S. S. Appadoo

Weighted possibilistic moments of fuzzy numbers with applications to GARCH modeling and option pricing

Carlsson and Fuller [C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315-326] have introduced possibilistic mean, variance and covariance of fuzzy numbers and Fuller and Majlender [R. Fuller, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363-374] have introduced the notion of crisp weighted possibilistic moments of fuzzy numbers. Recently, Thavaneswaran et al. [A. Thavaneswaran, K. Thiagarajah, S.S. Appadoo, Fuzzy coefficient volatility (FCV) models with applications, Mathematical and Computer Modelling 45 (2007) 777-786] have defined non-centered nth order possibilistic moments of fuzzy numbers. In this paper, we extend these results to centered moments and find the kurtosis for a class of FCA (Fuzzy Coefficient Autoregressive) and FCV (Fuzzy Coefficient Volatility) models. We also demonstrate the superiority of the fuzzy forecasts over the minimum square error forecast through a numerical example. Finally, we provide a description of option price specification errors using the fuzzy weighted possibilistic option valuation model.

Option valuation model with adaptive fuzzy numbers

In this paper, we consider moment properties for a class of quadratic adaptive fuzzy numbers defined in Dubois and Prade [D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980]. The corresponding moments of Trapezoidal Fuzzy Numbers (Tr.F.Ns) and Triangular Fuzzy Numbers (T.F.Ns) turn out to be special cases of the adaptive fuzzy number [S. Bodjanova, Median value and median interval of a fuzzy number, Information Sciences 172 (2005) 73-89]. A numerical example is presented based on the Black-Scholes option pricing formula with quadratic adaptive fuzzy numbers for the characteristics such as volatility parameter, interest rate and stock price. Our approach hinges on a characterization of imprecision by means of fuzzy set theory.

Fuzzy coefficient volatility (FCV) models with applications

Recently, Carlsson and Fuller [C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315-326] have introduced possibilistic mean, variance and covariance of fuzzy numbers and Fuller and Majlender [R. Fuller, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363-374] have introduced the notion of crisp weighted possibilistic moments of fuzzy numbers. In this paper, we propose a class of FCV (Fuzzy Coefficient Volatility) models and study the moment properties. The method used here is very similar to the method used in Appadoo et al. [S.S. Appadoo, M. Ghahramani, A. Thavaneswaran, Moment properties of some time series models, Math. Sci. 30 (1) (2005) 50-63]. The proposed models incorporate fuzziness, subjectivity, arbitrariness and uncertainty observed in most financial time series. The usual forecasting method does not incorporate parameter variability. Fuzzy numbers are used to model the parameters to incorporate parameter variability.

Application of possibility theory to investment decisions

Carlson and Fuller (2001, Fuzzy Sets and Systems, 122, 315–326) introduced the concept of possibilistic mean, variance and covariance of fuzzy numbers. In this paper, we extend some of these results to a nonlinear type of fuzzy numbers called adaptive fuzzy numbers (see Bodjanova (2005, Information Science, 172, 73–89) for detail). We then discuss the application of these results to decision making problems in which the parameters may involve uncertainty and vagueness. As an application, we develop expression for fuzzy net present value (FNPV) of future cash flows involving adaptive fuzzy numbers by using their possibilistic moments. An illustrative numerical example is given to illustrate the results.

Binary option pricing using fuzzy numbers

Abstract A binary option is a type of option where the payout is either fixed after the underlying stock exceeds the predetermined threshold (or strike price) or is nothing at all. Traditional option pricing models determine the option’s expected return without taking into account the uncertainty associated with the underlying asset price at maturity. Fuzzy set theory can be used to explicitly account for such uncertainty. Here we use fuzzy set theory to price binary options. Specifically, we study binary options by fuzzifying the maturity value of the stock price using trapezoidal, parabolic and adaptive fuzzy numbers.

Random coefficient GARCH models

Both volatility clustering and conditional nonormality can induce the leptokurtosis typically observed in financial data. An ARMA representation is used to derive the kurtosis of the various class of GARCH models such as power GARCH, non-Gaussian GARCH, nonstationary and random coefficient GARCH. Formula for autocorrelations of the power GARCH process |yt|^@d are given in terms of @j-weights. The kurtosis is also derived for random coefficient GARCH, nonstationary GARCH with possibly nonnormal errors and for hidden Markov GARCH models. The theoretical autocorrelation functions for various GARCH(1,1) models are also derived.

RCA models with GARCH innovations

Abstract Rapid developments of time series models and methods addressing volatility in computational finance and econometrics have been recently reported in the financial literature. The non-linear volatility theory either extends and complements existing time series methodology by introducing more general structures or provides an alternative framework (see Abraham and Thavaneswaran [B. Abraham, A. Thavaneswaran, A nonlinear time series model and estimation of missing observations, Ann. Inst. Statist. Math. 43 (1991) 493–504] and Granger [C.W.J. Granger, Overview of non-linear time series specification in Economics, Berkeley NSF-Symposia, 1998]). In this work, we consider Gaussian first-order linear autoregressive models with time varying volatility. General properties for process mean, variance and kurtosis are derived; examples illustrate the wide range of properties that can appear under the autoregressive assumptions. The results can be used in identifying some volatility models. The kurtosis of the classical RCA model of Nicholls and Quinn [D.F. Nicholls, B.G. Quinn, Random Coefficient Autoregressive Models: An Introduction, in: Lecture Notes in Statistics, vol. 11, Springer, New York, 1982] is shown to be a special case.

Option pricing for some stochastic volatility models

Purpose – To study stochastic volatility in the pricing of options. Design/methodology/approach – Random-coefficient autoregressive and generalized autoregressive conditional heteroscedastic models are studied. The option-pricing formula is viewed as a moment of a truncated normal distribution. Findings – Kurtosis for RCA and for GARCH process is derived. Application of random coefficient GARCH kurtosis in analytical approximation of option pricing is discussed. Originality/value – Findings are useful in financial modeling.

Recent developments in volatility modeling and applications

In financial modeling, it has been constantly pointed out that volatility clustering and conditional nonnormality induced leptokurtosis observed in high frequency data. Financial time series data are not adequately modeled by normal distribution, and empirical evidence on the non-normality assumption is well documented in the financial literature (details are illustrated by Engle (1982) and Bollerslev (1986)). An ARMA representation has been used by Thavaneswaran et al., in 2005, to derive the kurtosis of the various class of GARCH models such as power GARCH, non-Gaussian GARCH, nonstationary and random coefficient GARCH. Several empirical studies have shown that mixture distributions are more likely to capture heteroskedasticity observed in high frequency data than normal distribution. In this paper, some results on moment properties are generalized to stationary ARMA process with GARCH errors. Application to volatility forecasts and option pricing are also discussed in some detail.

Modified difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers

Li and Yang (D.F. Li and J. Yang, A difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers, Journal of Applied Mathematics 2013 (2013), 1-10) pointed out that there is no method in the literature for solving such bimatrix games in which payoffs are represented by intuitionistic fuzzy numbers and proposed a method for the same. In this paper, it is pointed out that Li and Yang have considered some mathematical incorrect assumptions in their proposed method. For resolving the shortcomings of Li and Yangs method, a new method (named as Mehar method) is proposed. Also, the exact optimal solution of the numerical problem, solved by Li and Yang by their proposed method, is obtained by the proposed Mehar method.