Option Pricing Model Research Articles

Analysis of Option Butterfly Portfolio Models Based on Nonparametric Estimation Deep Learning Method

The option butterfly portfolio is the commonly option arbitrage strategy. In reality, because the distribution of the option state price density (SPD) function is not normal and unknown, so the nonparametric deep learning methods to estimate option butterfly portfolio returns are proposed. This paper constructs the single-index nonparametric option pricing model which contains multiple influencing factors and presents the nonparametric estimation form for option butterfly portfolio returns. The empirical analysis shows that the SPD function estimated by using single-index nonparametric option model can effectively calculate the option butterfly portfolio returns with the minimum option strike price interval and provide an effective reference tool for risk-averse investors with limited risk preferences.

Introducing a real option framework for EVA/MVA analysis

The importance of including real option analysis (ROA) in traditional net present value (NPV) analysis has been demonstrated in the literature: ROA values the flexibility to defer, expand, contract, or abandon an investment project, allowing for dynamic decision making. However, methods to apply ROA for firms that utilize economic value added (EVA) and market value added (MVA) for project valuation purposes are not available. This article presents a novel framework that allows for an EVA/MVA analysis to be incorporated into a binomial tree option pricing model. This new EVA/MVA-embedded binomial tree framework not only allows for the consideration of real managerial options within EVA/MVA analysis but also demonstrates its equivalence to the traditional ROA used to adjust NPV. The benefits of this development are that EVA/MVA with an embedded binomial tree is deemed to be an equally valid and robust technique to traditional NPV analysis as opposed to simply being viewed as an alternative or competing measure of project value. This further validates the choice of some firms’ use of EVA/MVA and shows that strategic ROA can be incorporated into corporate decision making.

Calculations of fractional derivative option pricing models based on neural network

The work adopts the neural network algorithm to derive optimized series solutions of fractional option pricing equations. The studied models include Caputo time-fractional equation and space-time fractional differential equations. The solutions of fractional derivative models are made up of the time variable and the kernel functions of RBF neural network. According to the characteristics of the models, the information function, the test solution, the output function and the loss function are established in turn. And then the optimal parameters of the solution structures of fractional derivative models are calculated by the designed neural network in conjunction with Chinese market data. For Caputo time-fractional model, the pricing results under different kernel functions, with or without cluster analysis, are compared through numerical analysis and illustration. The results of cluster analysis are better than those without cluster analysis. For space-time fractional models, comparative studies between the pricing results under Caputo and conformable fractional derivatives and those from conformable fractional derivative are made and analyzed by the market data. The numerical simulations indicate that the space-time fractional derivative models have strong predictive abilities. Application analyses show that it is feasible and operable for fractional calculus tool and neural network algorithm to jointly act on the pricing problems of financial derivatives.

Calculation of a Class of Confluent Hypergeometric Equation and Analysis of its Roles in Option Pricing Models

Calculation of a Class of Confluent Hypergeometric Equation and Analysis of its Roles in Option Pricing Models

Limiting Cases of the Black-Scholes Type Asymptotics of Call Option Pricing in the Generalised CRR Model

The article concerns the generalised Cox‑Ross‑Rubinstein (CRR) option pricing model with new formulas for changes in upper and lower stock prices. The formula for option pricing in this model, which is the Black‑Scholes type formula, and its asymptotics are presented. The aim of the paper is to analyse limiting cases of the obtained asymptotics using probability theory and later data from the Warsaw Stock Exchange. Empirical analyses of option pricing in the generalised CRR model confirm the calculated limits.

Physical wave propagation and dynamics of the Ivancevic option pricing model

In this study, the Ivancevic option pricing model (IOPM), a nonlinear wave alternative to the traditional Black–Scholes option pricing model, is investigated. The Ivancevic option pricing model is formalized by adaptive nonlinear Schrodinger equations that characterize the option-pricing wave function in terms of stock price and time. This model exhibits the characteristic regulated Brownian behavior observed in financial markets. We use the unified auxiliary equation method, which is one of the most powerful methods to explore analytical solutions of the Ivancevic option pricing model. Via this method, we obtain several new exact solutions such as singular, periodic, bright, dark, hyperbolic, trigonometric, exponential and Jacobi elliptic functions solutions. In addition, we simulate 3D surface 2D graphs and counter plots graphics for some obtained solutions by giving specific values for the involved parameters. To further analyze the obtained solutions, we conduct simulations and generate visual representations in the form of 3D surface plots, 2D graphs, and counter plots. By assigning specific values to the parameters involved in the model, we are able to visualize and examine the dynamics of the solutions in a more tangible and intuitive manner. These graphical representations provide valuable insights into the behavior and patterns exhibited by the solutions, enhancing our understanding of the model’s implications and potential applications Furthermore, we present the option price wave functions of the dependent variable in 2D graphs under the effect of the time variable t.

Fast numerical scheme for the time-fractional option pricing model with asset-price-dependent variable order

We provide a fast numerical technique for a time-fractional option pricing model with asset-price-dependent variable order. Due to the complicated variable-order fractional derivative and its related fast approximations, the temporal coefficients are coupled with the inner product of the finite element method and lose monotonicity, which introduces uncommon difficulties in numerical analysis. In addition, the Riemann-Liouville fractional operators are often used in option pricing models, but its variable-order case gets far less attention than the corresponding Caputo-type problems. We prove error estimates for the proposed fast method and show that the computational cost is almost linear with respect to the temporal steps, which is much faster than the quadratic growth of the time-stepping solver. Numerical experiments are performed to illustrate the theoretical findings.

Tighter Robust Upper Bounds for Options via No-Regret Learning

Classic option pricing models, such as the Black-Scholes formula, often depend on some rigid assumptions on the dynamics of the underlying asset prices. These assumptions are inevitably violated in practice and thus induce the model risk. To mitigate this, robust option pricing that only requires the no-arbitrage principle has attracted a great deal of attention among researchers. In this paper, we give new robust upper bounds for option prices based on a novel η-momentum trading strategy. Our bounds for European options are tighter for most common moneyness, volatility, and expiration date setups than those presented in the existing literature. Our bounds for average strike Asian options are the first closed-form robust upper bounds for those options. Numerical simulations demonstrate that our bounds significantly outperform the benchmarks for both European and Asian options.

Modelling Asymmetric Dependence in Stochastic Volatility and Option Pricing: A Conditional Copula Approach

In this paper, stochastic volatility models with asymmetric dependence were presented and applied to pricing options. A dynamic conditional copula approach was proposed to capture this dependence asymmetry. This approach offered simplicity and flexibility, and yielded closed-form solutions for option pricing under different model constructions for the stochastic volatility based on a mean-reverting Gaussian, a square-root and a lognormal process. Empirical experimentation based on S&P 500 options showed that the developed dynamic option pricing models under asymmetric stochastic volatility significantly and consistently outperformed the basic Heston model across option maturities, strike prices and various copula function specifications. The square-root model combined with a Joe copula was the best ranked, having achieved 32.33% overall performance improvement. This superior empirical performance in option pricing, the unique flexibility to various dependence asymmetry considerations, and the analytical tractability added to the benefits of the proposed models framework.

Fast Filtering with Large Option Panels: Implications for Asset Pricing

Abstract The cross section of options holds great promise for identifying return distributions and risk premia, but estimating dynamic option valuation models with latent state variables is challenging when using large option panels. We propose a particle Markov Chain Monte Carlo framework with a novel filtering approach and illustrate our method by estimating index option pricing models. Estimates of variance risk premiums, variance mean reversion, and higher moments differ from the literature. We show that these differences are due to the composition of the option sample. Restricting the option sample’s maturity dimension has the strongest impact on parameter inference and option fit in these models.

Exotic option pricing model of the Black–Scholes formula: a proactive investment strategy

The option is an important derivative tool in financial market, and after decades of development, the option has emerged in various forms. This paper studies an exotic option with a proactive investment strategy. Compared with the classic option theory, we assume that the option holders continuously trade the underlying assets according to a predetermined investment strategy. Taking the European call option as an example, we first give some assumptions and define the loss function in terms of a logarithmic investment strategy. Then, the specific pricing expression of the exotic option is derived from the Black-Scholes option pricing formula. Next, numerical simulations are presented to visualize the mechanism of the exotic option in 2D and 3D forms by selecting appropriate parameters. The results indicate that the exotic option has a significant price advantage (up to 43.9% under specific parameter settings) over the classic option. Moreover, the empirical results illustrate a perfect fit between 50 ETF (issued by the Shanghai Stock Exchange) and our exotic option. The new proposed exotic option extends the Black-Scholes option theory from a no-trading condition (do not buy or sell underlying assets during the validity period) to a dynamic investment condition and has important practical significance in real life.

Analysis about the Black-Scholes asset price under the regime-switching framework

Assuming that the macroeconomic environment can be transformed into a two-district system, that is, the path of financial asset prices is uncertain, we track and study the motion of stocks and other asset price process under the conditional Black-Scholes model, and give the economical explanation of the mathematical formula. Further, we derive and analyze an option pricing formula for the Black-Scholes asset model under the condition that the risk-free interest rate is regime-switching too. The method in this article is applied to model the log rate of return of the Tencent stock in a two-district market environment. And the obtained parameter values are used to calculate the option price. In narrowing the gap with actual option prices, our method outperforms the classical option pricing model point by point. Compared with the general and pure mathematical model derived work and the empirical study work, our study does more work on the economic characteristics analysis and interpretation of the mathematical models, and plays a certain role in linking the results of mathematical models with empirical research.

The Application of Symbolic Regression on Identifying Implied Volatility Surface

One important parameter in the Black–Scholes option pricing model is the implied volatility. Implied volatility surface (IVS) is an important concept in finance that describes the variation of implied volatility across option strike price and time to maturity. Over the last few decades, economists and financialists have long tried to exploit the predictability in the IVS using various parametric models, which require deep understanding of financial practices in the area. In this paper, we explore how a data-driven machine learning method, symbolic regression, performs in identifying the implied volatility surface even without deep financial knowledge. Two different approaches of symbolic regression are explored through a simulation study and an empirical study using a large panel of option data in the United States options market.

A Study on the Development of Machine Learning Methods for Options Pricing Applications

The traditional option pricing model is of great importance, but has many defects in practical applications. With the development of machine learning theory, the powerful learning ability is applied to the option pricing problem, which can effectively solve the defects of the traditional option pricing model. This paper introduces the basic concepts of option pricing and machine learning, and describes the specific model of option pricing fused with BP neural network model and support vector machine model, expresses views on the option pricing methods of traditional model and fused with machine learning model, and presents the outlook on future research. Option pricing is an important topic in finance, and traditional option pricing models have many limitations. With the development of computers and artificial intelligence, machine learning theory can be applied to option pricing models with good results.

Pricing American Options under Azzalini Ito-McKean Skew Brownian Motions

In this paper, the pricing of American options whose asset price dynamics follow Azzalini Itô-McKean skew Brownian motions is considered. The corresponding optimal stopping time problem is then formulated, and the main properties of its value function are provided. We show that if the payoff function is positive and decreasing, then the value function and its partial derivatives are continuous and locally bounded, and therefore several variational inequalities are derived. Furthermore, the Feyman-Kac formula is calculated. Finally, under this more general as well as very versatile setting, the Black-Scholes option pricing model is nested as a special case.

Barrier Options and Greeks: Modeling with Neural Networks

This paper proposes a non-parametric technique of option valuation and hedging. Here, we replicate the extended Black–Scholes pricing model for the exotic barrier options and their corresponding Greeks using the fully connected feed-forward neural network. Our methodology involves some benchmarking experiments, which result in an optimal neural network hyperparameter that effectively prices the barrier options and facilitates their option Greeks extraction. We compare the results from the optimal NN model to those produced by other machine learning models, such as the random forest and the polynomial regression; the output highlights the accuracy and the efficiency of our proposed methodology in this option pricing problem. The results equally show that the artificial neural network can effectively and accurately learn the extended Black–Scholes model from a given simulated dataset, and this concept can similarly be applied in the valuation of complex financial derivatives without analytical solutions.

A Preconditioned Iterative Method for a Multi-State Time-Fractional Linear Complementary Problem in Option Pricing

Fractional derivatives and regime-switching models are widely used in various fields of finance because they can describe the nonlocal properties of the solutions and the changes in the market status, respectively. The regime-switching time-fractional diffusion equations that combine both advantages are also used in European option pricing; however, to our knowledge, American option pricing based on such models and their numerical methods is yet to be studied. Hence, a fast algorithm for solving the multi-state time-fractional linear complementary problem arising from the regime-switching time-fractional American option pricing models is developed in this paper. To construct the solution strategy, the original problem has been converted into a Hamilton–Jacobi–Bellman equation, and a nonlinear finite difference scheme has been proposed to discretize the problem with stability analysis. A policy-Krylov subspace method is developed to solve the nonlinear scheme. Further, to accelerate the convergence rate of the Krylov method, a tri-diagonal preconditioner is proposed with condition number analysis. Numerical experiments are presented to demonstrate the validity of the proposed nonlinear scheme and the efficiency of the proposed preconditioned policy-Krylov subspace method.

Option Contract Pricing Model for Real Option with Right to Sell/Purchase the Project

The study in the article aims to improve the methodology for calculating the price of an option agreement . The subject is calculating the price of the option agreement for a real option with the right to sell / purchase . The hypothesis is that using a more objective calculation formula will increase the efficiency of making the corporation’s strategic investment and management decisions . The method of mathematical modelling is used to refine the methodology for calculating the price of an option agreement within the framework of the ways of real options and discounting cash flows . The article justifies and clarifies the calculation formulas for the price of an option agreement for a real option with the right to sell / purchase . It has been shown that an objective assessment of the value of this option is a powerful tool for making a strategic investment decision . The article’s novelty is the combination of the practice of hedging standards for transactions and methods of real options and discounting cash flows when developing a model for calculating the price of an agreement . The application of a more objective pricing formula for an options contract , limited to the specified project profitability index , confirms the hypothesis . A more objective assessment of the effectiveness of projects with the application of real options is a competitive advantage of the corporation due to the ability to benefit further from risk and uncertainty . The direction of future research is the formalization of the option pricing model for calculating net assets .

Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model

We present a new nonlinear version of the well-known Black-Scholes model for option pricing in financial mathematics. The nonlinear Black-Scholes partial differential equation is based on the quasilinear diffusion term with the p-Laplace operator \(\Delta_p\) for \(1 < p < \infty\). The existence and uniqueness of a weak solution in a weighted Sobolev space is proved, first, by methods for nonlinear parabolic problems using the Gel'fand triplet and, alternatively, by a method based on nonlinear semigroups. Finally, possible choices of other weighted Sobolev spaces are discussed to produce a function space setting more realistic in financial mathematics. See also https://ejde.math.txstate.edu/special/02/t1/abstr.html

Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model

We present a fully-discrete finite element scheme to a generalized distributed-order time-fractional option pricing model, which adequately describes, e.g., the valuation of the European double barrier option. Due to the dependence of the density function on the stock price, the temporal discretization coefficients from the generalized distributed-order time-fractional derivative will be coupled with the inner product of the finite element method, which significantly complicates the analysis and traditional numerical analysis techniques do not apply. Novel techniques are developed to prove error estimates of this fully-discrete numerical scheme, which not only resolves the above difficulty, but indeed simplifies existing methods by avoiding the mathematical induction procedure. Based on the structure of the all-at-once coefficient matrix of the proposed numerical scheme, a fast divide and conquer algorithm is developed to reduce the computational cost of solving the numerical scheme from O(LNt2Nx) to O(LNtlogNtNx), where L, Nt and Nx refer to numbers of the degree of freedom of discretizations for the distributed-order integral, the spatial domain and the time period, respectively. Numerical experiments are performed to demonstrate the accuracy of the proposed numerical scheme and its applications in the valuation of the option price.