = 5 48.5*A + 108*B = 0 or: A*stock*Up+B*Interest=max(stock*Up-X,0) A*stock*Down+B*Interest=max(stock*Down-X,0) The solution is: check on state prices call price 2.16 q u q d

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Exercise 3 Page 3 call, X = 30 put, X = 40 5 5 2.25 12.25 0 20 stock bond: combine 3 calls and one put 35 20 ???? 19 20 20 Part a As shown above, combining 3 calls and one put gives a riskless security which costs 19 today and pays off 20 tomorrow. Therefore, the riskless interest rate is 1/19 = 5.2632%. Part b We need to combine two of the assets for which we know the price to get the stock's payoffs. At this point we will also know the price of the stock. Three calls + one bond = stock payoffs. This means that the stock price = 19 + 3*2.25 = 25.75. can be computed from any two assets, for example the put and the call). Using these prices to compute the value of the stock gives: stock price = 0.45*35 + 0.5*20 = 25.75. Check: If we compute the state prices, we get q u = 0.45 and q d = 0.50 (these state prices

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Exercise 4 Page 4 Stock Bond 65 1.06 50 1 45 1.06 Call option 15 ??? 0 Solving for the state prices gives: 0.38 ### 0.57 ### Using these state prices to value the call option gives: call 5.6604 ### Another way to do this problem is to solve the set of equations which give the call payoffs: 65*A + 1.06*B = 15 45*A + 1.06*B = 0 This solves to give: A 0.75 B -31.84 Thus, borrowing 31.8396 at the rate of 6% and buying 0.75 of a share will replicate the call payoffs. Calculating the cost of this strategy will give the cost of the call: cost 5.6604 The first thing to notice here is that the probabilities are irrelevant! They are already embedded in the price of the stock. q u q d
Page 5 Stock Bond 75 1.05 60 1 48 1.05 Put option 0 ### ??? 7 ### Solving for the state prices gives: 0.5291 ### 0.4233 ### Using these state prices to value the put option gives: put price 2.9630 ### Another way to do this problem is to solve the set of equations which give the put payoffs: 75*A + 1.05*B = 0 48*A + 1.05*B = 7 This solves to give: A -0.26 ### B 18.52 ### Thus, lending 18.51852 at the rate of 5% and shorting 0.25926 of a share will replicate the put payoffs. Calculating the cost of this strategy will give the cost of the call:

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## This note was uploaded on 01/23/2011 for the course FGB 780 taught by Professor Edwardchang during the Spring '09 term at Missouri State University-Springfield.

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fm3_problems17 - TWO-DATE BINOMIAL OPTION PRICING Up Down...

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