# Trading real options analysis course - business cases and software applic - Mun P.D.

ISBN 047-43001-3

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Correlation between Assets 0.00 0.00 0.00 0.00 0.00 0.00

Cost Multiplier 0.00 0.10 0.20 0.30 0.40 0.50

Time to Maturity 1.00 1.00 1.00 1.00 1.00 1.00

Risk-free Rate 0% 0% 0% 0% 0% 0%

Portfolio Volatility 0.14 0.14 0.14 0.14 0.14 0.14

Switching Option Value 5.64 2.21 0.72 0.20 0.05 0.01

Static NPV 0.00 -10.00 -20.00 -30.00 -40.00 -50.00

PANEL E: The longer the ability to switch, the higher the value of the ability to switch technology

PV First Asset 100.00 100.00 100.00 100.00 100.00 100.00

PV Second Asset 100.00 100.00 100.00 100.00 100.00 100.00

First Asset Volatility 10% 10% 10% 10% 10% 10%

Second Asset Volatility 10% 10% 10% 10% 10% 10%

Correlation between Assets 0.00 0.00 0.00 0.00 0.00 0.00

Cost Multiplier 0.00 0.00 0.00 0.00 0.00 0.00

Time to Maturity 1.00 2.00 3.00 4.00 5.00 6.00

Risk-free Rate 0% 0% 0% 0% 0% 0%

Portfolio Volatility 0.14 0.14 0.14 0.14 0.14 0.14

Switching Option Value 5.64 7.97 9.75 11.25 12.56 13.75

Static NPV 0.00 0.00 0.00 0.00 0.00 0.00

Closed-Form Equations and Binomial Convergence

183

CLOSED-FORM EQUATIONS AND BINOMIAL CONVERGENCE

A very interesting stability and convergence test can be performed on binomial lattices. In calculating the value of a European call option, the Black-Scholes model can be used. In addition, a binomial lattice also can be used to estimate the option value. Because the binomial lattice in the limit approaches the closed-form Black-Scholes model for a simple European call option, the higher the number of steps, the closer the results. In addition, because a binomial lattice is simply a discrete simulation of closed-form continuous models, how consistent, reliable, and stable are the results? This is a valid question because simulations do not always provide exact results as, by definition, simulations provide randomly assigned inputs and the outputs that may change from trial to trial.

To answer the question of stability, the Black-Scholes model is used, together with binomial lattices using 5, 50, 100, and 500 steps. Each model is then simulated 10,000 times using Crystal Ball Monte Carlo simulation software. Table 6.6 shows that the binomial lattice is indeed highly consistent, reliable, and stable, and that results from a binomial lattice are replicable.

TABLE 6.6 Convergence and Stability of Binomial Results

Mean Standard Deviation 10th Percentile 90th Percentile

Black-Scholes 32.41 7.89 22.86 43.17

5-Step Lattice 33.47 8.15 23.54 44.30

50-Step Lattice 32.68 8.04 23.02 43.53

100-Step Lattice 32.68 8.04 22.96 43.57

500-Step Lattice 32.67 8.04 22.95 43.56

Forecast: Black-Scholes Model with Dividends

Summary:

Certainty Level is 90.00%

Certainty Range is from 20.82 to 46.16

Display Range is from 13.33 to 53.12

Entire Range is from 10.15 to 63.04

After 1,000 Trials, the Std. Error of the Mean is 0.25

Statistics: Value

Trials 1000

Mean 32.41

Median 31.48

Mode —

Standard Deviation 7.89

Variance 62.27

Skewness 0.51

Kurtosis 3.46

Coeff. of Variability 0.24

Range Minimum 10.15

Range Maximum 63.04

Range Width 52.89

Mean Std. Error 0.25

Forecast: Black-Scholes Model (cont’d)

Percentiles:

Percentile

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Value

10.15

22.86

25.81

27.88

29.74

31.48

33.66

35.97

38.56

43.17

63.04

184

Forecast: American Binomial Call (5 Steps)

Summary:

Certainty Level is 90.00%

Certainty Range is from 21.19 to 47.59

Display Range is from 12.42 to 54.86

Entire Range is from 9.99 to 64.05

After 1,000 Trials, the Std. Error of the Mean is 0.26

Statistics: Value

Trials 1000

Mean 33.47

Median 32.65

Mode —

Standard Deviation 8.15

Variance 66.48

Skewness 0.40

Kurtosis 3.33

Coeff. of Variability 0.24

Range Minimum 9.99

Range Maximum 64.05

Range Width 54.06

Mean Std. Error 0.26

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1142 22 03 331* *425 5486

Ceeanr/1*3050*3 tom 21.1710 *765

Forecast: American Binomial Call (cont’d)

Percentiles:

Percentile

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Value

9.99

23.54

26.62

28.89

30.88 32.65 34.95

37.29

39.89

44.30 64.05

185

Forecast: American Binomial Call (50 Steps)

Summary:

Certainty Level is 90.00%

Certainty Range is from 20.97 to 46.75

Display Range is from 11.38 to 53.99

Entire Range is from 10.14 to 63.98

After 1,000 Trials, the Std. Error of the Mean is 0.25

Statistics: Value

Trials 1000

Mean 32.68

Median 31.67

Mode —

Standard Deviation 8.04

Variance 64.68

Skewness 0.53

Kurtosis 3.48

Coeff. of Variability 0.25

Range Minimum 10.14

Range Maximum 63.98

Range Width 53.84

Mean Std. Error 0.25

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MM 2104 1269 4114 5199

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Forecast: American Binomial Call (50 Steps) (cont’d)

Percentiles:

Percentile Value

0% 10.14

10% 23.02

20% 25.93

30% 28.06

40% 29.97

50% 31.67

60% 33.96

70% 36.26

80% 38.87

90% 43.53

100% 63.98

186

Forecast: American Binomial Call (100 Steps) Summary:

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