Fuzzy Levy-GJR-GARCH American Option Pricing Model Based on an Infinite Pure Jump Process

Abstract

SUMMARYThis paper focuses mainly on issues related to the pricing of American options under a fuzzy environment by taking into account the clustering of the underlying asset price volatility, leverage effect and stochastic jumps. By treating the volatility as a parabolic fuzzy number, we constructed a Levy-GJR-GARCH model based on an infinite pure jump process and combined the model with fuzzy simulation technology to perform numerical simulations based on the least squares Monte Carlo approach and the fuzzy binomial tree method. An empirical study was performed using American put option data from the Standard & Poor's 100 index. The findings are as follows: under a fuzzy environment, the result of the option valuation is more precise than the result under a clear environment, pricing simulations of short-term options have higher precision than those of medium-and long-term options, the least squares Monte Carlo approach yields more accurate valuation than the fuzzy binomial tree method, and the simulation effects of different Levy processes indicate that the NIG and CGMY models are superior to the VG model. Moreover, the option price increases as the time to expiration of options is extended and the exercise price increases, the membership function curve is asymmetric with an inclined left tendency, and the fuzzy interval narrows as the level set α and the exponent of membership function n increase. In addition, the results demonstrate that the quasi-random number and Brownian Bridge approaches can improve the convergence speed of the least squares Monte Carlo approach. key words: American option, fuzzy set theory, fuzzy simulation technology, Levy process, GJR-GARCH model, least squares Monte Carlo approach, binomial tree method, quasi-random number, Brownian Bridge approach

Fuzzy Random Option Pricing in Continuous Time: A Systematic Review and an Extension of Vasicek’s Equilibrium Model of the Term Structure

“…In most cases, the assumed form of the fuzzy magnitude is linear, i.e., triangular or trapezoidal. However, within type-1 fuzzy numbers, the literature has also used other shapes, such as adaptive fuzzy numbers [36,56,57], Gaussian fuzzy numbers [35], or parabolic fuzzy numbers [94]. The parameters that are considered crisp and those that are considered fuzzy are established ad hoc depending on the problem being addressed.…”

“…A great proportion of papers price European options. However, other types of options, such as American [47,71,72,94], binary [47,49,55,89], exchange [26,27,29,61], or compound [60,91,98] options, have been analysed. Sensitivity analysis of option prices from the perspective of the BSM model has been the subject of attention by various authors [35,36,41,46,69].…”

“…Papers [24,25] observe good adherence of the fuzzy version of the standard BSM formula to traded prices on the Spanish stock index IBEX-35. Furthermore, [77], in its extension of [104], and [93,94], in their fuzzy-Levy modelling, perform comprehensive empirical applications for options on Standard & Poor indexes.…”

“…Naturally, since the parameters that govern the movements of short-term interest rates have a clear economic interpretation, it could be assumed that they can be set intuitively by experts. Alternatively, following [31,32,36,52,77,94], we can adjust the parameters that are assumed to be fuzzy numbers based on existing evidence in financial markets. Likewise, it should be noted that the existence of studies that empirically apply FROPCP developments is relatively scarce [19,20].…”

A Fuzzy-Random Extension of Jamshidian’s Bond Option Pricing Model and Compatible One-Factor Term Structure Models

“…Developments in continuous time typically rely on conventional univariate geometric Brownian motion [25][26][27], which can be multivariate in the case of compound options [32]. However, the fuzzy-random literature has also extended to more complex continuous time modeling, such as Lévy processes [33,34] or fractional-type random movements [35][36][37][38]. In discrete time, the most common approach is provided by the binomial framework [31], with examples in this context found in [39][40][41][42][43].…”