Reliable Control of Convergence in Monte Carlo Pricing Methods for Options based on MSPE Technique MOSCA ROBERTO - CASSETTARI LUCIA University of Genoa Department of Production Engineering and Mathematical Modelling 15 Opera Pia, 16125 Genoa ITALY mosca@diptem.unige.it, cassettari@diptem.unige.it GIRIBONE PIER GIUSEPPE CARIGE Bank Financial Administration 15 Cassa di Risparmio, 16123 Genoa ITALY piergiuseppe.giribone@carige.it Abstract: In pricing complex derivatives, the research offices of banks adopt mathematical models comprising statistical frequency distributions in order to determine the value of said derivatives. Therefore, the correct appli- cation of the Monte Carlo simulation is an essential method for the pricing of the derivative under consideration. However, in order to obtain a correct final price, it is necessary to determine the number of repeated runs needed in the model. In this study the authors propose the application of the Mean Square Pure Error method in repeated runs, as a technique to control convergence with the fair value of the derivative. The inadequacy of relying on standard numbers of launches (1000 and 10000) as proposed in technical manuals for practitioners or in numerous examples of scientific literature in the field will also be underscored. Likewise it will be demonstrated that some of the mathematical methods proposed by academic literature cannot always be considered adequate to monitor errors made in the pricing of the derivative. Key–Words: Complex Derivatives, Pricing Convergence, Monte Carlo Simulation, Experimental Error, Mean Square Pure Error. 1 Introduction An aspect that is too often neglected in Monte Carlo simulation applications is the control of the so-called pure experimental error, [12]. This error affecting the model is generally distributed as a NID(0,σ 2 ). The value of σ 2 , which, according to Cochran’s theorem, can be estimated by calculating the Mean Square Pure Error quantity, its unbiased estimator, is an intrinsic characteristic of the model built and it is connected with the stochasticity affecting the real system, [10]. During the experimental phase what is really impor- tant is not to add to the noise characterizing the sys- tem under study a second source of experimental error due to a number of extractions from the probability distributions of the input variables inadequate to ob- tain a complete statistical description. As it is widely known, the larger the sample, the better the statisti- cal inference on the population. In simulation models using the Monte Carlo method, the Experimental Er- ror hence varies with the variation of the sample’s size and depends on the number of times the simulation is replicated, namely the function of the number of repli- cated runs, [11], [13], [14]. Many researchers study- ing these issues recommend a rather large number of replicated runs. This number of replications is gene- rally included between 1000 and 10000 runs without a suitable knowledge of the connected experimental error whose size will strongly impact on simulation output. Furthermore, it should be noted that gene- rally, in practice, only the mean value of the response is taken into account in these studies, while no im- portance is given to the variance, which instead could play a decisive role in stochastic systems. In fact, the knowledge of the average value of the response only may lead even to gross errors in the evaluation of the phenomenon studied. This article aims to illustrate pricing examples of derivatives using the Monte Carlo technique. They show, on the one hand, the validity of the method proposed by the authors for the optimal control of convergence with the fair-value and, on the other, underscore the lack of adequacy of traditional techniques in the literature. 2 Methodological Approach The proposed methodology makes it possible to ap- proach the problem in a scientific way since it allows for the identification of the number of replications ca- pable of minimizing the noise produced by an inade- quate overlapping of the probability density function of the input variables extracted using the Monte Carlo method. In order to obtain these results, it is ne-