Nonlinear Pricing Model Research Articles

AN INVESTIGATION ON NONLINEAR OPTION PRICING BEHAVIOURS THROUGH A NEW FRÉCHET DERIVATIVE-BASED QUADRATURE APPROACH

Abstract Complicated option pricing models attract much attention in financial industries, as they produce relatively better accurate values by taking into account more realistic assumptions such as market liquidity, uncertain volatility and so forth. We propose a new hybrid method to accurately explore the behaviour of the nonlinear pricing model in illiquid markets, which is important in financial risk management. Our method is based on the Newton iteration technique and the Fréchet derivative to linearize the model. The linearized equation is then discretized by a differential quadrature method in space and a quadratic trapezoid rule in time. It is observed through computations that the accurate solutions for the model emerge using very few grid points and time elements, compared with the finite difference method in the literature. Furthermore, this method also helps to avoid consideration of the convergence issues of the Newton approach applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. It is important to note that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.

Pricing strategies in the Silicon Valley housing market: an update on TOM and recent events

Purpose This study aims to explore the relationship between house prices and time-on-market (TOM) in Silicon Valley. Previous findings have been inconclusive due to variations in property characteristics. This paper highlights the discrepancy between listing and selling prices and identifies differences among housing types such as condominiums, detached houses and townhouses based on housing orientations and customer groups. Additionally, this study considers the impact of the COVID-19 pandemic and the Fed’s interest rate policies on the housing market. Design/methodology/approach The authors analyze 63,853 transactions from the Bay East Board of Realtors’ Multiple Listing Service during 2018 to 2022. The study uses a multiple-stage methodology, including a nonlinear hedonic pricing model, search theory and two-stage least squares method to address concerns relating to endogeneity. Findings The Silicon Valley housing market shows resilience, with low-end properties giving buyers more bargaining power without significant price drops. High-end properties, on the other hand, attract more attention over time, leading to aggressive bidding and higher final sale prices. The pandemic, despite reducing housing supply, did not dampen demand, leading to price surges. Post-COVID, price correlations with TOM changed, indicating a more cautious buyer approach toward high premiums. The Fed’s stringent monetary policies post-2022 intensified these effects, with longer listing times leading to greater price disparities due to financial pressures on buyers and shifting dynamics in buyer interest. Practical implications Results reveal a nonlinear positive correlation between TOM and the price formation process, indicating that the longer a listed property is on the market, the greater the price changes. For low-end properties, TOM becomes significantly negative, while for high-end properties, the coefficient becomes significantly positive, with effects and magnitudes varying by type of dwelling. Moreover, external environmental factors, especially those leading to financial strain, can significantly impact the housing market. Originality/value The experience of Silicon Valley is valuable for cities using it as a development model. The demand for talent in the tech industry will stimulate the housing market, especially as the housing supply will not improve in the short term. It is important for government entities to plan for this proactively.

Nonlinear pricing in multidimensional context: An empirical analysis of energy consumption

Modern business practices frequently employ a blend of pricing strategies to segment markets effectively. As a result, consumers may encounter pricing schedules that are nonlinear and multidimensional. This paper presents a structural approach for estimating multidimensional nonlinear pricing models involving multiple decision variables in an energy market. Using a unique, rich panel dataset of Chinese household electricity consumption, we structurally estimate consumer preferences under the influence of an Increasing Block Price (IBP) and a Time-of-Use (ToU) system. Our structural approach allows us to distinguish and evaluate household-level price elasticities of demand, presenting a novel explanation for consumers' feedback on marginal price changes. Through model-based simulations, we demonstrate that a 1% increase in price corresponds to a 0.7% reduction in total electricity demand. However, our analysis indicates that practical opportunities for optimization within multi-dimensional pricing systems are limited. Our findings offer distinct insights into the complex interplay between intricate pricing structures and energy consumption behavior, thereby providing valuable guidance for policymakers and regulators.

Game-Based Demand Feedback Reservation Parking Pricing

Reservation parking is a new type of parking that allows users to obtain a parking space for an appropriate time slot by reservation. This paper pioneers a high degree-of-freedom reservation strategy and rules for commercial districts in which users can arbitrarily select parking time slots. The nested logit is applied to build a parking revenue model by analyzing the cost, demand, and price factors sequentially according to the user’s choice behavior. The parking lot price game in the region is dissected and a nonlinear pricing model is proposed for maximizing regional returns. A model validation of a parking lot in an actual commercial area shows that compared to the traditional parking method, the parking lot revenue increased by 69.63% upon the application of this reservation strategy and pricing scheme, while the parking lot traffic increased by 22.4%. This model can provide strong support for reservation parking pricing in commercial areas. In the comparison of scenarios, the scientific validity of this model using dynamic pricing and regional game competition is verified. In the application, a reasonable price scheme can be formulated based on the parking lot traffic, cost, and surrounding parking strategies, which can result in higher parking revenue while providing users with great parking convenience.

Deep high-order splitting method for semilinear degenerate PDEs and application to high-dimensional nonlinear pricing models

Deep high-order splitting method for semilinear degenerate PDEs and application to high-dimensional nonlinear pricing models

Taxing sugar‐sweetened beverages: A nonlinear pricing approach

AbstractTaxation is frequently implemented to discourage the consumption of sugary beverages. Despite their popularity, little is known about the impacts of taxes when sellers practice price discrimination. To address this issue, we use a standard nonlinear pricing model with one product and two buyer types to study the effects of taxation on (i) consumption, (ii) consumer and producer surpluses, and (iii) the seller's choice of market segmentation scheme. We find that a tax would lead to reductions in consumption, consumer surplus, and expected profit. Additionally, the measure increases the likelihood that the sellers would exclude buyers with low preferences for the beverage to exclusively serve buyers with high willingness to pay for the product.

Bundling and nonlinear pricing in telecommunications

AbstractI develop a multiproduct nonlinear pricing model where a firm sells both discrete and continuous services to consumers with multidimensional heterogeneity. I derive the optimal selling mechanism and provide primitive conditions under which different bundling strategies arise. Exploiting both the firm and the consumer's optimality conditions, I show that the model structure is nonparametrically identified and propose a semiparametric estimation procedure. An application to China Telecom data shows that mixed bundling of internet and landline phone services is more beneficial to both the firm and the consumer relative to component pricing.

Omitted Budget Constraint Bias and Implications for Competitive Pricing

Standard choice-based conjoint models often ignore or insufficiently approximate consumers’ budget constraints, despite the prominent role of budget constraints in economic theory. The authors offer a theoretically motivated improvement to the choice-based conjoint model that is especially appropriate for high-ticket durable goods and develop a Bayesian method for the inference of unobserved budget constraints. The proposed method leverages respondents’ stated budget constraints that suffer from measurement error and respondents’ financial demographic variables as additional information to reduce the dependency on functional form assumptions in the estimation. The authors show that accounting for budget constraints substantially increases model fit and the accuracy of competitive pricing in an industry-grade discrete-choice experiment on consumer preferences for high-end laptops. The proposed model performs better than the canonical linear price benchmark model, which is not flexible enough to approximate budget constraints. In theory, more flexible utility specifications, such as the nonlinear dummy price model, can approximate consumers’ budget constraints. However, they perform poorly when only finite data are available. The authors conclude that applied researchers in industry and academia will benefit from having a better tool for estimating budgets in high-ticket categories.

Solitary Wave and Singular Wave Solutions for Ivancevic Option Pricing Model

The nonlinear option pricing model presented by Ivancevic is investigated. By using travelling wave transforming method, the nonlinear option pricing equation is transformed into a differential equation with constant coefficients. By solving the differential equation with F-expansion method, a series of exact solutions have been obtained for the Ivancevic option pricing model. By choosing appropriate parameter values, the dark-soliton and dark-soliton-like solutions, periodic wave solutions, and rogue wave solutions are obtained. These solutions will enrich the types of exact waves in the existing literature of the Ivancevic option pricing model. Furthermore, they may have potential uses in describing the possible physical mechanisms for wave phenomenon in financial markets.

A Fréchet derivative‐based novel approach to option pricing models in illiquid markets

Nonlinear option pricing models have been increasingly concerning in financial industries since they build more accurate values by regarding more realistic assumptions such as transaction cost, market liquidity, or uncertain volatility. This study defines a nonclassical numerical method to effectively capture the behavior of the nonlinear option pricing model in illiquid markets where the implementation of a dynamic hedging strategy affects the price of the underlying asset. Unlike the conventional numerical approaches, this study describes a numerical scheme based on the Newton iteration technique and the Fréchet derivative for linearization of the model. The linearized time‐dependent PDE is then discretized by a sixth‐order finite difference scheme in space and a second‐order trapezoidal rule in time. The computations revealed that the current approach appears to be somewhat more effective to some extent and at the same time economical for illustrative examples compared to the existing competitors. In addition, this method helps to prevent considering the convergence issues of the Newton approach applied to the nonlinear algebraic system.

Approach to the Delta Greek of nonlinear Black–Scholes equation governing European options

In this paper, we consider a class of a nonlinear option pricing models considering transaction costs. The focus is on the numerical investigation of the Delta equation, where the unknown is the first spatial derivative of the option value. A numerical algorithm for solving a generalized Black–Scholes partial differential equation, which arises in European option pricing considering transaction costs is studied. The Crank–Nicolson method used to discretize in the temporal direction and the FEM used to discretize in the spatial direction are discussed in terms of accuracy, convergence and efficiency. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behavior of the solutions, which are also found to be in good agreement with the exact solution of the linear case.

Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

An Analysis of Portion Cap Rules with a Multiproduct Seller

I study the impacts of limiting the quantity of one product in a two‐goods market with privately informed buyers. The goal is to explore effects on consumer surplus and consumption of both products. The main finding is that following moderate enough caps, a standard nonlinear pricing model predicts an increase in surplus for the buyer with low preference for the regulated good and high valuation for the unregulated product. Specific changes in allocation depend on the model's parameters. Consumption of the limited good is reduced, whereas consumption of the unregulated item often falls. Changes in screening are also interesting. Severe enough caps cause the seller to move from full separation to bunching some types. These results are applicable to food retail and other industries, where portion cap rules are a regulatory alternative and the impact on consumer surplus is an important consideration.

The competitive and welfare effects of long-term contracts with network externalities and bounded rationality

This paper compares the long-term and short-term contracts in terms of their competitive and welfare effects in a dynamic nonlinear pricing model with network externalities and bounded rationality. Contrary to the existing literature and traditional treatments adopted by competition authorities, we find that a long-term contract is at least as competition-friendly and socially efficient as a sequence of short-term contracts. If the consumers have constant types and pessimistic expectation regarding the network size, then for a certain range of parameters, a long-term contract facilitates entry of more efficient competitors and is socially more efficient than the short-term contracts. If the consumers’ types are independent across time, a long-term contract leads to the same competitive outcome as, but gives a higher social surplus than, its short-term counterpart.

Defining an intrinsic “stickiness” parameter of stock price returns

We introduce a non-linear pricing model of individual stock returns that defines a ”stickiness” parameter of the returns. The pricing model resembles the capital asset pricing model (CAPM) used in finance but has a non-linear component inspired from models of earth quake tectonic plate movements. The link to tectonic plate movements happens, since price movements of a given stock index is seen adding ”stress” to its components of individual stock returns, in order to follow the index. How closely individual stocks follow the index’s price movements, can then be used to define their ”stickiness”.

A Virtual Network Resource Allocation Model Based on Dynamic Resource Pricing

The key to network virtualization technology is virtual network mapping, which has been proven to be an NP-hard problem. At present, the methods to solve the problem of virtual network mapping still have the following defects. Most of the existing literature is limited to static virtual network (VN) mapping and static linear resource pricing, which rely on peak allocation and don't meet the user dynamic resource requirements. Therefore, this paper proposes a virtual network resource allocation model based on dynamic resource pricing named GSO-RBFDM. Firstly, group search optimization (GSO) is used to optimize the node mapping scheme during the network mapping process to reduce the cost of network mapping. Secondly, a dynamic nonlinear resource pricing model is established, and genetic algorithm (GA) is used to more accurately search a low-cost network mapping path instead of the traditional Dijkstra algorithm. Finally, virtual network dynamic modeling is performed according to the user dynamic resource requirements, and radial basis function (RBF) is used to predict resource requirements to realize the dynamic resource allocation to users. Simulation results show that, compared with traditional virtual network mapping algorithms, GSO-RBFDM can not only realize dynamic resource allocation, but also show good performance in terms of acceptance rate, network cost, link pressure and average network revenue.

An empirical analysis of competitive nonlinear pricing

We estimate a model of competitive nonlinear pricing with multidimensional preference heterogeneity using individual level data on advertisements bought by local businesses (e.g., doctors, electricians) from two Yellow Page Directories in one U.S. city-market. Variation in individual choices and payments allows us to identify the joint density of preferences, marginal costs of publishing and common utility parameters. Our estimates suggest substantial welfare loss due to asymmetric information. Comparing duopoly outcomes with (counterfactual) monopoly outcomes, we find that with less competition (i) producer surplus increases substantially; (ii) more “low-type” consumers are excluded; (iii) product variety increases, but benefits accrue only to the “high-type” consumers; (iv) total consumer surplus decreases; (v) but its distribution, across consumers, does not change.

An Analysis of Beverage Size Restrictions

Due to high levels of obesity, various government interventions have been proposed to curb the consumption of sugar‐sweetened beverages (SSBs). The New York City “soda ban,” which proposed to limit the size of SSBs is among the most well‐known and controversial. While public debates about beverage‐size‐restrictions tend to focus on how consumers are impacted, we use a nonlinear pricing model to show that, for all but extremely tight restrictions, consumer welfare would be unaffected by an enforceable restriction. However, sellers’ profit would decline. While consumption is predicted to decline overall, the magnitude of the decline will vary by consumer segment.

Bundling and Nonlinear Pricing in Telecommunications

I develop a multiproduct nonlinear pricing model where a firm sells both discrete and continuous goods/services to consumers with multidimensional heterogeneity. I derive the optimal selling mechanism and provide primitive conditions under which different bundling strategies arise. Exploiting the optimality conditions of both the firm and the consumer, I show that the model structure is nonparametrically identified and propose a three-step semiparametric estimation procedure. An application to China Telecom data shows that mixed bundling is beneficial to both the firm and the consumer relative to component pricing.

Inversión pública y plusvalías urbanas. Análisis espacial y efectos no lineales de la proximidad de la red de subterráneos sobre los precios de la vivienda en Buenos Aires

En relación a la discusión sobre la captación y recuperación de plusvalías urbanas, uno de los insumos necesarios para diseñar una política económica es la identificación no solo de la dimensión del efecto sino también del universo de unidades afectadas a partir del desarrollo urbano. En este trabajo se realiza tal análisis sobre los inmuebles ofertados para la venta en la Ciudad de Buenos Aires en relación a su localización respecto a la red de subterráneos. Asimismo, se implementa por primera vez un modelo no lineal de precios hedónicos con dependencia espacial a fin de captar la incidencia sobre este mercado mediante el uso de una base de datos propia de precios de oferta. Se constata la presencia de un efecto no lineal del precio por metro cuadrado en función de la distancia a las estaciones de subte, negativo para el rango de 0 a 200 metros (-5%) y positivo para el de 300 a 600 metros (+3,6%). Adicionalmente, se presenta un detalle para 2017-2018 del estado actual de las heterogeneidades entre líneas de subte, dando cuenta de diferencias en la distribución espacial de los precios de las propiedades, parcialmente determinada por los atributos (v.g. vigilancia, pileta, gimnasio) que poseen los edificios en los distintos barrios de la ciudad.