Answer
Monte Carlo valuation of European call option (instructions)
Cells shaded grey are parameter values that can be altered.
1. In cell B25, generate a random number from the standard normal distribution by entering the formula =NORM.S.INV(RAND()).
2. In cell C25, calculate the simulated stock price at date T for the standard normal random number in B25 by entering the formula =S_0*EXP((riskfree-sigma*sigma/2)*T+sigma*B25*SQRT(T)).
3. In cell D25, calculate the present value of the payoff to the European call option for the simulated stock price in cell C25 by entering the formula =EXP(-riskfree*T)*MAX(C25- K, 0).
4. Select and drag the formulas in cells B25:D25 down to B1024:D1024. You should now have 1000 simulated terminal stock prices and option payoff present values.
5. In cell F17, calculate the average of the simulated option payoff present values by entering the formula =average(D25:D1024). This is an estimate of the option price. The standard deviation of the estimate is reported in cell F18.
S_0 100 Mean (Call Price) 11.27 Monte Carlo Valuation of European Call Option (full implementation in VBA)
K 100 SD 14.87
r 0.06 This VBA function calculates the price of a European call option by Monte Carlo simulation with antithetic variables.
sigma 0.2 Parameter values are in grey and can be set independently of those in the illustration at left.
T 1
S_0 100 Note: The function permits a non-zero dividend yield q on the stock.
Standard Normal Simulated Option Payoff K 100 Leave this set equal to zero to compare results with the illutration at left.
Random Numbers Stock Price Present Value r 0.06
0.5563307842 116.33 15.38 sigma 0.2 Note: M is the number of trials in the Monte Carlo simulation. You may want to see how
0.8519236872 123.42 22.05 q 0 changing the value of M affects the estimated option price.
1.0906887837 129.45 27.74 T 1
0.245489673 109.32 8.78 M 10000
-1.1367160369 82.92 0.00
-0.222704557 99.55 0.00 Euorpean Call Monte Carlo 10.91
0.0282807378 104.67 4.40
-1.2876535038 80.45 0.00
0.7465291674 120.84 19.63 Black-Scholes model (VBA function for European call option)
-0.0374359388 103.30 3.11
0.1247951786 106.71 6.32 This VBA function uses the Black-Scholes formula to calculate the value of a European call option.
-1.3531689547 79.40 0.00 Parameter values are in grey and can be set indendently of those in the 3-step binomial example.
-1.8054288375 72.54 0.00
0.5609558684 116.44 15.48 S_0 100 Note: The function permits a non-zero dividend yield q on the stock.
-0.1488865564 101.03 0.97 K 100 Leave this set equal to zero to compare results with the illutration at left.
1.2237796909 132.94 31.03 r 0.06
0.0754016583 105.66 5.33 sigma 0.2 You may also wish to check how the results from the Monte Carlo simulation in the box above
0.243796876 109.28 8.74 q 0 compare to the values from the Black-Scholes formula for various numbers of trials.
-1.2254940884 81.46 0.00 T 1 number of time steps N becomes large.
-1.770504241 73.04 0.00
-1.4457873476 77.95 0.00 European Call Black-Scholes 10.99
0.5432377748 116.03 15.09
-0.1719934062 100.56 0.53
1.5451223589 141.77 39.34
-1.2012651496 81.85 0.00
1.0446683518 128.27 26.62
-1.3669796317 79.18 0.00
-0.4936327969 94.30 0.00
-0.4587172953 94.96 0.00
-0.4223829819 95.65 0.00
-0.2112642505 99.77 0.00
-0.1734883881 100.53 0.50
-1.5080786397 76.98 0.00
-0.3757463674 96.55 0.00
-0.9965629033 85.27 0.00
0.0584191184 105.30 5.00
2.0979678084 158.34 54.95
-0.6927371032 90.62 0.00
1.2502368918 133.65 31.69
-1.4617489796 77.70 0.00
0.0125862698 104.34 4.09
1.2363602764 133.28 31.34
-1.206719844 81.76 0.00
-1.1063139027 83.42 0.00
-1.2499594959 81.06 0.00
0.7742414709 121.51 20.26
0.3669333637 112.01 11.31
0.4808782705 114.59 13.74
0.6061736713 117.50 16.48
-1.1014935808 83.50 0.00
0.4693526989 114.32 13.49
0.511070084 115.28 14.39
0.6210393622 117.85 16.81
0.5656863777 116.55 15.58
-0.000077307 104.08 3.84
2.0326388039 156.29 53.01
-0.507261575 94.04 0.00
0.5182903351 115.45 14.55
-1.0887038115 83.72 0.00
0.6896266314 119.47 18.34
0.4752359928 114.46 13.62
2.0781681087 157.72 54.36
-0.6997493074 90.49 0.00
-2.8577778721 58.77 0.00
1.0498501979 128.40 26.74
-1.0050348465 85.13 0.00
-1.3086810213 80.11 0.00
-1.5016030375 77.08 0.00
-0.4003334197 96.07 0.00
-0.628261887 91.79 0.00
-0.1388639248 101.23 1.16
-0.268439635 98.64 0.00
2.3045504349 165.02 61.24
1.349071681 136.32 34.20
1.5947534848 143.18 40.67
-0.9842051441 85.48 0.00
-0.6332402336 91.70 0.00
-1.368912308 79.15 0.00
-0.7549306247 89.50 0.00
-2.139913704 67.84 0.00
1.6763760868 145.54 42.89
1.2340910871 133.22 31.28
-2.0940933609 68.47 0.00
0.3446416486 111.51 10.84
-0.2908586794 98.20 0.00
-0.1622618129 100.76 0.71
-0.5697938832 92.87 0.00
-1.5487557903 76.36 0.00
0.0658906174 105.46 5.14
2.064034561 157.27 53.94
-1.0701387509 84.03 0.00
-0.6097207006 92.13 0.00
-1.3183534975 79.96 0.00
0.497379915 114.97 14.10
1.2553664419 133.79 31.82
0.7149219527 120.08 18.91
-0.7606922736 89.39 0.00
-2.0048719307 69.70 0.00
1.7347565517 147.25 44.50
-0.4534103937 95.06 0.00
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0.260665729 109.65 9.09
0.4692697075 114.32 13.49
-0.850197921 87.81 0.00
-0.8688562048 87.48 0.00
0.6035111255 117.43 16.42
-0.6162076716 92.01 0.00
-1.6626563593 74.64 0.00
1.5518708096 141.96 39.52
0.6883067272 119.44 18.31
0.524051984 115.58 14.67
1.0892017599 129.41 27.70
-1.3793442122 78.99 0.00
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0.8286883713 122.84 21.51
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0.3013951755 110.55 9.93
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0.376941216 112.23 11.52
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-1.0393250705 84.55 0.00
0.6354628113 118.19 17.13
-1.2823852558 80.54 0.00
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0.0327918315 104.77 4.49
-0.4338949111 95.43 0.00
2.1660889615 160.52 56.99
-0.4302762591 95.50 0.00
-1.3286762623 79.79 0.00
1.0829739949 129.25 27.55
1.5091609384 140.75 38.38
1.8946957425 152.03 49.00
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0.1171679287 106.55 6.17
-0.4741787052 94.66 0.00
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0.0211150564 104.52 4.26
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-0.7749827091 89.14 0.00
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-0.8677648111 87.50 0.00
0.6555615073 118.66 17.58
0.0392788024 104.90 4.62
0.8243910088 122.74 21.41
1.534299372 141.46 39.05
1.4741181076 139.77 37.45
0.6267873687 117.98 16.93
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1.1736142369 131.62 29.78
0.1070350208 106.33 5.96
0.852426183 123.43 22.06
-0.8175425137 88.38 0.00
1.01944579 127.62 26.01
0.8126016837 122.45 21.14
-0.2603360372 98.80 0.00
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0.9405994206 125.62 24.13
0.6428126653 118.36 17.29
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0.3761169864 112.21 11.50
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-0.0205091055 103.66 3.44
2.361193765 166.90 63.01
1.3821568245 137.22 35.05
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0.2805086297 110.09 9.50
1.5226123651 141.13 38.74
0.1625767254 107.52 7.08
0.818049557 122.58 21.27
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0.1882733613 108.07 7.60
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1.3041562852 135.10 33.05
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0.3477032351 111.58 10.90
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1.1688916857 131.49 29.66
0.2375168151 109.14 8.61
-0.8256483852 88.24 0.00
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2.5426561479 173.07 68.82
0.1264220373 106.75 6.35
1.326243364 135.70 33.62
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0.9865061656 126.78 25.22
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2.2164113034 162.14 58.52
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0.9900486475 126.87 25.31
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1.783546395 148.69 45.86
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1.4497345546 139.09 36.81
0.1913986125 108.14 7.67
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