Assume that interest rate is 10% (=0.10), time to expiration is 180 days, and th
ID: 2750215 • Letter: A
Question
Assume that interest rate is 10% (=0.10), time to expiration is 180 days, and the current price of the July gasoline contract is USD 3.15/gal. Compute the values of a 3.00 call and a 3.00 put using Black’s option pricing model. Use the historical gasoline futures prices below to obtain the needed volatility estimate! You can solve this by hand or with a spreadsheet program such as Excel. If you chose to use a spreadsheet program, you need to turn in a printout of your work. Furthermore, indicate at each step what you did. Also, please write down the formula for each cell next to it so that it is clear what you calculated at each stage of the computation.
observation
futures closing price
ln(Rt)
(ln(Rt)-m)^2
1
3.070
2
3.044
3
3.071
4
3.011
5
3.099
6
3.067
7
3.048
8
3.022
9
3.021
10
3.032
11
3.020
12
3.021
13
3.020
14
3.059
15
3.050
16
3.064
17
3.005
18
3.000
19
3.040
20
3.011
21
3.069
22
3.039
23
3.068
24
3.011
25
3.098
26
3.033
27
3.091
28
3.055
29
3.088
30
3.055
31
3.003
observation
futures closing price
ln(Rt)
(ln(Rt)-m)^2
1
3.070
2
3.044
3
3.071
4
3.011
5
3.099
6
3.067
7
3.048
8
3.022
9
3.021
10
3.032
11
3.020
12
3.021
13
3.020
14
3.059
15
3.050
16
3.064
17
3.005
18
3.000
19
3.040
20
3.011
21
3.069
22
3.039
23
3.068
24
3.011
25
3.098
26
3.033
27
3.091
28
3.055
29
3.088
30
3.055
31
3.003
Explanation / Answer
observation futures closing price ln(Rt)(mean-closing price) (ln(Rt)-m)^2 1 3.07 -0.025 0.000625 2 3.044 0.001 0.000001 3 3.071 -0.026 0.000676 4 3.011 0.034 0.001156 5 3.099 -0.054 0.002916 6 3.067 -0.022 0.000484 7 3.048 -0.003 0.000009 8 3.022 0.023 0.000529 9 3.021 0.024 0.000576 10 3.032 0.013 0.000169 11 3.02 0.025 0.000625 12 3.021 0.024 0.000576 13 3.02 0.025 0.000625 14 3.059 -0.014 0.000196 15 3.05 -0.005 0.000025 16 3.064 -0.019 0.000361 17 3.005 0.04 0.001600 18 3 0.045 0.002025 19 3.04 0.005 0.000025 20 3.011 0.034 0.001156 21 3.069 -0.024 0.000576 22 3.039 0.006 0.000036 23 3.068 -0.023 0.000529 24 3.011 0.034 0.001156 25 3.098 -0.053 0.002809 26 3.033 0.012 0.000144 27 3.091 -0.046 0.002116 28 3.055 -0.01 0.000100 29 3.088 -0.043 0.001849 30 3.055 -0.01 0.000100 31 3.003 0.042 0.001764 94.385 0.025534 Mean 3.044677419 94.385/31 Variance 0.025534 Standard deviation= 0.16 SQRT of (0.025534) Black-Scholes Option Value Input Data Stock Price now (P) 3.15 Exercise Price of Option (EX) 3.308 Number of periods to Exercise in years (t) 0.5 Compounded Risk-Free Interest Rate (rf) 10.00% Standard Deviation (annualized s) 16.00% Output Data Present Value of Exercise Price (PV(EX)) 3.1462 s*t^.5 0.1131 d1 0.0673 d2 -0.0459 Delta N(d1) Normal Cumulative Density Function 0.5268 Bank Loan N(d2)*PV(EX) 1.5155 Value of Call 0.1439 Value of Put 0.1401