Black-Scholes Research Articles

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

Construction Evaluation of Three Outperformance Options

Locking the most promising company in a certain industry at an early stage is appealing for all investors, especially when the industry’s capacity and prospect can be confirmed. The article tries to create three different special options to help investors to achieve this goal. Specifically, the construction process is based on outperformance options, Black-Scholes model, and some special valuation methods. According to the calculation, the price and the power of each future options are demonstrated clearly. Investors can choose one of them to satisfy their requirement for investment return and preference for risk. Four factors related to the price of the future options are also assessed based on sensitive analysis, i.e., one can understand the internal structure of the future options clearly. However, the research is based on the hypothesis that only 2 companies exist in the industry and people can determine the future of the industry. Therefore, following research can focus on multi-companies’ industry and add more uncertain elements to make the results and process more general. These results shed light on eliminating the uncertainty of intra-industry competition, locking the industry leader early and increasing the investment return.

Exploration of JPMorgan Chooser Option Pricing

As the development of the capital market, the derivatives of financial instruments have been active in the financial market. Investors can trade customized derivatives designed to manage increasingly complex risk exposures. Examples of exotic options satisfying demands include the chooser option. Most contracts require the holder to commit to a particular payoff profile at the outset, which may often be unattractive for some potential investors. Therefore, we choose the stocks of JPMorgan and use the Black-Scholes model to simulate the value of the chooser option. This article suggests approximating the value on two periods and aims to set the value of the chooser option of JPMorgan, constructs the reasonable expectation for investors, and enables them to make the proper decision on the chooser options. We provide reliable option values and retains the flexibility of simulation, in that it allows great freedom in decision date when the investors decide which type of option for the underlying assets or a joint process for the asset price, its volatility, and other asset prices. From the simulation results, we conclude that although the chooser option is the same as the traditional option in nature, the chooser option puts off the decision date and makes the payoff less volatile to the movement of the stock market, thus partly hedge the risks of the stock market.

Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-orderαin the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

Artificial Neural Networks Performance in WIG20 Index Options Pricing.

In this paper, the performance of artificial neural networks in option pricing was analyzed and compared with the results obtained from the Black–Scholes–Merton model, based on the historical volatility. The results were compared based on various error metrics calculated separately between three moneyness ratios. The market data-driven approach was taken to train and test the neural network on the real-world options data from 2009 to 2019, quoted on the Warsaw Stock Exchange. The artificial neural network did not provide more accurate option prices, even though its hyperparameters were properly tuned. The Black–Scholes–Merton model turned out to be more precise and robust to various market conditions. In addition, the bias of the forecasts obtained from the neural network differed significantly between moneyness states. This study provides an initial insight into the application of deep learning methods to pricing options in emerging markets with low liquidity and high volatility.