case07 - Current stock price 25 sigma T n rf Stock tree...

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Current stock price 25 sigma 0.2 RNPs T 0.25 u 1.0594 pu 0.5208336 n 3 d 0.9439 pd 0.4791664 rf 1.05 r 1.0041 Stock tree 25.00 26.49 28.06 29.73 23.60 25.00 26.49 22.27 23.60 21.02 Call tree Strike 25 1.23 2.01 3.16 4.73 0.40 0.77 1.49 0.00 0.00 0.00 Prices of pure securities Value of call 0.1396 0.66 0.3852 0.57 0.3544 - 0.1087 - sum 0.9879 1.23 check 1
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Replicating Delta B t=0 0.57 -13.04 Stock price t=1 up 0.68 -16.03 25 t=1 down 0.44 -9.96 Option valu 1.13 Volalitity 0.2 sigma 0.2 RNPs T 0.25 u 1.0293 pu 0.5103958 0.5098772 n 12 d 0.9715 pd 0.4896042 0.4891068 rf 1.05 r 1.0010 Stock tree 1 2 3 4 5 6 7 8 9 10 11 12 25.00 25.73 26.49 27.26 28.06 28.88 29.73 30.60 31.49 32.42 33.37 34.34 35.35 0.000 24.29 25.00 25.73 26.49 27.26 28.06 28.88 29.73 30.60 31.49 32.42 33.37 23.60 24.29 25.00 25.73 26.49 27.26 28.06 28.88 29.73 30.60 31.49 0.000 22.93 23.60 24.29 25.00 25.73 26.49 27.26 28.06 28.88 29.73 22.27 22.93 23.60 24.29 25.00 25.73 26.49 27.26 28.06 21.64 22.27 22.93 23.60 24.29 25.00 25.73 26.49 21.02 21.64 22.27 22.93 23.60 24.29 25.00 20.43 21.02 21.64 22.27 22.93 23.60 19.84 20.43 21.02 21.64 22.27 19.28 19.84 20.43 21.02 18.73 19.28 19.84 18.20 18.73 17.68 Strike 25 Call price tree 1.13 1.53 2.03 2.62 3.31 4.07 4.88 5.73 6.60 7.49 8.42 9.37 10.35 0.71 1.02 1.41 1.92 2.53 3.23 4.01 4.83 5.67 6.55 7.44 8.37 0.40 0.60 0.89 1.28 1.80 2.43 3.16 3.96 4.78 5.62 6.49 0.19 0.30 0.48 0.75 1.14 1.67 2.34 3.11 3.91 4.73 0.07 0.12 0.20 0.35 0.59 0.97 1.54 2.29 3.06 0.01 0.03 0.05 0.10 0.20 0.39 0.76 1.49 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Delta tree 0.57 0.68 0.79 0.89 0.95 0.99 1.00 1.00 1.00 1.00 1.00 1.00 0.44 0.56 0.69 0.81 0.91 0.98 1.00 1.00 1.00 1.00 1.00 0.31 0.42 0.55 0.70 0.84 0.95 1.00 1.00 1.00 1.00 0.18 0.27 0.39 0.55 0.72 0.89 1.00 1.00 1.00 0.08 0.13 0.22 0.35 0.54 0.77 1.00 1.00 0.02 0.04 0.08 0.14 0.28 0.52 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Borrowing tree -13.04 -16.03 -18.96 -21.52 -23.41 -24.48 -24.85 -24.87 -24.90 -24.92 -24.95 -24.97 -9.96 -13.01 -16.33 -19.59 -22.34 -24.15 -24.87 -24.90 -24.92 -24.95 -24.97 -6.80 -9.57 -12.96 -16.77 -20.51 -23.44 -24.90 -24.92 -24.95 -24.97 -3.93 -6.05 -9.02 -12.91 -17.48 -21.98 -24.92 -24.95 -24.97 -1.73 -2.96 -4.99 -8.16 -12.84 -18.95 -24.95 -24.97 -0.44 -0.86 -1.69 -3.31 -6.49 -12.73 -24.97 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Pascal's triangle 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 1 3 6 10 15 21 28 36 45 55 66 1 4 10 20 35 56 84 120 165 220 1 5 15 35 70 126 210 330 495 1 6 21 56 126 252 462 792 1 7 28 84 210 462 924 1 8 36 120 330 792 1 9 45 165 495 1 10 55 220 1 11 66 1 12 1 Number of up moves RNPs =BINOMDI =COMBIN Payoff from Value 12 0.0003125 0.0003125 0.00031 0.0003125 1 10.35 0.00 11 0.0002998 0.0035975 0.00355 0.0035975 12 8.37 0.03 10 0.0002876 0.0189804 0.01875 0.0189804 66 6.49 0.12 9 0.0002759 0.0606906 0.05995 0.0606906 220 4.73 0.28 8 0.0002646 0.1309913 0.12940 0.1309913 495 3.06 0.40 7 0.0002538 0.2010483 0.19861 0.2010483 792 1.49 0.30 6 0.0002435 0.2250015 0.22227 0.2250015 924 0.00 - 5 0.0002336 0.1850021 0.18276 0.1850021 792 0.00 - 4 0.0002241 0.1109162 0.10957 0.1109162 495 0.00 - 3 0.0002149 0.0472879 0.04671 0.0472879 220 0.00 - 2 0.0002062 0.0136085 0.01344 0.0136085 66 0.00 - 1 0.0001978 0.0023735 0.00234 0.0023735 12 0.00 - 0 0.0001897 0.0001897 0.00019 0.0001897 1 0.00 - 1 1.13
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Volatility Page 3 0.15
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Stock price Strike 25 25 sigma 0.2 RNPs T 0.25 u 1.0130 pu 0.5046466192 n 60 d 0.9872 pd 0.4953533808 rf 1.05 r 1.00020 Call price Using the binomial formula 1.14565 Call price 1.14565 a 30.00 a' 31.00 This is the number of up moves so that the option is in the money Number of up moves S_T Call payoff RNPs Discount value 60 54.24 29.24 1.510922E-018 0.00 59 52.86 27.86 8.898588E-017 0.00 58 51.51 26.51 2.576742E-015 0.00 57 50.20 25.20 4.889961E-014 0.00 56 48.92 23.92 6.839873E-013 0.00 55 47.67 22.67 7.519584E-012 0.00 54 46.46 21.46 6.766016E-011 0.00 53 45.27 20.27 5.123379E-010 0.00 52 44.12 19.12 3.331733E-009 0.00 51 43.00 18.00 1.889551E-008 0.00 50 41.90 16.90 9.459249E-008 0.00 49 40.83 15.83 0.000000422 0.00 48 39.79 14.79 1.691626E-006 0.00 47 38.78 13.78 0.000006131 0.00 46 37.79 12.79 2.020354E-005 0.00 45 36.82 11.82 6.081657E-005 0.00 44 35.89 10.89 0.0001678967 0.00 43 34.97 9.97 0.0004265537 0.00 42 34.08 9.08 0.0010002244 0.01 41 33.21 8.21 0.0021703056 0.02 40 32.36 7.36 0.0043671943 0.03 39 31.54 6.54 0.008165278 0.05 38 30.74 5.74 0.0142082525 0.08 37 29.95 4.95 0.0230422132 0.11 36 29.19 4.19 0.0348692364 0.14 35 28.45 3.45 0.0492870349 0.17 34 27.72 2.72 0.0651261119 0.17 33 27.01 2.01 0.0805004054 0.16 32 26.32 1.32 0.0931283137 0.12 31 25.65 0.65 0.100869875 0.07 30 25.00 - 0.1023127329 - 29 24.36 - 0.0971889767 - 28 23.74 - 0.0864555329 - 27 23.14 - 0.0720053303 - 26 22.55 - 0.0561277014 - 25 21.97 - 0.0409270389 - 24 21.41 - 0.0278981622 - 23 20.87 - 0.0177628593 - 22 20.33 - 0.0105532173 - 21 19.82 - 0.0058434686 - 20 19.31 - 0.0030113261 - 19 18.82 - 0.0014418885 - 18 18.34 - 0.0006402709 - 17 17.87 - 0.0002630847 - 16 17.42 - 9.977450E-005 - 15 16.97 - 3.482209E-005 - 14 16.54 - 1.114592E-005 - 13 16.12 - 3.258922E-006 - 12 15.71 - 8.663708E-007 - 11 15.31 - 2.082652E-007 - 10 14.92 - 0.000000045 - 9 14.54 - 8.656150E-009 - 8 14.17 - 1.470590E-009 - 7 13.80 - 2.178881E-010 - 6 13.45 - 2.772462E-011 - 5 13.11 - 2.968807E-012 - 4 12.78 - 2.601907E-013 - 3 12.45 - 1.792275E-014 - 2 12.13 - 9.099670E-016 - 1 11.82 - 3.027829E-017 - 0 11.52 - 4.953451E-019 -
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The long-winded way to solve a 60-step binomial tree Stock price 25 sigma 0.2 RNPs Replicating portfolio T 0.25 u 1.0130 pu 0.504647 0.504544 Delta 0.567992 n 60 d 0.9872 pd 0.495353 0.495253 B -13.0542 rf 1.05 r 1.0002 Stock tree 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
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