Pricing Formula Research Articles

Closed-form pricing formulas for autocallable reverse convertible securities

Closed-form pricing formulas for autocallable reverse convertible securities

Pricing geometric average Asian options in the mixed sub-fractional Brownian motion environment with Vasicek interest rate model

<p>Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model.</p>

Fractal barrier option pricing under sub-mixed fractional Brownian motion with jump processes

<p>In this work, we mainly focused on the pricing formula for fractal barrier options where the underlying asset followed the sub-mixed fractional Brownian motion with jump, including the down-and-out call option, the down-and-out put option, the down-and-in call option, the down-and-in put option, and so on. To start, the fractal Black-Scholes type partial differential equation was established by using the fractal Itô's formula and a self-financing strategy. Then, by transforming the partial differential equation to the Cauchy problem, we obtained the explicit pricing formulae for fractal barrier options. Finally, the effects of barrier price, fractal dimension, Hurst index, jump intensity, and volatility on the value of fractal barrier options were exhibited through numerical experiments.</p>

Pricing forward-start style exotic options under uncertain stock models with periodic dividends

<p>In this study, we derived pricing formulas for various forward-start style exotic options based on an uncertain stock models with periodic dividends. Specifically, we present valuations for forward-start, Cliquet/Ratchet, and spread options. In addition, we conducted numerical simulations of these formulas and compared them to pricing formulas for the same options based on a dividend-paying stock model driven by standard Brownian motion.</p>

Pricing strategy and firms profitability

This paper summarizes the arguments and counterarguments within the scientific discussion on the issue price decision and its significantly impact on the firm profitability and market competitive position, pricing strategy considered to be fundamental tool for market growth, in today’s extremely competitive market, adopting a pricing plan is essential. In response to the demands of the industry, several pricing models have been developed under the influence of multiples factors (organization and management attitude, product cost structure, industry, and gap between local and target markets). The purpose of this paper is to review the price objectives and price methods that were adopted to set prices as well as their impact on the profitability of the firms. One of the most crucial managerial choices is the definition of the price policy since it has an impact on business profitability and market competitiveness. Despite the essential role that costs play in businesses. The objectives of this research were to develop and evaluate a theoretical model that would illustrate how pricing strategy affects business profitability. This study used the theoretical literature review technique to answer the following question: how product pricing methods impact on firm profitability? though we found that the pricing objectives are a function of pricing formula and methods. More important we found that firms are positively influenced by the value-based pricing strategy and high price, in contras its negatively influenced by the low price, this implies that prices policies affected the profitability of the company hence, this explained the importance of strategic consideration of the pricing, strategies, methods and process must be critically evaluated by management. The research results can be useful as baseline to further develop model that enable firms to quantify the impact of pricing strategies.

Mathematical Derivations of Actuarial Present Value for the Fully Continuous Whole Life Assurances from the Theoretical Market Price

Many current mortality tables are computationally Makehamised but technically devoid of continuous key life table functions such as and because of the underlying sophisticated mortality functions and the complex methods of computation which usually do not incorporate the market price. The computationally advanced Moore’s model, which does not endorse the market pricing mechanisms offers one of the most complex analytical results that seems not user friendly in actuarial literature. As a result of the Moore’s complexity, we offer the Gradshteyn and Ryzhik’s analytic integral as an alternative solution to examine the contingency issues involving the estimation of actuarial present values of the continuous whole life annuities. In this study, the objective are to (i) estimate the parameters of GM (1,2) through the method of equidistant points according to the natural order of human age (ii) Apply the mean value theorem to construct the survival probability function under the framework of policy modifications (iii) Develop a closed form pricing formula for the continuous whole life annuity and continuous whole life insurance. The Gradshteyn and Ryzhik’s analytic integral has eliminated the Moore’s complexities associated with symbolic computations such that the continuity assumption applied allows us to compute the continuous life annuity with expedience. Computational evidence further shows that the continuous life annuity reduces with age confirming that life annuities is a decreasing function.

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

Equilibrium pricing of European crude oil options with stochastic behaviour and jump risks

In this paper, we investigate the pricing of European crude oil options under nonlinear dynamics with stochastic behaviour and jump risks, incorporating the features of arising convenience yield of crude oil and potential extreme fluctuation in the dynamics of crude oil prices. We present a closed-form solution to European crude oil option prices under an incomplete market setting, after deriving the pricing kernel with the equilibrium method and determining the risk neutral dynamics of crude oil prices. We extend our model to a mean-reverting stochastic volatility case, which also admits an analytical formula for the equilibrium price of European crude-oil options. Finally, our model is shown to be consistent with a number of interesting facts documented in the recent literature.

Arbitrage problems with reflected geometric Brownian motion

Contrary to the claims made by several authors, a financial market model in which the price of a risky security follows a reflected geometric Brownian motion is not arbitrage-free. In fact, such models violate even the weakest no-arbitrage condition considered in the literature. Consequently, they do not admit numéraire portfolios or equivalent risk-neutral probability measures, which makes them unsuitable for contingent claim valuation. Unsurprisingly, the published option pricing formulae for such models violate classical no-arbitrage bounds.

Analytical valuation of vulnerable chained options

In this paper, we present analytical pricing formulae for vulnerable chained options. To derive the price, we provide the joint probability that one of the two-dimensional Brownian motions hits multiple barriers sequentially before a fixed time and the positions of both Brownian motions. Using the derived formulae, we perform numerical analysis to investigate the impacts of counterparty risk on option prices.

Valuation of Commodity-Linked Bond with Stochastic Convenience Yield, Stochastic Volatility, and Credit Risk in an Intensity-Based Model

In this study, we consider an intensity-based model for pricing a commodity-linked bond with credit risk. Recently, the pricing of a commodity-linked bond with credit risk under the structural model has been studied. We extend the result using an intensity-based model, stochastic volatility model, and stochastic convenience yield model. In the intensity-based model, the credit event by the counterparty occurs at the time of first jump in a stochastic Poisson process, in which intensity is modeled as the sum of two CIR prosesses. We assume that the underlying asset follows the stochastic volatility and convenience yield models. Using the measure change technique, we explicitly derive the commodity-linked bond pricing formula in the proposed model. As a result, we provide the explicit solution for the price of the commodity-linked bond with stochastic convenience yield, stochastic volatility, and credit risk as single integrations. In addition, we present several examples to demonstrate the effects of significant parameters on the value of commodity-linked bond using numerical integration. In particular, examples are provided, focusing on the behavior of prices based on effects of recovery rate.

A Simplified Approach to the Pricing of Vulnerable Options with Two Underlying Assets in an Intensity-Based Model

In this paper, we study a simplified approach to determine the pricing formula for vulnerable options involving two correlated underlying assets. We utilize an intensity-based model to describe the credit risk associated with these vulnerable options. Without the change of measure technique, we derive pricing formulas for vulnerable options involving two underlying assets based on the probabilistic approach. We provide closed-form pricing formulas for two specific types of options: the vulnerable exchange option and the vulnerable foreign equity option. Finally, we present numerical results to demonstrate the accuracy of our formulas using the Monte-Carlo method and the effect of various parameters on the price of options.

SOLUTION OF BLACK-SCHOLES EQUATION FOR STANDARD POWER EUROPEAN OPTIONS WITH DISCRETE DIVIDEND PAYMENT

The Nobel Prize celebrated option pricing formulas derived by Fischer Black and Myron Scholes for plain vanilla payoffs known as Black-Scholes formulas. They developed these formulas for pricing European call and put options based on certain assumptions in order to minimize risk factor. The underlying asset pays a constant dividend payment during the life of option was one of the assumption to derive these formulas. S. P. Zhu and X. J. He tried and succeed to improve this assumption by taking discrete dividend payments for underlying asset at fixed dividend date. They derived approximate Black-Scholes formulas of European options for plain vanilla payoffs. Note that standard power payoffs are the generalization of plain vanilla payoffs. This paper contributes to the derivation of approximate Black-Scholes formulas for standard power payoffs. Further, the accuracy of these formulas have been checked numerically.

Pricing European Options under a Fuzzy Mixed Weighted Fractional Brownian Motion Model with Jumps

This study investigates the pricing formula for European options when the underlying asset follows a fuzzy mixed weighted fractional Brownian motion within a jump environment. We construct a pricing model for European options driven by fuzzy mixed weighted fractional Brownian motion with jumps. By converting the partial differential equation (PDE) into a Cauchy problem, we derive explicit solutions for both European call options and European put options. The figures and tables demonstrating the effectiveness of the results highlight the suitability of the fuzzy mixed weighted fractional Brownian motion with jump model for option pricing.

Extreme values of solution of Caputo–Hadamard uncertain fractional differential equation and applications

Uncertain fractional differential equation (UFDE) is a useful tool for studying complex systems in uncertain environments. The mathematical characteristics of solution of an UFDE are also widely used in various fields. In this paper, we give the extreme value theorems of solution of Caputo–Hadamard UFDE and applications. A numerical algorithm for obtaining the inverse uncertainty distributions (IUDs) for extreme values of solution of Caputo–Hadamard UFDE is presented; the stability and feasibility of the proposed algorithm are validated by numerical experiments. As an application of extreme value theorems in uncertain financial market, the pricing formulas of the American option based on the new uncertain stock model are given. Besides, the algorithms for computing the price of the American option without explicit pricing formulas based on the Simpson's rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

A study of maximizing skew Brownian motion with applications to option pricing

The objective of this study is twofold. Firstly, to study the probabilistic distribution functions of maximum of skew-Brownian motion (SKBM) and stock price process driven by maximum of skew-Brownian motion. Secondly, to derive the pricing formulas of European style call options (with general payoff D(St)), contingent upon the maximum of skew-Brownian motion and stock price process driven by maximum of skew-Brownian motion. As an application of developed formulas, we have priced the binary (Cash or Nothing) options.

Prediction and Replenishment Decision Making for Automatic Pricing of Vegetable Commodities Based on LSTM Models

In this paper, the problem of automatic pricing and replenishment decision-making of vegetable commodities is studied, and the LSTM model is used to predict and optimize it, and the replenishment volume and pricing strategy to maximize the revenue of supermarkets are proposed. Calculating the total annual sales volume of each category and each vegetable, the distribution law of most vegetables with strong seasonal periodicity and high sales volume from September 2022 to January 2023 can be obtained, and finally the correlation coefficient between different categories and different single products of vegetable commodities is obtained through the Pearson correlation coefficient algorithm, among which the highest correlation coefficient between different categories is mosaic and cauliflower. The coefficient was 0.75, and the highest correlation coefficient between different single items was 0.99 for green eggplant (1) and purple round eggplant. According to the cost-plus pricing algorithm and formula, the cost-plus pricing of each vegetable category is calculated, and then the relationship between the total sales of each vegetable category and the cost-plus pricing can be obtained through visual analysis. Then, the LSTM time series model was constructed to predict the total daily replenishment of each vegetable category in the coming week, and finally the linear programming model was used to maximize the revenue as the objective function to finalize the total daily replenishment and pricing strategy, and the maximum revenue was 75,624 yuan.

Valuing European Option Under Double 3/2-Volatility Jump-Diffusion Model With Stochastic Interest Rate and Stochastic Intensity Under Approximative Fractional Brownian Motion

In this study, we propose a more comprehensive and realistic option pricing model based on approximative fractional Brownian motion, building upon recent advancements in this area. Specifically, we utilize the double 3/2-volatility Jump-Diffusion model, which incorporates approximative fractional Brownian motion with 3/2-volatility, stochastic interest rate, and stochastic intensity. To account for the stochastic interest rate, we employ a two-factor Vasicek model. Notably, our model accommodates negative interest rates. Consequently, we develop a multi-factor model with a stochastic interest rate structure for pricing European options and derive a closed-form pricing formula with an analytical solution by applying some algebraic calculations and Lie symmetries. In order to demonstrate the superiority of our proposed model over other classical approaches, we present numerical results that showcase the value of a European call option. This comparative analysis underscores the advantages of our model in comparison to traditional models.

Research on Vegetable Commodity Pricing and Replenishment based on Planning Models and Genetic Algorithm

This paper focuses on an in-depth study of supermarket vegetable pricing and replenishment problems, utilizing a variety of methods such as statistics, prediction models, planning models and other methods of analysis. First, the data were preprocessed, and frequency distribution histograms were drawn, revealing the distribution pattern and correlation between each category and each single product of vegetables through descriptive statistical analysis. Secondly, for the relationship between sales and pricing, the sales unit price was averaged through the cost-plus pricing formula, and MATLAB was used to nonlinearly fit the relationship between the total sales volume of the categories and the sales price, and the fitting result was further optimized through neural network, and a nonlinear planning model was established, and a genetic algorithm was used to solve the daily replenishment volume of supermarkets and the pricing strategy in order to achieve the maximization of revenue. Finally, the top-rated individual products were screened out by entropy weighting method, and a linear programming model was established to predict the replenishment quantity and pricing strategy for the coming day, which further provided effective decision support for the sales management of the supermarket.

Pricing first-touch digitals with a multi-step double boundary and American barrier options

A first-touch digital option delivers a payoff when an underlying variable first touches a given boundary, and most studies discuss its pricing by assuming a single flat boundary and a fixed payout amount. This paper derives explicit pricing formulas for first-touch digital options with a multi-step double boundary and customizable payoffs. To this end, we establish the probabilities of the first-hitting times of a Brownian motion for the multi-step double boundary. Furthermore, we explain how the first-touch digital option pricing formulas approximate an American knock-out multi-step barrier option.