GOTHENBURG UNIVERSITY
GRADUATE BUSINESS SCHOOL
ADVANCED DERIVATIVES
LAB EXERCISES 1 AND 2 (SPRING 2004)
SUPERVISORS: Charles Nodeau
Jianhua Zhang
AUTHORS:
Aijun Hou
Anders Johansson
Aránzazu Muñoz-Luengo
Lars Bjerkli
Table of Contents
1.0 INTRODUCTION.......................................................................................................................................... 2
1.1 PURPOSE AND OUTLINE OF THE PAPER .......................................................................................................... 2
1.2 METHODOLOGY ........................................................................................................................................... 2
1.2.1 Data and data collection ..................................................................................................................... 2
1.2.2 Data processing ................................................................................................................................... 2
2.0 COMPARISON BLACK-SCHOLES PRICE TO ACTUAL MARKET PRICE..................................... 3
2.1 THE BLACK-SCHOLES PRICE AND ACTUAL MARKET .................................................................................... 3
2.2 VOLATILITY AND IMPLIED VOLATILITY ........................................................................................................ 4
2.3 THE GREEKS ................................................................................................................................................ 5
3.0 COMPARISON OF BLACK-SCHOLES PRICE TO MONTE CARLO ESTIMATES......................... 7
3.1 BS FORMULA AND MONTE CARLO MODEL .................................................................................................. 7
3.2 BINOMINAL TREE FOR INVESTOR’S OPTION .................................................................................................. 8
4.0 CONCLUSION .............................................................................................................................................. 9
REFERENCES................................................................................................................................................... 10
APPENDIX 1. OTHER AMERICAN OPTIONS FROM INVESTOR B ..................................................... 11
1
1.0 Introduction
1.1 Purpose and outline of the paper
The main purpose of the report is to show our deep understanding of Black Scholes pricing
principle, which will be done by comparing one Swedish company’s (Investors B) actual
market price and Black Scholes price, as well as the comparison between Black Scholes and
Monte Carlo’s pricing. During the process, we will also compare the implied volatility and
standard deviation (based on the historical data). Binomial tree will be built up to examine
American options for Investor. Moreover, Greek letters will be presented and calculated.
After each comparison, the interpreting of the differences will be done by our analysis and
discussion. Finally, conclusion will be drawn based on our calculating, finding, and analysis.
1.2 Methodology
1.2.1 Data and data collection
We employ both primary data and secondary data in the report. Primary data are our
calculation such as Black Scholes option price, standard deviation, implied volatility and
time to maturity based on the secondary data. Two types of secondary data with different
maturity times (19 February, and 19 June), include Investor’s actual market price, option
exercise price, dividend yield are collected from Investor’s website and Gothenburg
University’s database1, however, the interest rate-- Swedish government bond, we obtain
from Internet2.
1.2.2 Data processing
Visual Basic Editor is the main method supports our data processing. Monte Carlo
simulation is another method employed to accomplish the comparison between BS price and
Monte Carlo price. The analysis and finding of this paper are based on these data processed
with these two methods.
1
2
http://www.ad.se/nyad/index.php?service=
http://www.afv.se
2
2.0 Comparison Black-Scholes Price to Actual market price
2.1 The Black-Scholes price and Actual market
In this section we will show the differences in Black-Scholes option price in comparison with
the actual traded market price for Investor B. The evaluation will be on Investor B options with
two different maturities, the 19th of March and 18th of June, 2004. This is an important matter,
since it will have an effect of both the maturity (T) of the expiration, the volatility (σ) of the
underlying asset and the dividends. At the time when the data was collected Investor B closed
at a stock price of 78 SEK the 12 of January, 2004. The actual market prices that have been
used are the mean value between ask and bid rate.
The first options that are investigated are those that mature the 19th of March, 2004. As can be
seen in table 1 both the call and the put are examined. To start with the call price, it can be seen
that the actual price is fairly priced in comparison to the result from the Black-Scholes formula.
By looking in the forth column the percentage is calculated from the difference between actual
price and BS price. It shows that the difference is between 1 and 7.5% until the exercise price
of 75 SEK (i.e. options that are in-the-money) and after that price, the actual price increases steadily
and rapidly in comparison to the BS price.
For the put option the reverse can be seen in the last column of table 1, where those options
that are in-the-money (80 to 90 SEK) actually are sold at a market price less than the theoretical
value of Black-Scholes. While for example the put option that has an exercise price of 70 SEK
are priced at 73% more than the theoretical value.
Exercise
price
BS Call
price
Actual Call
market price
55,00
57,50
60,00
65,00
70,00
75,00
80,00
85,00
90,00
23,178
20,686
18,194
13,214
8,322
4,046
1,335
0,277
0,036
23,500
21,000
18,500
13,500
8,750
4,375
1,475
0,400
0,280
Actual
minus
BS
1,37%
1,49%
1,65%
2,12%
4,90%
7,51%
9,48%
30,81%
87,29%
Exercise
price
BS Put
price
55,00
57,50
60,00
65,00
70,00
75,00
80,00
85,00
90,00
0,000
0,000
0,000
0,004
0,095
0,803
3,076
7,001
11,744
Actual Put
market
price
0
0
0
0
0,35
0,875
3,025
7
11,875
Actual
minus
BS
0
0
0
0
72,89%
8,20%
-1,68%
-0,02%
1,10%
Table 1. INVESTOR B, with an expiration date 19th of March, 2004
When investigating the second option (see table 2) that matures the 18th of June, 2004, the
dividend has to be considered and discounted back to present value as it occurs within the
maturity. It has been assumed that Investor pays the same dividends (2.25 SEK) at the 24th of
April in 2004 as they did previous year. For the call options it can be seen that the market price
is priced up to 10.50% more for all in-the-money options, whereas the out-of-money options
are sold at discount. For the put options there it is hard to make a valid evaluation, since the inthe-money put options are priced with a discount rate (80 to 85 SEK).
3
Exercise
price
BS Call
price
Actual Call
market price
Actual
minus BS
Exercise
Price
BS Put
Price
55,00
57,50
60,00
65,00
70,00
75,00
80,00
85,00
90,00
21,42
19,00
16,62
12,14
8,24
5,16
2,98
1,59
0,79
23,50
21,00
18,50
13,50
9,13
5,38
2,90
1,38
0,55
8,83%
9,52%
10,14%
10,10%
9,71%
3,94%
-2,80%
-15,60%
-43,00%
55,00
57,50
60,00
65,00
70,00
75,00
80,00
85,00
90,00
0,03
0,08
0,18
0,63
1,68
3,55
6,31
9,86
14,00
Actual Put
market
price
0,28
0,31
0,36
0,60
1,38
3,25
5,88
9,75
14
Actual
minus
BS
87,99%
73,25%
50,09%
-5,45%
-22,01%
-9,08%
-7,34%
-1,10%
0,02%
Table 2. INVESTOR B, with an expiration date 18th of June, 2004
As the volume traded on the underlying asset is very low or even zero in some cases it might
be a very good possibility that the bid and ask rate are inefficiently priced. For instance the
cases when the options are out of-the-money for the maturity 19 March (Call 85 to 90 SEK and Put
70 SEK) and 18 June (Call 90 SEK and Put 55 to 60 SEK) the BS model situates the price, but the
market maker has set another price and this price may not be effective due to the low trading
volumes, It should be recalled that the market price has been estimated by using the average
between the bids and ask rate, and this may be the reason for this difference. Moreover, the low
volume may be the cause that the current option prices were prior period’s prices, which may
be another reason for the difference between BS and actual price. Additionally, the stock
volatility that has been used in order to calculate the Black-Scholes prices might not be the
same that the market maker used with respect to the time period.
2.2 Volatility and Implied volatility
The volatility of a stock is a measure of the uncertainty about the returns of the stock. The
volatility of a stock can be calculated in two ways, so called, historical volatility and the
implied volatility. The historical volatility is based on the historical prices of the stock. When
calculating the historical volatility, we have based the calculation at the same time period as
the maturity date of the option. The exercise date for the first option is one month ahead so the
time period we have used to calculate the volatility is also one month. Since the exercise date
for the second period is four months ahead, the volatility is based on the last four month’s
stock prices. The annual standard deviation is calculated by taking the daily standard deviation
times 260 days. The reason for using 260 days is that it is the number of trading days per year,
see table 3.
Daily Return Statistics
Average Return
Return Variance
Return Standard Deviation
Annualised Return Statistics
Average Return
Return Variance
Return Standard Deviation
1 Month
4 Months
0,3335%
0,1495%
0,0172%
0,0230%
1,3128%
1,5153%
86,7129%
4,4809%
21,1681%
38,8739%
5,9698%
24,4332%
4
Table 3. Volatility measures
On the other hand the implied volatility shows the implicit volatilities of the option prices in
the market, so it is the market’s opinion about the volatility of the stock. When the options are
deep-in-the-money and deep-out-of-the-money, the prices are more insensitive to volatility and
the implied volatility is more unreliable. The implied volatility of actively traded options on the
stock is often used to calculate the appropriate volatility for options with lower volumes (Hull,
2003).
Exercise
price
55
57,5
60
65
70
75
80
85
90
Call options
Put options
March 19 June 18 March 19 June 18
69,66%
60,01%
14,79%
34,46%
61,99%
54,05%
12,98%
31,17%
54,59%
48,29%
11,13%
28,33%
40,47%
37,17%
7,87%
24,04%
32,48%
30,54%
28,80%
22,22%
25,25%
25,65%
21,76%
22,59%
22,66%
23,97%
20,62%
21,94%
23,59%
22,93%
21,14%
23,68%
31,13%
21,99%
27,72%
24,46%
Table 4. Implied volatility
It can be seen from table 4 above, the implied volatility is pretty close to the historical
volatility for the options with an exercise price close the stock price. The implied volatilities
for the exercise prices that are deep-out-of-the-money or deep-in-the-money are rather different
compared to the historical volatility. The reason can be that those options are not traded
actively and therefore the market’s opinion about the volatility is not reflected for those
options.
2.3 The Greeks
When calculating the Greeks, the exercise prices 55, 70 and 85 have been chosen. The Greeks
for the options are shown in the tables below.
Strike price
BS Value
55
23,18
Call
70
8,32
85
0,28
55
0,00
Put
70
0,09
85
7,00
Delta
Gamma
Rho
Theta
Vega
1,0000
0,0000
5,4071
-1,8036
0,0000
0,9564
0,0178
6,5366
-4,6116
2,2656
0,1130
0,0370
0,8420
-5,3199
4,6958
0,0000
0,0000
0,0000
0,0000
0,0000
-0,0436
0,0178
-0,3452
-2,3161
2,2656
-0,8870
0,0370
-7,5144
-2,5325
4,6958
Table 5. The Greeks for options maturing March 19
5
Since the ex-dividend date is assumed to be beyond the exercise date, the dividend yield is set
to zero for options with maturity date March 19. For the options maturing in June the dividend
has been taken into consideration and the corresponding Greeks are presented in table 6.
Strike price
BS Value
55
21,42
Delta
Gamma
Rho
Theta
Vega
0,9912
0,0022
18,6756
-2,1396
1,0645
Call
70
8,24
85
1,59
0,7582 0,2589
0,0286 0,0296
17,1220 6,2727
-6,5158 -5,6712
13,9487 14,4651
55
0,03
-0,0088
0,0022
-0,2435
-0,3507
1,0645
Put
70
1,68
85
9,86
-0,2418 -0,7411
0,0286
0,0296
-6,9569 -22,9660
-4,2390 -2,9066
13,9487 14,4651
Table 6. The Greeks for options maturing June 18
Delta measures the sensitivity of changes in the stock price; it measures the risk exposure of an
option position to sudden changes in the stock price. The delta 0,7582 for a call option with
exercise price 70 and maturity in June means that an increase in the stock price with 1 SEK
will lead to an increase in the price of the option with 0,7582 SEK. For the put option, delta is
negative which is natural since an increase in the stock price lead to a lower price of the option.
Gamma is the change of the delta with respect to the price of the stock. A small gamma means
that the delta changes slowly and a large gamma implies the opposite. Gamma is inversely
related to theta, when gamma is positive, theta is negative and vice versa. This is true for
Investor B.
Rho measures the sensitivity of the interest rates. The rho measures for the call options are
positive and the rhos for put options are negative. That is usual since a raise in interest rates
increases the value of call options and decreases the value of put options. For example, looking
at the call option maturing in March with exercise price 55, rho is 5,4071. An increase in the
interest rate with 1 % would lead to a 0,054071 (0,01*5,4071) higher price.
Theta is negative for most options because the value becomes lower when coming closer to
maturity. The theta should be more negative around the strike price for a call option and that is
also the case as we can observe in the table above.
Vega is the measure of sensitivity with respect to the volatility. Vega should be positive and
larger when the stock price is around the exercise price. As we can observe in table 6, the
vegas are higher for options with exercise price 70 than exercise price 55 since the used stock
price is 78 and (78-2,23). A vega of 13,95 means that if the volatility increases with 1%, the
price of the option increases with 0,01*13,95 = 0,1394.
6
3.0 Comparison of Black-Scholes Price to Monte Carlo Estimates
In the present section, we illustrate the procedures followed in order to perform the Monte
Carlo (MC) simulation as well as the findings of this model.
3.1 BS Formula and Monte Carlo Model
In this section, we applied MC simulation to value stock options, (Investor B), and then we will
compare these results with the stock options´ values obtained from the Black-Scholes (BS)
model over the same stock (for a stock price equal to 78). We performed the MC simulation to
value the two stock option of maturity 1 month where T = 0,099 and 4 months option with
present value of dividend where T = 0,348, and for which we used the historical volatilities
(21,17% and 24,4%) for each maturity respectively and for which we analyze four calls and
four puts, for the following exercise prices 60, 70, 80 and 90. Also we would like to note that
we assume we are in a risk free world (rf = 3,29%).
Next, we will detail the steps we took to proceed with the MC simulation:
1. Generation of 5000 random numbers between 0 and 1, and the inverse of these numbers
(  ) following a normal distribution, N (0,1).
2. Simulation of the stock prices following a geometrical Brownian motion of the type Ito
2
process: S  S 0 e ( r q 1 / 2 )T  T
3. Once we estimated the share prices (5000 sample), we calculate each payoff at maturity
date as follows, for a call option MAX (Share price – Exercise price ; 0) and for put
option MAX (Exercise price – Share price ; 0)
4. Then, we calculate the mean of the payoffs, which we will continuously discount in
order to get the MC estimate of the terminal values of the option.
σ (1 month) = 21,17%
Exercise Price
B-S Model
Call
M C Sim.
Std Error
B-S Model
Put
M C Sim.
Std Error
60
70
80
90
18,19
8,32
1,33
0,03
18,24
8,29
1,35
0,03
0,07
0,07
0,04
0,00
0,00
0,09
3,08
11,74
0,00
0,11
3,15
11,79
0,00
0,01
0,01
0,07
σ (4 month) = 24,4%
Exercise Price
B-S Model
Call
M C Sim.
Std Error
B-S Model
Put
M C Sim.
Std Error
60
70
80
90
16,62
8,24
2,97
0,79
16,58
8,12
3,09
0,77
0,15
0,12
0,08
0,04
0,18
1,68
6,31
14,00
0,17
1,68
6,29
14,15
0,01
0,05
0,10
0,13
Table 7. Monte Carlo prices vs. Black-Scholes prices, for maturities 1 month and 4 months.
7
In table 7 we gathered the results obtained from the MC simulation for 16 stock options and
which we face up to the BS model option prices.
As we can observe in the table, MC prices are very close to the BS prices. The reason for this
is that we performed a sample of 5000 normal random numbers in order to decrease the
uncertainty of the estimated values. We can affirm this by looking at the standard errors of the
estimated values, calculated as the standard deviation of the simulated payoffs divided by the
square root of the number of trials. Therefore, by observing the formula we conclude that if we
increase the number of trials the standard error of the values will decrease that is, the
uncertainty of the estimates is inversely proportional to the square root of the number of trials
(Hull, 2003). These numbers range from 0 to 0,15 and we believe these estimates indicate a good
approximation of the BS values.
3.2 Binominal tree for Investor’s option
According to Hull (2003), binomial tree is another effective method bedsides Monte Carlo
simulation to price options, specially, when the company pays out dividends. In this section,
we are going to examine Investor’s option price with the assumption that this is an American
option. The basic difference between American and European options is that American option
has the right to early exercise the options.
Figure 1. Binomial tree for American call and put option (maturity at 19th, June)
Figure 1 shows us Investor’s call and put option with exercise price 803 under the assumption
that this is American option. We divide the total maturity time into five equal intervals. There
are two crucial steps for tree building. 1. Calculation of stock price which is on the upper node
of the tree, 2.Calculation of option price which is on the lower node of the tree. As we know
Investor’s dividend (2,25) was paid out on 24th of April ( this is a known dividend case), and
3
The reason we choose this price as the trading volume at this price is comparatively high.
8
the stock price from 12th of February to 24th of April will be affected, hence, we consider the
dividend effect for the calculation of stock price, the formula is presented as below :
When it   ,
S 0 u j d i  j  De  r ( it ) , j  0,1,..., i
here,
S0=78, t = 24th of April
When it   ,
S 0u j d i j ,
here,
S0=78, t = 24th of April
j  0,1,..., i
There are dividend effects only for the first two time interval. For the first interval, we have 46
days to the time maturity, thus D e  r ( it ) =2,25*e(-0,0329*(46/260))=2,237, for the second
time interval, we have 21 days to maturity times, D e  r ( it ) =2,25*e(-0,0329*(46/260))=2,244.
The stock price changes with the changing of time to maturity, due to the change of the present
value of dividend.
On the left hand side of figure 1, American call option, the red number shows us the early
exercise price. With the chance of default, American call option is 4,66, which is higher than
actual European price 3.0. Same case with American put: 5,97 , which is higher than 5,88 of
European price. American option prices with one month maturity time4 are also higher than or
equal to the actual European option price. Additionally, all European option price we got from
binomial tree model is exactly the same as the actual price. Therefore we can say that
binomial tree is more effective and more suitable to price American options, especially, if the
company pays out dividend.
4.0 Conclusion
After comparing the Black-Scholes model with the actual price in different ways it is possible
to conclude that the model works better for options that are traded more frequently. That is
natural since a market with less trading is less efficient than a market with more trading. This
can be seen when looking at the implied volatility since that reflects the markets’ opinion about
the volatility. However, the volume has been extremely low for Investor B stock options,
therefore the prices are significantly different between the BS model and the actual market
price. The Greeks for the options evaluated are in accordance with the expected values.
Moreover, the Monte Carlo simulation has been performed for a 5000 sample, which yields
very approximate estimates to the BS prices. After building the binomial tree for Investor B,
we conclude that this model is more suitable to price American options, especially when
company pays out dividend.
4
Detail calculation see appendix 1. Binomial tree for American and European options
9
References
1. Book reference
HULL, J, C. (2003) Options, Futures and other Derivatives. 5th Edition. USA, Prentice Hall
2. Electronic Sources:
Affärsdata (database) [online]
Available from:
http://www.ad.se/nyad/index.php?service=
Accessed [2004-02-12]
Affärsvärlden [online]
Available from:
http://bors.affarsvarlden.se/mainoptionByIsin.asp?settings=afv&isin=SE0000107419&market=
sse&name=Investor+B
Accessed [2004-02-12]
INVESTOR HOMEPAGE [online]
Available from:
http://www.investorab.com/default.asp?lang=sv
Accessed [2004-02-12]
10
Appendix 1. Other American options from Investor B
11