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

ISBN 047-43001-3

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Certainty Level is 90.00%

Certainty Range is from 20.90 to 46.67

Display Range is from 11.38 to 53.98

Entire Range is from 10.12 to 64.15

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

Statistics: Value

Trials 1000

Mean 32.68

Median 31.69

Mode —

Standard Deviation 8.04

Variance 64.66

Skewness 0.53

Kurtosis 3.48

Coeff. of Variability 0.25

Range Minimum 10.12

Range Maximum 64.15

Range Width 54.03

Mean Std. Error 0.25

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1138 2203 32S8 *333 5398

Ccaamo 1*90009, tom 208910*6.70

Forecast: American Binomial Call (100 Steps) (cont’d)

Percentiles:

Percentile Value

0% 10.12

10% 22.96

20% 25.94

30% 28.04

40% 29.97

50% 31.69

60% 33.96

70% 36.30

80% 38.88

90% 43.57

100% 64.15

187

Forecast: American Binomial Call (500 Steps) Summary:

Certainty Level is 90.00%

Certainty Range is from 20.89 to 46.67

Display Range is from 11.38 to 53.97

Entire Range is from 10.17 to 64.10

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

Statistics: Value

Trials 1000

Mean 32.67

Median 31.68

Mode —

Standard Deviation 8.04

Variance 64.85

Skewness 0.53

Kurtosis 3.48

Coeff. of Variability 0.25

Range Minimum 10.17

Range Maximum 64.10

Range Width 53.93

Mean Std. Error 0.25

ro»cjet Amftu Dlnom HI 0*11(5® St «pH tOMTtkfc FrwjuenqfClwt 12 Out H*

1138 2102 3267 *332 S3S7

Cessna 1*90 00*3 tom 203710 *6.6*

Forecast: American Binomial Call (500 Steps) (cont’d)

Percentiles:

Percentile Value

0% 10.17

10% 22.95

20% 25.94

30% 28.03

40% 29.95

50% 31.68

60% 33.98

70% 36.31

80% 38.91

90% 43.56

100% 64.10

188

Nonrecombining Lattices

189

NONRECOMBINING LATTICES

This graphic illustrates a 5-step nonrecombining lattice for solving an American call option. Each node branches into two pathways that do not meet with other branches along the way (i.e., they do not recombine). The lattice shown here is the first lattice of the underlying asset.

Underlying Asset Lattice

222.6

149.2

100.0

100.0

67.0

Assumptions:

Asset = $100 Cost = $80 Maturity = 5 Years Risk-free Rate = 5% Volatility = 40% Steps = 5

44.9

332.0

149.2

149.2

67.0

149.2

67.0

67.0

30.1

190

REAL OPTIONS BUSINESS CASES

The lattice shown next is the valuation lattice of the American call option, obtained using the backward-induction approach and applying a riskneutral probability analysis.

Nonrecombining Lattices

191

The problem also can be solved using a recombining lattice as shown here. Notice the similar values along the nonrecombining and recombining lattices. In the recombining lattice, the amount of computational work is significantly reduced because identical values for a particular time period are collapsed and summarized as unique nodes.

Underlying Asset Lattice

738.9

495.3

332.0

332.0

222.6

222.6

100.0

149.2

67.0

100.0

149.2

67.0

100.0

149.2

67.0

Assumptions:

Asset = $100 Cost = $80 Maturity = 5 Years Risk-free Rate = 5% Volatility = 40% Steps = 5

Valuation Lattice

44.9

30.1

44.9

20.2

30.1

' 13.5 658.9

419.2

259.6

252.0

155.4

146.5

50.8

90.1

21.7

42.2

80.2

13.5

30.5

69.2

0.0

5.9

0.0

Intermediate Calculations:

Up Jump-Size = 1.4918 Down Jump-Size = 0.6703 Risk-Neutral Probability = 0.4637 Steps = 5

0.0

0.0

0.0

0.0

192

REAL OPTIONS BUSINESS CASES

Notice the similar results obtained using the recombining and nonrecombining lattice approaches.

738.9 (Frequency: 1)

495.3

332.0

332.0 (Frequency: 5)

222.6

222.6

100.0

149.2

67.0

100.0

149.2

67.0

100.0

149.2 (Frequency: 10)

67.0 (Frequency: 10)

44.9

44.9

30.1

30.1 (Frequency: 5)

20.2

13.5 (Frequency: 1)

Nonrecombining Lattices

193

However, there is a caveat in comparing the recombining and nonrecombining lattices. For instance, the six terminal nodes on a recombining tree are unique occurrences and a summary of the 32 terminal nodes on the nonrecombining tree. Therefore, it is incorrect to assume that there is a one-sixth probability of occurrence for each of the values 738, 332, 149, 67, 30, and 13. In reality, the distribution of the terminal nodes looks somewhat normal, with different outcome probabilities as seen in the chart. Depending on the input parameters, the distribution of the terminal nodes may change slightly (higher volatility means a higher frequency of occurrence in the extreme values).

Frequency

*

738.9 332 149.2 67 30.1 13.5

Future Value

194

REAL OPTIONS BUSINESS CASES

Although recombining lattices are easier to calculate and arrive at identical answers to nonrecombining lattices, there are conditions when nonrecombining lattices are required for the analysis. These conditions include when there are multiple sources of uncertainty or when volatility changes over time, as in the next example.

Nonrecombining Lattices

195

The next valuation lattice is on an American call option with changing volatilities using the risk-neutral probability approach.

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