Optimum Technology Product Life Cycle Technology Innovation Investment-Using Compound Binomial Options

The study considers the product life cycle in the stages of technological innovation, and focuses on how to evaluate the optimal investment strategy and the project value. It applies different product stages (three stages including production innovation, manufacture innovation, and business innovation) factors to different risks to build a technology innovation strategy model. This study of option premiums aims for the best strategy timing for each innovation stage. It shows that the variation of business cycle will affect the purchasing power under the uncertainty of Gross Domestic Product (GDP). In application, the compound binomial options for the manufacture innovation will only be considered after the execution of the production innovation, whereas the operation innovation will only be considered after the execution of the manufacture innovation. Thus, this paper constructs the dynamic investment sequential decision model, assesses the feasibility of an investment strategy, and makes a decision on the appropriate project value and option premiums for each stage under the possible change of GDP. Numerically, the result shows the equity value of the investment is greater than 0. Therefore, this paper recommends the case firm to invest in its innovation project known as one-time passwords. Sensitivity analysis shows when the risk-adjusted discounted rate r increases, the risk of the investment market increases accordingly, hence the equity value must also be higher in order to attract the case firm’s investment interest. Also, the average GDP growth rate u sensitivity analysis results in different phenomena. The equity value gradually decreases when the average GDP growth rate rises. When the average GDP growth rate u rises to a certain extent, however, its equity value is gradually growing. The study investigates the product life cycle innovation investment topic by using the compound binomial options method and therefore provide a more flexible strategy decision compared with other trend forecast criteria.


Introduction
In view of the short life cycles of technology products, businesses must focus on speed, innovation, and the extent of customer acceptance.Firms must have the ability to quickly respond to customer preferences and required changes in order to increase market share.De Kluyver (1977) proposed that innovation could extend product life cycles.So, product innovation was in fact an important part of extending product life cycles.In general, when assessing investment strategies, firms not only had to consider their own resources, but also had to consider the uncertainties of the investment risks of the external environment, emerging competitors, and economic cycles.Some studies confirm that product innovation has a positive impact on business performance.Lou et al. (2010) used a statistical regression model to study the impact of technology innovation on market value.Its empirical results indicated that technology innovation and market value had a positive correlation.Therefore, this paper utilizes the opportunity of assessing technology product innovation investment to offer references to decision makers in the presence of competitive behaviors for determining when to assume the leadership advantage, dominate the market, force competitors to drop out, and meanwhile consider how to avoid risks.
The paper applied the compound binomial options model to exploring innovation investment strategies of high-tech products.Three stages of technological innovation (product innovation, manufacturing innovation, and business innovation) are considered product life cycle characteristics.To determine whether to invest, the compound binomial options model to evaluate the value of an investment project, as well as the management of the elasticity of value, are utilized at each decision point.The high positive correlation between consumers purchasing technology products and Gross Domestic Product (GDP) is also taken into consideration.The sequence investment program is to complete the investment in the first stage of the introduction period (product innovation), and then have the opportunity to invest in the next stage.After completing the second stage of growth (manufacturing innovation) innovation investment, the program has the opportunity to enter the mature period (business innovation) investment strategy.This is different from other traditional financial assessment methods.The compound binomial options are options whose value is decided by the values of other options (Copeland and Antikarov 2001).While financial risk assessment is one of the important factors to consider when making investment decisions for products in the technology industry, in general, the net present value (NPV) method is the most commonly adopted assessment method used for investment strategies.However, the method is more suitable for a static investment environment because the NPV method lacks the considerations required for a dynamic investment environment and ignores the value of managerial flexibility.The real options approach (ROA), which takes managerial flexibility value into its strategic investment considerations, is a more applicable assessment method for a highly uncertain technology product investment project.In an uncertain and complex investment environment, a manager's investment strategy should include a dynamic strategic analysis model so the ROA is more adaptive than the conventional NPV method in a complex investment environment (Myers 1977;Dixit and Pindyck 1994).Cassimon et al. (2011) observed that the compound real options were permitted for phase-specific volatility estimates.This study reported that phase-specific volatilities can be estimated for project managers.Cheng et al. (2011) used the binomial model approach, which was modified from sequential compound options, to analyze future GDP changes based on past historical data, their resulting impact on electricity demand, and their influences on the development path of clean energy investment strategies.Liu et al. (2018) adopted the pricing of n-fold compound options with barriers.They developed a generalization of the compound barrier option with a stochastic interest rate to capture the interest rate risk.Detemple and Kitapbayev (2018) used the ROA when the investment cost changes with the random jump process.The performance of optimal investment boundaries and valuation components are inspected.Three valuation formulas are derived and all are in closed form.Their proposed binomial model allowed us to explore dynamic changes in various options for the development of small organizations to predict changes in financial results.With the help of the Bayes criterion, the calculation of the optimality of the adopted development strategy is proposed.Cucchiella et al. (2010) research products because of the rapid growth of the market and the rapid change of consumers, considering the introduction period and the growth period of the product life cycle by the real options approach (ROA).It is possible to consider the uncertainties of the consumer's fickle needs in a complex supply chain investment environment and construct a most appropriate investment management model.Koosha and Albadvi (2015) used the real options approach to construct consumer life cycle values.Their research points out that most measures of consumer life cycle value cannot be considered in an uncertain environment.Numerical analysis shows that traditional measurement methods ignore some considerable value, mainly the values of the decisive factors of management decision makers.
Lin (2010) used the ROA to determine if a company required a technology innovation investment project for market entry and market exit, evaluated the decision values and project values of the market exit phase, maintaining the existing technology phase, and adopted the advance technology phase.It also applied the market entry and exit critical values to reflect the need for technology innovation, offering a company the optimal investment strategies.Verdu et al. (2012) showed that the ROA improved the level of product technological innovation.This paper found that this development would increase when the level of uncertainty of the environment is higher.Wang and Yang (2012) used the ROA to assess research and development projects by dividing a drug development process into three stages, and then studying the differences in the risks presented in each stage.The study assesses the value of each stage of the research and development (R&D) process by using the ROA.Lo Nigro et al. (2014) used the ROA in the open innovation field in order to push firms to take on the innovation model.Daming et al. (2014) pointed out that many researchers pay more attention to new technology development modes, which is completely different from the incremental technological innovation of the past.This paper constructed a real option game model for the analysis on game equilibrium that a company will have a strong motivation for implementing radical innovation only when market reaction to its products is moderate.Ball et al. (2015) used the ROA to establish a flexible decision model under agent uncertainty.Morreale et al. (2017) used game options to shape the timing of biopharmaceuticals and biotechnology alliances.Lee and Lin (2018) used the real option approach method to construct a three-stage management decision model.They analyzed the frequency of fires from a dynamic point of view, examined the fire damage data of an international city for ten years, and proposed the best time to purchase fire hazards or install fire safety systems.Morozko et al. (2018) suggested that the method of choice of ROA can be used to make the most reasonable substitution decision.
The arrangement of this article is as follows.Section 2 describes the research hypothesis and uses the compound binomial options to construct technology investment products based on product life cycle characteristics and innovative investment evaluation models.Section 3 numerical example, for the construction of the model in Section 2, analyzes the cost of innovation in each stage of the evaluation of the case company's One-Time Password product.Finally, Section 4 concludes the paper.

The Model
This paper aims to explore the process of deciding whether to adopt technology innovation on a product when facing the uncertainty of environmental changes.During this process, a change in the economic cycle, which is a part of the overall economy, affects consumer purchasing power, while the GDP, which represents future economic growth, is uncertain.The study takes into account the characteristics of product life cycles in the introductory, growth, and maturity stages, and assesses the optimal investment strategies of each stage and project values, as well as the options premiums of the decision-making points.

Assumptions
The paper assumes sustainable development of high-tech products.According to the product life cycle characteristics, the introduction of technological innovation is divided into three stages (product innovation, manufacturing innovation, and business innovation).It is in the early stage of the three stages to assess whether to invest in innovation cost investment decisions.The investment process must be a sequential investment style.The paper also assumes that consumer purchasing power is highly positively correlated with GDP.According to the historical data of GDP, it is possible to analyze the future GDP that may affect the purchasing power of consumers.At the same time, GDP affects the firm's revenue, and in time affects the development path of the firm's investment strategy for product development and innovation technology.Firm's revenue will also increase when GDP grows.Therefore, the paper is based on the trend of GDP changes in the past, which corresponds to the trend of similar firm's revenue.The compound binomial options model is used to assume a perfectly efficient market.The change in the price of the target (GDP or revenue) during the duration of the option is used to evaluate the reasonable value of the option.This study split the selection period into T periods, with only two options for each period.One is to increase the fixed range u, and the probability is p, while the other is to fall a fixed range d, and the probability is 1 − p.The current price of the target (GDP or revenue), after the first period, rises to GDP(revenue) × u or falls to GDP(revenue) × d. u represents the price increase of the target (u ≥ 1), and d the price decrease of the target (0 < d ≤ 1).
The study assumes that the changes in GDP affect the purchase and consumption abilities of its citizens, and in turn impact production and revenues.The study analyzed historical data to find possible future GDP occurrences and determine their impact on consumer purchasing power and their further impact on investment strategy development paths concerning their product development in technology innovation.Assume that t = 0 is the GDP base stage.u and d are defined as the upward and downward multipliers of the variable GDP in the next period, respectively, and u × d = 1 is satisfied.Let s denote the three stages of technology innovation and s = I, II, III denote the first stage, the second stage, and the third stage respectively.Let r denote the risk-adjusted discounted rate (Lin 2009;Lin and Huang 2011), complying with the natural constraints of u > 1 + r > d.The possible growth path of risk-adjusted probability is p = (e n×r − d) ÷ (u − d) (Copeland and Antikarov 2001); the decline path of risk-adjusted probability is 1 − p (0 ≤ p ≤ 1).This paper shows the GDP change path, the relevant path ratio distribution, and the corresponding GDP nodes of each time frame as shown in Figure 1 (Copeland and Antikarov 2001;Cheng et al. 2011 GDP affects the firm's revenue, and in time affects the development path of the firm's investment strategy for product development and innovation technology.Firm's revenue will also increase when GDP grows.Therefore, the paper is based on the trend of GDP changes in the past, which corresponds to the trend of similar firm's revenue.The compound binomial options model is used to assume a perfectly efficient market.The change in the price of the target (GDP or revenue) during the duration of the option is used to evaluate the reasonable value of the option.This study split the selection period into T periods, with only two options for each period.One is to increase the fixed range u, and the probability is p, while the other is to fall a fixed range d, and the probability is 1 p  .The current price of the target (GDP or revenue), after the first period, rises to GDP( ) revenue u  or falls to GDP( ) revenue d  .u represents the price increase of the target ( 1 u  ), and d the price decrease of the target ( 01 d  ).
The study assumes that the changes in GDP affect the purchase and consumption abilities of its citizens, and in turn impact production and revenues.The study analyzed historical data to find possible future GDP occurrences and determine their impact on consumer purchasing power and their further impact on investment strategy development paths concerning their product development in technology innovation.Assume that 0 t  is the GDP base stage.u and d are defined as the upward and downward multipliers of the variable GDP in the next period, respectively, and 1 ud  is satisfied.Let s denote the three stages of technology innovation and s = I, II, III denote the first stage, the second stage, and the third stage respectively.Let r denote the risk-adjusted discounted rate (Lin 2009;Lin and Huang 2011), complying with the natural constraints of In Figure 1 GDP change path, this study assumes GDP i is the GDP of i node, where , , , , , , , , ,  In Figure 1 GDP change path, this study assumes GDP i is the GDP of i node, where i = A, B, C, D, E, F, G, H, I, J and GDP J = GDP 0 is the GDP of J node.Then the GDP of H node is GDP H = u × GDP 0 .In addition, the GDP of In Figure 2 revenue change path, this paper assumes that i SR is the revenue of i node, where , , , , , , , , , is the revenue of J node.Following the GDP variation path in the next step, where the economic growth adjusted-risk probability is p and the average growth rate of each stage is u , the revenue of H node is 0 H

SR u SR 
. In addition, the recession adjusted-risk probability is 1 p  and the average recession rate is d ; therefore, the revenue of I node is . Similarly, the revenue of each node is , and , respectively.That is, there are two possible situations for each node.In one case, the GDP will increase in the case of economic growth, and the corresponding GDP will increase the purchasing power of consumers, and the revenue of firm will also increase.In another case, when the economy is in a recession, the GDP at the node is declining, and the purchasing power of the corresponding consumer is declining, which also results in a decrease in firm revenue.

Compound Binomial Options
This study utilizes the compound binomial options model (Copeland and Antikarov 2001;Cheng et al. 2011) and the three stages of the product life cycle characteristics: introductory, growth, and maturity.For the sustainable operation of the product, each stage requires innovation cost investments: product innovation, manufacturing innovation, and business innovation.According to product life cycle characteristics, the sequence investment program primarily completes the first stage of the introduction period (product innovation) innovation investment, and subsequently moves in the next stage of investment (manufacturing innovation).After completing the second stage of innovation investment, it finally enters the mature period (business innovation) investment strategy.
Due to the long lead time of technology innovation, this paper assumes that the investment strategy in each stage adopts the early assessment investment strategy method.That is, the investment strategy in the E, F, and G nodes of the second stage is to consider whether to invest in business innovation costs, or to abandon the investment project.Similarly, the investment strategy in the H and I nodes of the first stage is to consider whether to invest in manufacturing innovation, while in the stage 0 t  , the decision is to determine whether product innovation is to be continued or abandoned.Suppose  In Figure 2 revenue change path, this paper assumes that SR i is the revenue of i node, where i = A, B, C, D, E, F, G, H, I, J and SR J = SR 0 is the revenue of J node.Following the GDP variation path in the next step, where the economic growth adjusted-risk probability is p and the average growth rate of each stage is u, the revenue of H node is SR H = u × SR 0 .In addition, the recession adjusted-risk probability is 1 − p and the average recession rate is d; therefore, the revenue of I node is respectively.That is, there are two possible situations for each node.In one case, the GDP will increase in the case of economic growth, and the corresponding GDP will increase the purchasing power of consumers, and the revenue of firm will also increase.In another case, when the economy is in a recession, the GDP at the node is declining, and the purchasing power of the corresponding consumer is declining, which also results in a decrease in firm revenue.

Compound Binomial Options
This study utilizes the compound binomial options model (Copeland and Antikarov 2001;Cheng et al. 2011) and the three stages of the product life cycle characteristics: introductory, growth, and maturity.For the sustainable operation of the product, each stage requires innovation cost investments: product innovation, manufacturing innovation, and business innovation.According to product life cycle characteristics, the sequence investment program primarily completes the first stage of the introduction period (product innovation) innovation investment, and subsequently moves in the next stage of investment (manufacturing innovation).After completing the second stage of innovation investment, it finally enters the mature period (business innovation) investment strategy.
Due to the long lead time of technology innovation, this paper assumes that the investment strategy in each stage adopts the early assessment investment strategy method.That is, the investment strategy in the E, F, and G nodes of the second stage is to consider whether to invest in business innovation costs, or to abandon the investment project.Similarly, the investment strategy in the H and I nodes of the first stage is to consider whether to invest in manufacturing innovation, while in the stage t = 0, the decision is to determine whether product innovation is to be continued or abandoned.Suppose C s is the innovation cost of the s stage.At the t = 0 stage, it is to decide whether to invest in the first stage product innovation input cost.At the first stage of the introduction period t = n, it is to decide whether to invest in the second stage of growth and innovation costs.The second stage t = 2n of the growth phase determines the third stage of business innovation investment strategy.
This study assumes that the investment strategy in each stage adopts the early assessment investment strategy method.Therefore, the consideration factors affecting whether to proceed with the next stage of investment at each revenue node of each stage include the decision point, the expected revenue of the previous stage, and the discounted present value of the revenue of the expected invested innovation cost.That is, at E node of the 2nd stage, the revenue at the decision point SR I I E is included in the added total of the expected revenue of the 1st stage for n years and the discounted present value of the expected revenue of the 3rd stage after the business innovation cost is invested (Cheng et al. 2011): From Equation (1), SR I I E denotes the revenue of E node decision point, which is the added total of the discounted present value of A and B nodes revenues of the 3rd stage (p × SR A + (1 − p) × SR B )/e n×r , and the expected revenue SR E .This paper constructs an investment model that incorporates the three stages of innovation strategies of new product development, and uses the early strategic decision method.That is, at the decision point of the 2nd stage, the revenue resulting from the 1st stage strategic options and the present value of the expected revenue of the corresponding expected results of the 3rd stage are considered, which are the results of the adopted strategies at each node of the 2nd stage.
Similarly, SR I I F denotes the revenue of F node decision point as shown in Equation (2) (Cheng et al. 2011): Then, SR I I G denotes the revenue of G node decision point as shown in Equation (3) (Cheng et al. 2011): The revenue of J node decision point is denoted as SR 0 J as shown in Equation ( 6) (Cheng et al. 2011): This paper continues to calculate the equity values of nodes E, F, G, H, I, J. Assuming NP s i is the equity value of i node of s stage, the rule of the decision is that when the equity value is greater than 0, i.e., NP s i > 0, the option is to invest; if the equity value is smaller than 0, i.e., NP s i < 0, the option is to abandon the investment or to wait for the next appropriate investment opportunities.
The option of whether to invest in the business innovation cost is at E node.Its equity value NP I I E equals the decision point revenue of the 2nd stage SR I I E minus the business innovation cost C I I I invested during the 2nd stage.When the value is greater than 0, the plan of investing in the business innovation cost will be implemented; otherwise, the choice will be to abandon the plan or to wait for the next investment opportunity when the equity value equals 0. The E node option of whether to invest in the business innovation cost and its equity value are shown in Equation ( 7): Additionally, the equity value of F node NP I I F equals the decision point revenue of the 2nd stage SR I I F minus the business innovation cost C I I I invested in the 2nd stage.When the value is greater than 0, i.e., NP I I E > 0, the option will be to invest in the business innovation cost; otherwise, the option will be to defer the investment.The F node option of whether to invest in the business innovation cost and the equity value are shown in Equation ( 8): Likewise, G node equity value NP I I G equals the decision point revenue of the 2nd stage SR I I G minus the business innovation cost C I I I invested in the 2nd stage.When the value is greater than 0, i.e., NP I I G > 0, the option will be to invest in the business innovation cost; otherwise, the option will be to defer the investment.The G node option of whether to invest in the business innovation cost and the equity value are shown in Equation ( 9): The equity value of H node NP I H must include the results of the decision point revenue of the 1st stage SR I H minus the invested manufacturing innovation cost C I I of the 1st stage.It means that when SR I H − C I I is greater than 0, it is worth proceeding with the investment.Because the investment decision is correlated to the decision of business innovation investment of the 3rd stage, this consideration must also include the discounted present value of the expected equity return of E and F nodes of the 2nd stage p × NP . If the value is greater than 0, it is one of the consideration factors in the manufacturing innovation cost investment decision, that is, when NP I H > 0, investing in the manufacturing innovation cost will be considered.But if NP I H = 0, the option will be to abandon the investment in the manufacturing innovation cost and wait for another opportunity as shown in Equation ( 10 .If the value is greater than 0, it is one of the consideration factors in the manufacturing innovation cost investment decision, that is, when NP I I > 0, investing in the manufacturing innovation cost will be considered.But if NP I I = 0, the option will be to abandon the investment in the manufacturing innovation cost and wait for another opportunity as shown in Equation (11): The equity value of J node NP 0 J includes the results of expected revenue after the product innovation cost is invested SR 0 J minus the results of whether to invest in the product innovation cost C I in stage 0, i.e., SR 0 J − C I .In addition, this calculation must also include the expected equity value of H and I nodes of the 1st stage p × NP I H + (1 − p) × NP I I e n×r because if the innovation cost is not invested at J node, neither manufacturing innovation nor business innovation is possible.Therefore, the consideration factor in stage 0 includes the decision to invest the profit in innovation costs.If losses are incurred in stage 0, the equity value of manufacturing innovation of the 1st stage is still one of the consideration factors.That is, when NP 0 J > 0, investing in the product innovation cost will be considered.But when NP 0 J = 0, the option is to abandon the investment or to defer to the next investment opportunity as shown in Equation ( 12): The equity value NP s i of nodes E, F, G, H, I, J and the equity values of binomial options are shown in Figure 3:    Figure 3 is the equity values of binomial options.This model employs a compound binomial options approach that is more flexible in strategic thinking than other approach to construct an innovation investment strategy, which is product life cycle specific.Its strategic principle is that when the equity value is greater than 0, i.e., NP s i > 0, the option is to proceed with the investment; if the value equals 0, i.e., NP i = 0, the option is to abandon the investment or to defer to the next appropriate investment opportunity.This model serves as a reference for corporate decision-making in product investment strategies.Section 3 below uses a firm case study, where One-Time Password's investment is studied and simulation analysis is performed, providing a basis for managerial reference when considering the product innovation investment.

Numerical Examples
By using the constructed model in Section 2 as a base, this paper analyzes a case study of a firm with the product, One-Time Password, in an uncertain investment environment scenario, and assesses the innovation cost investment worthiness of each stage.
Based on the historical GDP data from 1995 to 2017, this paper estimates the future GDP changes, and assesses the changes in the future consumer purchasing power, which in turn affects the revenue.That is, the changes in the future revenue are dependent on the GDP change model.This paper assumes when t = 0 of the case firm, the estimated One-Time Password's revenue is NT$18.00(million), i.e., J node revenue is SR J = 18.00 (million).Its relevant parameter assumptions are shown in Table 1:  ).
This paper assumes that the investment strategy for each stage adopts the early assessment investment strategy method.Therefore, the consideration factors affecting whether to proceed with the next phase of investment or to abandon the project at each revenue node of each stage include the decision point, the expected revenue of the previous stage, and the discounted present value of the revenue of the next stage after the innovation cost is invested.Each decision point revenue This paper assumes that the investment strategy for each stage adopts the early assessment investment strategy method.Therefore, the consideration factors affecting whether to proceed with the next phase of investment or to abandon the project at each revenue node of each stage include the decision point, the expected revenue of the previous stage, and the discounted present value of the revenue of the next stage after the innovation cost is invested.Each decision point revenue SR s i is shown in Figure 5: Risks 2018, 6, 98 10 of 14 investment strategy method.Therefore, the consideration factors affecting whether to proceed with the next phase of investment or to abandon the project at each revenue node of each stage include the decision point, the expected revenue of the previous stage, and the discounted present value of the revenue of the next stage after the innovation cost is invested.Each decision point revenue
This paper continues to calculate the equity value of E, F, G, H, I, and J nodes.s i NP denotes the equity value of i node of t period.Its equity value of each node of compound binomial options is shown in Figure 6.This paper continues to calculate the equity value of E, F, G, H, I, and J nodes.NP s i denotes the equity value of i node of t period.Its equity value of each node of compound binomial options is shown in Figure 6.
9.89( ) 16.70( )   Figure 6 shows that C I I I = 50 million dollars of business innovation cost is invested at E node, which means the resulting value derived from the expected revenue of the 2nd stage SR I I E minus the invested business innovation cost of the 2nd stage C I I I is greater than 0 so it is worth investing.But the F and G nodes of the 2nd stage, the respective resulting value derived from SR I I F and SR I I G minus the invested business innovation cost of the 2nd stage C I I I is less than 0 so the option is to wait and not to proceed with the investment.As both equity values of H and I nodes NP I H and NP I I derived from the expected sales revenue of the 1st stage SR I H and SR I I minus the invested manufacturing innovation cost of the 1st stage C I I = 20 million are greater than 0 so the option is to proceed with the investment.
At J node, the product innovation cost is invested and the expected five-year sales revenue minus the considered product innovation cost C I = 20 million dollars is 16.01 million dollars that is greater than 0 so it is worth investing.At the same time, the investment plan considers three stages, the product innovation in the first stage of the introduction period, the manufacturing innovation in the second stage, and the business innovation in the third stage.Therefore, the decision point must also consider 9.89( ) 16.70( ) This paper utilizes a more flexible strategic thinking method, the compound binomial options, to construct the product life cycle innovation investment strategy model, and divides the decisionmaking process into three phases.It is only after the investment of the 1st stage introductory period (product innovation) is made, the option to invest in the second stage growth period (manufacturing innovation) will become available, and then the last stage maturity period (business innovation) will follow.
After a numerical analysis, the results show the equity value of the investment is greater than 0, which means it is worth investing for the case firm.This result is offered to the manager of the case firm as a reference for the product innovation investment.
Based on the changes of risk-adjusted discounted rate r , which affects the revenue growth path risk-adjusted probability p and the decline path risk-adjusted probability 1 p  , this paper analyzes the changes of expected revenue of binomial options and their influence on equity values.The changes are shown in Table 2:  This paper utilizes a more flexible strategic thinking method, the compound binomial options, to construct the product life cycle innovation investment strategy model, and divides the decision-making process into three phases.It is only after the investment of the 1st stage introductory period (product innovation) is made, the option to invest in the second stage growth period (manufacturing innovation) will become available, and then the last stage maturity period (business innovation) will follow.
After a numerical analysis, the results show the equity value of the investment is greater than 0, which means it is worth investing for the case firm.This result is offered to the manager of the case firm as a reference for the product innovation investment.
Based on the changes of risk-adjusted discounted rate r, which affects the revenue growth path risk-adjusted probability p and the decline path risk-adjusted probability 1 − p, this paper analyzes the changes of expected revenue of binomial options and their influence on equity values.The changes are shown in Table 2: As shown in Table 2, when the risk-adjusted discounted rate r rises, the revenue growth path risk-adjusted probability p also rises, which means the probability of future economic growth is higher.When the revenue decline path risk-adjusted probability 1 − p is lower, the future economic recession probability is lower and the equity value of the case firm's project-one-time passwords' technology innovation investment is higher.Because when the risk-adjusted discounted rate r rises, the case firm's investment risk in the project increases; therefore, a higher equity value must be present in order to attract the case firm's investment interest.
Then, we will change the average growth rate of GDP every five years u.Linkage affects the average annual decline rate of GDP every five years d, and for the impact of the rising profit path risk adjustment probability p and the decline path risk adjustment probability 1 − p change situation.Analysis of changes in expected returns of binomial options and their impact on equity values are shown in Table 3.As shown in Table 3, if the average GDP growth rate u rises, at the same time, GDP average decline rate in every five years d is gradually decreasing.The revenue growth path risk-adjusted probability p will be lower.Due to the economic cycle, the probability of prosperity will decline; and revenue decline path risk-adjusted probability 1 − p will increase.That is, the probability of future economic recession is gradually rising.The case firm's project studied the equity value of the one-time passwords' technology innovation investment case, In addition to considering the growth rate of consumption (the average GDP growth rate u), we must also consider the revenue growth path risk-adjusted probability p or revenue decline path risk-adjusted probability 1 − p.When the average GDP growth rate u = 1.10~1.40,its equity value gradually declines.When the average GDP growth rate u = 1.40~1.60,its equity value is gradually growing.At the same time of analysis, when u = 1.10, in the third stage of business innovation, the E, F, G nodes have the option value of 0, NP I I E = 0, NP I I F = 0, NP I I G = 0. Therefore, when u = 1.10, you should choose to abandon the investment in business innovation.

Conclusions
The paper utilizes a more flexible management method, the compound binomial options, to construct an investment strategy model.This paper takes into account the circumstance that confronts the policy maker: the changes of external environment delineated by the uncertainty of GDP growth, which affects consumer purchasing power and in turn affects business revenues.This paper divides the product life cycle based on its characteristics into three stages including introductory, growth, and maturity; and explores the decision-making process of whether to invest in technology innovation in three periods: product innovation in the introductory period, manufacturing innovation in the growth period, and business innovation in the maturity period.The strategic principle of this model shows that only after the investment of the 1st stage introductory period (product innovation) is made, the option to invest in the next stage will become available; while after entering the 2nd stage growth period (manufacturing innovation), the investment of the 3rd stage maturity period (business innovation) will be considered.This paper uses this model to assess the optimal investment strategy of each stage and the project values as well as options premiums of the decision-making points.
The paper applies this model to the technology innovation decision-making process of the case firm's product known as one-time passwords.Based on the product's characteristics, the paper divides its life cycle into three stages: product innovation, manufacturing innovation, and business innovation.Various stages beget various scenarios that require various policy models.After applying a numerical analysis, the result shows that the equity value of the investment is greater than 0. Thus, this paper recommends the case firm to invest in one-time passwords innovation project.This result differs from the net present value (NPV) method, originally used by the firm, which assumes a static investment environment and only considers continuous economic growth.The assessment is perhaps unduly optimistic.Whereas, this paper constructs an investment model that considers the uncertainty of the investment environment and takes into account future economic growth and decline.This paper also concludes that when the risk-adjusted discounted rate increases, the risk of the investment market increases, and that the equity value must also be higher in order to attract the case firm's investment interest.Then, if the average GDP growth rate u rises, at the same time, GDP average decline rate in every five years d is gradually decreasing.The revenue growth path risk-adjusted probability p will be lower.Due to the economic cycle, the probability of prosperity will decline; and revenue decline path risk-adjusted probability 1 − p will increase.Different phenomena will occur.When the average GDP growth rate u rises, the equity value gradually decreases.However, when the average GDP growth rat u rises to a certain extent, its equity value is gradually growing.This result is offered to the manager of the case firm as a reference for the product innovation investment.
This model is designed specifically for the technology products with short life cycles, which require constant technology innovation investment to extend product life cycles in an uncertain market.The model utilizes the compound binomial options assessment model, which has management flexibility and includes an investment that is decision relevant and has options to choose in each phase.Using this model to assess the innovation investment issues of each phase of the product life cycle offers more flexibility in decision-making than other trend forecasting standards.The results can also be used as a reference for business policy makers in extending product life cycles.

(
Copeland and Antikarov 2001); the decline path of risk-adjusted probability is 1 p  ( 01 p  ).This paper shows the GDP change path, the relevant path ratio distribution, and the corresponding GDP nodes of each time frame as shown in Figure 1(Copeland and Antikarov 2001;Cheng et al. 2011): The expected future revenue of the product corresponding to each node of Figure 1 is shown in Figure 2 revenue change path (Copeland and Antikarov 2001; Cheng et al. 2011):
and GDP D = d 3 × GDP 0 , respectively.The expected future revenue of the product corresponding to each node of Figure1is shown in Figure2revenue change path(Copeland and Antikarov 2001;Cheng et al. 2011):

sC
is the innovation cost of the s stage.At the 0 t  stage, it is to decide whether to invest in the first stage product innovation input cost.At the first stage of the introduction period tn  , it is to decide whether to invest in the second stage of growth and innovation costs.The second stage 2 tn  of the growth phase determines the third stage of business innovation investment strategy.
Then, at H node in the 1st stage, the consideration factors, when facing the decision of investing in the manufacturing innovation cost, include the discounted present value of the expected revenues of E and F nodes of the 2nd stage p × SR I I E + (1 − p) × SR I I F /e n×r , and the expected revenue SR H(Cheng et al. 2011): revenue of I node decision point as shown in Equation (5)(Cheng et al. 2011): equity value of I node NP I I must include the results of the decision point revenue of the 1st stage SR I I minus the invested manufacturing innovation cost C I I of the 1st stage.It means that when SR I I − C I I is greater than 0, it is worth proceeding with the investment.Because the investment decision is correlated to the decision of business innovation investment of the 3rd stage, this consideration must also include the expected equity values of F and G nodes of the 2nd stage p × NP I I F + (1 − p) × NP I I G e n×r of nodes E, F, G, H, I, J and the equity values of binomial options are shown in Figure 3:

Figure 3 .
Figure 3. Equity values of binomial options.Figure 3. Equity values of binomial options.

Figure 3 .
Figure 3. Equity values of binomial options.Figure 3. Equity values of binomial options.

Figure 5 .
Figure 5. Revenue SR s i value of each node of compound binomial options (r = 0.68% ).

Figure
Figure6shows that
the expected equity of the first stage node H and I node p × NP I H + (1 − p) × NP I I e n×r = 16.70 million dollars.So, take a larger value according to Equation (12) NP 0 J = 16.70 million dollars.Its compound options values are shown in Figure 7.

Author
Contributions: Conceptualization, C.-C.K., T.T.L., F.-M.Z. and C.-Y.L.; Methodology, C.-C.K., T.T.L., F.-M.Z. and C.-Y.L.; Validation, C.-C.K. and T.T.L.; Formal Analysis, C.-C.K., T.T.L., F.-M.Z. and C.-Y.L.; Resources, F.-M.Z.; Data Curation, C.-C.K., T.T.L., F.-M.Z. and C.-Y.L.; Writing-Original Draft Preparation, C.-C.K., T.T.L. and F.-M.Z.; Writing-Review & Editing, C.-C.K, T.T.L. and C.-Y.L.; Funding Acquisition, T.T.L. Funding: This research was funded by the Ministry of Science and Technology of the Republic of China, Taiwan grant number MOST 104-2410-H-259-035-MY3. ): the results of whether to invest in the product innovation because if the innovation cost is not invested at J node, neither manufacturing innovation nor business innovation is possible.Therefore, the consideration factor in stage 0 includes the decision to invest the profit in innovation costs.If losses are incurred in stage 0, the equity value of manufacturing innovation of the 1st stage is still one of the consideration factors.That is, when

Table 1 .
Parameter assumptions.After incorporating values to Figure 2 revenue change path, each node's revenue is shown in Figure 4 revenue (SR i ) change path.

Table 2 .
The influence of the risk-adjusted discounted rate r on equity values.

Table 2 .
The influence of the risk-adjusted discounted rate r on equity values.

Table 3 .
The influence of the GDP average growth rate in every five years u on equity values.