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Question 2 [10 Marks] A stock price per share moves according to geometric Brownian motion, S(t)=S 8,120 Suppose that S = 4, S(t) = 4e". What is the probability that the stock price will reach high of 7 before a low of 2? Question 3 [10 Marks) Show that B (t) is an Ito process and find the d(B(t))'. [10 Marks) Prove that (B6) dB() = 86.) = }(BO) Question 4 2 Question 5 [10 Marks) Let X (t) = je dB(s). What the distribution is of X(t)? 96 971 e TABLE A.1 Cumulative Standardized Normal Distribution A(z) is the integral of the standardized normal distribution from 0 to 2 in other words, the area under the curve to the left of 2). It gives the probability of a normal random variable not being more than z standard deviations above its mean. Values of 2 of particular importance: ACEI 1.645 0.9500 Lower limit of rights Tail 1.90 0.9750 Lower limit of right 2.5 til 23 0.990 Luer limit of rid I lil 2576 09990 Lower limit of right 0.5% til 300 09990 Lower limit of right stall 3.291 199495 |rlimsaini inil 1 2 2 0.00 0.01 0.03 0.04 0.05 0.01 0.0 0.DR 0.00 0.0 0.5000 05000 03080 05120 3100 0.5199 0.3239 05319 0.1 0.5398 05438 0.5478 0.5517 0.5587 0.5596 0.5636 0.5675 0.5714 0.5753 0.5793 0.2 0.5832 0.5871 0.9910 5948 0.9987 0.6064 L6103 0.6141 0.6179 0.6217 0.6255 0.6293 0.6331 0,6363 0.6406 0.6441 0.4 0.6554 0.6817 0.6.591 0.6628 0.6664 0.6700 0.6772 0.6541 06 OS 0.750 0.65 0.7019 0.7054 0.7123 0.7157 0.7257 0.7190 0.7224 0.6 0.7291 Q.7357 0.7399 0.7454 0.7540 0.7611 0.7642 0774 0.7734 0.7794 0.8 0.7852 GEN 0.7910 0.79 OM 07902 0.7993 08031 ONITS 8106 0.9 0.85 0.8133 OSING 08254 0.99 08315 0.8340 30s 1.0 0.8413 0843 03461 0885 DSSON 0.85$ . 0.8572 08599 0.8021 1.1 9505 0.866 0.8705 08729 0.8909 08770 en 0.8790 08810 0.30 1.2 0.89 Q8509 1891 08925 0.1964 0892 7.1999 09015 1.3 0.9012 0.0049 0.9066 09039 0.9115 0.9131 0.9147 0.9162 0.9177 0.9192 0.9207 022 09236 0.9222 massi 0.9251 mar 0.9279 0.9265 0.9292 00 0.9301 0.9319 1.5 0.9312 09345 0.9357 09370 0.9394 0.9392 0.94061 0.9418 0.9429 0.9441 0.94 09463 0.9074 09484 . 19.095 0.9505 019515 to! 0.9525 09505 0.9545 1.7 0.955 0.9564 0.9873 0.9582 0.959 09ROR 0.941 0.9616 0.9625 0.9633 1.R 0.9656 0.9621 0.9678 0.986 0.01 0.00 1.9 0.9706 0.9713 0.9719 0.9721 09782 0.973 0.9744 0.9750 2.0 0.9772 0.9756 0.9761 0.9702 0.9778 0.973 0.97 0.9793 0.9798 09803 0.5908 2.1 SRI2 0.9817 0921 0.9526 0.9830 0.93 1.9846 0.6850 0.9854 0.9857 0.9861 0.986 novas 0.78 9978 0988 0.4 0.80 0900 2.3 0.9993 09596 0.9898 0.9901 994 0.9905 19909 09913 09111 0.9910 2.4 0.9918 09920 0.9922 09923 0.9927 0.9929 19931 0.9932 0.9934 0.9930 2.5 0.9918 09940 0.9941 0.9945 19948 0.9919 We 09951 GO 2.6 0.9952 0.9953 09955 09987 0.9939 0.9900 0.9951 We 0.9962 0993 agen 0.9964 2.7 0.9966 0.9967 09905 1999 0.9972 09973 2. 0.9974 09975 WA 0.9976 0.9977 0.9977 con 0.9978 11.9979 0.979 0.9980 0.99RI 2.9 0.99RI 0.9982 0.996 993 0.9954 0.9985 0.9984 0.995 0.9986 0.99 3.0 0.997 AN 09987 0.997 99 0.99 0.99 01.9999 0.999 0.990 0.990 3.1 0.9990 0.9991 0.9991 0.9991 0.99992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.900 Q.9904 0.9004 0.004 0.9994 0.9095 0.9005 3.3 0905 0.9005 0.000 0.999 0.9996 0.999% 0.999 1.4 0.9997 (9947 0.9997 0.9999 0.9997 0.9999 0.9997 0.9997 09987 0.999 3.5 0.9999 0948 0.999 09995 1999 0.99 0901 0.9999 9998 0.9998 3.6 0.90 0.00 0.976 070 0233 0.7517 097 0.99% 1.99 0.9995 09990 Question 2 [10 Marks] A stock price per share moves according to geometric Brownian motion, S(t)=S 8,120 Suppose that S = 4, S(t) = 4e". What is the probability that the stock price will reach high of 7 before a low of 2? Question 3 [10 Marks) Show that B (t) is an Ito process and find the d(B(t))'. [10 Marks) Prove that (B6) dB() = 86.) = }(BO) Question 4 2 Question 5 [10 Marks) Let X (t) = je dB(s). What the distribution is of X(t)? 96 971 e TABLE A.1 Cumulative Standardized Normal Distribution A(z) is the integral of the standardized normal distribution from 0 to 2 in other words, the area under the curve to the left of 2). It gives the probability of a normal random variable not being more than z standard deviations above its mean. Values of 2 of particular importance: ACEI 1.645 0.9500 Lower limit of rights Tail 1.90 0.9750 Lower limit of right 2.5 til 23 0.990 Luer limit of rid I lil 2576 09990 Lower limit of right 0.5% til 300 09990 Lower limit of right stall 3.291 199495 |rlimsaini inil 1 2 2 0.00 0.01 0.03 0.04 0.05 0.01 0.0 0.DR 0.00 0.0 0.5000 05000 03080 05120 3100 0.5199 0.3239 05319 0.1 0.5398 05438 0.5478 0.5517 0.5587 0.5596 0.5636 0.5675 0.5714 0.5753 0.5793 0.2 0.5832 0.5871 0.9910 5948 0.9987 0.6064 L6103 0.6141 0.6179 0.6217 0.6255 0.6293 0.6331 0,6363 0.6406 0.6441 0.4 0.6554 0.6817 0.6.591 0.6628 0.6664 0.6700 0.6772 0.6541 06 OS 0.750 0.65 0.7019 0.7054 0.7123 0.7157 0.7257 0.7190 0.7224 0.6 0.7291 Q.7357 0.7399 0.7454 0.7540 0.7611 0.7642 0774 0.7734 0.7794 0.8 0.7852 GEN 0.7910 0.79 OM 07902 0.7993 08031 ONITS 8106 0.9 0.85 0.8133 OSING 08254 0.99 08315 0.8340 30s 1.0 0.8413 0843 03461 0885 DSSON 0.85$ . 0.8572 08599 0.8021 1.1 9505 0.866 0.8705 08729 0.8909 08770 en 0.8790 08810 0.30 1.2 0.89 Q8509 1891 08925 0.1964 0892 7.1999 09015 1.3 0.9012 0.0049 0.9066 09039 0.9115 0.9131 0.9147 0.9162 0.9177 0.9192 0.9207 022 09236 0.9222 massi 0.9251 mar 0.9279 0.9265 0.9292 00 0.9301 0.9319 1.5 0.9312 09345 0.9357 09370 0.9394 0.9392 0.94061 0.9418 0.9429 0.9441 0.94 09463 0.9074 09484 . 19.095 0.9505 019515 to! 0.9525 09505 0.9545 1.7 0.955 0.9564 0.9873 0.9582 0.959 09ROR 0.941 0.9616 0.9625 0.9633 1.R 0.9656 0.9621 0.9678 0.986 0.01 0.00 1.9 0.9706 0.9713 0.9719 0.9721 09782 0.973 0.9744 0.9750 2.0 0.9772 0.9756 0.9761 0.9702 0.9778 0.973 0.97 0.9793 0.9798 09803 0.5908 2.1 SRI2 0.9817 0921 0.9526 0.9830 0.93 1.9846 0.6850 0.9854 0.9857 0.9861 0.986 novas 0.78 9978 0988 0.4 0.80 0900 2.3 0.9993 09596 0.9898 0.9901 994 0.9905 19909 09913 09111 0.9910 2.4 0.9918 09920 0.9922 09923 0.9927 0.9929 19931 0.9932 0.9934 0.9930 2.5 0.9918 09940 0.9941 0.9945 19948 0.9919 We 09951 GO 2.6 0.9952 0.9953 09955 09987 0.9939 0.9900 0.9951 We 0.9962 0993 agen 0.9964 2.7 0.9966 0.9967 09905 1999 0.9972 09973 2. 0.9974 09975 WA 0.9976 0.9977 0.9977 con 0.9978 11.9979 0.979 0.9980 0.99RI 2.9 0.99RI 0.9982 0.996 993 0.9954 0.9985 0.9984 0.995 0.9986 0.99 3.0 0.997 AN 09987 0.997 99 0.99 0.99 01.9999 0.999 0.990 0.990 3.1 0.9990 0.9991 0.9991 0.9991 0.99992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.900 Q.9904 0.9004 0.004 0.9994 0.9095 0.9005 3.3 0905 0.9005 0.000 0.999 0.9996 0.999% 0.999 1.4 0.9997 (9947 0.9997 0.9999 0.9997 0.9999 0.9997 0.9997 09987 0.999 3.5 0.9999 0948 0.999 09995 1999 0.99 0901 0.9999 9998 0.9998 3.6 0.90 0.00 0.976 070 0233 0.7517 097 0.99% 1.99 0.9995 09990