PS_8_11 - Derivatives(3 credits Professor Michel Robe...

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1 Derivatives (3 credits) Professor Michel Robe Practice Set #8: Binomial trees and Continuous-time option pricing. What to do with this practice set? To help students with the material, eight practice sets with solutions shall be handed out. These sets contain mostly problems of my own design as well as a few carefully chosen, worked- out end-of-chapter problems from Hull. None of these Practice Sets will be graded: the number of "points" for a question solely indicates its difficulty in terms of the number of minutes needed to provide an answer. Students are strongly encouraged to try hard to solve the practice sets and to use office hours to discuss any problems they may have doing so. The best self-test for a student of her/his command of the material is whether s/he can handle the questions of the relevant practice sets. The questions on the mid-term and final exams will cover the material covered in class. Their format, in particular, shall in large part reflect questions such as the numerical exercises solved in class and/or the questions in the practice sets. Question 1 Consider a 3-month European put option on a non-dividend-paying stock. The current stock price is \$60, the strike price is \$50, the risk-free interest rate is 5% per annum, and the volatility (measured in terms of the standard deviation) is 30% per annum. (a) Using a binomial tree with a time interval of one month between nodes, calculate the option premium (i.e., its price). What are the risk-neutral probabilities on the tree? (b) If you use 10 intervals (of approximately 1.5 weeks) instead, does the price change? Why? (c) Do the probabilities change with the sampling period? Why? (d) Now consider a similar American put option. Is it optimal to exercise the option early? Does the conclusion change if you use 10 steps instead of 3 steps? Question 2 A market maker is considering introducing a nine-month American call option on corn futures.
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