Usually, traditional methods for investment project appraisal such as the net present value (hereinafter NPV) do not incorporate in their values the operational flexibility offered by including a real option included in the project. In this paper, real options, and more specifically the option to abandon, are analysed as a complement to cash flow sequence which quantifies the project. In this way, by considering the existing analogy with financial options, a mathematical expression is derived by using the binomial options pricing model. This methodology provides the value of the option to abandon the project within one, two, and in general

Real options were introduced as a complement to the information provided by the cash flow sequence presented by an investment project. In this paper, we will use the NPV model to evaluate the level of operational flexibility provided by a real option. This is a very important feature of investment projects because the information provided by a real option can modify the strategy of a company [

In other words, real options are managerial tools which make easier the dynamic management of investment projects when dealing with abstract concepts [

Real options can affect several parameters of the project and can appear at any moment. Table

Parameters of a project and related real options.

Parameter | Related real option |
---|---|

Cash flow | To expand/to reduce |

Residual value | To abandon |

Duration | To defer |

The consideration of real options in the assessment of a project is useful both in offering an alternative approach and in providing more concrete material for calculation [

The real option approach is not widely used in business practice, despite the advice given in the academic literature to incorporate it in the assessment of a project. It is in fact normally used only by companies dealing with very large capital such us those supplying energy or healthcare, or in the technology sector. The findings of previous surveys cite the complexity of its implementation, together with lack of familiarity, as the reasons for the infrequent use of real options. Many managers consider the models used to calculate the real option value as a “black box,” whose implementation requires a high degree of sophistication. This leads them to believe that any error in use may be very difficult to detect [

Option pricing constitutes one of the most challenging problems in computational finance and derivative modelling [

The existing literature about real option valuation indicates two identifiable approaches, namely, discrete and continuous, depending on the complexity and nature of the analysed option [

In particular, the option to abandon provides the investor with the opportunity of liquidating the entire investment project in exchange for an amount called the residual value [

Therefore, the aim of this paper is to obtain a mathematical expression for the value of the option to abandon a project, terminating the business activity and liquidating assets, after one, two, and

In general, the total value of an investment project can be divided into two components: the static NPV and the value of the real option [

First, the value of an investment project is calculated by including the flexibility represented by the possession of an option to abandon.

Second, it must be shown mathematically whether this value is greater than the project value without the option to abandon.

Finally, the value of the option to abandon is obtained by the difference between the values of the project when the real option is included and when it is not.

The value of the option to abandon a project at any time prior to the expiry of the first period (denoted by

Nevertheless and for the sake of simplicity, we have opted for the use of the discrete stochastic model called the binomial options pricing model. Accordingly, starting from the present value of cash flows (

By using the binomial options pricing distribution, these factors will allow us to foresee the value of the project in a favourable (

The net present value of an investment project with the option to abandon within a period is always greater than or equal to

In effect, we are going to consider the following two cases:

If

which confirms the expected equality/inequality.

If

Since

To summarise, the possible results for the net present value of a project with the option to abandon within a period are as follows:

If

Contrarily, if

XHMobile is a mobile phone company which was established in 2011 and currently operates at national level. XHMobile managers are considering the expansion of the business into a foreign market with a view to increasing its turnover. In order to analyse the project the following numerical information has been taken into account [

To carry out the project the initial investment required

The present value of expected cash flows

The riskless discount rate

Upward and downward factors affecting cash flows are

Upward and downward factors affecting the residual value are

Finally, we are going to derive the value of the option to abandon a project within a period (denoted by

Project value with the option to abandon within a period.

The value of the option to abandon an investment project after one period is

The proof of the first case is obvious whilst, in the second case, if

In Example

Value of the option to abandon within a period.

The present value of an investment project with the option to abandon after two periods, denoted by

Evolution of the initial payment, cash flows, and residual values over a period of two years.

In this way, the possible project values after two periods are given by the following expressions:

the project value with the option to abandon within a period and

the project value without this option.

If the market discount rate (

Let us start from expression (

Hereinafter, we are going to suppose that the logical hypothesis of Lemma

For a better understanding of this demonstration, Figure

To understand the meaning of Figure

If

If

Finally, if

If

which is the expected equality/inequality.

If

Now we are going to compare this value with the project value without the option to abandon,

Hence, since

Thus, it remains to compare the project value including the option to abandon within a period when

with the project value including the option to abandon within two periods. To do this, we will take into account that

and that, from Lemma

In this case,

since

Finally, if

Taking into account that

since

since

Possible intervals for

In Example

Project value with the option to abandon within one and two periods.

The value of the option to abandon the project after two periods, denoted by

In effect, the value of the option to abandon the project within two periods can be calculated as the difference between the project values with the option,

If

which, once simplified, remains as

If

In Example

Option value to abandon within one and two periods.

In this section, we are going to determine the general mathematical expression for the present value of an investment project which includes the option to abandon within

In Example

Finally, the mathematical expression of

Project value with the option to abandon within one, two, and five periods.

In Example

Option value to abandon within one, two, and five periods.

Real options constitute an important part of project assessment; they provide a means to assess the value of flexibility in business operations. Given the limited use of the methodologies to quantify real options in business practice, as a result of their high degree of sophistication, this paper aims to provide a simple formula to quantify the value of one of the types of real options most common in practice: the option to abandon.

Obviously, the right to exercise the option to abandon allows the sale of the project in exchange for its residual value, since this operation is similar to a put option. In this way, the methodology employed to develop the expression to obtain the value of the option to abandon is based on the implementation of the binomial options pricing model. The procedure consists in an accurate reconstruction of all possible future scenarios with their respective probabilities of occurrence. This analysis has been initially performed for options to abandon whose maturity is within one period. In the next step, we deduce the expression of those options whose expiry is within two periods and, finally, the corresponding expression for options expiring within

The main contribution of this paper is the introduction of the mathematical expression, complementary to the net present value expression, in order to quantify in an intuitive way the value of a project including the option to abandon within a given period. Its use contributes to a more accurate assessment, thus giving more control over the element of uncertainty.

In all cases, it has been shown that the option value to abandon is greater than or equal to zero and that, as the maturity of the option increases, its value also increases. This analysis has been completed with the graphic representation of the value of the project and the option in every one of the expiry periods considered (one, two, and

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are very grateful for the comments and suggestions made by an anonymous referee.