Option Pricing with Black Scholes Model
VerifiedAdded on 2020/04/13
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AI Summary
This assignment focuses on calculating option prices using the Black-Scholes model. It provides input data such as stock price, exercise price, interest rates, and volatility. The student then applies the Black-Scholes formula to determine the present value of the exercise price, calculate d1 and d2 values, and find the delta using the normal cumulative density function. The assignment also includes calculating the value of call and put options, computing percentage errors between model results and actual market prices, and exploring a plotting strategy for a 'pout' strategy.
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Running head: FINANCIAL MODELLING
Financial Modelling
Name of the Student:
Name of the University:
Authors Note:
Financial Modelling
Name of the Student:
Name of the University:
Authors Note:
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FINANCIAL MODELLING
1
Table of Contents
1. Computing mean, variance and standard deviation of monthly returns:...............................2
2. Computing beta of stocks:......................................................................................................2
3. Computing covariance matrix of the stocks:..........................................................................5
4. Computing two envelope portfolios:......................................................................................5
5. Plotting optimal portfolio in graph:........................................................................................8
6. Computing mean, variance and beta of monthly returns of the portfolio:.............................8
7. Computing optimum portfolio with short sale constraint:.....................................................9
8a. Computing binomial model for the stock:............................................................................9
8b. Computing option using Black Scholes model:...................................................................9
8c. Computing percentage error:..............................................................................................10
9. Plotting graph with a pout strategy:.....................................................................................11
Bibliography:............................................................................................................................12
1
Table of Contents
1. Computing mean, variance and standard deviation of monthly returns:...............................2
2. Computing beta of stocks:......................................................................................................2
3. Computing covariance matrix of the stocks:..........................................................................5
4. Computing two envelope portfolios:......................................................................................5
5. Plotting optimal portfolio in graph:........................................................................................8
6. Computing mean, variance and beta of monthly returns of the portfolio:.............................8
7. Computing optimum portfolio with short sale constraint:.....................................................9
8a. Computing binomial model for the stock:............................................................................9
8b. Computing option using Black Scholes model:...................................................................9
8c. Computing percentage error:..............................................................................................10
9. Plotting graph with a pout strategy:.....................................................................................11
Bibliography:............................................................................................................................12
FINANCIAL MODELLING
2
1. Computing mean, variance and standard deviation of monthly returns:
Particulars Amazon.com
, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Monthly Mean 2.620% 0.899% 1.382% -0.838% 2.435%
Monthly Variance 0.0
055 0.0035
0.0
048
0.
0093 0.0019
Monthly Standard
deviation
0.0
740 0.0590
0.0
693
0.
0965 0.0439
2. Computing beta of stocks:
Date Amazon.
com, Inc
America
n
Express
Co
Apple
Inc
Macy's
Inc
United
Health
Group Inc
S&P 500 Portfolio
Returns
01-01-13 5.67% 2.29% -15.56% 1.25% 1.77% 4.92% 7.74%
01-02-13 -0.46% 5.52% -3.14% 3.95% -3.24% 1.10% -12.61%
01-03-13 0.84% 8.20% 0.29% 1.78% 6.80% 3.54% 8.08%
01-04-13 -4.88% 1.40% 0.03% 6.39% 4.64% 1.79% 1.71%
01-05-13 5.89% 10.14% 1.56% 8.05% 4.41% 2.06% -0.46%
01-06-13 3.11% -1.26% -12.59% -0.71% 4.45% -1.51% 15.01%
01-07-13 8.13% -1.33% 13.21% 0.71% 10.67% 4.83% 23.61%
01-08-13 -6.96% -2.55% 7.39% -8.43% -1.54% -3.18% -0.62%
01-09-13 10.68% 4.90% -2.17% -2.65% -0.18% 2.93% 4.30%
01-10-13 15.22% 7.99% 9.20% 6.36% -4.79% 4.36% -14.13%
2
1. Computing mean, variance and standard deviation of monthly returns:
Particulars Amazon.com
, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Monthly Mean 2.620% 0.899% 1.382% -0.838% 2.435%
Monthly Variance 0.0
055 0.0035
0.0
048
0.
0093 0.0019
Monthly Standard
deviation
0.0
740 0.0590
0.0
693
0.
0965 0.0439
2. Computing beta of stocks:
Date Amazon.
com, Inc
America
n
Express
Co
Apple
Inc
Macy's
Inc
United
Health
Group Inc
S&P 500 Portfolio
Returns
01-01-13 5.67% 2.29% -15.56% 1.25% 1.77% 4.92% 7.74%
01-02-13 -0.46% 5.52% -3.14% 3.95% -3.24% 1.10% -12.61%
01-03-13 0.84% 8.20% 0.29% 1.78% 6.80% 3.54% 8.08%
01-04-13 -4.88% 1.40% 0.03% 6.39% 4.64% 1.79% 1.71%
01-05-13 5.89% 10.14% 1.56% 8.05% 4.41% 2.06% -0.46%
01-06-13 3.11% -1.26% -12.59% -0.71% 4.45% -1.51% 15.01%
01-07-13 8.13% -1.33% 13.21% 0.71% 10.67% 4.83% 23.61%
01-08-13 -6.96% -2.55% 7.39% -8.43% -1.54% -3.18% -0.62%
01-09-13 10.68% 4.90% -2.17% -2.65% -0.18% 2.93% 4.30%
01-10-13 15.22% 7.99% 9.20% 6.36% -4.79% 4.36% -14.13%
FINANCIAL MODELLING
3
01-11-13 7.81% 4.77% 6.19% 14.42% 8.72% 2.77% 7.18%
01-12-13 1.30% 5.59% 0.89% 0.26% 1.09% 2.33% -0.97%
01-01-14 -10.60% -6.50% -11.39% -0.38% -4.09% -3.62% -6.87%
01-02-14 0.95% 7.10% 5.00% 8.40% 6.68% 4.22% 2.68%
01-03-14 -7.37% -1.38% 1.98% 2.44% 5.93% 0.69% 7.32%
01-04-14 -10.08% -2.93% 9.48% -3.19% -8.86% 0.62% -21.50%
01-05-14 2.73% 4.55% 7.02% 4.19% 5.94% 2.08% 6.23%
01-06-14 3.84% 3.62% 2.73% -3.17% 2.63% 1.89% 6.81%
01-07-14 -3.70% -7.52% 2.83% -0.40% -0.86% -1.52% 0.72%
01-08-14 7.99% 1.75% 6.97% 7.50% 6.72% 3.70% 9.85%
01-09-14 -5.02% -2.27% -1.72% -6.83% -0.50% -1.56% 3.24%
01-10-14 -5.41% 2.72% 6.95% -0.62% 9.67% 2.29% 14.62%
01-11-14 10.31% 2.71% 9.64% 11.57% 3.74% 2.42% 0.63%
01-12-14 -8.72% 0.67% -7.46% 1.29% 2.46% -0.42% 1.13%
01-01-15 13.31% -14.24% 5.96% -2.89% 4.98% -3.15% 26.71%
01-02-15 6.98% 1.11% 9.21% -0.25% 6.72% 5.34% 14.87%
01-03-15 -2.14% -4.35% -3.19% 1.85% 4.02% -1.75% 9.42%
01-04-15 12.53% -0.86% 0.58% -0.43% -6.00% 0.85% -5.53%
01-05-15 1.75% 2.89% 4.02% 3.53% 7.61% 1.04% 11.44%
01-06-15 1.13% -2.54% -3.79% 0.77% 1.48% -2.12% 5.54%
01-07-15 21.12% -2.16% -3.35% 2.33% -0.49% 1.95% 9.95%
01-08-15 -4.44% 0.86% -7.16% -16.41% -4.81% -6.46% 0.79%
01-09-15 -0.20% -3.43% -2.35% -13.28% 0.27% -2.68% 12.82%
01-10-15 20.11% -1.18% 8.01% -0.66% 1.51% 7.97% 12.60%
01-11-15 6.03% -2.24% -1.01% -26.58% -4.40% 0.05% 14.82%
01-12-15 1.65% -2.96% -11.68% -11.08% 4.28% -1.77% 22.35%
01-01-16 -14.10% -26.24% -7.82% 14.43% -2.13% -5.21% -3.72%
3
01-11-13 7.81% 4.77% 6.19% 14.42% 8.72% 2.77% 7.18%
01-12-13 1.30% 5.59% 0.89% 0.26% 1.09% 2.33% -0.97%
01-01-14 -10.60% -6.50% -11.39% -0.38% -4.09% -3.62% -6.87%
01-02-14 0.95% 7.10% 5.00% 8.40% 6.68% 4.22% 2.68%
01-03-14 -7.37% -1.38% 1.98% 2.44% 5.93% 0.69% 7.32%
01-04-14 -10.08% -2.93% 9.48% -3.19% -8.86% 0.62% -21.50%
01-05-14 2.73% 4.55% 7.02% 4.19% 5.94% 2.08% 6.23%
01-06-14 3.84% 3.62% 2.73% -3.17% 2.63% 1.89% 6.81%
01-07-14 -3.70% -7.52% 2.83% -0.40% -0.86% -1.52% 0.72%
01-08-14 7.99% 1.75% 6.97% 7.50% 6.72% 3.70% 9.85%
01-09-14 -5.02% -2.27% -1.72% -6.83% -0.50% -1.56% 3.24%
01-10-14 -5.41% 2.72% 6.95% -0.62% 9.67% 2.29% 14.62%
01-11-14 10.31% 2.71% 9.64% 11.57% 3.74% 2.42% 0.63%
01-12-14 -8.72% 0.67% -7.46% 1.29% 2.46% -0.42% 1.13%
01-01-15 13.31% -14.24% 5.96% -2.89% 4.98% -3.15% 26.71%
01-02-15 6.98% 1.11% 9.21% -0.25% 6.72% 5.34% 14.87%
01-03-15 -2.14% -4.35% -3.19% 1.85% 4.02% -1.75% 9.42%
01-04-15 12.53% -0.86% 0.58% -0.43% -6.00% 0.85% -5.53%
01-05-15 1.75% 2.89% 4.02% 3.53% 7.61% 1.04% 11.44%
01-06-15 1.13% -2.54% -3.79% 0.77% 1.48% -2.12% 5.54%
01-07-15 21.12% -2.16% -3.35% 2.33% -0.49% 1.95% 9.95%
01-08-15 -4.44% 0.86% -7.16% -16.41% -4.81% -6.46% 0.79%
01-09-15 -0.20% -3.43% -2.35% -13.28% 0.27% -2.68% 12.82%
01-10-15 20.11% -1.18% 8.01% -0.66% 1.51% 7.97% 12.60%
01-11-15 6.03% -2.24% -1.01% -26.58% -4.40% 0.05% 14.82%
01-12-15 1.65% -2.96% -11.68% -11.08% 4.28% -1.77% 22.35%
01-01-16 -14.10% -26.24% -7.82% 14.43% -2.13% -5.21% -3.72%
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FINANCIAL MODELLING
4
01-02-16 -6.05% 3.81% -0.67% 6.70% 3.36% -0.41% -3.13%
01-03-16 7.18% 9.96% 11.97% 2.02% 7.91% 6.39% 9.61%
01-04-16 10.53% 6.36% -15.07% -10.77% 2.13% 0.27% 17.00%
01-05-16 9.15% 0.50% 6.32% -17.57% 1.50% 1.52% 18.73%
01-06-16 -1.00% -7.91% -4.36% 1.20% 5.48% 0.09% 16.02%
01-07-16 5.86% 5.91% 8.62% 6.40% 1.41% 3.50% -4.48%
01-08-16 1.35% 1.72% 1.80% 0.97% -5.12% -0.12% -12.20%
01-09-16 8.49% -2.38% 6.34% 2.38% 2.86% -0.12% 8.52%
01-10-16 -5.84% 3.65% 0.43% -1.52% 0.95% -1.96% -2.23%
01-11-16 -5.10% 8.12% -2.70% 14.54% 11.35% 3.36% 6.05%
01-12-16 2.38% 2.79% 4.68% -16.42% 1.08% 1.80% 12.57%
01-01-17 6.89% 3.06% 4.66% -19.25% 1.28% 1.77% 17.13%
01-02-17 2.58% 4.71% 12.12% 11.74% 2.00% 3.65% -8.80%
01-03-17 4.79% -1.19% 4.75% -11.40% -0.83% -0.04% 8.62%
01-04-17 4.25% 0.18% -0.01% -1.43% 6.42% 0.91% 16.43%
01-05-17 7.26% -2.96% 6.15% -21.79% 0.17% 1.15% 20.29%
01-06-17 -2.71% 9.06% -5.89% -1.11% 5.68% 0.48% 6.95%
01-07-17 2.02% 1.17% 3.22% 2.17% 3.39% 1.92% 5.03%
01-08-17 -0.73% 1.02% 9.77% -13.41% 3.63% 0.05% 14.09%
01-09-17 -1.98% 4.94% -6.21% 4.93% -1.55% 1.91% -9.49%
01-10-17 13.95% 5.44% 9.24% -15.11% 7.08% 2.19% 27.20%
01-11-17 6.27% 2.27% 1.65% 23.80% 8.19% 2.77% 1.09%
Particulars Amazon.co
m, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Beta
4
01-02-16 -6.05% 3.81% -0.67% 6.70% 3.36% -0.41% -3.13%
01-03-16 7.18% 9.96% 11.97% 2.02% 7.91% 6.39% 9.61%
01-04-16 10.53% 6.36% -15.07% -10.77% 2.13% 0.27% 17.00%
01-05-16 9.15% 0.50% 6.32% -17.57% 1.50% 1.52% 18.73%
01-06-16 -1.00% -7.91% -4.36% 1.20% 5.48% 0.09% 16.02%
01-07-16 5.86% 5.91% 8.62% 6.40% 1.41% 3.50% -4.48%
01-08-16 1.35% 1.72% 1.80% 0.97% -5.12% -0.12% -12.20%
01-09-16 8.49% -2.38% 6.34% 2.38% 2.86% -0.12% 8.52%
01-10-16 -5.84% 3.65% 0.43% -1.52% 0.95% -1.96% -2.23%
01-11-16 -5.10% 8.12% -2.70% 14.54% 11.35% 3.36% 6.05%
01-12-16 2.38% 2.79% 4.68% -16.42% 1.08% 1.80% 12.57%
01-01-17 6.89% 3.06% 4.66% -19.25% 1.28% 1.77% 17.13%
01-02-17 2.58% 4.71% 12.12% 11.74% 2.00% 3.65% -8.80%
01-03-17 4.79% -1.19% 4.75% -11.40% -0.83% -0.04% 8.62%
01-04-17 4.25% 0.18% -0.01% -1.43% 6.42% 0.91% 16.43%
01-05-17 7.26% -2.96% 6.15% -21.79% 0.17% 1.15% 20.29%
01-06-17 -2.71% 9.06% -5.89% -1.11% 5.68% 0.48% 6.95%
01-07-17 2.02% 1.17% 3.22% 2.17% 3.39% 1.92% 5.03%
01-08-17 -0.73% 1.02% 9.77% -13.41% 3.63% 0.05% 14.09%
01-09-17 -1.98% 4.94% -6.21% 4.93% -1.55% 1.91% -9.49%
01-10-17 13.95% 5.44% 9.24% -15.11% 7.08% 2.19% 27.20%
01-11-17 6.27% 2.27% 1.65% 23.80% 8.19% 2.77% 1.09%
Particulars Amazon.co
m, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Beta
FINANCIAL MODELLING
5
1.45 1.28 1.18 1.00 0.65
3. Computing covariance matrix of the stocks:
Variance-Covariance Matrix
Amazon.co
m, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Amazon.com, Inc 0.00548 0.00104 0.001
56
-
0.0007
0
0.00035
American
Express Co
0.00104 0.00348 0.000
77
0.0005
7
0.00071
Apple Inc 0.00156 0.00077 0.004
80
0.0002
8
0.00061
Macy's Inc -0.00070 0.00057 0.000
28
0.0093
2
0.00142
United Health
Group Inc
0.00035 0.00071 0.000
61
0.0014
2
0.00192
4. Computing two envelope portfolios:
Computing an envelope portfolio with constant 0%
z Envelope portfolio X
3.638279741 25.26%
-1.003380874 -6.97%
5
1.45 1.28 1.18 1.00 0.65
3. Computing covariance matrix of the stocks:
Variance-Covariance Matrix
Amazon.co
m, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Amazon.com, Inc 0.00548 0.00104 0.001
56
-
0.0007
0
0.00035
American
Express Co
0.00104 0.00348 0.000
77
0.0005
7
0.00071
Apple Inc 0.00156 0.00077 0.004
80
0.0002
8
0.00061
Macy's Inc -0.00070 0.00057 0.000
28
0.0093
2
0.00142
United Health
Group Inc
0.00035 0.00071 0.000
61
0.0014
2
0.00192
4. Computing two envelope portfolios:
Computing an envelope portfolio with constant 0%
z Envelope portfolio X
3.638279741 25.26%
-1.003380874 -6.97%
FINANCIAL MODELLING
6
0.195186814 1.35%
-2.761517194 -19.17%
14.33689575 99.52%
100.00%
Computing an envelope portfolio with constant 1%
z Envelope portfolio Y
2.564877578 38.84%
-2.488044159 -37.68%
-0.789162338 -11.95%
-3.212129067 -48.64%
10.52777521 159.43%
100.00%
E (x) 0.4612
Var (x) 0.4612
Sigma (x) 67.91%
E (y) 0.3172
Var (y) 0.2511
Sigma (y) 50.11%
Cov(x,y) 0.3172
Corr(x,y) 0.9319
Portion of 0.3000
6
0.195186814 1.35%
-2.761517194 -19.17%
14.33689575 99.52%
100.00%
Computing an envelope portfolio with constant 1%
z Envelope portfolio Y
2.564877578 38.84%
-2.488044159 -37.68%
-0.789162338 -11.95%
-3.212129067 -48.64%
10.52777521 159.43%
100.00%
E (x) 0.4612
Var (x) 0.4612
Sigma (x) 67.91%
E (y) 0.3172
Var (y) 0.2511
Sigma (y) 50.11%
Cov(x,y) 0.3172
Corr(x,y) 0.9319
Portion of 0.3000
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FINANCIAL MODELLING
7
E(rp) 36.04%
σp2 0.2978
σp 54.57%
Proportion of x Sigma Return
54.57% 36.04%
-2.2 58.16% 0.03%
-1.9 53.09% 4.35%
-1.6 48.95% 8.67%
-1.3 45.97% 12.99%
-1 44.40% 17.31%
-0.7 44.38% 21.63%
-0.4 45.91% 25.96%
-0.1 48.86% 30.28%
0.2 52.98% 34.60%
0.5 58.02% 38.92%
0.8 63.78% 43.24%
1.1 70.06% 47.56%
1.4 76.74% 51.89%
1.7 83.73% 56.21%
2 90.96% 60.53%
2.3 98.37% 64.85%
7
E(rp) 36.04%
σp2 0.2978
σp 54.57%
Proportion of x Sigma Return
54.57% 36.04%
-2.2 58.16% 0.03%
-1.9 53.09% 4.35%
-1.6 48.95% 8.67%
-1.3 45.97% 12.99%
-1 44.40% 17.31%
-0.7 44.38% 21.63%
-0.4 45.91% 25.96%
-0.1 48.86% 30.28%
0.2 52.98% 34.60%
0.5 58.02% 38.92%
0.8 63.78% 43.24%
1.1 70.06% 47.56%
1.4 76.74% 51.89%
1.7 83.73% 56.21%
2 90.96% 60.53%
2.3 98.37% 64.85%
FINANCIAL MODELLING
8
40.00% 50.00% 60.00% 70.00% 80.00% 90.00% 100.00% 110.00%
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
17.31%
Efficient Frontier
Series2
5. Plotting optimal portfolio in graph:
0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00%
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
17.31%
Efficient Frontier
Series2
Portfolio Z
6. Computing mean, variance and beta of monthly returns of the portfolio:
Portfolio mean 6.11%
Portfolio sigma 10.30%
Portfolio beta 26.76%
8
40.00% 50.00% 60.00% 70.00% 80.00% 90.00% 100.00% 110.00%
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
17.31%
Efficient Frontier
Series2
5. Plotting optimal portfolio in graph:
0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00%
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
17.31%
Efficient Frontier
Series2
Portfolio Z
6. Computing mean, variance and beta of monthly returns of the portfolio:
Portfolio mean 6.11%
Portfolio sigma 10.30%
Portfolio beta 26.76%
FINANCIAL MODELLING
9
7. Computing optimum portfolio with short sale constraint:
Portfolio Z
0.2697
0.0000
0.0000
0.0000
0.7303
1.0000
Portfolio mean 2.48%
Portfolio sigma 3.95%
e=Theta 30.97%
8a. Computing binomial model for the stock:
175.61
174.19
172.78 172.78
171.38 171.38
170 170.00 170.00
168.63 168.63
167.26 167.26
165.91
164.57
8b. Computing option using Black Scholes model:
Input Data Value
Stock Price now (P) 171.5
9
7. Computing optimum portfolio with short sale constraint:
Portfolio Z
0.2697
0.0000
0.0000
0.0000
0.7303
1.0000
Portfolio mean 2.48%
Portfolio sigma 3.95%
e=Theta 30.97%
8a. Computing binomial model for the stock:
175.61
174.19
172.78 172.78
171.38 171.38
170 170.00 170.00
168.63 168.63
167.26 167.26
165.91
164.57
8b. Computing option using Black Scholes model:
Input Data Value
Stock Price now (P) 171.5
1 out of 10
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