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Unformatted text preview: Introduction to derivatives, Fall 2011 BUSI 588, Homework 3 solutions Homework 3 solutions 1. (*) The up and down movements in the tree are u = 1 . 35 and d = 0 . 8. Moreover, r = 1 . 05 1 / 6 = 1 . 00816. From these are the riskfree rate one can back out the riskneutral probabilities ˆ p u = r d u d = 1 . 00816 . 8 1 . 35 . 8 = 0 . 3785; ˆ p d = 0 . 6215; as well as the binomial evolution of the S&P500 over the next four months. 100 135 182.25 80 108 64 Using these, one can work backwards through the tree and backout the value of the call option with a strike of $100. 15.30 35.81 82.25 3.00 8.00 0.00 (a) A good estimate of the call value seems $15.30. (b) If the S&P 500 went up in value of the first two months I would value the call at $35.81; whereas if the S&P 500 went down in value the estimate would be $3.00. (c) The replicating portfolio at date t = 0 is comprised of Δ units of the S&P500 and B dollars in cash, such that Δ135 + B 1 . 00816 = 35 . 81; Δ80 + B 1 . 00816 = 3 . 00; so that Δ = 0 . 596 and B = 44 . 35. If we were long the call and wanted to hedge (i.e. eliminate all exposure to the S&P500), then we would like to take a short position in the S&P500 (for 0 . 596 shares), and lend 44 . 35. This portfolio would have to be rebalanced in two months time depending on whether the S&P 500 went up in value or not: • If the S&P 500 went up in value in the first two months, then in order to replicate the payoffs at maturity of the call option we would need to invest in Δ u units of the S&P500 and B u dollars in cash, such that Δ u 182 . 25 + B u 1 . 00816 = 82 . 25; Δ u 108 + B u 1 . 00816 = 8 . 00; so that Δ u = 1 and B u = 99 . 19. • If the S&P 500 went down in value in the first two months, then in order to replicate the payoffs at maturity of the call option we would need to invest in Δ d units of the S&P500 and B d dollars in cash, such that Δ d 108 + B d 1 . 00816 = 8; Δ d 64 + B d 1 . 00816 = 0; so that Δ d = 0 . 18 and B d = 11 . 5....
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 Fall '10
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 Derivatives, Derivative, Rational pricing, KenanFlagler Business School, ıa

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