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hw4sol - Introduction to derivatives Fall 2011 BUSI 588...

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Introduction to derivatives, Fall 2011 BUSI 588, Homework 4 solutions Solutions to Homework 4 1. (*) See hw4.xls for details on the calculations. (a) Using the Black-Scholes formula (including dividends) one can back out the value for the volatility from the expression 76 = C (1313 . 5 , 1350 , 0 . 05 , 0 . 02 , 1 , σ ) either by trial-and-error, bisection, or using solver. My calculations suggest σ = 14 . 62%. (b) Using the previuosly calculated implied volatility in our Black-Scholes formula we get that the price of the put should be P = 51 . 41. 2. (*) The following table presents the values of the deltas and gammas for the call options under consideration. Strike 90.0 100 110 Delta 0.7708 0.6256 0.4756 Gamma 0.0121 0.0152 0.0159 A position that is long thirty calls with a strike of $100 and ten puts with a strike of $90 has a Δ of Δ = 30(0 . 6256) + 10(0 . 7708 - 1) = 16 . 48 and a Γ of Γ = 30(0 . 0152) + 10(0 . 0121) = 0 . 5760 (a) Let n c denote the number of calls we need to buy in order to have a delta-neutral position. We would like for n c to solve n c 0 . 7708 + 16 . 48 = 0 or n c = - 21 . 37. The Γ of the combined position would be Γ = - 21 . 37(0 . 0121) + 0 . 5760 = 0 . 3169 (b) Let n i denote the number of calls with a strike of 90 and n o denote hte number of calls with a strike of 110 (for in-the-money and out-of-the-money). We would like to choose n i and n o so that n i 0 . 7708 + n o 0 . 4756 + 16 . 48 = 0 n i 0 . 0121 + n o 0 . 0159 + 0 . 5760 = 0 My calculations suggest that n i = 1 . 7753 and n o = - 37 . 52 make the portfolio delta- and gamma-neutral. 3. (**) Figure 1 plots the gammas of the call options as a function of the underlying asset. The solid line in particular deals with the case where the option as a one-year maturity.
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