Pricing European Options with Triangular Fuzzy Parameters: Assessing Alternative Triangular Approximations in the Spanish Stock Option Market Jorge de Andre ´s-Sa ´nchez 1 Received: 12 May 2017 / Revised: 24 January 2018 / Accepted: 15 February 2018 Ó Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract The aim of this paper is contributing from a practical and empirical perspective to option pricing under fuzziness. When we evaluate Black–Scholes option pricing formula with triangular fuzzy numbers, we obtain a non- triangular price that may be slightly difficult to use in practical applications. We improve the applicability of the fuzzy version of that formula by proposing and testing three triangular approximations when the subjacent asset price, its volatility and free interest rate are triangular fuzzy numbers. To check the goodness of these approximations, we firstly evaluate their closeness to the actual values of fuzzy Black and Scholes model. We find that the quality of those approximations depends on options maturity and moneyness grade and if we are pricing call or put options. We also assess the capability of those approximating methods to reflect satisfactorily real market prices and obtain good results. To develop all empirical applications, we use a sample of options on IBEX35 traded in the Spanish derivatives market on 3/1/2017. Keywords Fuzzy numbers  Fuzzy number approximation  Finance  Option pricing  Black–Scholes formula 1 Introduction Since the publication of Black–Scholes (BS) formula [6], option pricing (OP) has become a prominent branch of financial economics. Although stochastic modeling is at the heart of OP, in areas related to economics and finance such as asset pricing, a great deal of information is imprecise and vague. Fuzzy set theory (FST) is undoubtedly a suit- able way to model this kind of information. That explains why OP with fuzzy parameters is nowadays an active research field. So [2, 12, 51] apply FST in real OP and [8, 11, 13, 15, 23, 28, 31, 33, 46–49, 52] use FST to extend BS formula and related concepts as the Greeks or implied volatility to the use of imprecise parameters. FST also has been applied to other OP models that are based on more complex hypothesis about the behavior of subjacent asset than geometrical Brownian motion. So, [50] assumes a double exponential diffusion jump and [34–37] extend Levy processes hypothesis to fuzziness in parameters. Likewise, [18] fuzzifies Heston’s stochastic volatility model [25]. Notice that not only options of European or American style have been object of fuzzy versions. Thus, [40, 42, 44, 45] develop exotic option pricing with fuzzy parameters. A reader interested in a deep review on fuzzy OP can consult [32]. The strike price and maturity of financial options are always crisp parameters. So, in these options the uncer- tainty comes from the price and volatility of the underlying asset and the discount rate. Fuzzy OP models quantify those parameters by fuzzy numbers (FNs) that in this paper, as usually, are assumed to be triangular fuzzy numbers (TFNs). Unfortunately, under this hypothesis, option prices are not TFNs. This fact makes fuzzy version of BS formula (FBS) slightly difficult to use in practical market situations and motivates us developing and testing three alternative & Jorge de Andre ´s-Sa ´nchez jorge.deandres@urv.cat 1 University Rovira i Virgili, Social and Business Research Laboratory, Department of Business Management, Avinguda Universitat 1, 43204 Reus, Spain 123 Int. J. Fuzzy Syst. https://doi.org/10.1007/s40815-018-0468-5