Fuzzy Volatility Research Articles

Fuzzy Fractional Brownian Motion: Review and Extension

In traditional finance, option prices are typically calculated using crisp sets of variables. However, as reported in the literature novel, these parameters possess a degree of fuzziness or uncertainty. This allows participants to estimate option prices based on their risk preferences and beliefs, considering a range of possible values for the parameters. This paper presents a comprehensive review of existing work on fuzzy fractional Brownian motion and proposes an extension in the context of financial option pricing. In this paper, we define a unified framework combining fractional Brownian motion with fuzzy processes, creating a joint product measure space that captures both randomness and fuzziness. The approach allows for the consideration of individual risk preferences and beliefs about parameter uncertainties. By extending Merton’s jump-diffusion model to include fuzzy fractional Brownian motion, this paper addresses the modelling needs of hybrid systems with uncertain variables. The proposed model, which includes fuzzy Poisson processes and fuzzy volatility, demonstrates advantageous properties such as long-range dependence and self-similarity, providing a robust tool for modelling financial markets. By incorporating fuzzy numbers and the belief degree, this approach provides a more flexible framework for practitioners to make their investment decisions.

Intelligent financial management of company based on neural network and fuzzy volatility evaluation

By integrating business processes, financial intelligent management system can provide effective data information with high quality and low cost for strategic decision-making. In this paper, the authors analyze the intelligent financial management of

SMART-or and SMART-and fuzzy average operators: A generalized proposal

In this paper we introduce new average operators for merging any number of fuzzy numbers, without any exogenous components. The proposed n-ary operators are based on a specific adaptation of Marzullo's algorithm, and depart from the usual fuzzy arithmetic mean according to the degree of agreement or disagreement among the memberships of input fuzzy numbers. Such merging operators are suitable to be applied in any model where the same quantity (usually a parameter) can be measured (estimated) through different fuzzy memberships stemming by different sources of information. The special case of two fuzzy memberships was the focus of our previous contributions that were elicited in order to estimate the fuzzy volatility parameter in an hybrid fuzzy-stochastic model for option pricing. In this paper we generalize the setting to the case of n fuzzy inputs to be merged and also remove exogenous factors from the definition of the operators. In order to have an application at hand we consider the same example treated in the quoted paper and we compare the outcomes obtained via the new operators, named SMART, with the fuzzy arithmetic mean as a canonical benchmark.

Indices for Financial Market Volatility Obtained Through Fuzzy Regression

The measurement of volatility is of fundamental importance in finance. Standard market practice adopted for volatility estimation from option prices leads to a considerable loss of information and the introduction of an element of arbitrariness in the volatility index computation. We propose to adopt fuzzy regression methods in order to include all the available information from option prices, and to obtain an informative volatility index. In fact, the obtained fuzzy volatility indices not only offer a most possible value, but also a lower and an upper bound for the interval of possible values, providing investors with an additional source of information. We also propose a defuzzification procedure to select a representative value within this interval. Moreover, we investigate the occurrence of truncation and discretization errors in volatility index computation by adopting an interpolation-extrapolation method. We also test the forecasting power of each volatility index on future realized volatility.

Pricing of minimum guarantees in life insurance contracts with fuzzy volatility

We propose a model for the pricing of the minimum guarantee option embedded in equity-linked life insurance policies under uncertainty of randomness and fuzziness. The future lifetime of the insured is modelled as a random variable and the asset price evolution is described using a fuzzy binomial-tree model. In order to deal with both randomness and fuzziness, we model the present value of liabilities as a fuzzy random variable. Our results can be used by the actuary to understand the incidence of the minimum guarantee on the premium and to define the appropriate coverage strategies. A numerical example illustrates how our methodology works.

Numerical Methods to Calculate Fuzzy Boundaries for Brownfield Redevelopment Negotiations

A numerical method is proposed to represent the likelihood of contamination of a brownfield using fuzzy boundaries, and then to estimate the parameters in a fuzzy real options model for brownfield evaluation from different decision maker perspectives. These different values can be used to facilitate negotiations on redevelopment projects. Linguistic quantifiers and ordered weighted averaging (OWA) techniques are utilized to determine the pollution likelihood at sample locations based on multiple environmental indicators. Risk preferences of decision makers are expressed as different “orness” levels of OWA operators, which affect likelihood estimates. When the fuzzy boundary of a brownfield is generated by interpolation of sample points, the parameters of fuzzy real options, drift rate and volatility, can be calculated as fuzzy numbers. Hence, this proposed method can act as an intermediate between decision makers and the fuzzy real options models, making this model much easier to apply. A potential negotiation support system (NSS) implementing these numerical methods is discussed in the context of negotiating brownfield redevelopment projects. A public–private-partnership will be enhanced through information sharing, scenario generation, and conflict analysis provided by the NSS, encouraging more efficient brownfield redevelopment and leading to greater regional sustainability.

Sustainable risk management: fuzzy approach to volatility and application on FTSE 100 index

In this paper, a fuzzy volatility labeling algorithm is offered to detect the periods with abnormal activities on daily share returns. Considering the vagueness in the switches of the time periods, the membership functions of high and normal volatility classes are introduced. In the assignments, both the density structure and membership degree are used. It is believed that this algorithm may be helpful to construct different estimation models for the time periods with normal and abnormal activities. Authors offer algorithm, which can be used as a tool for sustainable risk management.

Compound Option Pricing under Fuzzy Environment

Considering the uncertainty of a financial market includes two aspects: risk and vagueness; in this paper, fuzzy sets theory is applied to model the imprecise input parameters (interest rate and volatility). We present the fuzzy price of compound option by fuzzing the interest and volatility in Geske’s compound option pricing formula. For eachα, theα-level set of fuzzy prices is obtained according to the fuzzy arithmetics and the definition of fuzzy-valued function. We apply a defuzzification method based on crisp possibilistic mean values of the fuzzy interest rate and fuzzy volatility to obtain the crisp possibilistic mean value of compound option price. Finally, we present a numerical analysis to illustrate the compound option pricing under fuzzy environment.

On an implicit assessment of fuzzy volatility in the Black and Scholes environment

In this work we suggest a methodology to obtain the membership of a non-observable parameter through implicit information. To this aim we profit from the interpretation of membership functions as coherent conditional probabilities. We develop full details for the well known Black and Scholes pricing model where the membership of the volatility parameter is obtained from a sample of either asset prices or market prices for options written on that asset.

The double exponential jump diffusion model for pricing European options under fuzzy environments

The interest rate and volatility may have different values in the different commercial banks and financial institutions. Moreover, the fluctuations of the underlying assets are rare events, and there are not enough historical data to estimate the jump intensity in a precise sense. This paper considers European option pricing problems with the fuzzy interest rate, fuzzy drift, fuzzy volatility and fuzzy jump intensity. We present the fuzzy pricing formula of European options based on the Kou's double exponential jump diffusion model. We also obtain the crisp possibilistic mean option pricing formula in fuzzy double exponential jump diffusion model by using the crisp possibilistic mean values of the fuzzy numbers. Comparing with B-S formula, numerical analysis and empirical results show that the fuzzy double exponential jump diffusion formula and the crisp possibilistic mean option pricing formula are reasonable and can be taken as reference pricing tools for the financial investors.

MODEL CONSTRUCTION OF OPTION PRICING BASED ON FUZZY THEORY

Option pricing is a tool that investors often use for the purpose of arbitrage or hedging. However, both the BlackScholes model and the CRR model can only provide a theoretical reference value. The volatility in the CRR model cannot always appear in the precise sense because the financial markets fluctuate from time to time. Hence, the fuzzy volatility is naturally to be considered. The main purpose of this paper is the application of fuzzy sets theory to the CRR model. It is expected that fuzzy volatility, instead of the crisp values conventionally used in the CRR model, can provide reasonable ranges and corresponding memberships for option prices, as a result of which, investors can interpret optimal value differently for different risk preferences. This paper shows a new method for option valuation using fuzzy set theory that is based on findings from earlier option valuation methods and from fuzzy membership function. In conclusion, the empirical evidence indicates the effectiveness of the proposed fuzzy model.

Pricing European Call Currency option based on Adaptive Fuzzy Numbers with Possibilistic Mean

In this paper we use the fuzzy estimators based on confidence intervals in order to estimate the volatility of currency exchange rate having sample data. We model the uncertainty of the characteristics such as interest rates and volatility using adaptive fuzzy numbers and replace the fuzzy currency option price and the fuzzy volatility by the possibilistic mean value. Furthermore, a numerical example is presented, and we get the expected prices which depend on given degree of confidence. Key words: Currency Option; Option Pricing; Possibilistic Mean Value; Fuzzy Volatility; G-K Model

A Fuzzy LP Approach to Option Portfolio

Abstract: Owing to the fluctuation of financial market from time to time,the volatility and stock price may occur imprecisely in the real world. Therefore, it is natural to consider the fuzzy volatility and fuzzy stock price in the financial market. Under these assumptions,the theoretical price deduced by Black–Scholes formula are will turn into the fuzzy numbers, and the derivatives, called the Greek parameters delta, gamma, of the BS model are also fuzzy numbers. An option portfolio considering these fuzzy numbers will be more accord with actual situations. In this paper, we propose a fuzzy programming model of option portfolio based a ranking criterion of fuzzy numbers, which the fuzzy option portfolio model is converted into a classical linear programming problem. Finally, a numerical example is given to illustrate the validity of the method. Key words: Ranking Criterion; Option portfolio; Fuzzy Linear Programming; Delta-Gamma Neutral

A study of Greek letters of currency option under uncertainty environments

Owing to the fluctuations in the financial markets from time to time, some input variables, in particular the interest rate, spot exchange rate and volatility, in the Garman–Kohlhagen model [M.B. Garman, S.W. Kohlhagen, Foreign currency option values, Journal of International Money and Finance 2 (1983) 231–237] cannot be expected in a precise sense. Therefore, the fuzzy set theory introduced by Zadeh [L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353] can provide a useful tool to deal with this kind of impreciseness. In this paper, three different versions of the G–K model, the put-call parity relationship and the calculation formulas of the Greek letters according to these three different G–K models are then proposed based on fuzzy set theory. Under the considerations of fuzzy foreign and domestic interest rates, fuzzy spot exchange rate and fuzzy volatility, the European currency option prices and the Greek letters turn into fuzzy numbers or closed intervals. This means that the investors can pick any European currency option price and Greek letters with an acceptable belief degree or belief degree weighting parameter for his later use. The empirical results indicate that the Greeks calculated under fuzzy environment can be considered as a useful tool for managing option risk for an option writer.

On theoretical pricing of options with fuzzy estimators

In this paper we present an application of a new method of constructing fuzzy estimators for the parameters of a given probability distribution function, using statistical data. This application belongs to the financial field and especially to the section of financial engineering. In financial markets there are great fluctuations, thus the element of vagueness and uncertainty is frequent. This application concerns Theoretical Pricing of Options and in particular the Black and Scholes Options Pricing formula. We make use of fuzzy estimators for the volatility of stock returns and we consider the stock price as a symmetric triangular fuzzy number. Furthermore we apply the Black and Scholes formula by using adaptive fuzzy numbers introduced by Thiagarajah et al. [K. Thiagarajah, S.S. Appadoo, A. Thavaneswaran, Option valuation model with adaptive fuzzy numbers, Computers and Mathematics with Applications 53 (2007) 831–841] for the stock price and the volatility and we replace the fuzzy volatility and the fuzzy stock price by possibilistic mean value. We refer to both cases of call and put option prices according to the Black & Scholes model and also analyze the results to Greek parameters. Finally, a numerical example is presented for both methods and a comparison is realized based on the results.

Using fuzzy sets theory and Black–Scholes formula to generate pricing boundaries of European options

The application by using extension principle in fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the fluctuation of financial market from time to time, the risk-free interest rate, volatility and stock price may occur imprecisely in the real world. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price in the financial market. Under these assumptions, the European call and put option price will turn into the fuzzy numbers, and the extension principle will be invoked to generate the pricing boundaries of the European call and put options. This will make the financial analyst who can pick any reasonable European option price with an acceptable belief degree for his/her later financial analysis. In order to obtain the belief degree, an optimization problem has to be solved.

A Fuzzy Set Approach for Generalized CRR Model: An Empirical Analysis of S&P 500 Index Options

This paper applies fuzzy set theory to the Cox, Ross and Rubinstein (CRR) model to set up the fuzzy binomial option pricing model (OPM). The model can provide reasonable ranges of option prices, which many investors can use it for arbitrage or hedge. Because of the CRR model can provide only theoretical reference values for a generalized CRR model in this article we use fuzzy volatility and fuzzy riskless interest rate to replace the corresponding crisp values. In the fuzzy binomial OPM, investors can correct their portfolio strategy according to the right and left value of triangular fuzzy number and they can interpret the optimal difference, according to their individual risk preferences. Finally, in this study an empirical analysis of S&P 500 index options is used to find that the fuzzy binomial OPM is much closer to the reality than the generalized CRR model.

Pricing European options based on the fuzzy pattern of Black–Scholes formula

The application of fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the fluctuation of financial market from time to time, some input parameters in the Black–Scholes formula cannot always be expected in the precise sense. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price. The fuzzy pattern of Black–Scholes formula and put–call parity relationship are then proposed in this paper. Under these assumptions, the European option price will turn into a fuzzy number. This makes the financial analyst who can pick any European option price with an acceptable belief degree for the later use. In order to obtain the belief degree, an optimization problem has to be solved.