math5621-f11-final solutions

math5621-f11-final solutions - Math 5621 Financial Math II...

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Math 5621 Financial Math II Fall 2011 Final Exam Solutions - With corrections Dec. 19, 2011 December 9 to December 14, 2011 This is an open book take-home exam. You may consult any books, notes, websites or other printed material that you wish. Having so consulted then submit your own answers as written by you. Do NOT under any circumstances consult with any other person. Do NOT under any circumstances cut and paste any material from another source elec- tronically into your answer. Do NOT under any circumstances electronically copy a spreadsheet that was not created by you. Failure to follow these rules will be grounds for a failing grade for the course. Put your name on all papers submitted and please show all of your work so that I can see your reasoning. The eight questions will be equally weighted in the grading. Please return the completed exams by 5:30 PM Wednesday, December 14 to my mailbox in the department o¢ ce, under my o¢ ce door MSB408, or by email. 1. Build a binomial pricing model using the following assumptions: r f = : 02 , = : 22 , T = 2 , N = 4 , S 0 = 50 , and q u = 1 = 2 . (Do NOT use any other choice for q u ). Use the model to price an American Put option on S with strike price 65 expiring at T = 2 . What is the value of the put? What is the value of the position held in S 0 at time 0 in the replicating portfolio? Solution See spreadsheet on course website: The value of the put is 15 : 69 which is derived from the tree. There are several instances of early exercise of the American put noted on the tree. The value of the position in S 0 at time 0 is a short position 41 : 36 which is equal to the value of S 0 multiplied by the delta in the tree at time 0 , namely the di/erence in the V values one step ahead divided by the di/erence in the S values one step ahead. delta = V u V d S u S d position = V u V d S u S d S 0 Since the V values rise as S declines, the delta is negative, indicating a short position. 2. Using the same assumptions as question #1, according to the Black- Scholes formula what is the value of the position held in S 0 at time 0 in the hedging portfolio for a European Put expiring at T = 2 ? Solution 1
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See spreadsheet on the course website: The value of the position in S 0 at time 0 is a short position 35 : 60 . It is equal to the value of S 0 multiplied by the delta in the Black-Scholes formula. The delta is the part of the Black-Scholes formula multiplied by S 0 . For a European put option use put-call parity to see that the delta is the delta of the European call option minus 1. The delta of the European call option is the usual d 1 ) = ln S 0 K + ( r f + 1 2 2 ) T p T ! delta
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This note was uploaded on 02/06/2012 for the course MATH 5421 taught by Professor Jamesbridgeman during the Spring '11 term at UConn.

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math5621-f11-final solutions - Math 5621 Financial Math II...

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