_{1}

Motivated by previous papers with conventional models of Geometric Brownian Motion (Hereafter GBM) or Mean-Reverting (Hereafter MR), we discuss the classical investment timing problem in this paper by assuming the output price follows Heston-GBM process. That is, constant volatility in the classical GBM or MR framework is replaced by stochastic volatility in Heston-GBM model. We first derive the asymptotic solution for the investment timing problem. Then the impacts of stochastic volatility on trigger prices and the range of inaction are demonstrated by numerical simulation. Lastly, we examine the analytical properties of trigger prices and the range of inaction quantitatively as well as qualitatively.

In the real world, most investment decisions are made in an uncertain background and are costly to reverse in the future. Such as firms enter and exit foreign markets in responding to real exchange rate fluctuations. And corporations decide whether to initiate new or terminate existing products based on uncertain profitability prospects of these actions. There are many other similar examples can be offered, but we won’t enumerate them one by one here in order to free from repeating. In view of these facts, a number of investment models have been studied within stochastic calculus framework in the past decades. Nowadays, the interest has been motivated and great progresses have been made in this field due to the recent advances in the theory of mathematical finance and of stochastic optimal control. Absolutely, considerable results have been obtained.

Usually, industries often go through successive phases of entry and exit with the change of economic environment. For example, excess entry may be observed when the market demand is unknown and a wave of exit is expected to follow. Moreover, there may be exit by either too many or too few firms because demand uncertainty is not completely resolved, then in this case, a new wave of entry or exit respectively will follow in turn. It’s well known that much work on investment, specifically that exploits an analogy between real and financial investment decisions, which known as “real options” approach has been done before. In this paper, however, we still present a dynamic model of entry and exit under price uncertainty to establish optimum time for entry and exit as strategic decisions for a firm, decisions that can be modeled through real options.

We begin by making an overview as well as stating the relation of this paper to some recent literature on this topic. Eriksson [

In the past decade, many researchers spent much time and energy to investigate this area from different aspects, and a number of new and useful results have been achieved. Within these papers, Amir and Lambson [

As we all know, firms in developing industries decide whether and when to enter the market depending on the state of demand, existing firms in the industry, and the firm’s capabilities. Thus, a model of increasing demand, in which firms decide when to enter the market anticipating the strategic behavior of other potential entrants, and the effects of entry on future potential entrants has been examined in Shen and Villas-Boas [

In the end, we mention that the problem we study in this paper has various applications in many aspects. One potential application, for instance, appears in the area of valuating off-shore petroleum leases, and specially in pricing the right to exploit an oilfield with given reserves. And the other application occurs in the field of valuating “know-how”. Further, by modifying the model framework slightly, one can apply to the analysis of decisions to buy or scrap a durable consumption good, to hire or fire labor and so forth. In view of this point, we will use the analysis of decisions to study a modified model that differs from the previous literatures.

The contribution of this paper is that we make a generalization to previous papers such as Dixit and Tsekrekos, which are two standard model to study entry and exit decisions under uncertainty. The difference between ours and theirs is that we depict uncertainty by Heston-GBM, rather than conventional GBM or MR. In other words, we replace constant volatility in their settings by allowing volatility process itself to be stochastic. More specifically, we assume the instantaneous variance in the price follows a Cox-Ingersoll-Ross process, which is often referred to as Heston stochastic volatility in the theory of option pricing under stochastic volatility. Nevertheless, our improvements are not just by using complicated and profound mathematical knowledge to examine or repeat one commonplace topic. The study shows that the extension and generalization have theoretical significance and application value as well. What’s more important, the results we obtained are not only consistent with our anticipation, but also in line with some economic phenomena in reality, and further can make new interpretations to them from different perspective. Of course, the problem presented in this paper is still an open topic, so far to our knowledge.

The remainder of this paper is organized as follows. We first introduce the model at length in Section 2. Then in Section 3, we solve and analyze the model. Subsequently, we illustrate numerical simulations in Section 4. Lastly, Section 5 concludes the results.

In this section, we will introduce our model in detail for the sake of intactness though it is an extension to the standard models to study entry and exit decisions under uncertainty, which proposed by Dixit and Tsekrekos.

A firm possesses an exclusive production technology and consider to enter some market by making a fixed scale investment of one lump sum entry cost

The uncertainty originates from the output price P, which evolves exogenously to the firm. Namely, the firm is a price-taker. Unlike Dixit and Tsekrekos suppose the price P therein follows classical GBM or MR process, we suppose the price P herein follows a Heston-GBM process. According to Heston [

Here,

The firm is assumed to be risk-neutral as well as competitive, and its objective is to maximize the expected net present value. Clearly, its strategy problem has three state variables, one is the current price P, one is current variance Q, and the other is a discrete variable that indicates the firm is active or inactive. This setting, in some sense, corresponding to the firm possess a call-option to enter the market when it is inactive, and a put-option to exit when it is active. The two options must be valued at the same time because they are interlinked by exercising one the firm acquires the other.

Again, we emphasize that suppose the price follows Heston-GBM not to just apply complicated and profound mathematical knowledge to make the framework hard to understand, but because the following reasons such as: 1) Stochastic volatility, for one thing, can explain many empirical findings like “Volatility Clusters” observed in financial markets, and for another thing, can replicate the observed “Volatility Smile” and thereby is essential for pricing and hedging financial derivatives; 2) In the form of implied volatility, further proof shows that constant volatility hypothesis in Black-Scholes framework is not realistic; 3) The necessity of including stochastic volatility into a real option model is strongly indicated by commercial property literature; 4) Whenever changes in surrounding economic environment affect the level of fluctuations in project value, then stochastic volatility will undoubtedly be relevant for the timing of investment project; 5) So far as we know, the issue of entry and exit strategies under uncertainty has not been referred in stochastic volatility setting.

As we mentioned, the risk neutral and competitive firm has two discrete states, idle or active. When the firm’s state is idle, then it decides whether to keep idle or to enter. Likewise, when the firm’s state is active, then it decides whether to keep active or to exit. Thus, the decision problem of the firm is to switch optimally from one state to the other state. For explicitness, we denote by

Entry Problem. We assume an idle firm starts at time 0 with current price P as well as variance Q, and choose to enter into the market at time t. Then it must pays one lump-sum entry cost

where E is the symbol of expectation operator.

We point that if the idle firm keeps its state, then it will not has any other profit besides its expected capital gain

with boundary condition

and value-matching condition

smooth-pasting condition

Here,

Exit Problem. We assume an active firm starts at time 0 with current price P as well as variance Q, and choose to exit from the market at time t. Then it must pays one lump-sum exit cost

Similar to the computation of Equation (1), there yields the following PDE

with boundary condition

and value-matching condition

smooth-pasting condition

Here,

Remark 1. As proposed in Dixit’s,

Now, we try to solve the two PDEs shown in (1)-(2) and (5)-(6), respectively. For our purpose, we rewrite PDE (1) into compact form as

where

Then we can expand

Keeping terms up to

To determine functions

Adopting the approaches presented in Ting et al. [

since

which are the two roots of quadratic equation

Remark 2. The argument processes of solutions

Remark 3. The solution

value of

Recalling value-matching condition and smooth-pasting condition (3)-(4) for

Obviously,

Although the analytical expressions of the two trigger prices cannot be derived from Equations (17) and (18), we can verify that there holds

Remark 4. When the variance process

In this section, we investigate the analytical properties for trigger prices by numerical simulation. Specifically, we focus on the effects of the drift rate

From

8.5891 | 1.4978 | 8.3202 | 1.4700 | 8.1590 | 1.4558 | |

8.2522 | 1.2987 | 8.0099 | 1.2725 | 7.8964 | 1.2583 | |

7.9342 | 1.0800 | 7.7184 | 1.0563 | 7.6163 | 1.0440 | |

7.6418 | 0.8408 | 7.4489 | 0.8210 | 7.3570 | 0.8105 | |

7.3660 | 0.5810 | 7.1975 | 0.5763 | 7.1148 | 0.5685 |

Following

8.4215 | 1.1238 | 7.7184 | 1.0563 | 7.4229 | 1.0183 | |

8.0484 | 1.0914 | 7.7184 | 1.0563 | 7.5667 | 1.0377 | |

7.9342 | 1.0800 | 7.7184 | 1.0563 | 7.6163 | 1.0440 | |

7.8787 | 1.0742 | 7.7184 | 1.0563 | 7.6415 | 1.0471 | |

7.8466 | 1.0707 | 7.7184 | 1.0563 | 7.6564 | 1.0489 |

In terms of

7.2852 | 1.1759 | 7.1017 | 1.1481 | 7.0148 | 1.1335 | |

7.6134 | 1.1255 | 7.4133 | 1.0999 | 7.3180 | 1.0866 | |

7.9342 | 1.0800 | 7.7184 | 1.0563 | 7.6163 | 1.0440 | |

8.2521 | 1.0386 | 8.0212 | 1.0165 | 7.9112 | 1.0051 | |

8.5678 | 1.0006 | 8.3206 | 0.9800 | 8.2030 | 0.9694 |

According to

7.7880 | 1.0644 | 7.7184 | 1.0563 | 7.6834 | 1.0523 | |

7.8610 | 1.0722 | 7.7184 | 1.0563 | 7.6504 | 1.0481 | |

7.9342 | 1.0800 | 7.7184 | 1.0563 | 7.6163 | 1.0440 | |

8.0110 | 1.0876 | 7.7184 | 1.0563 | 7.5826 | 1.0398 | |

8.0881 | 1.0951 | 7.7184 | 1.0563 | 7.5494 | 1.0356 |

Finally, throughout Tables 1-4, it’s not difficult to find two common places, no matter for GBM model with constant volatility or for Heston-GBM model with stochastic volatility. The one is our numerical results confirmed that uncertainty would widen Marshallian range of inaction, namely

In the past decades, many investment decisions of firms such as when to invest in an emerging market or whether to expand the capacity have been studied by many researches, which involve irreversible investment and uncertainty about demand, cost or competition, etc. Naturally, considerable results and conclusions have been achieved. In spite of this, there still exist some issues that can be studied further. In view of this point, a model of optimal inertia in investment decisions under uncertainty with stochastic volatility is established in this paper to deepen understanding of the issue and open up the way for treating further problems of this topic.

Unsurprisingly, our conclusions show that stochastic volatility undoubtedly be relevant for the timing of investment decisions. For Heston-GBM model, as demonstrated in the previous section, the change of the drift rate of price process plays negative role in trigger prices for any given

The major contribution of this paper is that we study a classical and hot topic from a new perspective. Unlike previous literatures, we here construct one stochastic volatility models, Heston-GBM, to study firm’s entry and exit strategy under uncertainty. The findings show that stochastic volatility has potent and significant impacts on firm’s investment decisions, which implies that we replace constant volatility by stochastic volatility not to apply profound mathematical knowledge to make the problem complicated, but to say the conventional GBM and MR process is not always the most appropriate for some economic variables. Indeed, relatively speaking, stochastic volatility models are able to capture the so called “Leverage Effect”. In brief, our extension not only has theoretical significance, but also has application value.

The author would like to thank the anonymous referees for their helpful comments, which greatly improved the presentation of this paper. The author also would like to thank the editors for their help.

The author declares no conflicts of interest regarding the publication of this paper.

Huang, J.W. (2020) Optimal Entry and Exit Strategy under Uncertainty with Stochastic Volatility. Journal of Mathematical Finance, 10, 157-172. https://doi.org/10.4236/jmf.2020.101011