Solution of Derivative

19 2079 034 4212 032 3543 74 k100 023 2989 046 5390

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Unformatted text preview: turn: K=70 110.83 K=90 110.83 K=100 110.83 45.95 100.00 K=80 110.83 37.38 29.21 22.73 0.24 0.27 0.31 36.14 0.26 100.00 100.00 28.34 91.68 0.30 21.16 0.34 91.68 100.00 0.34 15.96 91.68 0.376 91.68 27.77 20.56 14.17 10.06 0.29 0.33 0.39 0.42 73 Part 3 Options Delta and B: K=110 0.67 -57.32 0.54 -42.79 0.40 -30.24 K=120 0.55 -48.16 0.43 -34.46 0.39 0.41 0.19 -14.83 K=120 110.83 100.00 K=130 110.83 12.52 16.63 11.14 0.31 -26.13 0.29 -22.54 Call option price and gamma: K=110 110.83 100.00 K=130 0.43 -39.00 8.41 0.41 100.00 8.18 91.68 0.44 0.47 5.22 0.49 91.68 91.68 6.30 4.36 2.41 0.48 0.50 0.57 We clearly see that the more the option is out-of-the-money, the higher the required expected return is. This is a consequence of the option becoming more and more leveraged as it moves out of the money. Question 11.10. For the following questions, we will, due to space limitations, only report the first two of the 10 nodes. We have for the European put options of strike 70, 80, 90 and 100: Delta and B: K=70 -0.02 2.29 -0.05 5.85 K=80 -0.06 7.63 K=90 -0.12 15.40 -0.12 14.41 -0.09 9.18 -0.21 25.68 -0.19 20.79 -0.34 42.12 -0.32 35.43 74 K=100 -0.23 29.89 -0.46 53.90 Chapter 11 Binomial Option Pricing: II Put option price and gamma, the required rate of return: K=70 110.83 K=80 110.83 K=90 110.83 K=100 110.83 0.25 100.00 0.99 2.13 4.94 -0.51 -0.41 -0.37 -0.28 100.00 0.75 -0.41 100.00 2.19 -0.32 91.68 4.24 -0.28 91.68 100.00 8.27 91.68 -0.21 91.68 1.23 3.32 6.23 11.43 -0.39 -0.30 -0.25 -0.18 Delta and B: K=110 -0.33 45.04 -0.46 58.75 K=120 -0.45 63.51 K=130 -0.57 81.97 -0.57 76.31 -0.60 72.12 -0.69 93.87 -0.71 89.13 -0.81 106.14 Call option price and gamma: K=110 110.83 K=130 110.83 8.16 100.00 K=120 110.83 13.35 18.55 -0.24 -0.19 100.00 12.68 18.95 -0.18 -0.13 100.00 91.68 -0.16 25.23 -0.11 91.68 91.68 16.98 24.34 31.69 -0.15 -0.11 -0.09 We clearly see that the more the put option is out-of-the-money, the lower the required expected return is. Question 11.12. We first calculate u and d to be: u = e (r −δ )h +σ d = e (r −δ )h −σ h h Now we can calculate: p * = = e (0.08 )×1+0.3× 1 = 1.46228 = e (0.08 )×1−0.3× 1 = 0.80252 e (r −δ )h − d 1.08329 − 0.80252 0.28077 = = = 0.42556 u−d 1.46228 − 0.80252 0.65976 75 Part 3 Options This finally enables us to calculate the probabilities using equation 11.17 of the main text: n = 3: node i 0 1 2 3 stock price at node i proba of reaching node i 312.676837 0.07706811 171.600686 0.31209319 94.1764534 0.42128174 51.6851334 0.18955696 0 1 2 3 4 5 6 7 8 9 10 stock price at node i proba of reaching node i 4470.11845 0.0001948 2453.25302 0.00262948 1346.3738 0.01597245 738.90561 0.0574948 405.519997 0.13581727 222.554093 0.22000096 122.140276 0.2474752 67.0320046 0.19088947 36.7879441 0.09662769 20.1896518 0.02898527 11.0803158 0.0039126 n = 10 node i Probability Approximation of the distribution of the stock price (n=3) 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 50 100 150 200 Stock Price 76 250 300 350 Chapter 11 Binomial Option Pricing: II Approximation of the distribution of the stock price (n=10) 0.3 Probability 0.25 0.2 0.15 0.1 0.05 0 0 1000 2000 3000 4000 5000 Stock price Question 11.14. a) We can calculate the forward prices as: F0, 4 months = 100e 0.08×1 / 3 = 102.7025 F0,8months = 100e 0.08×2 / 3 = 105.4781 F0,1 year = 100e 0.08 = 108.3287 b) after t = 1/3 years, we have: Stock price probability 122.124611 0.45680665 86.3692561 0.54319335 This yields an expectation of 102.70254, equivalent to the forward price of part a). After t = 2/3 years, we have: Stock price probability 149.144206 0.20867232 105.478118 0.49626867 74.596484 0.29505901 77 Part 3 Options This yields an expectation of 105.4781, equivalent to the forward price of part a). After 1 year, we have: Stock price probability 182.141781 0.0953229 128.814741 0.34004825 91.1006658 0.40435476 64.4284283 0.16027409 This yields an expectation of 108.3287, equivalent to the forward price of part a). Question 11.16. Stock tree 233.621 210.110 188.966 169.949 152.847 137.465 123.631 111.190 100.000 123.631 95.000 89.937 152.847 137.465 111.190 123.631 100.000 80.886 123.631 111.190 100.000 89.937 80.886 72.746 152.847 137.465 111.190 89.937 188.966 169.949 100.000 89.937 80.886 72.746 65.425 80.886 72.746 65.425 58.841 65.425 58.841 52.920 52.920 47.594 42.804 European Call 138.621 114.746 93.500 74.600 57.798 43.327 31.432 22.118 57.829 42.719 30.213 20.571 57.847 42.554 28.976 18.736 78 93.966 74.836 28.631 16.442 Chapter 11 Binomial Option Pricing: II 15.140 13.563 8.703 11.670 7.065 9.221 5.080 4.183 2.760 1.483 5.000 2.431 1.182 0.575 0.280 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 American Call 138.621 115.110 93.966 74.949 57.981 43.422 31.482 22.144 15.153 42.746 20.578 8.704 57.884 30.226 13.566 42.554 18.736 7.065 28.631 16.442 9.221 5.080 2.760 1.483 57.847 28.976 11.670 4.183 93.966 74.949 5.000 2.431 1.182 0.575 0.280 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 European Put 0.000 0.000 0.000 0.000 0.000 0.458 4.265 0.000 0.000...
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