A. Thavaneswaran

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.

Optimal estimation for semimartingales

This paper extends a result of Godambes theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. Let P = {P} be a family of probability measures such that (fI, F,P) is complete, (F, t > 0) is a standard filtration, and X = (X,, F, t > 0) is a semimartingale for every P E P. For a parameter 0(P), suppose X, = V,, + H,.e where the Vo process is predictable and locally of bounded variation and the He process is a local martingale. Consider estimating equations for 0 of the form f, au,odHue =0 where the a, process is predictable. Under regularity conditions, an optimal form for ao in the sense of Godambe (1960) is determined. When V,,o is linear in 0 the optimal 0, corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of 6, are discussed.

Non-parametric estimation of the conditional mode

The problem of estimating the mode of a conditional probability density function is considered. It is shown that under some regularity conditions the estimate of the conditional mode obtained by maximizing a kernel estimate of the conditional probability density function is strongly consistent and asymptotically normally distributed.

Prediction via estimating functions

This paper is concerned with prediction methods for linear as well as non-linear non-Gaussian models. Recursive formulas are obtained by combining the information associated with the predictive functions. Nonlinear predictors are obtained for linear and nonlinear time series models. The innovation algorithm is shown to be a special case of the proposed algorithm for linear processes with known autocovariances. Least absolute deviation predictors are shown to be a special case as well. Recursive prediction incorporating the variability due to parameter estimation is also discussed in some detail.

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.

A nonlinear time series model and estimation of missing observations

This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.

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.