Pricing European Option Under Fuzzy Mixed Fractional Brownian Motion Model with Jumps
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Pricing European Option Under Fuzzy Mixed Fractional Brownian Motion Model with Jumps Wei‑Guo Zhang1 · Zhe Li2 · Yong‑Jun Liu1 · Yue Zhang3 Accepted: 27 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract As we all know, the financial environment on which option prices depend is very complex and fuzzy, which is mainly affected by the risk preferences of investors, economic policies, markets and other non-random uncertainty. Thus, the input data in the options pricing formula cannot be expected to be precise. However, fuzzy set theory has been introduced as a main method for modeling the uncertainties of the input parameters in the option pricing model. In this paper, we discuss the pricing problem of European options under the fuzzy environment. Specifically, to capture the features of long memory and jump behaviour in financial assets, we propose a fuzzy mixed fractional Brownian motion model with jumps. Subsequently, we present the fuzzy prices of European options under the assumption that the underlying stock price, the risk-free interest rate, the volatility, the jump intensity and the mean value and variance of jump magnitudes are all fuzzy numbers. This assumption allows the financial investors to pick any option price with an acceptable belief degree to make investment decisions based on their risk preferences. In order to obtain the belief degree, the interpolation search algorithm has been proposed. Numerical analysis and examples are also presented to illustrate the performance of our proposed model and the designed algorithm. Finally, empirically studies are performed by utilizing the underlying SSE 50 ETF returns and European options written on SSE 50 ETF. The empirical results indicate that the proposed pricing model is reasonable and can be treated as a reference pricing tool for financial analysts or investors. Keywords Fuzzy stochastic differential equation · Mixed fractional Brownian motion · European option pricing · Fuzzy jump-diffusion · Interpolation search algorithm
* Zhe Li [email protected] Extended author information available on the last page of the article
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1 Introduction The pricing, hedging and risk management of derivative products are very important since derivative products are now widely used to transfer risk in financial markets. However, the pricing problem of derivatives is probably one of the most challenging topics in modern financial theory. Since Black and Scholes (1973) originally developed their option pricing formula, there have been considerable researches on the option pricing theory. Although the success of the Black–Scholes model under the premise of the stock price follows a geometric Brownian motion, a lot of empirical studies have indicated that it is inadequate to precisely model the dynamics of the underlying asset price. In real financial markets, the log returns of the financial assets exhibit the asymmetric leptokurtic features, i.e., the asset return distribution has a hig
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