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Pricing European Options with Triangular Fuzzy Parameters: Assessing Alternative Triangular Approximations in the Spanish Stock Option Market

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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.

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Notes

  1. Notice that since 2016, in Eurozone, interest rates have been negative in all short-term monetary markets due to the expansive monetary policy of the European Central Bank.

  2. In [22], it is developed a non-triangular approximating method for fuzzy option prices that also uses the Greeks. This paper proposes a flexible polynomial approximation to the fuzzy price of options, which is built up with the value of the extremes of the α-cuts in certain predefined knots and the value of the Greeks in these knots.

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Acknowledgements

The author thanks anonymous reviewers for their constructive comments and suggestions.

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Authors and Affiliations

  1. University Rovira i Virgili, Social and Business Research Laboratory, Department of Business Management, Avinguda Universitat 1, 43204, Reus, Spain

    Jorge de Andrés-Sánchez

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Correspondence to Jorge de Andrés-Sánchez.

Appendix: Relative Deviations of Three Alternative Triangular Approximations to the Fuzzy Black and Scholes Formula

Appendix: Relative Deviations of Three Alternative Triangular Approximations to the Fuzzy Black and Scholes Formula

See Tables 9, 10, 11, 12, 13 and 14.

Table 9 \(\underline{wae} \left( {\alpha^{*} } \right)\) and \(\overline{wae} \left( {\alpha^{*} } \right)\) of TFNAs to call prices with expiration date 3/24/2017
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Table 10 \(\underline{wae} \left( {\alpha^{*} } \right)\) and \(\overline{wae} \left( {\alpha^{*} } \right)\) of TFNAs to call prices with expiration date 4/21/2017
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Table 11 \(\underline{wae} \left( {\alpha^{*} } \right)\) and \(\overline{wae} \left( {\alpha^{*} } \right)\) of TFNAs to call prices with expiration date 5/19/2017
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Table 12 \(\underline{wae} \left( {\alpha^{*} } \right)\) and \(\overline{wae} \left( {\alpha^{*} } \right)\) of TFNAs to put prices with expiration date 3/24/2017
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Table 13 \(\underline{wae} \left( {\alpha^{*} } \right)\) and \(\overline{wae} \left( {\alpha^{*} } \right)\) of TFNAs to put prices with expiration date 4/21/2017
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Table 14 \(\underline{wae} \left( {\alpha^{*} } \right)\) and \(\overline{wae} \left( {\alpha^{*} } \right)\) of TFNAs to put prices with expiration date 5/19/2017
Full size table

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de Andrés-Sánchez, J. Pricing European Options with Triangular Fuzzy Parameters: Assessing Alternative Triangular Approximations in the Spanish Stock Option Market. Int. J. Fuzzy Syst. 20, 1624–1643 (2018). https://doi.org/10.1007/s40815-018-0468-5

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