binomial option pricing

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FEW2355

Seminar Risk ManagementECB 2

BINOMIAL OPTION PRICING

Friday February 2, 2007



Table of contents Page

1 Introduction 2

2 Data and methodology 2

2.1 Data 2

2.2 Methodology 2

3 Results 5

4 Summary and conclusions 8

List of references 9

Appendices

A.1 Binomial tree for the European call option with N = 12 10




B.1 Binomial tree for the American call option with N = 12 21

B.2 Binomial tree for the American call option with N = 24 22

B.3 Binomial tree for the American call option with N = 36 24B.4 Binomial tree for the American call option with N = 48 28



1. Introduction

In this paper we apply the binomial model to calculate the price of a hypothetical European

call option with one year to maturity. The model's accuracy is evaluated by comparing its

outcomes with the Black-Scholes model. In addition, the value of an American call option

with identical specifications is calculated, using the control variate technique to adjust for the

approximation error of the binomial method. With regard to the American option, we also

compute the critical value of the index at the end of each month.

We find that the binomial model option price converges to the Black-Scholes price in

an oscillatory pattern as we increase the number of intervals. For each binomial tree, the value

of the American option is higher which we would expect since American options can be

exercised early. The critical value of the option decreases as the option nears expiration.

The structure of this paper is as follows. In section 2, we provide all the relevant data

and methodologies. Section 3 presents the results of our analysis. Section 4 gives a brief

summary and our conclusions.

2. Data and methodology

2.1 Data

The data used for the calculations of the option prices are all hypothetical. We consider a

European and an American call option on a stock index. This index has a current price of 110

and a volatility of 30% per annum. The continuously compounded dividend yield is 10% per

annum. Both call options have an exercise price of €90 and a remaining maturity of 1 year.

The continuously compounded risk-free interest rate is 9% per annum for all maturities.

2 2 Methodology



(2.1) )()( 210 d N Ked N eScrT qT −−

−=

(2.2)T

T qr K

S

d σ

σ )2

()ln(2

0

1

+−+

=

(2.3)T

T qr K

S

d σ

σ )2

()ln(2

0

2

−−+

=

Secondly, the binomial model is used to value the European call option. The value of

the option today depends on the probability distribution of its pay-offs at expiration. The

binomial model can therefore be regarded as a method to calculate the expected future pay-off

discounted back to the present. The value of the call option f at expiration date T is therefore

equal to its pay-off and depends directly on the underlying stock index price at that moment.

(2.4) )0,max( K S f T T −=

The binomial model is a discrete-time model, which means that the time to expiration

of the option is split up into a specific number of intervals, N . To create a stock index price

distribution, the model assumes that the underlying asset’s price will either increase with a

factor u or decrease with a factor d during each interval. The two possible stock index price

movements are the same for each interval and are defined as follows.

(2.5) N T eu

/ σ

=



these possible future stock index prices, we have corresponding call option prices given by

equation (2.4). Using backward induction, the option value f E at the beginning of each interval

after i up movements and j down movements is calculated as

(2.7) ])1([ 1,,1

/

,

E

ji

E

ji

N rT E

ji f p pf e f ++

−

−+=

where p can be regarded as the risk-neutral probability that an up movement will occur in the

stock index price1.

(2.8)d u

d e p

N T qr

−

−=

− / )(

Our analysis is presented in so-called binomial trees, displaying the possible stock

index price paths and corresponding call option prices. We analyse the binomial model for

, , , and6= N 12= N 24= N 36= N 48= N . These trees are produced by using a spreadsheet.

To analyse the effects of an increasing number of intervals, we used DerivaGem.

We also want to value an American call option with the same specifications as theEuropean call option. Including the possibility of early exercise, the value of the American

call option in the binomial tree at each moment t is calculated by using backward induction,

starting from (2.4) and using (2.9) in this process.

(2.9) }],)1([max{ 1,,1

/

, k S f p pf e f t

A

ji

A

ji

N rT A

ji −−+= ++

−

Since the Black-Scholes model cannot work and the binomial model only gives an

approximation of a continuous-time model, we need to adjust the outcome of the binomial



With respect to our analysis of the American option, we also compute the critical value

of the index now and at the end of each month until the maturity date. The critical value at

time t is the lowest index value for which the call option will be exercised early. We do this

by using the binomial tree with 48 intervals.

3. Results

The relevant outcomes of all intermediate calculations are shown in Table 1. The value of the

European call option is calculated using the Black-Scholes model. This results in the value of

21.571.

Table 1 Intermediate calculations for the option prices calculations

The call option prices are calculated for a hypothetical European and American call option on a stock index with

K = 90, S0 = 110, T = 1, σ = 0.30 p/a, q = 0.10 p/a and r = 0.09 p/a.

Variable c d 1 N(d 1) d 2 N(d 2) u d p

Value 21.571 0.7856 0.7839 0.4856 0.6864 1.1303 0.8847 0.4626

Next we use the binomial model. Using a different number of intervals leads to different

results as can be seen from Table 2. This is also true for the calculated value of the American

call option, as is shown in the same table. The value of the American option using the control

variate technique is 22.779 using the binomial model with 48 intervals. As should be

expected, we find that the American option has a higher value than its European equivalent,

irrespective of the number of intervals.

Table 2 Call option prices with the binomial method



Figure 1 Binomial tree for the European call option with N = 6

The binomial prices are calculated for a hypothetical European call option on a stock index with K = 90,

S0 = 110, T = 1, σ = 0.30 p/a, q = 0.10 p/a and r = 0.09 p/a. At each node, the upper value represents a possible

spot price St of the stock index at t while the lower value represents the corresponding call option value.

0 1 2 3 4 5 6

229.37

139.37

202.93

110.91

179.54 179.54

86.31 89.54

158.84 158.84

65.05 67.56

140.53 140.53 140.53

47.02 48.58 50.53

124.33 124.33 124.3332.53 32.81 33.62

110.00 110.00 110.00 110.00

21.62 20.98 20.15 20.00

97.32 97.32 97.32

12.84 11.38 9.12

86.10 86.10 86.10

6.19 4.15 0.00

76.18 76.181.89 0.00

67.40 67.40

0.00 0.00

59.63

0.00

52.75

0.00

Figure 2 shows the binomial tree using six intervals ( 6= N ) for the American call option.

Red option values indicate it is optimal to exercise the American option. The trees of 12, 24,



Figure 2 Binomial tree for the American call option with N = 6

The binomial prices are calculated for a hypothetical American call option on a stock index with K = 90,

S0 = 110, T = 1, σ = 0.30 p/a, q = 0.10 p/a and r = 0.09 p/a. At each node, the upper value represents a possible

spot price St of the stock index at t while the lower value represents the corresponding call option value. A red

option value indicates exercise at that node.

t 0 1 2 3 4 5 6

229.37

139.37

202.93112.93

179.54 179.54

89.54 89.54

158.84 158.84

68.84 68.84

140.53 140.53 140.53

50.53 50.53 50.53

124.33 124.33 124.3334.54 34.33 34.33

110.00 110.00 110.00 110.00

22.74 21.75 20.47 20.00

97.32 97.32 97.32

13.22 11.53 9.12

86.10 86.10 86.10

6.26 4.15 0.00

76.18 76.181.89 0.00

67.40 67.40

0.00 0.00

59.63

0.00

52.75

0.00

The critical values are calculated using the binomial tree with 48 intervals. At the beginning

of the first and the second month it is not optimal to exercise. The following months it can be



Increasing the number of time steps results in oscillatory conversion of the binomial price to

the Black-Scholes price, as can be seen from Figure 3.

Figure 3 Binomial price versus the number of steps and the Black-

Scholes price

21

21.2

21.4

21.6

21.8

22

1 12 23 34 45 56 67 78 89 100 111 122 133 144

N

P r i c e

Binomial Black-Scholes

4. Summary and conclusions

We can conclude that the binominal price of a call option converges to the Black & Scholes

price in an oscillatory pattern. The critical value of the American call option decreases as

expiration nears.



List of references

Cox, J.C., S.A. Ross and M. Rubinstein (1979). "Option Pricing: A Simplified Approach",

Journal of Financial Economics, 7, 229-264.

Hull, J.C. (2005). "Fundamentals of Futures and Options Markets", 5th edition, Pearson

Prentice Hall, Upper Saddle River, New Jersey.

Hull, J.C. and A. White (1988). "The Use of the Control Variate Technique in Option

Pricing", Journal of Financial and Quantitative Analysis, 23, 237-251.



Appendix A.1 Binomial tree for the European call option with N = 12

t = 0 1 2 3 4 5 6 7 8 9 10 11 12

310,97

220,97

285,18193,48

261,52 261,52

168,54 171,52

239,82 239,82145,90 148,51

219,93 219,93 219,93125,38 127,63 129,93

201,68 201,68 201,68

106,76 108,71 110,68184,95 184,95 184,95 184,95

89,89 91,55 93,23 94,95

169,61 169,61 169,61 169,6174,60 76,00 77,42 78,87

155,54 155,54 155,54 155,54 155,54

60,82 61,91 63,10 64,31 65,54142,63 142,63 142,63 142,63 142,63

48,56 49,29 50,13 51,12 52,12

130,80 130,80 130,80 130,80 130,80 130,80

37,91 38,23 38,63 39,17 39,98 40,80119,95 119,95 119,95 119,95 119,95 119,95

28,91 28,87 28,84 28,84 28,99 29,63110,00 110,00 110,00 110,00 110,00 110,00 110,00

21,54 21,23 20,86 20,44 19,96 19,52 20,00

100,87 100,87 100,87 100,87 100,87 100,8715,21 14,65 13,98 13,17 12,13 10,71

92,51 92,51 92,51 92,51 92,51 92,51

10,02 9,28 8,38 7,24 5,65 2,5184,83 84,83 84,83 84,83 84,83

6,00 5,18 4,20 2,94 1,18

77,79 77,79 77,79 77,79 77,793,13 2,38 1,52 0,55 0,00

71,34 71,34 71,34 71,34

1,33 0,78 0,26 0,00

65,42 65,42 65,42 65,420,40 0,12 0,00 0,00

60,00 60,00 60,000,06 0,00 0,00

55,02 55,02 55,020,00 0,00 0,00

50,45 50,45

0,00 0,00

46,27 46,270,00 0,00

42,430,00

38,91

0,00



21

Appendix B.1 Binomial tree for the American call option with N = 12

310,97220,97

285,18

195,18261,52 261,52

171,52 171,52

239,82 239,82

149,82 149,82219,93 219,93 219,93

129,93 129,93 129,93201,68 201,68 201,68

111,68 111,68 111,68

184,95 184,95 184,95 184,9594,95 94,95 94,95 94,95

169,61 169,61 169,61 169,61

79,61 79,61 79,61 79,61155,54 155,54 155,54 155,54 155,54

65,54 65,54 65,54 65,54 65,54

142,63 142,63 142,63 142,63 142,6352,63 52,63 52,63 52,63 52,63

130,80 130,80 130,80 130,80 130,80 130,80

40,80 40,80 40,80 40,80 40,80 40,80

119,95 119,95 119,95 119,95 119,95 119,9530,83 30,54 30,22 29,95 29,95 29,95

110,00 110,00 110,00 110,00 110,00 110,00 110,00

22,80 22,30 21,74 21,14 20,55 20,00 20,00

100,87 100,87 100,87 100,87 100,87 100,87

15,90 15,21 14,42 0,00 13,52 12,39 10,8792,51 92,51 92,51 92,51 92,51 92,51

10,37 9,55 8,59 7,38 5,73 2,51

84,83 84,83 84,83 84,83 84,836,16 5,30 4,27 2,98 1,18

77,79 77,79 77,79 77,79 77,79

3,19 2,42 1,54 0,55 0,0071,34 71,34 71,34 71,34

1,35 0,79 0,26 0,00

65,42 65,42 65,42 65,42

0,40 0,12 0,00 0,0060,00 60,00 60,00

0,06 0,00 0,0055,02 55,02 55,02

0,00 0,00 0,0050,45 50,45

0,00 0,00

46,27 46,27

0,00 0,0042,43

0,0038,91

0,00

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