This document contains a group project report for Group 28. It includes 5 parts: (1) calculating option values using binomial and Black-Scholes models, (2) confirming the Black-Scholes option value and calculating Greeks, (3) generating a random stock price path, (4) calculating option values and Greeks along the path over time, and (5) plotting the changing option value as it approaches expiration. The group members and their student numbers are listed at the top.

1. GROUP PROJECT 2
GROUP 28
Group Member Student No. Email
C Rohith Thatchan 3035324950 rohithchokka@gmail.com
GUO, Gunan 3035324106 ggn1994@sina.cn
HU, Zeyu 3035323932 ryan.hzy@outlook.com
JIANG, Wei 3035236335 samuel.w.jiang@gmail.com
YAN, Yu Sze 3035268455 yanyusze@gmail.com
OCTOBER 1, 2016

2. Part (a)
Use n = 1, 5, 10, 25, or 50 (correspondingly, h = 50/365, 10/365, 5/365, 2/365, or 1/365) in binomial option model
to calculate the option value. Take the risk-neutral pricing approach instead of constructing the complete
binomial trees.
The option value is calculated as follows using risk-neutral pricing approach:
Number of binomial steps (n) 1 5 10 25 50
Length of one period (h) 0.1370 0.0274 0.0137 0.0055 0.0027
Risk-neutral probability (p) 0.4539 0.4793 0.4854 0.4908 0.4935
Size of up move (u) 1.2116 1.0878 1.0610 1.0380 1.0267
Size of down move (d) 0.8368 0.9218 0.9438 0.9639 0.9743
Option Value (C) 1.4565 1.1540 1.1455 1.1083 1.1181
Part (b)
Use the Black-Scholes formula to calculate the option value.
Given Black-Scholes formula, the option value is calculated as follow:
. ∗ . ∗
.
0.13411 N(d ) = 0.44666
0.5 = -0.31917 N(d ) = 0.37480
, , . , . ,
,
. ∗
1.11609
Part (c)
Using the functions for option values and option Greeks available in the CD-ROM accompanying the textbook,
calculate the option value (to confirm the answer above) and four option Greeks (delta, gamma, theta and vega).
Given 6 parameters: 1) price of underlying asset; 2) strike price; 3) volatility; 4) continuously compounded risk-free
interest rate; 5) continuously compounded dividend yield; and 6) time to expiration. The option values and 4 option
Greeks are as follows:
Item Function (S, K, σ, r, T, δ) Outputs
Option Value BSCall (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 1.116093
Delta BSCallDelta (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 0.446659
Gamma BSCallGamma (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 0.106823
Theta BSCallTheta (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) -0.015704
Vega BSCallVega (S = 20, K = 21, σ = 0.5, r = 0.05, T = 50/365, δ = 0) 0.029267

3. Part (d)
Generate a random sequence of stock prices for 50 days.
A set of 50 random numbers (z) following standard normal distribution (i.e. mean=0, standard deviation=1) is generated
with “Generate Random Number” function in Microsoft Excel.
50 random numbers (z) following standard normal distribution
1.168005 1.742483 -0.926711 0.449127 -0.330140
0.685358 -0.560838 -0.983819 0.248989 0.241342
-0.605913 -1.142803 -0.571795 -1.234511 -0.054072
-0.509372 0.378445 0.030412 -0.597661 -1.321650
0.292239 0.422042 -0.799669 0.565680 -0.239610
0.617906 0.014574 -2.168090 0.811629 -1.044061
0.947120 1.269937 0.512509 0.962325 0.518362
0.240318 0.958441 -1.682820 1.465792 1.624012
-1.200706 -0.153560 0.676869 0.902423 -0.900011
0.639250 1.140747 -0.638594 0.670534 1.754834
Given defined parameters = 20%, = 0, = 50%, = 1/365, a sequence of stock prices for 50 days can be
generated by inputting 50 random numbers (z) in the following equation:
Stock prices for 50 days with random numbers (z)
20.625039 22.267185 23.145855 20.255206 22.223524
21.002636 21.947248 22.562145 20.391816 22.368933
20.676462 21.304935 22.231594 19.747571 22.341891
20.406849 21.521417 22.253868 19.445088 21.586751
20.567749 21.764918 21.797450 19.739160 21.456215
20.907357 21.777695 20.599298 20.167073 20.882166
21.436474 22.518282 20.881747 20.685685 21.171737
21.576155 23.095009 19.986150 21.499054 22.095522
20.912986 23.007107 20.347529 22.017375 21.585592
21.270172 23.709205 20.014403 22.411765 22.604695

4. Part (e)
Based on the stock price series generated in (d), calculate the option value and the four option Greeks from day
1 to day 50. Note that as time goes by, the maturity of the option drops to zero. Plot the option value from now to
maturity.
The option values can be obtained with Black-Scholes option pricing models, given parameter S from Part (d), K = 21,
T = (50 − )/365, = 50%, r = 5% and = 0. Option values and 4 option Greeks are as follows:
Random
number z
Stock
price
Strike
price
Time to
maturity
Option
value
Delta Gamma Vega Theta
N/A 20 21.000000 0.136986 1.116093 0.446659 0.106823 0.029267 -0.015704
1.168005 20.625039 21.000000 0.134247 1.399480 0.511924 0.105536 0.030134 -0.016629
0.685358 21.002636 21.000000 0.131507 1.583263 0.550773 0.103910 0.030139 -0.017065
-0.605913 20.676462 21.000000 0.128767 1.392270 0.515578 0.107456 0.029577 -0.017002
-0.509372 20.406849 21.000000 0.126027 1.240388 0.485176 0.110061 0.028881 -0.016883
0.292239 20.567749 21.000000 0.123288 1.302721 0.501766 0.110481 0.028811 -0.017241
0.617906 20.907357 21.000000 0.120548 1.461829 0.538260 0.109410 0.028826 -0.017720
0.947120 21.436474 21.000000 0.117808 1.743732 0.594835 0.105364 0.028520 -0.018089
0.240318 21.576155 21.000000 0.115068 1.809666 0.609612 0.104874 0.028089 -0.018274
-1.200706 20.912986 21.000000 0.112329 1.410809 0.536860 0.113349 0.027843 -0.018322
0.639250 21.270172 21.000000 0.109589 1.591143 0.576558 0.111222 0.027572 -0.018694
1.742483 22.267185 21.000000 0.106849 2.200790 0.681858 0.098023 0.025966 -0.018423
-0.560838 21.947248 21.000000 0.104110 1.968920 0.650401 0.104567 0.026219 -0.018935
-1.142803 21.304935 21.000000 0.101370 1.554129 0.580039 0.115251 0.026515 -0.019395
0.378445 21.521417 21.000000 0.098630 1.662871 0.604923 0.113943 0.026026 -0.019629
0.422042 21.764918 21.000000 0.095890 1.793866 0.632866 0.111757 0.025382 -0.019771
0.014574 21.777695 21.000000 0.093151 1.782071 0.634995 0.113103 0.024983 -0.020020
1.269937 22.518282 21.000000 0.090411 2.262944 0.715508 0.100197 0.022968 -0.019297
0.958441 23.095009 21.000000 0.087671 2.673433 0.772154 0.088341 0.020655 -0.018213
-0.153560 23.007107 21.000000 0.084932 2.587383 0.766826 0.091269 0.020516 -0.018607
1.140747 23.709205 21.000000 0.082192 3.130237 0.828138 0.074979 0.017321 -0.016695
-0.926711 23.145855 21.000000 0.079452 2.657848 0.784941 0.089586 0.019066 -0.018561
-0.983819 22.562145 21.000000 0.076712 2.195519 0.730740 0.105677 0.020634 -0.020381
-0.571795 22.231594 21.000000 0.073973 1.938626 0.696473 0.115612 0.021134 -0.021424
0.030412 22.253868 21.000000 0.071233 1.932635 0.701247 0.116858 0.020612 -0.021692
-0.799669 21.797450 21.000000 0.068493 1.602428 0.646698 0.130299 0.021202 -0.022913
-2.168090 20.599298 21.000000 0.065753 0.903729 0.475873 0.150776 0.021034 -0.023130
0.512509 20.881747 21.000000 0.063014 1.020328 0.517096 0.152075 0.020893 -0.024049
-1.682820 19.986150 21.000000 0.060274 0.596879 0.375554 0.154632 0.018615 -0.022100
0.676869 20.347529 21.000000 0.057534 0.719110 0.428881 0.160876 0.019161 -0.023907
-0.638594 20.014403 21.000000 0.054795 0.562276 0.371160 0.161345 0.017707 -0.023074
0.449127 20.255206 21.000000 0.052055 0.632001 0.406449 0.167883 0.017927 -0.024630
0.248989 20.391816 21.000000 0.049315 0.663650 0.425848 0.173143 0.017753 -0.025755
-1.234511 19.747571 21.000000 0.046575 0.402028 0.310537 0.165686 0.015047 -0.022912
-0.597661 19.445088 21.000000 0.043836 0.294639 0.254122 0.157462 0.013049 -0.021026
0.565680 19.739160 21.000000 0.041096 0.352834 0.294628 0.172350 0.013799 -0.023746
0.811629 20.167073 21.000000 0.038356 0.468047 0.365142 0.190358 0.014848 -0.027459

5. 0.962325 20.685685 21.000000 0.035616 0.652920 0.462649 0.203487 0.015506 -0.031041
1.465792 21.499054 21.000000 0.032877 1.063722 0.626472 0.194307 0.014763 -0.032456
0.902423 22.017375 21.000000 0.030137 1.383083 0.727676 0.173753 0.012692 -0.030851
0.670534 22.411765 21.000000 0.027397 1.655050 0.800693 0.150623 0.010364 -0.028141
-0.330140 22.223524 21.000000 0.024658 1.476689 0.781191 0.169167 0.010301 -0.030789
0.241342 22.368933 21.000000 0.021918 1.562591 0.817250 0.159983 0.008773 -0.029705
-0.054072 22.341891 21.000000 0.019178 1.510217 0.827165 0.165314 0.007913 -0.030584
-1.321650 21.586751 21.000000 0.016438 0.898712 0.682516 0.257562 0.009865 -0.042998
-0.239610 21.456215 21.000000 0.013699 0.765794 0.658443 0.292323 0.009218 -0.047918
-1.044061 20.882166 21.000000 0.010959 0.386004 0.471753 0.364075 0.008699 -0.055667
0.518362 21.171737 21.000000 0.008219 0.478157 0.583715 0.406504 0.007488 -0.064029
1.624012 22.095522 21.000000 0.005479 1.131721 0.919223 0.183121 0.002449 -0.033244
-0.900011 21.585592 21.000000 0.002740 0.630132 0.857517 0.398719 0.002545 -0.066072
1.754834 22.604695 21.000000 0.000000 1.604695 1.000000 N/A N/A N/A
If option values in above table are plotted from now to maturity, we can get the overall picture of option value movement
in the 50-days period:

6. Part (f)
Assume that you are the market maker who just issued 1 million units of call option. To hedge your short option
position, you take a long position on the synthetic call and rebalance your account daily (daily delta hedging).
Since hedging is not continuous but daily, there will be resulted hedging profit or loss every day. Prepare a
spreadsheet showing the following information over the entire period: stock price, option value, option delta,
your hedging position (stock position and money market position), and the hedging profit or loss. Compute the
cumulative hedging profit or loss (with interest) over the entire period and express it as a percentage of the
original call premium.
In this case, Delta is changing on a daily basis, and Delta-hedging position has to be rebalanced accordingly. Hedging
profit and loss may be a result of these adjustment. Given daily stock prices, call option value and Delta, we may
calculate the position of stock, amount of borrowed money, gain or loss from stock and option, overnight interest. The
overnight net profit can be calculated by subtracting interest payment from gain of stock and option position. By adding
up the future value of overnight net profit, we can get the cumulative net hedging profit.
Over 50-days period, the cumulative net hedging profit for the is $ 67,671.50, which is 6.06% of original call premium.
Day Gain/(loss)from
stock position
Gain/(loss) From
option position
Overnight
Interest paid
Over Night Net
Profit
Net Profit at
Maturity
1 279,179.50 (283,387.19) (1,070.91) (5,278.60) (5,314.15)
2 193,300.84 (183,783.23) (1,254.74) 8,262.88 8,317.39
3 (179,647.93) 190,993.45 (1,367.82) 9,977.70 10,042.14
4 (139,006.61) 151,881.82 (1,269.68) 11,605.53 11,678.89
5 78,065.18 (62,333.25) (1,186.46) 14,545.47 14,635.41
6 170,403.59 (159,108.22) (1,235.36) 10,060.01 10,120.83
7 284,802.61 (281,902.07) (1,341.43) 1,559.11 1,568.32
8 83,086.94 (65,934.33) (1,507.97) 15,644.63 15,734.90
9 (404,275.09) 398,857.19 (1,554.00) (6,971.89) (7,011.16)
10 191,758.98 (180,334.07) (1,344.82) 10,080.09 10,135.47
11 574,835.48 (609,647.29) (1,462.06) (36,273.87) (36,468.18)
12 (218,151.44) 231,870.12 (1,778.51) 11,940.16 12,002.48
13 (417,761.30) 414,790.65 (1,685.81) (4,656.46) (4,680.12)
14 125,568.07 (108,741.93) (1,480.04) 15,346.10 15,421.97
15 147,299.14 (130,995.05) (1,555.71) 14,748.38 14,819.26
16 8,086.30 11,795.36 (1,641.26) 18,240.40 18,325.55
17 470,269.22 (480,872.92) (1,650.34) (12,254.04) (12,309.56)
18 412,652.93 (410,489.65) (1,897.26) 266.02 267.19
19 (67,874.02) 86,050.16 (2,076.78) 16,099.35 16,167.86
20 538,386.98 (542,854.13) (2,062.48) (6,529.62) (6,556.51)
21 (466,532.04) 472,389.80 (2,261.01) 3,596.75 3,611.07
22 (458,178.42) 462,328.54 (2,124.84) 2,025.28 2,033.06
23 (241,547.00) 256,892.89 (1,957.88) 13,388.01 13,437.62
24 15,513.27 5,991.52 (1,855.62) 19,649.17 19,719.28
25 (320,061.34) 330,206.40 (1,873.12) 8,271.94 8,300.32
26 (774,842.76) 698,699.32 (1,711.62) (77,855.06) (78,111.44)
27 134,410.23 (116,598.58) (1,219.11) 16,592.54 16,644.90
28 (463,110.35) 423,448.96 (1,339.48) (41,000.87) (41,124.62)
29 135,717.21 (122,231.78) (946.50) 12,538.92 12,575.05

7. 30 (142,871.23) 156,834.75 (1,097.00) 12,866.52 12,901.82
31 89,376.19 (69,725.38) (940.65) 18,710.16 18,758.92
32 55,525.13 (31,648.96) (1,041.26) 22,834.91 22,891.28
33 (274,350.22) 261,621.64 (1,098.73) (13,827.30) (13,859.54)
34 (93,932.35) 107,389.64 (785.03) 12,672.26 12,700.06
35 74,730.24 (58,195.58) (636.59) 15,898.07 15,930.77
36 126,075.12 (115,212.92) (748.39) 10,113.81 10,133.23
37 189,367.11 (184,873.32) (944.70) 3,549.10 3,555.42
38 376,303.90 (410,801.30) (1,221.63) (35,719.03) (35,777.79)
39 324,713.61 (319,360.74) (1,699.41) 3,653.46 3,658.97
40 286,988.64 (271,967.48) (2,005.40) 13,015.76 13,033.60
41 (150,723.19) 178,361.11 (2,231.65) 25,406.28 25,437.63
42 113,592.04 (85,902.36) (2,176.06) 25,513.62 25,541.60
43 (22,100.12) 52,373.83 (2,290.35) 27,983.36 28,010.21
44 (624,625.55) 611,505.00 (2,324.85) (15,445.39) (15,458.09)
45 (89,092.78) 132,917.94 (1,895.28) 41,929.88 41,958.61
46 (377,978.73) 379,790.61 (1,830.52) (18.65) (18.66)
47 136,606.06 (92,153.50) (1,296.70) 43,155.87 43,173.61
48 539,227.14 (653,564.10) (1,627.52) (115,964.49) (115,996.26)
49 (468,739.13) 501,589.07 (2,627.44) 30,222.51 30,226.65
50 873,898.63 (974,562.88) (2,449.47) (103,113.72) (103,113.72)

8. Part (g)
Repeat steps (d), (e), and (f) 50 times. Each time, you get the cumulative hedging profit or loss as a percentage of
the original call premium. Compile a list of these numbers and prepare a table of summary statistics (mean,
standard deviation, maximum, etc.). For this repetition job, show me the list and the table only (not the 50
spreadsheets). Discuss what you have learned about pricing and risk management from the simulation.
We have obtained the following Summary Statistics by repeating steps (d), (e), and (f) for 50 times.
Summary Statistics
Mean 0.036497
Standard Error 0.024155
Median 0.032494
Standard Deviation 0.170799
Sample Variance 0.029172
Kurtosis 0.437200
Skewness -0.076208
Range 0.803712
Minimum -0.406759
Maximum 0.396954
Sum 1.824844
Count 50
t-value 1.510971
t critical (two-tailed) 2.009575
Implications on option pricing
The calculation of option price with Black-Scholes option pricing model requires input of 6 parameters: 1) price of
underlying asset; 2) strike price; 3) volatility; 4) continuously compounded risk-free interest rate; 5) continuously
compounded dividend yield; and 6) time to expiration. Options prices are affected by the 6 parameters.
1) Price of underlying asset
The underlying asset is stock in this case. From the simulation in Part (e), we may find that as the price of stock
increases, call prices increase. Conversely, as the price of stock, call prices decrease. An increase in stock price
increases possibility of exercising the call option and also increase the profit of exercising it. This relationship can
also be demonstrated by the positive deltas in Part (e).
2) Strike price
The call option prices increase as strike price decrease. For otherwise identical call options, the lower strike price
call option has a higher option price since the call option are more likely to be exercised and the profit if exercised
is larger.
3) Volatility of underlying asset
Call option price increases as volatility increases. The call option eliminates downside risk of the stock price by
locking the buy price. In this case, a higher degree of volatility is favorable to investors since the potential profits
can be larger but potential loss remains the same. This relationship can also be demonstrated by the positive vegas
in Part (e).

9. 4) Continuously compounded risk-free interest rate
If interest rates rise, call option price will increase. Conversely, if interest rate drop, call option prices will increase.
We may calculate Rho to demonstrate this relationship. Rho should be positive in this case.
5) Continuously compounded dividend yield
If the underlying's dividend increases, call prices will decrease. Conversely, if the underlying's dividend decreases,
call prices will increase. Dividends can affect option prices because the underlying stock's price typically drops by
the amount of any cash dividend on the ex-dividend date. We may calculate Psi to demonstrate this relationship. Psi
should be negative in this case.
6) Time to expiration
The call option price decreases as expiration approaches. The shorter period of time an option has until expiration,
the less the chance that it will end up in-the-money, or profitable. This relationship can also be demonstrated by the
negative thetas in Part (e).
Implications on risk management
Market makers would hedge a short position in call options by using delta hedging strategy, aiming to hedge the risk
associated with price movements in stocks. The short call option position can be hedged, for one unit call option, by
longing delta units of stocks and borrowing money, amount of which is equal to the difference between one unit of
option and delta units of stocks. We notice that interest cost is important in delta hedging analysis. Stock purchase is
funded by borrowed money and interest expense incurred decreases overall profit. The magnitude depends on the
amount of money borrowed and interest rate.
We also run a t-test, setting the null hypothesis that the cumulative return is zero. The t value equals to 1.510971 and
critical value of 95% confidence level with 49 degrees of freedom in t-test is ± 2.009575, which means we cannot reject
the null hypothesis at 95% confidence level. This conclusion is consistent with the conception that market makers should
break even in average.
In addition, we notice that if stock price increases, the market maker, who takes a short position in call option and uses
delta hedging strategy in this case, will have loss on option position and gain on stock position. When stock price changes
significantly, gamma increase and delta hedging is not enough to hedge risks, therefore, the market maker might consider
gamma hedging by purchasing certain amount of deep-out-of money calls and puts.