خطای قیمت‌گذاری اختیار معامله: شواهدی از بازارهای غیرخطی مبتنی بر شبکه‌های عصبی احتمالی و پرسپترون چندلایه

هدف: هدف از انجام این پژوهش مقایسۀ خطای قیمت‌گذاری دو مدل بارلز‌سونر و باکستین‌‌هاویسون در بازار اختیار معاملۀ شاخص s p500 است. این مقایسه برای مشخص‌کردن نزدیک‌ترین مدل به بازار اختیار معامله انجام شده است.روش: در ابتدا پس از معرفی دو مدل بارلز‌سونر و باکستین‌‌هاویسون به حل معادلات دیفرانسیل جزئی با گروه‌های لی پرداخته شده است. سپس با داده‌های تاریخی شاخص s p500 از  آگوست  تا  آگوست ، قیمت اختیار معامله این دارایی تحت هر دو مدل محاسبه شده است. سپس داده‌های به‌دست‌آمده با شبکه‌های عصبی پرسپترون چند‌لایه و احتمالی دسته‌بندی شده‌اند. بعد از آموزش، شبکه‌ها با ارائۀ داده‌های آزمون بررسی شده و نشان داده‌اند که بازار به کدام مدل نزدیک‌تر است. در ادامه، علاوه‌بر دسته‌بندی داده‌ها با شبکه‌های عصبی، به روش جبرلی به قیمت‌گذاری اختیار معاملۀ s p500 پرداخته و جواب حاصل با مقادیر واقعی اختیار معامله در بازار مقایسه شده است.یافته‌ها و نوآوری: با داده‌های آماری بعد از 18 آگوست 2023، شبکه‌های عصبی احتمالی و پرسپترون چند‌لایه آزموده شده‌اند. سپس به‌کمک همان داده‌ها معادلات بارلز‌سونر و باکستین‌‌هاویسون قیمت‌گذاری و اختلاف آنها از قیمت واقعی بازار محاسبه شده است. در آزمودن شبکه‌ها، شبکۀ عصبی پرسپترون چند‌لایه 60 درصد از داده‌های آزمون و شبکۀ عصبی احتمالی تمامی داده‌ها را در دسته بارلز‌سونر قرار داد. در محاسبۀ اختلاف جواب‌های گروه‌های لی با داده‌های واقعی بازار، 80 درصد داده‌ها اختلاف کمتری با مدل بارلز‌سونر داشت. درنتیجه، قیمت واقعی اختیار ‌معاملۀ s p500 در بازار به مدل بارلز‌سونر نزدیک‌تر بوده است. به‌عبارت دیگر، در این بازار مدل بارلز‌سونر خطای کمتری نسبت‌به مدل باکستین‌‌هاویسون داشته است.

option pricing error: evidence from nonlinear markets based on probabilistic neural networks and multilayer perceptron

this sudy aims to compare the pricing error of barles-soner and bakstein-howison equation, in the s p500 index option market. option pricing equations are solved using lie algebra. using the historical data of the s p500 index from august 18, 2022, to august 18, 2023, the price of this asset has been calculated with each model considered. in the sequel, the obtained data are classified using multilayer perceptron and probabilistic neural networks. the networks show which model is closest to the real market. in addition, the prices obtained from lie algebra have been compared with the actual values of the options in the market. pnn and mlp have been tested with statistical data after august 18, 2023. two assumed models were priced with the same data and then compared with the real market. in testing the networks, mlp put 60% of the test data and pnn put all the data into the barels-soner category. by calculating the difference between the results of the lie groups and the real data, 80% of the data were less different from the barrels-soner model. with the results obtained in s p option pricing, the barels-soner model has less error than other models.keywords: nonlinear markets, lie groups, neural networks, options. introductionoptions are created to manage risk, control, and prevent loss. in trading, one party’s gain means the other party’s loss. to eliminate these gains and losses, option pricing must be fair. in other words, the price must be determined so that neither party suffers a loss. in this study, two nonlinear models barles-soner (barles, 1998) and bakstein-howison (bakstein, 2003) were used. by including transaction costs, these two models are closer to market reality than the black-scholes model. according to the two models, european option pricing was performed using lie and symmetric groups. in this method, by order reduction, the partial differential equation is converted into a solvable ordinary differential equation. after evaluating two nonlinear black-scholes models, their responses are classified using a probabilistic neural network (specht, 1990) and a multilayer perceptron, and it is predicted that the market is closer to which model.in the section part, after introducing the two nonlinear black-scholes models, the basic concepts, and theoretical foundations of lie groups, probabilistic neural networks, and perceptron are stated. in the second part, the method used in the research is explained. in this section, after finding the exact solutions of the equations of barles-soner (1998) and bakstein-howison (2003), the european the s p500 index option is priced. the findings obtained from this pricing are presented in the next section and finally, the results obtained in this research are presented in the last section. materials and methodsthe pricing of the european option on the s p500 index has been done using false groups, based on the nonlinear equations of barles-soner (1998) and bakstein-howison (2003). next, perceptron and probabilistic neural networks were explained. to find the exact solution, lie and symmetry groups were used. in this method, by order reduction, the partial differential equations are converted into ordinary differential equations. then, an exact solution of this equation is provided by solving ordinary differential equations. then by flowing the solutions, other new solutions were found (dastranj hejazi, 2017).neural networks consist of three layers. the three main layers of all models are the input layer, hidden layer, and output layer. in some models, the hidden layer is divided into multiple layers. each layer has several nodes whose number is tested in the hidden layer by trial and error. all nodes in each layer are connected to the next and previous layers. this relationship is established by the weights being multiplied by the previous layer’s output, and the result is considered as the input of the current layer. the most important process in a neural network is learning and training the network. the goal of training the network is to accurately determine the weights so that the network output is closer to the true value of the output (hagan, 1997).after pricing with the solutions obtained from lie algebra, we classify the data using neural networks to determine which model is closest to the actual market price. findingstable 1 presents the results of market model prediction by comparing lie algebra pricing, pnn, and mlp.table 1. market model prediction by comparing lie algebra pricing, pnn, and mlpxtmarket pricebarles-sonererrorbakstein-howisonerrorlie algebrapnnmlp4467.710.0876492488.092597.26109.172644.69156.600004404.330.0873022446.572596.79150.222607.17160.600004518.440.1343872511.832602.9191.082674.72162.890004399.770.1141732379.872599.80219.932604.47224.600014405.710.18650824052608.06203.062607.99202.99101 in table 1, x is the s p500 index and t is the remaining time until the option expires. using the following equation and for the values of x and t presented in table 1, pricing was done under the barles-soner (1998) model, the result of which is presented in the column related to barles-soner (1998).  in the bakstein-howison column, the pricing solutions of the bakstein-howison (2003) are described with the following equation.  finally, the differences between the solutions under the two mentioned models have been compared with the real market prices, and the results are presented in the column related to lie algebra. when the barles-soner (1998) price is closer to the market price, zero is placed in the table, and if the bakstein-howison (2003) price is close to it, one is placed. pnn and mlp columns show the results of probabilistic neural networks and multilayer perceptron, respectively. in these two columns, the one in the table indicates that the data is in the bakstein-howison (2003) category, and the zero is used to indicate that the data belongs to the barles-soner (1998) category. discussion and conclusionsfirst, using lie and symmetric groups for two nonlinear models, barles-soner (1998) and bakstein-howison (2003), european option pricing was performed. after pricing the s p500 index between august 18, 2022, and august 18, 2023, a multi-layer perceptron and probabilistic neural network were trained and the data were classified into two classes, barles -soner (1998) and bakstein-howison (2003). then, to find the model closest to the real market, the network is tested with 5 data points. additionally, the answers obtained from the lie algebra are evaluated using test data, and their differences from the real data are calculated to determine which answer is closest to the real data. the perceptron neural network assumes that 3 out of 5 data items belong to the barles-soner (1998) model and the probabilistic neural network places those 5 data items into the barles-soner (1998) category. by calculating the difference between the lie group’s response and the actual market price, 4 out of 5 data are close to the barles-soner (1998) model. therefore, according to the results obtained, the s p500 options market with a maturity of one year and an exercise price of 2,000 usd is close to the barles-soner (1998) model.