^{1}

^{2}

^{3}

^{*}

In this study, we develop an option-based model to valuate New Product Development (NPD) projects in which management has the flexibility to abandon the project upon completion if the value of the established product falls below the required investment outlay. In the analysis, we explicitly consider the fact that the level of product volatility changes across development stages, as well as the stochastic nature of competition erosion. A closed-form solution is derived under a simplifying assumption of independence between product volatility and other stochastic processes considered in the model. The complete model is solved numerically by using Monte Carlo simulation. Our result indicates that ignoring the stochastic natures of product development uncertainty and competition erosion introduces a severe undervaluation bias. Such a bias worsens when 1) current product value is close to the required investment cost (so that the NPD project is nearly “at-the-money”); 2) development duration lengthens; 3) competition is intense; 4) the window of profitable opportunity lengthens, and 5) the market and the developing firm are more risk-prone (less risk-averse).

New Product Development (NPD) is fundamental to stimulating economic growth for business organizations. Successful NPDs not only support important business activities that over time contribute to long-run business profitability, but also provide firms with sufficient cutting edges in competitive battles.

Recognition of the importance of NPD to business prosperity has triggered considerable research interests from a variety of domains including marketing, strategic management and economics. Of all widely studied research topics, none is more challenging and has received more attention than the valuation of NPD investments. When corporate resources are limited, competition is intense, and the cost associated with NPD investment is significant; therefore, the accuracy with which NPD investments can be evaluated becomes critically important.

Traditionally, the Net Present Value (NPV) rule is recommended for analyzing investment decisions. The NPV rule helps decision makers choose between two alternatives: accepting or rejecting an investment opportunity. However, it is widely accepted that the majority of the investments firms make are dynamic in that decision makers do more than just accept or reject an investment opportunity using information available at the time of the decision. Upon arrival of additional information, decision makers update their beliefs about the profitability of the investment and may choose to defer, expand, contract, or shut down temporarily and later restart the investment project (Trigeorgis and Mason [

Alternatively, academic researchers adopt the principles of option pricing in analyzing investment decisions. Strategic flexibilities that allow decision makers to alter the course of investment at later dates resemble options written upon the underlying assets (Caballero [

In this study, we develop an option-based model to evaluate a NPD project that allows management to abandon the project upon completion. NPD is a risky process. Previous studies document high product failure rates and significant costs to sponsoring company upon product failure (Booz-Allen and Hamilton [

Our model differs from other real option models in two ways. Previously, literature generally assumed that uncertainty associated with product development remained constant throughout the development process. However, there exists abundant evidence, from both marketing and economic literature, that suggests NPD uncertainty varies across development stage progresses (Sahal [

The dynamics underneath our stochastic (product) volatility with stochastic competition model (SVSCM) is very general in which previously studied project dynamics are special cases of ours. For example, by assuming a constant level of product volatility, our model reduces to a stochastic competition with Constant (product) Volatility Model (CVM). By further restricting that change in the rate of competition erosion to zero, our model is further reduced to the Constant Competition with constant product volatility Model (CCM).

We derive a closed-form solution to SVSCM under a simplifying assumption that a change in product volatility is independent of changes in new product value and competition erosion. We further provide a numerical solution to the full-scale SVSCM using Monte Carlo simulations. We design a control variate methodology to obtain accurate (as measured by standard deviation of simulation results) estimations of project value with low computation cost (as measured by the number of simulation repetitions). We then examine the valuation benefit of admitting the stochastic nature of development uncertainty and industrial competition based on simulation results. Our results suggest valuation under SVSCM is much higher than those under CVM and CCM under various scenarios. We find that the undervaluation bias in CVM and CCM becomes severe when 1) current product value is close to required investment cost (so that the NPD project is nearly “at-the-money”); 2) development duration lengthens; 3) competition is intense; 4) the window of profitable opportunities lengthens, and 5) markets and developing firms are more risk-averse.

The major contribution of this study is that it highlights the importance of additional volatility from changing project uncertainty and competition erosion in the valuation of NPD projects. Previous studies assume constant product uncertainty and/or constant competition erosion. We argue that these assumptions are inadequate to capture the dynamic nature of NPD development. Real option theory suggests that under dynamic management the value of an NPD project increases with the uncertainty. Changing competition erosion and product volatility adds to a project’s uncertainty, which increases the likelihoods of both significant gain and loss. The option to abandon effectively limits downside losses, but allows a developing firm to reap potential gains; consequently additional uncertainty implies higher valuation. Our results indicate that ignoring the stochastic nature of competition erosion, and especially changing level product volatility will introduce severe undervaluation bias, and lead to an underinvestment problem.

The study is structured as follows. Section 2 briefly reviews real option literature. Section 3 describes the dynamics of the NPD project, derives a valuation model, and provides a solution to under risk-neutral representation. In Section 4, a closed-form solution is provided assuming independence between change in product volatility and other stochastic variables. Section 5 utilizes a Monte Carlo simulation technique to numerically solve the value of the NPD project. We also examine the marginal contribution of our derived model under various scenarios. We then conclude our study in Section 6.

The conventional (static) DCF technique assumes that an investment, once started, will be operated continuously until the end of its expected useful life. The valuation criterion is based on present values of expected outputs/inputs discounted at a risk-adjusted rate, i.e.

where

Standard NPV analysis suffers from several limitations. First, it assumes a static operating strategy. NPV analysis recommends either acceptance or rejection of an investment based on information available at the time of the initial decision, but it fails to consider that management may act upon the arrival of new information in the future of the project’s operating life to defer, expand, contract, or abandon investment. These flexibilities enable management to amplify future gain or reduce loss upon favorable or unfavorable future events; consequently, they create real value. NPV fails to account for these flexibilities and may result in an undervaluation problem. Secondly, NPV analysis relies on estimates of future cash flow distribution that is inherently subjective. Investors hold heterogeneous beliefs about cash flow perspectives. The heterogeneity of expectations can generate significant variations in project valuations among market participants. Consequently, NPV analysis usually results in inconsistent valuations across users. Finally, consistent estimation of the risk-adjusted discount rate may also be difficult to reach in NPV analysis. The most popular technique used is the CAPM. However, it has been pointed out that an asset’s true beta depends on the asset’s growth opportunities (Myers and Turnbull [

Since seminal work by Myers [

The problems with static NPV easily can be solved with a real option approach. First of all, the real option approach relies on contingent claim analysis, and replicates future cash flows from a potential investment using those of a portfolio of existing assets. This “cash flow equivalent” portfolio has the same payoff as the investment under consideration. In the absence of arbitrage opportunities, the value of investment is the value of this replicating portfolio. All individuals, regardless of their subjective estimates of future cash flow distributions, agree on the valuation. Secondly, as future cash flow estimates are independent of individual risk preference, risk-neutral probabilities and a risk-free rate are used in the valuation. Finally, operating flexibilities are models as a set of boundary conditions, so that initial investment strategies (and investment outcomes) can be altered upon realization of future events.

So far, the real option approach has been adopted in the valuation of various types of managerial flexibilities. Examples include (but not limited to) 1) option to defer (Paddock, Siegel and Smith [

Option to abandon a project at a later stage is an important strategic tool especially when project development faces high level of uncertainty and severe competition, for example a NPD project. A NPD project typically involves technology whose future is virtually unknown. As evidenced by unexceptionally high failure rates among NPD projects (Mansfield et al. [

Financial Options | Real Options | |
---|---|---|

- Current stock price | - Current value of asset | |

- Stock return volatility | - Variance of rate of change in asset value | |

- Exercise price | - (Per unit) development cost | |

- Time to maturity | - Life of the project | |

- Risk free rate of interest | - Risk free rate of interest | |

- Dividend yield | - Convenience yield |

of similar technologies, reducing the potential market for the new product. The ability to abandon the project when product value at a future time (e.g. upon completion) falls below required thresholds would effectively curb downside loss but enable firms to reap upside potential gain if the product turns out to be a success; consequently, the ability to abandon represents real value. Intuitively, the value of this strategic tool increases with uncertainty.

Chen, Ho, Ik and Lee [

In this section, we propose an option-based valuation model to capture the value of flexibility to abandon NPD projects at later stage. We explicitly consider the impacts of varying project volatility and stochastic competition on values of NPD projects.

Suppose the firm under consideration is developing a new product. The development will last T months. Upon product completion, if value of the established product falls below required production costs K, the firm will choose to abandon the project. The value of NPD project at time t,

Donate time t (t ≤ T) value of the new product as

where μ is the expected value appreciation for the product;

We model instantaneous product volatility,

where

Loss in value due to competition represents value outflows that will not accrue to the developing firm. This resembles dividend-like convenience yield on real assets (Fama and French [

where

It is assumed competition erosion is positively correlated with product value, i.e.

Managers have the flexibility to abandon the investment if the value of developed product falls below the required investment outlay, K, upon product complete at time T. Project value upon considering this flexibility can be modeled as:

where

Risk neutral representation of NPD dynamics (2), (3) and (4) could be established by adjusting the actual movements of underlying state variables with corresponding risk premiums (Cox, Ingersoll and Ross [

where r is the risk free rate;

With Ito’s Lemma, it can be shown that in the absence of arbitrage opportunities NPD value following the risk-adjusted processes (2)*, (3)* and (4)* must satisfy the following partial differential equation (for simplicity, we drop the “^” from the notation):

subject to boundary condition (5).

Equation (6) doesn’t depend on investor risk preference. Therefore, its solution has a risk-neutral representation (Cox and Ross [

where

NPD value at evaluation can be calculated by determining the time T distribution

Equation (8) implies that with a stochastic process governed by a risk adjusted pro- cess (2)*, (3)* and (4)*, NPD value

^{1}That is,

^{2}Note

^{3}This is based on the fact that

With movement of project value governed by processes specified in Section 3, there exists no closed-form solution to Equation (8). However, under the simplifying solution that volatility change is independent of NPD change (therefore

Define Θ as the point in the probability space that labels the stochastic path. For each

path Θ, we define ^{2}.

Under the assumptions that NPD volatility is independent of both NPD value change and industrial competition defined in (2)* and (4)*, (8) can be expressed as3:

where

trality, where

will use

The inner integral in (9) represents NPD value under a path Θ. It represents solution to the following partial differential equation (Cox, Ingersoll and Ross [

with boundary condition

Appendix A shows that with a selected sample path, the resulting distribution of new product’s value at time T is log-normal distributed, with a density function given by:

in which

Given that the NPD value at time T under each sample path of

In Appendix B, we show that NPD value distribution (11) is given by:

where

With time T product value distribution

Even under the simplifying assumption, the closed-form solution given in (13) requires integration with imaginary functions; therefore the calculations are very complex and time-consuming. To solve for NPD value under the full model (8), we implement a Monte Carlo simulation to numerically find its solution. There are two major concerns to any Monte Carlo simulation, namely accuracy (as measured by standard deviation of estimates) and computational cost (as measured by number of simulation repetitions). In order to improve the accuracy of Monte Carlo estimates while reducing the number of simulation repetitions performed, a control variate methodology is adopted. The basic idea is to replace the integration Equation (8) with one that has an analytic solution, under the assumption that product volatility,

We first present the base case simulation results.

Variable | Base Case Value | Variable | Base Case Value |
---|---|---|---|

S(0) (millions $) | 120 | K (millions $) | 100 |

T (month) | 12 | r | 8% |

40% | 0 | ||

12% | 20% | ||

5.0 | 10.0 | ||

0.5 | 0.5 | ||

0.0 | 0.0 | ||

0.3 | 0.1 |

Parameter values are based on the simulations by Chen et al. (2001) and Schwartz (2004).

and its long-term level, α, is set to be 12%. Initially, the product volatility,

^{1/2}); consequently 1000 replications offer an accurate result at reasonable computation cost. Panel A indicates that when the development duration increases from 3 months to 24 months (S(0) = $120 million), standard deviations for simulated NPD values increase from 0.0162 to 0.0372. Consistent with previous studies (Boyle [

In Panel A, we observe that NPD value decreases as development duration lengthens. Intuitively, the longer it takes to establish the final product, the more potential customers will be diverted to products of competitors. Our result suggests being able to bring out products faster will provide companies with significant competitive advantage over competitors.

In the next section, we explore the marginal contribution of our SCSVM model (i.e. Stochastic Competition with Stochastic Volatility model) relative to CCM (i.e. Constant Competition Model) and CVM (Constant Volatility Model). We allow several parameters that govern dynamic processes (1), (2) and (3) to vary, and examine changes in the marginal contribution of SCSVM over CCM and CVM.

T | NPD Value under SCSVM | S.t.d. of Estimates |
---|---|---|

3 | 20.97 | 0.0162 |

6 | 20.85 | 0.0183 |

9 | 20.56 | 0.0215 |

12 | 20.27 | 0.0232 |

15 | 20.02 | 0.0289 |

18 | 19.79 | 0.0319 |

21 | 19.59 | 0.0337 |

24 | 19.41 | 0.0372 |

S(0) | S/K | NPD Value under SCSVM | S.t.d. of Estimates |
---|---|---|---|

100 | 1.00 | 8.457 | 0.0193 |

120 | 1.20 | 20.27 | 0.0232 |

150 | 1.50 | 44.27 | 0.0290 |

180 | 1.80 | 70.83 | 0.0348 |

In CCM, we set competition erosion (δ) and product volatility (

In

For the base case, with the current value of the new product at $120 million and investment cost at $100 million, the value of the NPD project under SVSCM is $20.27 million, which is $3.38 million and $2.34 million(correspond to increases of 20.01% and 13.06%, respectively) higher than that under CVM and CCM, respectively. Within the range of current product values, SCSVM consistently outperforms CCM and CVM. The result indicates that failure to capture the stochastic nature of competition and product volatility will undervalue the NPD project, and will introduce an undervaluation bias.

CCM | CVM | SCSVM | |
---|---|---|---|

Assumptions | 1. Competition erosion is constant 2. NPD volatility is constant | 1. Competition erosion is stochastic 2. NPD volatility is constant | 1. Competition erosion is stochastic 2. NPD volatility is stochastic |

Governing stochastic process | 1. S(t) follows (2) 2. σ_{S} is constant 3. δ is constant | 1. S(t) follows (2) 2. σ_{S} is constant 3. δ(t) follows (4) | 1. S(t) follows (2) 2. σ_{S}(t) follows (3) 3. δ(t) follows (4) |

S(0) | V_{CCM}^{*} | V_{CVM}^{*} | V_{SCSVM}^{*} |
---|---|---|---|

70 | 0.13 | 0.16 | 0.46 |

80 | 0.71 | 0.81 | 1.57 |

90 | 2.34 | 2.61 | 3.87 |

100 | 5.54 | 6.01 | 8.46 |

110 | 10.47 | 11.26 | 13.75 |

120 | 16.89 | 17.93 | 20.27 |

130 | 24.37 | 25.63 | 27.71 |

140 | 32.51 | 33.95 | 35.79 |

150 | 41.02 | 42.61 | 44.27 |

160 | 49.72 | 51.44 | 53.00 |

170 | 58.52 | 60.36 | 61.87 |

^{*}V_{SCSVM}, V_{CCM}, V_{CVM} are NPD values derived under SCSVM, CCM and CVM, respectively.

We compute the differences in values estimated under SCSVM and those under CVM and CCM. We plot value differences against current product value S(0) in

protects firms from downside loss when product volatility increases; therefore, it carries more value. In the meantime, increasing volatility is more likely to result in an increase in project value. Consequently, the difference between values derived from SCSVM and CVM/CCM will widen, and will peak when project is at the money (ATM) (i.e. S(0) is nearly equal to K).

Increasing the required investment outlay K reduces project value. In

The effect of development duration, T, on NPD value is two-folded. On one hand, information accumulates with time. Firms gradually adjust product developments by the arrival of information over time, therefore would benefit from a high learning effect. Also lengthened development duration carries more flexibility. It is argued that managerial flexibility is more valuable for investment of longer duration (Teisberg [

K | V_{CCM} | V_{CVM} | V_{SCSVM} |
---|---|---|---|

50 | 60.27 | 61.58 | 62.53 |

60 | 51.05 | 52.35 | 53.32 |

70 | 41.85 | 43.14 | 44.21 |

80 | 32.82 | 34.09 | 35.41 |

90 | 24.32 | 25.51 | 27.31 |

100 | 16.89 | 17.93 | 20.27 |

110 | 10.95 | 11.79 | 14.54 |

120 | 6.65 | 7.27 | 10.15 |

130 | 3.80 | 4.22 | 6.93 |

140 | 2.06 | 2.32 | 3.80 |

150 | 1.07 | 1.22 | 2.35 |

that growth opportunities of longer duration are more valuable and a firm should avoid to exercise them sooner than necessary (Kester [

We at first vary only the development duration T within 3 to 24 months, but leave all other parameters at their base case values.

The marginal contribution of SCSVM over CCM/CVM increases with development duration. For example, when T = 3 months, project value under SCSVM exceeds those under CCM and CVM by $2.35 million and $1.11 million, respectively. However, when T = 24 months, the differences widen to $4.07 million and $3.25 million respectively. This is not surprising, as the marginal contribution by SCSVM comes from additional uncertainty being recognized. As project uncertainty increases as development duration lengthens, so do the value differences.

We then allow current product value to vary within a range (thus create a full list of ITM, ATM, and OTM projects) while development duration T varies at the same time. It is interesting to note that the relation between “competition erosion effect” and “value increase effect” varies with project moneyness. In

T | V_{CCM} | V_{SCM} | V_{SCSVM} |
---|---|---|---|

3 | 18.62 | 19.86 | 20.97 |

6 | 17.85 | 19.07 | 20.85 |

9 | 17.33 | 18.45 | 20.56 |

12 | 16.89 | 17.93 | 20.27 |

15 | 16.48 | 17.46 | 20.02 |

18 | 16.10 | 17.01 | 19.79 |

21 | 15.71 | 16.58 | 19.59 |

24 | 15.34 | 16.16 | 19.41 |

projects, “competition erosion effect” dominates. This suggests competition intensifies when underlying innovation becomes more profitable.

Project uncertainty comes from two sources, namely, competition uncertainty and uncertainty associated with the product under development. The essence of real option literature is that uncertainty creates value (Trigeorgis [

Consistent with real option literature,

S(0) | T = 6 Months | T = 12 Months | T = 15 Months | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

V_{CCM} | V_{CVM} | V_{SCSVM} | V_{CCM} | V_{CVM} | V_{SCSVM} | V_{CCM} | V_{CVM} | V_{SCSVM} | ||||

70 | 0.01 | 0.02 | 0.19 | 0.13 | 0.16 | 0.46 | 0.22 | 0.26 | 0.58 | |||

80 | 0.21 | 0.26 | 0.98 | 0.71 | 0.81 | 1.57 | 0.94 | 1.06 | 1.79 | |||

90 | 1.31 | 1.54 | 3.09 | 2.34 | 2.61 | 3.87 | 2.69 | 2.97 | 4.48 | |||

100 | 4.47 | 5.02 | 7.48 | 5.54 | 6.06 | 8.46 | 5.85 | 6.34 | 8.80 | |||

110 | 10.16 | 11.09 | 13.40 | 10.47 | 11.26 | 13.75 | 10.51 | 11.24 | 13.85 | |||

120 | 17.85 | 19.07 | 20.85 | 16.89 | 17.93 | 20.27 | 16.48 | 17.46 | 20.02 | |||

130 | 26.62 | 28.04 | 29.29 | 24.37 | 25.63 | 27.71 | 23.46 | 24.65 | 27.06 | |||

140 | 35.84 | 37.41 | 38.31 | 32.51 | 33.95 | 35.79 | 31.12 | 32.48 | 34.73 | |||

150 | 45.20 | 46.90 | 47.63 | 41.02 | 42.61 | 44.27 | 39.20 | 40.72 | 42.83 | |||

160 | 54.61 | 56.42 | 57.09 | 49.72 | 51.44 | 53.00 | 47.52 | 49.18 | 51.20 | |||

170 | 64.02 | 65.95 | 66.62 | 58.52 | 60.36 | 61.87 | 55.98 | 57.76 | 59.75 |

s_{δ} | V_{CVM} | V_{SCSVM} | s_{s} | V_{CVM} | V_{SCSVM} | |
---|---|---|---|---|---|---|

0.35 | 17.91 | 20.23 | 0.35 | 17.93 | 19.96 | |

0.40 | 17.91 | 20.24 | 0.40 | 17.93 | 20.05 | |

0.45 | 17.92 | 20.25 | 0.45 | 17.93 | 20.10 | |

0.50 | 17.93 | 20.27 | 0.50 | 17.93 | 20.27 | |

0.55 | 17.94 | 20.30 | 0.55 | 17.93 | 20.30 | |

0.60 | 17.96 | 20.33 | 0.60 | 17.93 | 20.37 | |

0.65 | 17.97 | 20.36 | 0.65 | 17.93 | 20.45 | |

0.70 | 17.99 | 20.40 | 0.70 | 17.93 | 20.54 |

^{*}NPD value under CCM remains at $16.89 in all scenarios.

CCM, and value difference increases as volatility becomes higher. Also, the marginal contribution of SCSVM increases as uncertainty associated with competition, s_{δ}, and with product, s_{s}, increase. This is consistent with the fact that the marginal contribution of SCSVM comes from added uncertainty recognized.

Adjustment rates k_{d} and k_{s} determine the average time needed to absorb random shocks generated by changes in competition and product uncertainty. _{d} imply that competition is more intense. Increased competitive activities tend to devour more product value, making NPD values low. Increase in k_{s}, on the other hand, suggests shorter windows of opportunities opened by unexpected changes in project uncertainty. As profitable investment opportunities disappear quickly, the value of projects tends to be lower.

The marginal contribution of SCSVM over CCM/CVM increases with k_{d}. The value differences between SCSVM and CCM/CVM is $3.69 million and $2.36 million, respectively, at k_{d} = 6.0, but rise to $4.05 million and $3.33 million when k_{d} is 6.0. The result suggests that the importance of uncertainty to value creation strengthens as more value is devoured by competition. The marginal contribution of SCSVM decreases with k_{s}. High k_{s} value implies shorter windows of opportunities opened by project uncertainty; consequently reducing the advantage of SCSVM over CCM and CVM.

Prices of risk, l_{d} and l_{s}, represent prices developing firms would be willing to pay to reduce uncertainty associated with competition erosion and changing product volatility. They also measure levels of risk aversion: when developing firms are more risk-averse, they would accept a higher risk premium, resulting in higher values for l_{d} and l_{s}.

k_{d} | V_{CVM} | V_{SCSVM} | k_{s} | V_{CVM} | V_{SCSVM} | |
---|---|---|---|---|---|---|

6.0 | 18.43 | 20.82 | 3.0 | 17.93 | 21.03 | |

7.0 | 18.22 | 20.58 | 3.5 | 17.93 | 20.77 | |

8.0 | 18.06 | 20.41 | 4.0 | 17.93 | 20.57 | |

9.0 | 17.93 | 20.27 | 4.5 | 17.93 | 20.41 | |

10.0 | 17.83 | 20.16 | 5.0 | 17.93 | 20.27 | |

11.0 | 17.74 | 20.08 | 5.5 | 17.93 | 20.16 | |

12.0 | 17.67 | 20.00 | 6.0 | 17.93 | 20.06 | |

13.0 | 17.61 | 20.94 | 6.5 | 17.93 | 19.98 | |

14.0 | 17.56 | 20.89 | 7.0 | 17.93 | 19.91 | |

15.0 | 17.52 | 20.85 | 7.5 | 17.93 | 19.84 |

^{*}NPD value under CCM remain at $16.89 in all scenarios.

Varying l_{d} and l_{s} allows us to examine the effect of firms’ risk altitudes towards NPD valuations.

Notice that competition erosion represents value outflows. As competitors erode the value of the product, they simultaneously bear part of a product’s risk. Consequently competition erosion represents risk sharing. To developing companies, an increase in competition erosion reduces the risk they eventually bear; therefore risk premium requested for competition erosion is negative, i.e. developing firms get paid by competitors to reduce part of a project’s risk.

We develop a stochastic volatility real option model to value NPD projects that carry managerial flexibility to abandon the project if upon completion, product value falls below the required development cost. Real option theory suggests uncertainty is important to value creation under dynamic management. We explicitly consider the stochastic nature of competition erosion and changing project uncertainty across development stages. We derive a close-form solution under a simplifying assumption of independence between product development uncertainty and other stochastic processes considered in the model. We then solve the full-scale model numerically with Monte

l_{d} | V_{CVM} | V_{SCSVM} | l_{s} | V_{CVM} | V_{SCSVM} | |
---|---|---|---|---|---|---|

0.0 | 17.93 | 20.27 | −0.5 | 17.93 | 21.74 | |

−0.1 | 17.54 | 19.90 | −0.4 | 17.93 | 21.43 | |

−0.2 | 17.16 | 19.53 | −0.3 | 17.93 | 21.13 | |

−0.3 | 16.78 | 19.17 | −0.2 | 17.93 | 20.84 | |

−0.4 | 16.40 | 18.81 | −0.1 | 17.93 | 20.55 | |

−0.5 | 16.03 | 18.45 | 0.0 | 17.93 | 20.27 | |

−0.6 | 15.66 | 18.10 | 0.1 | 17.93 | 20.00 | |

−0.7 | 15.30 | 17.75 | 0.2 | 17.93 | 19.73 | |

−0.8 | 14.95 | 17.41 | 0.3 | 17.93 | 19.48 | |

−0.9 | 14.60 | 17.06 | 0.4 | 17.93 | 19.23 | |

−1.0 | 14.25 | 16.74 | 0.5 | 17.93 | 18.99 | |

−1.1 | 13.91 | 16.41 | 0.6 | 17.93 | 18.77 |

^{*}NPD value under CCM remain at $16.89 in all scenarios

Carlo simulation. Our result is consistent with real option theory. We find a significant undervaluation bias if uncertainties association with competition and changing project volatility are ignored. We examine the marginal contribution (therefore the size of undervaluation bias) of our stochastic volatility with stochastic competition model. We find undervaluation bias is more severe when 1) The NPD project is either out-of-the- money or at-the-money; 2) when development duration lengthens; 3) when competition is more intense; 4) when the window of profitable opportunity shortens; and 5) when developing firms are more risk-prone.

Hu, C.R., Jun, C. and Foley, M. (2016) Valuating New Product Development Project with a Stochastic Volatility Model. Journal of Mathematical Fi- nance, 6, 975-1001. http://dx.doi.org/10.4236/jmf.2016.65064

To owner (i.e. the firm) of the new product, competition erosion creates a dividend-like value outflow. Upon product completion, firm obtains spot new product value. Assume there exists a contract F, which requires the transfer of completed new product to contract owner upon product completion. Under a risk-neutral economy, future payment streams from these two positions are identical, therefore values of the contract and NPD project must be the same.

Donate F as the value of the contract. By construction, it should be the function of time t, new product value

with a deterministic continuous sample path of

where

It can be shown (Cox, Ingersoll and Ross [

with boundary condition given in (A.1).

Duffie [

with Ito’s lemma, given selected stochastic volatility path and risk-neutral processes described in (2)* and (4)*, instantaneous value change of the contract is:

Denote:

with (A.4) and (A.6), (A.5) can be simplified as:

Since

Denote:

(A.6a) could be simplified to:

It is obvious that change in contract value

The resulting distribution of the contract value at time T is given by:

with

We follow Stein and Stei [

Define

Apply the Fourier transformation to (B.1)

Using definition in (A.6b),

And (B.2) can be re-expressed as:

where

Apply Fourier inversion formula to (B.3) to get:

Now make the change of variables of

In this section, we derive a closed-form solution under the assumption that NPD volatility follows a deterministic process.

Assume σ_{σ} = 0. Partial deferential Equation (6) becomes

Solution to Equation (C.1), with the boundary condition (5), admits the following special representation (Duffi [

where

and

Our estimate of the NPD project (7) can be rewritten as:

Starting with time 0, for a small time interval (0, 0 + Δt), we randomly generate values for, dω_{S}, dω_{δ} and dω_{σ}, from a joint normal distribution such that

and

and

We then simulate incremental changes^{*}, (3)^{*} and (4)^{*}. Provided with initial values of S(0), δ(0), and σ(0), values of S, δ and σ at the end of the interval (i.e. S(0 + Δt), δ(0 + Δt), and δ(0 + Δt)) can be found.

We repeat the above operation for all the following intervals

By time T, a sample path for S(T), δ(T), and σ(T) can be generated, respectively. A value of the project upon time T, as defined by boundary condition (5), can be computed.

We utilize the same series of dω_{S}, dω_{δ} and dω_{σ} values generated in the above simulation and repeat the above procedure to simulate sample paths of S(T) and δ(T). However, we restrict σ_{σ} = 0 in (3)^{*} when simulating δ(T) (such that δ(T) is actually deterministic). We compute the value of the project upon time T using boundary condition (5), based on the value of the project simulated under the restriction σ_{σ} = 0.

We compute the difference between the project values obtained from these two simulations. We repeat the above process. Givena large quantity of repetitions, time T distributions of S(T), δ(T), and σ(T), with and without the assumption σ_{σ} = 0, as well as a sample of the differences between project values under these assumptions, can be obtained.

We compute B^{*}[S(0), δ(0), σ(0)] with solutions provided by (C.3) to (C.5). The value of the integral on the right-hand-side of (C.6) is computed as the present value (discounted at risk-free rate r) of the sample average of the differences obtained from above simulations. Based on the law of large numbers, when a number of repetitions is large enough, the value of the integral should converge to its true value, regardless of the true distribution

Because the value of the integral on the right-hand-side of (C.6) is simulated using the same random variable series of dω_{S}, dω_{δ} and dω_{σ}, and the underlying dynamics are similar except we assume σ_{σ} = 0 in the second round of simulation, the value of project from both simulations should be closely correlated. By utilizing only their differences, the control variate methodology should achieve a significant variance reduction.

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