Application of the fuzzy–stochastic methodology to appraising the firm value as a European call option

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Abstract

The valuing of a firm equity as a call option is a crucial problem in financial decision-making. There are two basic aspects that are studied; contingent claim features (payoff functions) and risk (stochastic process of underlying assets). However, non-preciseness (vagueness, uncertainty) of input data is often neglected. Thus, a combination of risk (stochastic) and uncertainty (fuzzy instruments) could be a useful approach in calculating a firm value as a call option. The Black–Scholes methodology of appraising equity as a European call option is applied. Fuzzy–stochastic methodology under fuzzy numbers (T-numbers) is proposed and described. Fuzzy–stochastic model of appraising a firm equity is proposed. Input data are in a form of fuzzy numbers and result, firm possibility-expected equity value is also determined vaguely as a fuzzy set. Illustrative example is introduced.

Introduction

Option pricing theory has many applications in finance. A number of modifications have been applied to the basic Black–Scholes model with different stochastic diffusion process and stochastic variables applied (see Black and Scholes, 1973, Duffie, 1988, Kariya, 1993, Boyle et al., 1995, Campbell et al., 1997, Musiela and Rutkowski, 1997, Hull, 2000). Most of the models are considered as short term. Applications of the option methodology in corporate finance decision-making, which have some special features, are called as real options. A survey of the real option literature is available in Dixit and Pindyck, 1994, Sick, 1995, Trigeorgis, 1998, Brennan and Tigeorgis, 1999. One possible and tractable usage of this methodology is stating the firm value as an option.

Black and Scholes show that the equity in levered firm is really call option value on the firm equity; see Black and Scholes, 1973, Copeland and Weston, 1988, Damodaran, 1994, Boyle et al., 1995. In the case of application of an option methodology the estimation of an option value is determined by input data precision, mainly concerning validity, quality and availability of input data. There are also problems with data frequency and stochastic error as well. A survey of input data vagueness problems in finance and accounting decision-making is in Zebda (1995). One of the suitable approaches for solving the problem is to apply a fuzzy–stochastic methodology and create a fuzzy–stochastic model. In the paper applying the proposed fuzzy–stochastic option pricing methodology the process of valuing an equity value as a European call option will be solved.

Assume a firm has only two sources of capital, equity (F) and debt (D). Debt has a face value that matures dt years from now (dt=Tt). Creditors may not force the firm into bankruptcy until the maturity day. The firm pays no dividends. Under these assumptions the equity value can be computed as a European call option. If there is a possibility to force the firm into bankruptcy prior to expiration date, it is a case of an American call option. However, since the value of an American call option without dividends is the same as a European call option, the process of valuing should be identical. Proof of the statement is for example in Musiela and Rutkowski (1997).

The underlying assets is market value of the assets (A), equity (F) is call option premium. At maturity (T) the shareholders wealth isFT=max(0,AT−DT).If we depict the volatility of the assets value σA, continuously compounded risk-free interest rate r and expected equity value , then Black and Scholes pricing formulae for equity value as European call option is expressed as follows:F̄=f(A,D,dt,r,σA)=AN(d1)−e−rdtDN(d2),whered1=lnAD+r+σA22dtσAdt,d2=d1−σAdt,and N(·) is a standard normal cumulative distribution function.

Assume an asset's value is sum of equity and debt. With respect to the feedback relations the expected equity value () must be estimated by non-linear programming method as follows:

(Problem P1)MaxF̄accordingto(2)s.t.σA2=wD2σD2+wE2σE2+2wDwEσEσDρED,wD+wE=1,wE=F̄A.Simultaneously wDandwE are market percentages of debt and equity (wE=F̄/A), further σE2andσD2 are variances of equity and debt, and ρED is correlation coefficient.

Section snippets

Fuzzy–stochastic model development

In the case of applying contingent claim methodology for long-term valuing, it means the case of equity value calculation as well, the situation is somewhat difficult and different than a short-term financial security valuation, because the estimation of the equity value is determined by input data precision. Forecasting is more risky and vague and there are problems with data availability, quality, frequency and stochastic validity.

Thus one of the suitable approaches for solving this problem

Description of the B-S fuzzy–stochastic model of valuing firm equity as a call option

In the Problem P1 (, , , ) input data for B-S model are to be determined. It is traditionally supposed that these data are random or deterministic. It is often difficult to have crisp or statistically valid data for long-term decision-making and real option valuation. These data are in real options application, from practical and implementation respect, non-precise. Thus, it is reasonable to solve the problem under assumption of non-precise input data.

Illustrative example

We now give the example of calculating a contingent equity value. The value will be calculated by two ways as the stochastic and fuzzy stochastic problem. The stochastic approach is adapted from Damodaran (1994), example of valuing airline corporation.

Conclusion

Appraising a firm property contains aspects of contingent claim, risk (randomness) and indeterminacy (fuzziness). It is apparent that estimation of the financial terms (cash flow, assets and liability pattern) is determined by input data precision. The valuation process is usually made under deterministic or stochastic environment but uncertainty (vagueness) is mostly neglected and not considered. However, it is not easy to get statistically valid and quality data for long-term financial

Acknowledgements

The research was supported by Grant Agency of the Czech Republic (GAR) CEZ: J 17/98: 75100015. This support is gratefully acknowledged.

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