key: cord-0737636-3996xuif authors: Kharrat, Mohamed; Arfaoui, Hassen title: A New Stabled Relaxation Method for Pricing European Options Under the Time-Fractional Vasicek Model date: 2022-05-06 journal: Comput Econ DOI: 10.1007/s10614-022-10264-4 sha: 8deaf2830e4d0e8edf510ef6dba98ef98e8ef1e2 doc_id: 737636 cord_uid: 3996xuif Our objective is to solve the time-fractional Vasicek model for European options with a new stabled relaxation method. This new approach is based on the splitting method. Some numerical tests are presented to show the stability and the reliability of our approach with the theory of options. Pricing derivatives and especially options is one of the most popular problems in mathematical financial literature. For instance, European options are very popular in the worldwide financial markets. Over the last few decades, several papers investigated the problem of pricing options generated by different models using many methods for instance (Black & Scholes, 1973; Bensoussan, 1984; Heston, 1993; Kharrat, 2014) . The most famous are the Black and Scholes model (Black & Scholes, 1973) and Heston model (Heston, 1993) , which the first one rests upon the concept that the stock price of the underlying asset is log-normally distributed conditional on the current stock price with constant volatility. As compared to the case of the Black and Scholes model, where the volatility is constant, the Heston model (Heston, 1993) is more important since the volatility is stochastic, as the dynamics of the volatility is fundamental to elaborate strategies for hedging and arbitrage, a model based on constant volatility cannot explain the reality of the financial markets. But for the case where the interest rate is stochastic we are forced to use the Vasicek Model (Vasicek, 1977) . So, pricing option under stochastic model is then more important and required. In the following we introduce the standard Vasiček model. Let S t be the asset price generated by the following dynamic: dS t ¼ r t S t dt þ rS t dW S t ð1:1Þ and r t be the interest rate process which follows the following process: where the volatility r supposed to be constant, W S t and W r t are two correlated Brownian motion i.e. W S t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À q 2 p B 1 t þ B 2 t and W r t ¼ B 2 t where B is Standard 2dimensional Brownian motion and q 2 À1; 1 ½ and the parameters Àl 1 l 2 , Àl 2 and g represent respectively the long-term mean level, the speed of the reversion, and the volatility of the interest rate r t . Let # the price of European option, using the standard hedging and the application of Ito's lemma we get: The fractional calculus is invested in several fields (Amit et al., 2019; Benchohra et al., 2011; Daftardar-Gejji & Bhalekar, 2008; Dumitru et al., 2020; Podlubny, 1999; Srivastava et al., 2020) and (Yu et al., 2011) . For example, fractional derivation models have shown an ability to describe shape-memory materials better than full derivation models. When a material is purely elastic, it is described by an integer derivation of order zero while when it is purely viscous it is described by an integer derivation of order one. Immediately, we can describe a viscous-elastic material by a derivation between 0 and 1. This justifies the use of fractional derivation for this kind of material. So out of mathematical curiosity and to get closer to the reality of the financial market we find ourselves obliged to use models based on fractional derivatives. Recently, it has been integrated in the Mathematical finance field (Yu et al., 2011; Xiaozhong et al., 2016; Kharrat, 2021) especially designed to resolve the pricing option problem. For instance (Kharrat, 2018; Zhang et al., 2016) which are devoted for the evaluation of the European option. From this perspective, using the splitting method, we present a new resolution for the pricing European option under the fractional Vasicek model. The aforesaid method allowing to solve a mixed problem Parabolic/Hyperbolic by decoupling the parabolic and hyperbolic operators, (for more details see Arfaoui, 2020) . A nonlinear mixed problem generated by two completely different operators, (Parabolic/ Hyperbolic), can cause difficulties in the numerical simulations. During discretization, the splitting method makes it possible to treat each operator Parabolic and Hyperbolic by an adequate numerical scheme. This method preserves the numerical properties (stability, consistency, Á Á Á) of each scheme used for each operator. This new method allowed us to give relevant numerical results besides we found in the literature that the coefficient of correlation it's always between À 0:7 and 0.7. With our new numerical method, we can extend the aforesaid coefficient between À 0:9 and 0.9. In the following definition, we present the Caputo time-fractional derivative. where CðÁÞ is the Gamma function given by This definition of fractional derivative is interesting, among other reasons, because it holds properties of the non-fractional derivatives as to make nullthe derivative of a constant (see Podlubny, 1999) . In our work, we will use the definition when m ¼ 1. Definition 1.2 (Erdelyi etal., 1981) The Mittag-Leffler function of one parameter is defined as: where CðÁÞ is the gamma function. Definition 1.3 (Erdelyi etal., 1981) The Mittag-Leffler function of two parameters is defined as: where CðÁÞ is the gamma function. The outline of this work is as follows. In Sect. 2, we introduce the time fractional Vasicek model. The Splitting method and the discretization of the Model are derived respectively in Sects. 3 and 4. In Sect. 5, we present the numerical analysis and discuss the stability of the solution. In Sect. 6, we present some numerical results. 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