La acción extra cambiaria de enriquecimiento injusto

The Black-Scholes option pricing model is, to the best of our knowledge, the best known and most widely used model for real option valuation. For these reasons we select the Black-Scholes model (Black and Scholes, 1993), as enhanced by Merton (Merton, 1973), as the starting point for our fuzzy real option valuation model (FROV). We use the construct of the Black-Scholes model, however, we have three enhancements to the construct and the application of the model:

i) We use fuzzy variables to replace the crisp variables for the present value of the expected cash flows (S0), and for the present value of

the expected costs (X), used in the original model.

ii) We use a new approach in calculating the volatility (standard deviation) used in the formula, and in the treatment of the fuzzy variables within the model.

We calculate the standard deviation used in our model from the fuzzy present value of the free cash flows from the giga- investment, using a method developed in (Carlsson, Fuller, and Majlender, 2001) for calculating possibilistic variance and standard deviation for fuzzy numbers. This is different from the calculation of standard deviation used in the original Black-Scholes formula. This method allows the market information from the experts, who have contributed to the cash flow estimates, to be included in the calculation of the standard deviation. This may enhance the usability of the model for giga-investments.

We use the possibilistic mean value of S0 and X (crisp), as defined

in (Carlsson and Fuller, 2001a), within the model for calculation of d1 and d2.

iii) We suggest the estimation of (fuzzy) present values of the expected revenues and costs be done by using judgmental forecasting (expert-opinion), because, in our opinion, it suits giga-investments better, than using (only) stochastic methods.

By this we mean that judgement is used for estimation and adjustment of the fuzzy future cash flow estimates for the giga- investment. From these fuzzy future estimated cash flows the

present value is calculated, and aggregated to form the S0 and X

used in the model.

We base our belief that using judgemental methods suits giga- investments better, than using only stochastic methods, on the fact that giga-investments have long (or very long) economic lives, and that stochastic (econometric) methods commonly fail to produce reliable results on the long term, e.g., (Shnaider and Kandel, 1989). Indeed, "... some economists, based on human reasoning and only relatively limited data and without the support of econometric models, were more accurate in predicting the timing and the intensity of the turning points of the economy. This is possibly because their reasoning was not constrained by the results generated by the econometric models" (Shnaider and Kandel, 1989). It is also our experience that the firms making giga- investments have the best experts on their planned giga-investments in their employment, and as "quite often in finance future cash amounts and interest rates are estimated. One usually employs educated guesses, based on expected values or other statistical techniques..." (Buckley, 1987), it is not unreasonable to expect that the result obtained with judgmental methods may be better, than the result obtained with stochastic methods.

Furthermore, if future cash flows for giga-investments are estimated by experts, taking into consideration all information about the future they possess, then the estimates reflect the future information and are forward looking, even if the experts would base their estimates on their past experience, or on econometric models. Stochastic methods rely only on past data.

There are also some practical considerations that speak for the approach: simplicity of the approach makes it usable in the industry, unlike, e.g., jump models that "... are more difficult to implement in the everyday industry practice (Keppo and Lu, 2003)".

Under these circumstances we define our model as (Carlsson, Fuller, and Majlender, 2001) and (Collan, Carlsson, and Majlender, 2003):

DEFINITION. We suggest the following formula for computing the fuzzy real option value (FROV):

) ( ) ( FROV=S₀e−δTN d₁ −Xe−rTN d_{2 [9]} where,

(

)

(

)

and S0 = Present value of the expected free cash flows

(fuzzy)

X = Present value of the expected costs (fuzzy) E(S0) = The possibilistic mean value of the present

value of expected cash flows (crisp) E(X) = The possibilistic mean value of expected

costs (crisp)

σ = Possibilistic standard deviation of the present value of expected cash flows (crisp)

T = Time to expiry of the real option (crisp)

δ = The value lost over the duration of the option (crisp)

r = The annualized continuously compounded

rate on a safe asset (crisp)

The output from the model is a fuzzy number, which can be used together with the fuzzy NPV value to assist in the profitability analysis of giga- investments.

If the estimation of the expected fuzzy cash flows are done by judgmental methods, and the standard deviation used in the model is calculated from the present value of the fuzzy expected cash flows, then the standard deviation used will reflect the volatility of the cash flows, as it is seen by the experts. This makes also the calculation of standard deviation a forward-looking exercise.

It is interesting to compare the FROV model with the presented option valuation methods using fuzzy sets; for a short comparison we select the fuzzy-stochastic model presented in (Zmeskal, 2001). The two approaches differ quite a lot from each other, indeed, FROV is designed to value real options and the fuzzy-stochastic model for valuing the firm equity. Still,

both use fuzzy sets in option valuation, and some choices of modelling can be compared.

The treatment and derivation of the standard deviation used in the models is different; (Zmeskal, 2001) uses a fuzzification of the commonly used implied volatility, or historical volatility (actually it is not stated which, only that the volatility measure is fuzzy). The FROV model expects that the market uncertainty is captured by the originally fuzzy expert opinions and is found in the fuzzy cash flow forecasts, deriving volatility from them. This is, (Zmeskal, 2001) model relies on past data, and FROV model relies on forward-looking data. The use of the models is different, valuation of firm equity and valuation of a future investment, hence the difference in volatility measures may only reflect the different realities of these two situations.

(Zmeskal, 2001) replaces the geometric Brownian motion (GBM) used in the original Black-Scholes formula (and in the FROV model) with a fuzzy- stochastic methodology to include the market uncertainty in the valuation. A fuzzy-stochastic methodology allows the use of fuzzy variables also in the modelling of the markets, this may enhance the usability and credibility of the model.

The FROV model uses two fuzzy variables and derives volatility internally, the (Zmeskal, 2001) model has all variables fuzzy. The models are different, and without empirical testing it is not possible to draw definitive conclusions about their valuation abilities. The more forward looking approach of FROV model would seems to capture the uncertainty of giga-investments better, whereas, the methods used in (Zmeskal, 2001) may be more suitable for valuation of firm equity as an option. Applying a fuzzy-stochastic method to enhance the FRIV is an interesting future research opportunity.

The FROV model is presented, together with a numerical example, in (Collan, Carlsson, and Majlender, 2003), and is based on earlier work developed in, e.g., (Carlsson and Fuller, 2000) and (Carlsson, Fuller, and Majlender, 2001).

The FROV model can be used together with fuzzy NPV to aid in the real investment decision making. It enables the use of the same fuzzy cash flow estimates that are used for the fuzzy NPV calculation and provides a forward looking approach to calculating the standard deviation used in the real option valuation. The FROV model brings more realistic support to

giga-investments than the traditionally non-fuzzy real option valuation methods, because there is less need to make oversimplifying assumptions about the uncertainty of variable values, see appendix 2 for an example on how fuzzy numbers capture uncertainty.