Black-Scholes Research Articles

Practical finite difference method for solving multi-dimensional black-Scholes model in fractal market

In this paper, we employ a practical finite difference method to research the multi-dimensional fractional Balck-Scholes model under one asset and three assets. In the case of one asset, we establish explicit scheme and Crank-Nicolson scheme to study the effect of different Hurst exponent (H) on numerical results and Greeks. With the increase of H, the numerical figures of the finite difference scheme also change. In addition, we also verify the effectiveness of Crank-Nicolson scheme in numerical solution of Greeks. We observe that when H=0.5, the results of Delta, Gamma and Theta are consistent with the accurate results. In the case of three assets, we use operator splitting method (OSM) and establish semi-implicit scheme. We hold that H will affect the numerical results and Greeks results in fractional Black-Scholes model. If the effect of H is not considered in option hedging, the result will deviate greatly from the actual result.

Non-equilibrium skewness, market crises, and option pricing: Non-linear Langevin model of markets with supersymmetry

This paper presents a tractable model of non-linear dynamics of market returns using a Langevin approach. Due to non-linearity of an interaction potential, the model admits regimes of both small and large return fluctuations. Langevin dynamics are mapped onto an equivalent quantum mechanical (QM) system. Borrowing ideas from supersymmetric quantum mechanics (SUSY QM), a parameterized ground state wave function (WF) of this QM system is used as a direct input to the model, which also fixes a non-linear Langevin potential. Using a two-component Gaussian mixture as a ground state WF with an asymmetric double well potential produces a tractable low-parametric model with interpretable parameters, referred to as the NES (Non-Equilibrium Skew) model. Supersymmetry (SUSY) is then used to find time-dependent solutions of the model in an analytically tractable way. Additional approximations give rise to a final practical version of the NES model, where real-measure and risk-neutral return distributions are given by three component Gaussian mixtures. This produces a closed-form approximation for option pricing in the NES model by a mixture of three Black–Scholes prices, providing accurate calibration to option prices for either benign or distressed market environments, while using only a single volatility parameter. These results stand in stark contrast to the most of other option pricing models such as local, stochastic, or rough volatility models that need more complex specifications of noise to fit the market data.

Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions

Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions

A Black–Scholes user's guide to the Bachelier model

AbstractTo cope with the negative oil futures price caused by the COVID–19 recession, global commodity futures exchanges temporarily switched the option model from Black–Scholes to Bachelier in 2020. This study reviews the literature on Bachelier's pioneering option pricing model and summarizes the practical results on volatility conversion, risk management, stochastic volatility, and barrier options pricing to facilitate the model transition. In particular, using the displaced Black–Scholes model as a model family with the Black–Scholes and Bachelier models as special cases, we not only connect the two models but also present a continuous spectrum of model choices.

Valuing lookback options with barrier

In this paper, we introduce a new class of exotic options, termed lookback-barrier options, which literally combine lookback and barrier options by incorporating an activating barrier condition into the European lookback payoff. A prototype of lookback-barrier option was first proposed by Bermin (1998), where he intended to reduce the expensive cost of lookback option by considering lookback options with barrier. However, despite his novel trial, it has not attracted much attention yet. Thus, in this paper, we revisit the idea and extend the horizon of lookback-barrier option in order to enhance the marketability and applicability to equity-linked investments. Devising a variety of payoffs, this paper develops a complete valuation framework which allows for closed-form pricing formulas under the Black–Scholes model. Our closed-form pricing formulas provide a substantial advantage over the method of Monte Carlo simulation, because the extrema appearing in both of the lookback payoff and barrier condition would require a large number of simulations for exact calculation. Complexities involved in the derivation process would be resolved by the Esscher transform and the reflection principle of the Brownian motion. We illustrate our results with numerical examples.

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.

A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

Multi‐step reflection principle and barrier options

AbstractThis paper examines a class of barrier options, multi‐step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression for option prices of this type under the Black–Scholes model by deriving the multi‐step reflection principle, that is, the multi‐step boundary‐crossing probability of Brownian motion. Multi‐step barrier options are not only useful in that they can handle barriers of different levels and time steps but can also approximate options with arbitrary barriers. Moreover, they can be applied to pricing barrier options under jump‐diffusion models.

Pricing equity warrants with jumps, stochastic volatility, and stochastic interest rates

A warrant is a derivative that gives the right, but not the obligation, to buy or sell a security at a certain price before the expiration. The warrant valuation method was inspired by option valuation because of the certain similarities between these two derivatives. The warrant price formula under the Black–Scholes is available in the literature. However, the Black–Scholes formula is known to have a number of flaws; hence, this study aims to develop a pricing formula for warrants by incorporating jumps, stochastic volatility, and stochastic interest rates into the Black–Scholes model. The closed-form pricing formula is presented in this study, where the derivation involves stochastic differential equations (SDE), which include the Cauchy problem and heat equation.

An alternative valuation of public-private partnerships by using the Black-Scholes model: the Portuguese highway case

An alternative valuation of public-private partnerships by using the Black-Scholes model: the Portuguese highway case

An alternative valuation of public-private partnerships by using the Black-Scholes model: the Portuguese highway case

This study evaluates Portuguese highway PPPs' initial valuation using real options methods to assess the potential mispricing of these projects that is model dependent. Our findings suggest that some of these PPPs could be completed without resorting to outside financing for the initial investment by selling a right to ownership of projects. This different approach to real investments using options could also enable hedging future operational risks without changing the project operationally. Overall, our study addresses a relevant policy implication - whether governments should protect their interest by adopting PPP valuation as real options.

A New Parallel Difference Method for Solving Time Fractional Black-Scholes Model

A modified Black-Scholes (B-S) model with time fractional derivative is studied when the price change of the underlying is considered as a fractal transmission system. It is very practical in application to study the numerical computation of this time fractional B-S model (TFBSM) governing European options. This paper constructs mixed alternative segment Crank-Nicolson (MASC-N) parallel difference scheme, which accuracy is spatially 2 order and temporally 2-α order. In addition, we have provided a theoretical proof for the stability and convergence of the MASC-N scheme, and demonstrated the accuracy and effectiveness of the proposed method through numerical experiments. The numerical results show that the MASC-N scheme is easy to implement when applied to time fractional B-S model.

High‐Accurate Numerical Schemes for Black–Scholes Models with Sensitivity Analysis

The significance of both the linear and nonlinear Black‐Scholes partial differential equation model is huge in the field of financial analysis. In most cases, the exact solution to such a nonlinear problem is very hard to obtain, and in some cases, it is impossible to get an exact solution to such models. In this study, both the linear and the nonlinear Black‐Scholes models are investigated. This research mainly focuses on the numerical approximations of the Black‐Scholes (BS) model with sensitivity analysis of the parameters. It is to note that most applied researchers use finite difference and finite element‐based schemes to approximate the BS model. Thus, an urge for a high accurate numerical scheme that needs fewer grids/nodes is huge. In this study, we aim to approximate and analyze the models using two such higher‐order schemes. To be specific, the Chebyshev spectral method and the differential quadrature method are employed to approximate the BS models to see the efficiency of such highly accurate schemes for the option pricing model. First, we approximate the model using the mentioned methods. Then, we move on to use the numerical results to analyze different aspects of stock market through sensitivity analysis. All the numerical schemes have been illustrated through some graphics and relevant discussions. We finish the study with some concluding remarks.

Option Pricing Based on GA-BP neural network

Machine Learning has received widespread attention in the financial field, especially in the issue of option pricing. Compared with the original BSM (Black-Scholes-Merton) option pricing method, this paper introduces the genetic algorithm into the option pricing process based on the characteristics of financial data, and constructs a GA-BP neural network (Neural Networks) option pricing Model and empirical analysis of European call options based on the Shanghai and Shenzhen 300 Index. The research results show that the option price calculation result of the GA-BP neural network algorithm has better accuracy than the BP neural network model or the classic BSM method, while taking into account efficiency, which is helpful to predicting option prices in financial practice to a certain extent.

Analytical Solutions of Time Space Fractional Black-Scholes Option Pricing Model with Their Applications

Analytical Solutions of Time Space Fractional Black-Scholes Option Pricing Model with Their Applications

Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model

The Ivancevic option pricing model comes as an alternative to the Black‐Scholes model and depicts a controlled Brownian motion associated with the nonlinear Schrodinger equation. The applicability and practicality of this model have been studied by many researchers, but the analytical approach has been virtually absent from the literature. This study intends to examine some dynamic features of this model. By using the well‐known ARS algorithm, it is demonstrated that this model is not integrable in the Painlevé sense. He’s variational method is utilized to create new abundant solutions, which contain the bright soliton, bright‐like soliton, kinky‐bright soliton, and periodic solution. The bifurcation theory is applied to investigate the phase portrait and to study some dynamical behavior of this model. Furthermore, we introduce a classification of the wave solutions into periodic, super periodic, kink, and solitary solutions according to the type of the phase plane orbits. Some 3D‐graphical representations of some of the obtained solutions are displayed. The influence of the model’s parameters on the obtained wave solutions is discussed and clarified graphically.

Derivation of black Scholes equation using Heston’s model with dividend yielding asset

Black Scholes formula is crucial in modern applied finance. Since the introduction of Black – Scholes concept model that assumes volatility is constant; several studies have proposed models that address the shortcomings of Black-Scholes model. Heston’s models stands out amongst most volatility models because the process of volatility is positive and is a process that obeys mean reversion and this is what is observed in the real market world. One of the shortcomings of Heston’s model is that it doesn’t incorporate dividend yielding asset. Black Scholes partial differential equation revolves around Geometric Brownian motion and its extensions. We, therefore, incorporate dividend yielding assets on Heston’s model and use it to model a new Black Scholes equation using the knowledge of partial differential equations.

Application of Continous Time Model in Prediction of Loss Reserves in Credit Insurance for Asset-Based Lending Companies

Asset-based lending companies and other loan providers are exposed to risk of loan defaults by borrowers. To reduce this risk, these companies acquire credit insurance. Thus when the borrower defaults in payment, the insurance company covers a percentage of the outstanding balance which generates a way to lessen and spread credit risk that the lender incurs. Therefore there are a number of methods put in place such as frequency-severity and hazard rate models used to value credit insurance. Valuing of credit insurance for asset-based lending companies is a challenging task especially in Kenyan market, where in the case of a borrower’s default, the process for recovering of the collateral will last a longer period of more than a year and where data on the borrower’s behavior of payment is of poor quality or generally unavailable. The existing methods do not consider the time to repossession of the collateral in case of loan default. Our proposed model takes into account time to repossession of the collateral and can be used in emerging market economies where other available methods may be either unsuitable or are too complex to implement due to lack of enough data. Therefore, this paper incorporates a continuous time model to forecast loss reserves in credit insurance for asset-based lending companies. First, we establish a discrete-time model to describe delinquency of credits in loan insurance product. Martingale properties, Replicating of asset portfolio strategy and Itô’s calculus are used to obtain results on expected values of future losses of credit insurance products. Secondly, we used the Black-Scholes model to develop a continuous-time model to forecast future losses in credit insurances. This is constructed by linking the latter from the discrete-time (Binomial) model using the methods of stochastic calculus. We estimated the loss reserves by applying the Geometric Brownian Motion on data of outstanding balances of 30% of 100 loanees from car loan business with assets valued at between 1 million and 10 million, to predict the probability of default of the borrower.

Option pricing with neural networks vs. Black-Scholes under different volatility forecasting approaches for BIST 30 index options

This study compares the performances of neural network and Black-Scholes models in pricing BIST30 (Borsa Istanbul) index call and put options with different volatility forecasting approaches. Since the volatility is the key parameter in pricing options, GARCH (Generalized Autoregressive Conditional Heteroskedasticity), implied volatility, historical volatility, and implied volatility index (VBI) are used to determine the best volatility approach for pricing options according to moneyness and time-to-maturity dimensions. The paper also includes a subsample analysis in which the pricing performance of the models are evaluated during the turbulent periods. Overall results indicate that neural network outperforms Black-Scholes during tranquil times while Black-Scholes outperforms neural network during turbulent periods for call options. For put options, the Black-Scholes model is the best model during tranquil periods while neural network is the best model during turbulent periods.

PENGARUH PEMBAGIAN DIVIDEN MELALUI MODEL BLACK-SCHOLES

Stock trading has a risk that can be said to be quite large due to fluctuations in stock prices. In stock trading, one alternative to reduce the amount of risk is options. The focus of this research is on European options which are financial contracts by giving the holder the right, not the obligation, to sell or buy the principal asset from the writer when it expires at a predetermined price. The Black-Scholes model is an option pricing model commonly used in the financial sector. This study aims to determine the effect of dividend distribution through the Black-Scholes model on stock prices. The effect of dividend distribution through the Black-Scholes model on stock prices results in the stock price immediately after the dividend distribution being lower than the stock price shortly before the dividend distribution