Faculty of Business, Law and the Built Environment
Derivatives
Mr. Roger Adkins
Assignment:
Option pricing by Binomial Option Pricing Model
Ugur Demir
MSc Investment Banking
Roll no @00036345
I have chosen FTSE 350 Software and Computer Services index for my assignment. I have
gathered the index data from Perfect Analysis database. I have 1304 data daily price of the index
observed over five years from 03 Feb 2003 to 31 Jan 2008. I have derived 1303 of daily return
data calculated continuously by the following formula:
𝐷𝑎𝑖𝑙𝑦 𝑟𝑒𝑡𝑢𝑟𝑛 = ln (
index(t + 1)
)
index(t)
Daily return of the index is widely scattered. Chart 1 shows the daily returns over five years. The
maximum daily return reaches to 9.28% and drops to -6.49% over this period. Daily returns have
an average of 0.0278% which carries index from 313.78 to 450.77 in five year time (Chart 2).
High volatility can be seen from Chart 1 as the area between 3% to -3% is highly populated.
When we compare average return to standard deviation of daily returns it can be seen
numerically too, which is 1.5231% for sample of 1303 compared to 0.0278%.
Chart 1
10.00%
9.00%
8.00%
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
-1.00%
-2.00%
-3.00%
-4.00%
-5.00%
-6.00%
01/09/2002
Daily rates
14/01/2004
28/05/2005
1
10/10/2006
22/02/2008
Daily rates’ volatility drops and raises during five year as can be seen in Chart 1 as it gets wider
and narrower in some areas. In order to show the change in volatility I calculated ‘moving
standard deviation’. Resulted Chart 3 showing the raise and fall of the standard deviation
calculated using one year data as sample. In last two years the volatility is on the average when
compared with the years between 2003 and 2006, when the index has relatively high and
relatively low volatilities in those days. While the previous years movements doesn’t match
much with the recent ones I included them to calculations as the index has risen these price
levels with the help of price movements in those years.
Chart 2
FTSE 350 Software & Computer Services
600.00
550.00
500.00
450.00
400.00
350.00
300.00
250.00
200.00
03/02/2003
03/02/2004
03/02/2005
03/02/2006
03/02/2007
Chart 3
Standard Deviation of daily returns over a
year
0.021
0.019
0.017
0.015
0.013
0.011
0.009
04/02/2003
04/02/2004
04/02/2005
04/02/2006
2
04/02/2007
I will use the daily rates standard deviation in calculating annual volatility resulting 24.60%.
2
𝐴𝑛𝑛𝑢𝑎𝑙 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 = 1.5231% ∗ √(1303⁄5)
It is hard to expect this happen to be true in the future as the movement in the prices will depend
on many prospective up comings in the market. But in short term the historic data can supply a
sound base to make calculation as it is depending on relatively high number of data compared
with our projection.
By using binomial option pricing model I have drawn a lattice chart for predicting the movement
of the index for 12 months from the end of Jan 2008. To draw the lattice I used the following
formulas in order to find the ratios that the index can go up or down for the next month:
2
𝑢 = exp(𝜎 √𝜏) − 1
2
𝑑 = exp(−𝜎 √𝜏) − 1
τ = 1/12; as I will calculate the movements month by month
σ = annual volatility of the index
Resulting;
u = 7.36%: expected rise in index for next month
d = -6.85%: expected fall in index for next month
I will use LIBOR rate as risk free rate (r). I acquired GBP LIBOR rates from British Bankers’
Association web site from historic LIBOR rates. It is 5.36375% at 31st of Jan for 12 months.
Figure 1 shows the lattice calculated by given variables. I have chosen monthly period lengths to
obtain sensitivity and convenience in my calculations. As period length gets smaller the
calculations are getting more precise but adopting smaller period will make calculations harder.
Also getting a smaller period doesn’t effect calculations so much while we take into account
other assumptions we made.
3
By using u, d and r I have calculated p, and 1-p which will give the probability for movements of
index which is denoted by p and 1-p. ‘p’ shows the probability of a price increase for coming
month and ‘1-p’ show the probability of a decrease. Calculated as:
𝑝=
𝑟−𝑑
𝑢−𝑑
p = 0.514562872: Probability of an increase in price for next month
1-p = 0.485437128: Probability of a decrease in price for next month
Figure 1
Lattice
Month11
Month10
984.39
Month9
916.92
Month8
854.07
854.07
Month7
795.53
795.53
Month6
741.00
741.00
741.00
Month5
690.21
690.21
690.21
Month4
642.90
642.90
642.90
642.90
Month3
598.83
598.83
598.83
598.83
Month2
557.79
557.79
557.79
557.79
557.79
Month1
519.55
519.55
519.55
519.55
519.55
483.94
483.94
483.94
483.94
483.94
483.94
450.77
450.77
450.77
450.77
450.77
450.77
419.87
419.87
419.87
419.87
419.87
419.87
391.09
391.09
391.09
391.09
391.09
364.29
364.29
364.29
364.29
364.29
339.32
339.32
339.32
339.32
316.06
316.06
316.06
316.06
294.39
294.39
294.39
274.22
274.22
274.22
255.42
255.42
237.91
237.91
221.60
206.42
Month12
1056.83
916.92
795.53
690.21
598.83
519.55
450.77
391.09
339.32
294.39
255.42
221.60
192.27
During the calculations, another assumption is made in risk free rate. Risk free rate assumed to
be constant for the following year and the value of the options are deducted by this ratio, which
is subject to change over time. When compared with the index it is not subject to a substantial
change as it is expected to be in band, so it is not a big assumption, but still it affects the option
prices through calculations. It affects probability calculations and it affects the pricing also in
discounting the value of the option. Especially long term options are more effected from risk free
rate.
4
After constructing the lattice and the probabilities, I have calculated several call and put option
prices for 6 and 12 month maturities. I also tested my European options by Black and Scholes
options pricing model showed in Figure 2 and Figure 3.
Figure 2 Options and prices’ with a maturity of 6 months:
European Call
European Put
American Call
American Put
Exercise Binomial
B&S
Exercise Binomial
B&S
Exercise Binomial Exercise Binomial
350
111.83
111.81
350
1.57
1.53
350
111.83
350
1.62
400
69.78
69.70
400
8.17
8.06
400
69.78
400
8.57
450
36.55
37.72
450
23.58
24.72
450
36.55
450
25.24
500
18.25
17.70
500
53.92
53.34
500
18.25
500
56.94
550
7.52
7.30
550
91.84
91.58
550
7.52
550
99.23
600
1.63
2.69
600
134.59
135.62
600
1.63
600
149.23
650
0.73
0.91
650
182.33
182.47
650
0.73
650
199.23
Figure 3 Options and prices’ with a maturity of 12 months:
European Call
European Put
Exercise Binomial
B&S
Exercise Binomial
300
167.77
167.87
300
0.95
400
86.47
86.43
400
14.3
500
35.25
34.88
500
57.74
600
10.5
11.58
600
127.63
700
2.75
3.37
700
214.53
800
0.52
0.90
800
306.95
900
0.11
0.23
900
401.2
American Call
American Put
B&S
Exercise Binomial Exercise Binomial
1.02
300
167.77
300
1.00
14.22
400
86.47
400
15.32
57.31
500
35.25
500
63.43
128.65
600
10.5
600
149.23
215.07
700
2.75
700
249.23
307.24
800
0.52
800
349.23
401.21
900
0.11
900
449.23
Constructing the binomial tree and discounting them is a detailed and confusing procedure
especially if you have lots of branches. Thus it is hard to decrease period length. Assumptions
have great importance and crude estimates don’t shed enough light for the future. Pricing for
longer maturities therefore is not much reassuring. Binomial and Black-Scholes supports each
other by giving near to same results.
5