Option Pricing Research Articles

Advanced Option Pricing and Hedging with Q-Learning: Performance Evaluation of the QLBS Algorithm

Advanced Option Pricing and Hedging with Q-Learning: Performance Evaluation of the QLBS Algorithm

A Compact Difference Scheme for Mixed‐Type Time‐Fractional Black‐Scholes Equation in European Option Pricing

A Compact Difference Scheme for Mixed‐Type Time‐Fractional Black‐Scholes Equation in European Option Pricing

The Application of Stochastic Processes in Financial Mathematics

Stochastic processes are crucial in financial mathematics, providing a framework to model the inherent uncertainties of financial markets. This paper explores stochastic process application across various financial domains, such as option pricing, risk management, and financial engineering. Through case studies, literature review, and case analysis, the study demonstrates the practical effectiveness of stochastic processes in finance. The findings highlight the adaptability and robustness of these models in capturing market dynamics and optimizing financial strategies. This research particularly focuses on the implementation of Brownian motion and Its processes in derivative pricing, revealing significant improvements in accuracy compared to traditional methods. Additionally, we examine the integration of machine learning techniques with stochastic models to enhance predictive capabilities in risk assessment. This study also addresses the challenges and limitations of current stochastic approaches, proposing innovative solutions for practical implementation. These insights contribute to both theoretical understanding and practical applications in quantitative finance, offering valuable guidance for practitioners and researchers in the field

High order semi-IMEX BDF schemes for nonlinear partial integro-differential equations arising in finance

In this paper, semi-implicit-explicit (Semi-IMEX) and semi-implicit multistep methods are proposed to solve nonlinear partial integro-differential equations (PIDEs), which describe the option pricing models with transaction costs or illiquid markets under Merton's jump-diffusion process. After spatial differential operators are treated by using finite difference methods and the jump integral is computed by using the composite trapezoidal rule, the consistency error and global error bounds for the semi-IMEX and semi-implicit multistep methods for abstract PIDEs are provided when the nonlinear operator satisfies the boundedness and coercivity conditions. A numerical study is carried out for different option pricing models based on the convergence properties of the schemes and the comparison of the different Greek values. Numerical experiments verify our theoretical results and illustrate the intrinsic nature of the proposed option pricing models under jump-diffusion models with transaction costs or illiquid markets.

BINARY OPTION PRICING USING LATTICE METHOD

One way to minimize risks due to uncertainty in stock price movements is by using derivative products, one of which is an option. Binary options, a type of exotic option, provide a fixed payout if certain conditions are met at maturity, but are difficult to solve analytically. In this study, we utilize binomial and trinomial lattice methods, specifically the Cox-Ross-Rubinstein Binomial, Hull-White Trinomial, and Kamrad-Ritchken Trinomial models, to determine the price of binary options. Results indicate that all three methods converge towards the exact solution, demonstrating their applicability for pricing binary options, with the Kamrad-Ritchken Trinomial method showing superior accuracy due to the lowest mean relative error. Additionally, we analyze factors influencing binary option prices, including initial price, strike price, maturity time, volatility, and risk-free interest rate. The study’s originality lies in the comparative analysis of these methods under the same market conditions. However, limitations include model assumptions and potential data variability that may affect generalizability. Future research could extend these methods to various stock data or other financial instruments to test robustness. This research provides insights into optimal lattice method selection for practitioners in binary option pricing.

A Model-Free Lattice

Predicting future price movements has always been one of the major topics in financial research, and there is no better method to predict the future prices of an asset than using its derivatives. In this paper, we propose a model-free lattice model that describes the complete price evolution of the underlying asset and simultaneously re-prices all of its European options. Given that such a lattice is consistent with market option prices, it must embed all necessary risk factors (e.g., random volatility, random interest rates, and jumps) and market restrictions (e.g., mean-reversion and liquidity) that are priced into the European options.

An Analytically Modified Finite Difference Scheme for Pricing Discretely Monitored Options

Finite difference methods are commonly used in the pricing of discretely monitored exotic options in the Black–Scholes framework, but they tend to converge slowly due to discontinuities contained in terminal conditions. We present an effective analytical modification to existing finite difference methods that greatly enhances their performance on discretely monitored options with non-smooth terminal conditions. We apply this modification to the popular Crank–Nicolson method and obtain highly accurate option pricing results with significantly reduced CPU cost. We also introduce an adaptive mesh refinement technique that further improves the computational speed of the modified finite difference method. The proposed method is especially useful for options with high monitoring frequencies, which are difficult to price using other existing methods.

NN de-Americanization: an efficient method to facilitate calibration of American-style options

Neural network (NN) de-Americanization produces fast and accurate pseudo-European option prices from listed American option prices, facilitating the calibration of derivative models. The industry approach binomial de-Americanization takes a flat volatility surface as input for each strike and expiration. In contrast, the NN de-Americanization method takes the detailed shape of the volatility surface as an input; this is critical for accurately evaluating the early exercise premium (EEP) when interest rates are not close to zero.

Quantum walk option pricing model based on binary tree

In this paper, we present a novel quantum model for financial markets constructed within the risk-neutral framework. We innovatively integrate the theory of quantum walks into the binomial tree model for option pricing, thereby achieving a more precise simulation of asset price dynamics. Furthermore, we explore the multi-step quantum binomial tree model and enhance the traditional approach by incorporating the concept of quantum superposition. This advancement enables our model to simultaneously consider multiple states of asset prices at various nodes, effectively reducing computational complexity as time steps increase while significantly improving option pricing accuracy. The superior performance of our quantum model is demonstrated through simulation experiments.

Lockdowns and Leverage: Option Pricing during the Covid Pandemic

Lockdowns and Leverage: Option Pricing during the Covid Pandemic

Responsible investing: Upside potential and downside protection?

Conventional risk proxies are measured assuming that investors have symmetric risk preferences, with upside and downside deviations from the expectation being equivalently undesirable. Responsible investors, however, have dual financial aims of enhancing upside potential while reducing downside risk by actively incorporating environmental, social, and governance (ESG) aspects into the investment process. We utilize a non-symmetric option pricing research design to test whether responsible investors could live up to their ambitions. We find that those who are simply Principles for Responsible Investment (PRI) members do not deliver the desirable asymmetric performance, while financial firms with highly rated responsible investment processes can actually achieve both aims for their own shareholders: enhancing upside potentials and protecting themselves from downside risks.

Valuation of Currency Options with Stochastic Interest Rates and Uncertain Exchange Rate

There are numerous currency models, but only a few are suitable for practical use. The Liu-Chen-Ralescu (L-C-R) model, proposed by Liu, Chen, and Ralescu, is one of them, where the exchange rate abides by an uncertain differential equation. However, the model treats interest rates as constants, which is unrealistic given the fluctuations in financial markets caused by various human-caused or natural disasters. Our preliminary intention was to explore a model that incorporates additional parameters such as log drift, log diffusion, and variance of the observed data. We discovered a new currency model, referred to as the X-W model, which was proposed by Xiao Wang in his 2019 research article. He proposed that an uncertain differential equation governs the exchange rate, while stochastic differential equations govern both domestic and foreign interest rates. Since the X-W model considers various circumstances and calculates more parameters, the option prices are assumed to be higher. We find the call and put currency option prices for the X-W model using Euler’s method and trapezoidal rule in this article and then the option prices have been compared with the currency option prices obtained from the L-C-R model. Finally, we present some numerical and graphical results for both models using MATLAB coding for better observation. We were successful in finding our expected results and can finally conclude that the X-W model is a better fit for the real market. J. Bangladesh Math. Soc. 44.2 (2024) 065–076

A Research on the Application of Option Pricing Strategy Framework Based on BS Model

This paper first reviews the development history of option modeling and its application in corporate strategy, providing a comprehensive background for understanding its evolution. It introduces the European option pricing model studied by scholars in the past, particularly focusing on the Black-Scholes (BS) model, a foundational tool in financial modeling. After establishing the research significance, this paper discusses several basic properties and influencing factors of the BS model while introducing real options as a dynamic framework for strategic decision-making. The core focus of this paper lies in the modification and adaptation of the BS model at the corporate strategy level. Specifically, it develops a combined model that integrates the BS real option method to address the limitations of traditional models, enhancing their applicability to complex business scenarios. Xinhua Wenxuan Company is employed as a case study to validate the feasibility of this refined model, providing a practical evaluation of its results. Lastly, this paper highlights the future research direction of this model, underlining its adaptability and potential for broader applications in corporate strategic management, particularly in industries characterized by high uncertainty and innovation.

A Study of Option Pricing Models with Market Price Adjustments: Empirical Analysis Beyond the Black-Scholes Model

A Study of Option Pricing Models with Market Price Adjustments: Empirical Analysis Beyond the Black-Scholes Model

Research Progress in Option Pricing Methods

In recent years, option pricing has garnered significant attention as a critical topic in financial engineering. Classical models, such as Black-Scholes, exhibit limitations in accurately capturing the dynamic changes in complex market environments. Consequently, machine learning techniques and neural network methods have emerged as pivotal research directions in option pricing. This paper provides a systematic review of the advances in option pricing based on optimization methods, machine learning techniques, and neural network models. Optimization methods, incorporating techniques like genetic algorithms and particle swarm optimization, significantly enhance the adaptability and accuracy of classical models. Machine learning methods, such as support vector machines and decision trees, demonstrate their advantages in nonlinear data modeling. Neural networks and deep learning models, with their robust nonlinear fitting capabilities, have become central tools in pricing complex financial products. Additionally, this review summarizes the application of hybrid models and evaluates the strengths and weaknesses of various approaches in practical applications. Finally, the paper discusses future research directions, including the integration of reinforcement learning techniques, improvement in model interpretability, and deep fusion of heterogeneous multi-source data. This work aims to provide theoretical references and technical support for further research in the field of option pricing.

Вычисление цен опционов в модели Хестона с помощью искусственных нейронных сетей

The article considers numerical methods for pricing options in the Heston model. The main attention is paid to solving the Cauchy problem for the partial differential equation for option prices using artificial neural networks. The possibility of approximating the solution using feedforward neural networks with one hidden layer and sigmoid activation function is substantiated. The advantage of the proposed method is the ability to explicitly calculate loss functions when choosing the logistic regression as an activation function. The loss function of neural networks tracks errors in the execution of equations for option prices and approximation of initial conditions. For comparison method of solving the boundary value problem, Monte-Carlo method and method based on recurrent formulas on a binary tree are used. Computational experiments show that even with a moderate number of neurons in the hidden mode, the neural network approximates the prices of European in-the-money options quite well.

Deep learning of optimal exercise boundaries for American options

Efficiently determining a price and optimal exercise boundary for an American option is a critical subject in the financial sector. This study introduces a novel application of long short-term memory neural networks to solve a relevant Volterra equation, enhancing the accuracy and efficiency of American option pricing. The proposed approach outperforms traditional numerical techniques, including finite difference methods, binomial trees, and Monte Carlo methods, delivering an impressive speed improvement by a factor of thousands while maintaining industry-accepted accuracy levels. It exhibits computational speeds up to about a hundred times faster than the state-of-the-art method used by Andersen et al. [High-performance American option pricing, J. Comput. Finance 20(1) (2016), pp. 39–87] when evaluating numerous options. The proposed network, trained on a reasonable range of parameters related to the Black–Scholes model, swiftly determines option prices and exercise boundaries, serving as a practical closed-form solution.

Natural Cubic Spline Approximation of Risk-Neutral Density

The risk-neutral density is a fundamental concept in pricing financial derivatives, risk management, and assessing financial markets’ perceptions over significant political or economic events. In this paper, we propose a new nonparametric method for estimating the risk-neutral density using natural cubic splines (NCS). The estimated density is twice continuously differentiable with linear tails at both ends. Our method targets the logarithm of the underlying asset price, releasing the restriction to the positive domain. We theoretically prove the consistency of our NCS method. We conduct a comprehensive empirical study comparing the proposed NCS method with a piecewise constant method, a uniform quartic B-spline method, and a cubic spline method from the literature using 20 years of S&P 500 index option data. The empirical results show that our NCS method is more robust than the piecewise constant method, which can only produce a discontinuous density, especially for options with maturities longer than six months. Moreover, our NCS method outperforms other historical continuous methods in terms of optimization feasibility and option price estimation.

The Price Dynamism during REIT Acquisitions

This study creates, models and calibrates the prices of real estate investment trusts (REITs) within an option pricing framework. Unlike Barraclough et al. [(2013). Using option prices overpayments and synergies in M&A transactions. The Review of Financial Studies, 26(3), 695-772] and Marcato and Sebehela [(2022). The paradoxical prices of options. Review of Pacific Basin Financial Markets & Policies, 25(2), 2250009], this study went further and incorporated new information in the form of news. The results show that for a target (acquiring) firm, the exponential risk premium is made up of the cost of carry plus (less) new news. The same illustration applies equally in a merged entity.

Power Law Amplitude Estimation for Option Pricing with NISQ Devices

AbstractThe flourishing development of quantum computing has brought breakthroughs to many classical algorithms, particularly in fields like quantum finance, where quantum computers are employed for estimating the pricing of financial options. Compared to traditional Monte Carlo methods, quantum amplitude estimation (AE) algorithms offer exponential acceleration in estimating the pricing of financial options. However, the execution of quantum AE algorithms is limited by the quantum depth available in current Noisy Intermediate Scale Quantum (NISQ) devices. To explore more applications of quantum algorithms with NISQ devices, a power law AE algorithm for option pricing with NISQ devices is proposed and simulated on Qiskit to validate the theoretical performance. Compared with other AE algorithms, the power law AE algorithm achieves an overall accuracy improved by about 13.12%.