Solution of Derivative

# 004 0012 0013 0013 0012 0012 0012 perc change 1114

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Unformatted text preview: ercise is not always correct. We assume S = 50, sigma = 0.3, r = 0.08, delta = 0, K = 40, 50 and 60, and T = 1 month, 3 months, ..., 13 months. 85 Part 3 Options We can calculate: K=40 Time to theta expiration 1 month -0.010 3 months -0.012 5 months -0.012 7 months -0.012 9 months -0.012 11 months -0.011 13 months -0.011 Call price dollar change perc. change 10.271 10.939 11.678 12.409 13.115 13.792 14.443 -0.010 -0.012 -0.012 -0.012 -0.012 -0.011 -0.011 -0.09% -0.11% -0.11% -0.10% -0.09% -0.08% -0.07% theta -0.034 -0.022 -0.018 -0.016 -0.015 -0.014 -0.013 Call price 1.892 3.481 4.669 5.688 6.606 7.453 8.247 Dollar change -0.034 -0.022 -0.018 -0.016 -0.015 -0.014 -0.013 perc. change -1.82% -0.63% -0.39% -0.28% -0.22% -0.18% -0.16% theta -0.004 -0.012 -0.013 -0.013 -0.012 -0.012 -0.012 call price 0.037 0.577 1.319 2.088 2.846 3.586 4.306 Dollar change -0.004 -0.012 -0.013 -0.013 -0.012 -0.012 -0.012 perc. change -11.14% -2.01% -0.97% -0.61% -0.44% -0.34% -0.27% K=50 1 month 3 months 5 months 7 months 9 months 11 months 13 months K=60 1 month 3 months 5 months 7 months 9 months 11 months 13 months Please note that we measure theta as the dollar change in the call value per day. Therefore, we divided the returned value of the Excel function BSTheta by 360. We can see that in fact the statement of the exercise is not correct. Only the at the money call option (K = 50) has a monotonically decreasing theta (in time) and thus the greatest time decay for short expirations (i.e., a decreasing dollar and percentage price change if we reduce the time to maturity by one day). Both the out of the money and in the money option have thetas that are not monotonically decreasing in time to maturity, and neither the dollar change nor the percentage change are necessarily greater the shorter the time to expiration is. In the money and out of the money options can have thetas that are increasing in time to maturity, as the following figure, graphing the theta of the above options, depending on time to maturity, shows: 86 Chapter 12 The Black-Scholes Formula Question 12.12 epsilon 0.0010 0.0100 0.1000 call_u 14.3364 13.8923 9.9457 call_d 14.4363 14.8917 19.9526 div_appr. -0.4997 -0.4997 -0.5003 Let’s do a quick check: C(..., delta = 0.03) = 14.3863, C(..., delta = 0.04) = 13.8923 The difference is -0.4940, which is very close to our approximation of -.4997. Question 12.14 The greeks of the bull spread are simply the sum of the greeks of the individual options. The greeks of the call with a strike of 45 enter with a negative sign because this option was sold. Price Delta Gamma Vega Theta Rho Bought Call(40) 4.1553 0.6159 0.0450 0.1081 -0.0136 0.1024 87 Sold Call(45) -2.1304 -0.3972 -0.0454 -0.1091 0.0121 -0.0688 Bull spread 2.0249 0.2187 -0.0004 -0.0010 -0.0014 0.0336 Part 3 Options b) Bought call (40) Price 7.7342 Delta 0.8023 Gamma 0.0291 Vega 0.0885 Theta -0.0137 Rho 0.1418 Sold call(45) -4.6747 -0.6159 -0.0400 -0.0122 0.0152 -0.1152 bull spread 3.0596 0.1864 -0.0109 -0.0331 0.0016 0.0267 c) Because we simultaneously buy and sell an option, the graphs of gamma, vega and theta have inflection points (see figures below). Therefore, the initial intuition one may have had – that the greeks should be symmetric at S = \$ 40 and S = \$45 – is not correct. 88 Chapter 12 The Black-Scholes Formula Question 12.16 aa) 1 day to expiration S call delta put delta call vega put vega call theta put theta call rho put rho 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.509 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 ab) S 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -0.999 -0.491 -0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.021 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.021 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.002 -0.326 -0.025 -0.022 -0.022 -0.022 -0.022 -0.022 -0.022 -0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.021 -0.304 -0.003 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.003 -0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 call vega 0.105 0.154 0.207 0.258 0.303 0.336 0.358 0.368 0.366 0.354 0.335 0.311 0.283 0.254 0.225 0.197 0.171 Put vega 0.105 0.154 0.207 0.258 0.303 0.336 0.358 0.368 0.366 0.354 0.335 0.311 0.283 0.254 0.225 0.197 0.171 call theta -0.006 -0.008 -0.012 -0.015 -0.018 -0.021 -0.023 -0.025 -0.026 -0.027 -0.028 -0.028 -0.028 -0.027 -0.027 -0.026 -0.025 put theta 0.015 0.012 0.009 0.006 0.002 0.000 -0.003 -0.005 -0.006 -0.007 -0.007 -0.007 -0.007 -0.007 -0.006 -0.006 -0.005 call rho 0.052 0.086 0.131 0.184 0.245 0.310 0.376 0.442 0.504 0.563 0.617 0.665 0.707 0.743 0.775 0.801 0.824 put rho -0.871 -0.837 -0.792 -0.739 -0.678 -0.614 -0.547 -0.482 -0.419 -0.360 -0.306 -0.259 -0.216 -0.180 -0.148 -0.122 -0.100 1 year to expiration call delt...
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## This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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