Option Pricing Model Research Articles

Computation and Applications of Gaussian Integrals in Mathematics and Applied Sciences

Gaussian integrals, particularly those involving the function e−x2 , play a central role in various fields, ranging from physics to finance. This paper explores the computation of Gaussian integrals, beginning with the fundamental integral in the interval −∞ to ∞, which yields π . The author extends this analysis to integrals involving powers of x multiplied by e−x2 , showing their relevance in calculating moments in quantum mechanics and probability theory. Additionally, the author discusses the applications of multivariate Gaussian integrals in machine learning and statistical mechanics, where they are key to solving problems in high-dimensional spaces. Practical examples are provided from quantum mechanics (path integrals), statistical mechanics (partition functions), finance (option pricing models), and machine learning (Gaussian processes). Through these examples, the paper highlights the versatility and universality of Gaussian integrals as essential tools in both theoretical and applied contexts. The integral’s widespread applicability reflects its importance in connecting mathematical theory with real-world phenomena. This work highlights the role played by the Gaussian integral.

SDEs with two reflecting barriers driven by optional processes with regulated trajectories

We study the existence, uniqueness, and approximation of solutions of general stochastic differential equations (SDEs) with two time-dependent reflecting barriers driven by optional semimartingales. We do not assume that the probability space has to satisfy the usual conditions. We define and solve an appropriate version of the deterministic Skorokhod problem for regulated functions. Applications to currency option pricing in financial models are given.

Calibration of European option pricing model in uncertain environment using an artificial neural network

Calibration of European option pricing model in uncertain environment using an artificial neural network

Debt Risk Analysis of Automotive Enterprises

In order to maintain their financial stability, businesses must recognize and manage the financial risks that are necessarily involved in their operations and output. Businesses may better comprehend their financial status and undertake efficient risk early warning by examining financial indicators. Academic research on the detection and mitigation of financial risks is extensive, and in an effort to increase the accuracy of assessments, an increasing number of academics are using multi-indicator systems for thorough analysis. Furthermore, non-statistical techniques including hierarchical analysis, B-S option pricing models, and artificial neural networks have shown extremely effective at handling complicated data and non-linear connections. As the economy transitions from high to medium-high growth, traditional and new energy vehicles in the automobile sector confront significant possibilities and problems. Traditional automakers aggressively converting to new energy cars include SAIC Motor, BAIC Group, and GAC Group. Companies that produce new energy vehicles, such XPeng, BYD, Tesla, Li Auto, and NIO, are becoming more competitive by developing new technologies and tapping into new markets. To enhance market competitiveness and sustainability, businesses must address issues like excessive inventory and price volatility that have been brought about by the new energy vehicle market’s explosive expansion through efficient financial risk management. The short- and long-term debt repayment capacities of traditional and new energy vehicle (NEV) enterprises are the main subjects of this study’s analysis of their financial statements. According to the findings, traditional automakers have more stable financial circumstances whereas NEV enterprises exhibit stronger short-term solvency and lower long-term financial risk in specific years with superior liquidity and lower debt ratios. The study attempts to offer resources for the automobile industry’s deleveraging, sustainable development, and inventory reduction.

Analytical solution of the Dupire-like equation in calibration to the generalized stochastic volatility jump-diffusion model for option pricing

Analytical solution of the Dupire-like equation in calibration to the generalized stochastic volatility jump-diffusion model for option pricing

Comparative analysis of machine learning and traditional models for financial option pricing

This study conducts a comparative analysis of traditional and machine learning models for financial option pricing, using historical stock prices and interest rates data. Traditional models such as the Black-Scholes, Heston, Merton Jump-Diffusion, and GARCH are evaluated against machine learning models including Multi-Layer Perceptrons (MLPs) and Long Short-Term Memory (LSTM) networks. The analysis employs performance metrics like Mean Squared Error (MSE), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and R value. Results indicate that the GARCH model excels in predictive accuracy due to its ability to capture volatility clustering, while machine learning models, especially the Tuned Neural Network, demonstrate superior flexibility and adaptability in managing complex non-linear relationships in financial data. Traditional models, although theoretically robust, show limitations under varying market conditions. The study underscores the potential of hybrid approaches combining traditional and machine learning techniques to leverage their respective strengths for more accurate and reliable option pricing. Future research directions include exploring advanced machine learning architectures and improving model transparency through explainable AI.

Current Developments and Limitations for Pricing Options

Valuing options holds significant importance within the derivative market, serving as a cornerstone for investors, traders, and financial institutions. Over time, economists have dedicated efforts to develop models that accurately reflect the complexities of this market environment. The evolution of economic theories can be observed through the examination of key option pricing models, which have undergone refinements and enhancements over the decades. By exploring models such as the Black-Scholes model, the Merton model, and the Cox-Ross-Rubinstein model, it is possible to gain valuable insights into the progression of economic thought in this field. Each of these models offers unique perspectives on option pricing, illustrating various principles and methodologies employed to assess and value options. Moreover, understanding the extensions and limitations of these models provides crucial insights into their applicability and effectiveness in real-world scenarios. By conducting an in-depth examination of these models, the objective of this article is to offer a nuanced comprehension of how economic theories have developed and adjusted to the ever-changing conditions of the derivative market.

Deterministic modelling of implied volatility in cryptocurrency options with underlying multiple resolution momentum indicator and non-linear machine learning regression algorithm

Modeling implied volatility (IV) is important for option pricing, hedging, and risk management. Previous studies of deterministic implied volatility functions (DIVFs) propose two parameters, moneyness and time to maturity, to estimate implied volatility. Recent DIVF models have included factors such as a moving average ratio and relative bid-ask spread but fail to enhance modeling accuracy. The current study offers a generalized DIVF model by including a momentum indicator for the underlying asset using a relative strength index (RSI) covering multiple time resolutions as a factor, as momentum is often used by investors and speculators in their trading decisions, and in contrast to volatility, RSI can distinguish between bull and bear markets. To the best of our knowledge, prior studies have not included RSI as a predictive factor in modeling IV. Instead of using a simple linear regression as in previous studies, we use a machine learning regression algorithm, namely random forest, to model a nonlinear IV. Previous studies apply DVIF modeling to options on traditional financial assets, such as stock and foreign exchange markets. Here, we study options on the largest cryptocurrency, Bitcoin, which poses greater modeling challenges due to its extreme volatility and the fact that it is not as well studied as traditional financial assets. Recent Bitcoin option chain data were collected from a leading cryptocurrency option exchange over a four-month period for model development and validation. Our dataset includes short-maturity options with expiry in less than six days, as well as a full range of moneyness, both of which are often excluded in existing studies as prices for options with these characteristics are often highly volatile and pose challenges to model building. Our in-sample and out-sample results indicate that including our proposed momentum indicator significantly enhances the model’s accuracy in pricing options. The nonlinear machine learning random forest algorithm also performed better than a simple linear regression. Compared to prevailing option pricing models that employ stochastic variables, our DIVF model does not include stochastic factors but exhibits reasonably good performance. It is also easy to compute due to the availability of real-time RSIs. Our findings indicate our enhanced DIVF model offers significant improvements and may be an excellent alternative to existing option pricing models that are primarily stochastic in nature.

Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model

We present a unified, market-complete model that integrates both Bachelier and Black–Scholes–Merton frameworks for asset pricing. The model allows for the study, within a unified framework, of asset pricing in a natural world that experiences the possibility of negative security prices or riskless rates. Unlike the classical Black–Scholes–Merton, we show that option pricing in the unified model differs depending on whether the replicating, self-financing portfolio uses riskless bonds or a single riskless bank account. We derive option price formulas and extend our analysis to the term structure of interest rates by deriving the pricing of zero-coupon bonds, forward contracts, and futures contracts. We identify a necessary condition for the unified model to support a perpetual derivative. Discrete binomial pricing under the unified model is also developed. In every scenario analyzed, we show that the unified model simplifies to the standard Black–Scholes–Merton pricing under specific limits and provides pricing in the Bachelier model limit. We note that the Bachelier limit within the unified model allows for positive riskless rates. The unified model prompts us to speculate on the possibility of a mixed multiplicative and additive deflator model for risk-neutral option pricing.

The Impact of Stochastic Volatility on Option Pricing Using the Black-Scholes Model: Empirical Evidence from NVIDIA

This paper investigates the impact of volatility on option pricing using the Black-Scholes model. Utilizing historical stock and option data from NVIDIA, this study demonstrates the limitations of the Black-Scholes model in capturing dynamic market conditions. The empirical analysis reveals significant discrepancies between model predictions and actual market prices, highlighting the importance of accurate volatility estimation. The findings suggest that incorporating better volatility measures can improve the accuracy of option pricing models. Additionally, the implications of these findings for traders and risk managers are discussed, emphasizing the need for more sophisticated approaches to volatility estimation in financial markets. This study adds to the body of knowledge by presenting actual data regarding the efficiency of different volatility metrics in option pricing. Moreover, it underscores the necessity of continuous improvement in financial modeling techniques to adapt to changing market dynamics. The study's results align with previous research that has shown the limitations of the Black-Scholes model and the benefits of alternative volatility estimation methods. This paper also discusses the potential application of these findings to improving trading strategies and risk management practices. The practical implications are significant, suggesting that traders and risk managers should consider incorporating advanced volatility measures into their models to enhance decision-making processes and financial outcomes. The conclusions drawn from this study highlight the critical role of accurate volatility estimation in achieving more reliable option pricing and underscore the need for ongoing research and innovation in this field.

An inter-provincial coordinate model under Renewable Portfolio Standards policy based on tradable green certificate options trading

The current territorial management model under Renewable Portfolio Standards in China cannot fully develop renewable energy due to the mismatch between high energy-consumption regions and renewable energy-rich regions. Therefore, we proposed an inter-provincial coordinate model based on tradable green certificate options trading (CMTOT), which includes four components: (1) A dual-objective optimization model is established to minimize the RPS target realization cost and maximize the pollutant emission reduction based on Blacke-Scholes option pricing model. (2) The identification of buyer and seller provinces of tradable green certificate (TGC) options based on the above model. (3) An optimization model in buyers' or sellers' cooperation alliance is established. (4) The cost allocation models by Shapley value and Nucleolus are established to ensure the fairness in different scenarios. A case study of Tianjin, Shandong, Shanxi and Inner Mongolia shows that the CMTOT can effectively reduce the RPS target realization cost by 16.79 % and increase the pollutant emission reduction by 4.32 %. A sensitivity analysis confirms the robustness of the CMTOT. The results show that the CMTOT can both fully develop renewable energy and effectively reduce pollutant emission. Therefore, we proposed some policy recommendations to support the implementation of the CMTOT.

Persistent and transient variance components in option pricing models with variance-dependent Kernel

This paper examines theoretically and empirically a variance-dependent pricing kernel in the continuous-time two-factor stochastic volatility (SV) model. We investigate the relevance of such a kernel in the joint modeling of index returns and option prices. We contrast the pricing performance of this model in capturing the term structure effects and smile/smirk patterns to discrete-time GARCH models with similar variance-dependent kernels. We find negative and significant risk premium for both volatility factors, implying that investors are willing to pay for insurance against increases in volatility risk, even if it has little persistence. In-sample, the component GARCH model exhibits a slightly better fit overall and across all maturity buckets than the two-factor SV model. However, the two-factor SV model reduces strike price bias, giving rise to the model’s ability in reconciling the physical and risk-neutral distribution. Out-of-sample, the two-factor SV model has better fit to data.

Considering Appropriate Input Features of Neural Network to Calibrate Option Pricing Models

Considering Appropriate Input Features of Neural Network to Calibrate Option Pricing Models

The Role of FPGAs in Modern Option Pricing Techniques: A Survey

In financial computation, Field Programmable Gate Arrays (FPGAs) have emerged as a transformative technology, particularly in the domain of option pricing. This study presents the impact of Field Programmable Gate Arrays (FPGAs) on computational methods in finance, with an emphasis on option pricing. Our review examined 99 selected studies from an initial pool of 131, revealing how FPGAs substantially enhance both the speed and energy efficiency of various financial models, particularly Black–Scholes and Monte Carlo simulations. Notably, the performance gains—ranging from 270- to 5400-times faster than conventional CPU implementations—are highly dependent on the specific option pricing model employed. These findings illustrate FPGAs’ capability to efficiently process complex financial computations while consuming less energy. Despite these benefits, this paper highlights persistent challenges in FPGA design optimization and programming complexity. This study not only emphasises the potential of FPGAs to further innovate financial computing but also outlines the critical areas for future research to overcome existing barriers and fully leverage FPGA technology in future financial applications.

The study of phase portraits, multistability visualization, Lyapunov exponents and chaos identification of coupled nonlinear volatility and option pricing model

The study of phase portraits, multistability visualization, Lyapunov exponents and chaos identification of coupled nonlinear volatility and option pricing model

Evolution of option pricing models: From Black-Scholes to Heston and beyond

This essay explores the evolution of option pricing models, tracing their development from the foundational Black-Scholes model to more advanced frameworks such as the Heston model and beyond. Beginning with an introduction to option pricing theory, the essay discusses the origins of the Black-Scholes model and its assumptions, as well as the challenges and limitations it faces. It then examines the extension of the Black-Scholes model, so-called the Black-Scholes-Merton model. It incorporates dividends, and lays the groundwork for further research into options pricing and financial derivatives. Then various stochastic volatility models emerge, and the essay chooses the Heston model as a typical example for analysis, highlighting its advantages and applications in option pricing. Furthermore, the essay compares the Heston model with other option pricing models, including the SABR model and Bates model. At the end of the essay, recent advances and future directions in option pricing are introduced and discussed. Through this comprehensive exploration, readers can gain a deeper understanding of the evolution of option pricing models and their significance in modern finance.

Application of Stochastic Processes in Financial Market Models

This paper explores the significant role of stochastic processes in financial modeling, tracing the evolution from basic Brownian motion to sophisticated stochastic differential equations used in modern financial markets. Beginning with the historical development of Brownian motion, identified by Robert Brown and later mathematically modeled by Louis Bachelier for stock price fluctuations, the paper outlines its foundational influence on the Efficient Market Hypothesis and the random walk theory. The extension of these concepts in the Black-Scholes model for option pricing highlights the practical applications of these theories in predicting financial outcomes. The discussion progresses to geometric Brownian motion (GBM) and its crucial role in stock price modeling, emphasizing its use in Monte Carlo simulations for option pricing. The limitations of the Black-Scholes model and GBM in dealing with real market conditions such as stochastic volatility and jump-diffusion processes are addressed, showcasing the evolution of more complex models like the Heston model and GARCH. Interest rate models like the Vasicek and Cox-Ingersoll-Ross models are evaluated for their real-world applicability, particularly in scenarios of low and negative interest rates. This comprehensive review not only underscores the theoretical advancements in financial modeling but also its practical implications in contemporary financial markets.

Barrier Option Pricing in Regime Switching Models with Rebates

Barrier Option Pricing in Regime Switching Models with Rebates

Cross-Sectional Variation of Option-Implied Volatility Skew

The stock option-implied volatility skew reflects both the structural risk characteristics of the underlying company and the short-term information flow about the stock price movement. This paper builds a semistructural, cross-sectional option pricing model to separate the structural risk contributions from the information flow. The model identifies two structural risk sources that contribute to the cross-sectional variation of the skew: the company’s business cyclicality and its default risk. The model can explain as much as 44% of the cross-sectional variation in implied volatility skew and is particularly informative during and after recessions. The remaining skew variation reflects mainly short-term information flow and can be used to construct stock portfolios with much better investment performance and without hidden structural risk exposures. This paper was accepted by Agostino Capponi, finance. Funding: L. Wu gratefully acknowledges support by a grant from the City University of New York PSC-CUNY Research Award Program. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4872 .

Analysis of Option Prices Using Black Scholes Model

A mathematical formula used in finance to calculate the theoretical price of an option and ascertain its option premium is known as the Black Scholes option pricing model, which aids option traders in making informed decisions. This article estimates the option premium of various call and put options using the Black Scholes Model. The three distinct option chains chosen for this essay are all Mid-Cap companies that are listed on the Indian National Stock Exchange. The companies are Suzlon Energy, Kalyan jewellers India, and Exide Industries Ltd. The analysis demonstrates that the options are expensive because the Black Scholes model's computation of the option premium is lower than the actual premium in the market. Keywords: Option, Black Scholes Option Pricing Model, Call, Put, Option Premium