Black-Scholes Research Articles

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.

Min–max multi-step barrier options and their variants

This paper studies a new type of barrier option, min–max multi-step barrier options with diverse multiple up or down barrier levels placed in the sub-periods of the option’s lifetime. We develop the explicit pricing formula of this type of option under the Black–Scholes model and explore its applications and possible extensions. In particular, the min–max multi-step barrier option pricing formula can be used to approximate double barrier option prices and compute prices of complex barrier options such as discrete geometric Asian barrier options. As a practical example of directly applying the pricing formula, we introduce and evaluate a re-bouncing equity-linked security. The main theorem of this work is capable of handling the general payoff function, from which we obtain the pricing formulas of various min–max multi-step barrier options. The min–max multi-step reflection principle, the boundary-crossing probability of min–max multi-step barriers with icicles, is also derived.

An Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process

After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2+k2).

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Asymptotics for the Laplace transform of the time integral of the geometric Brownian motion

We present an asymptotic result for the Laplace transform of the time integral of the geometric Brownian motion F(θ,T)=E[e−θXT] with XT=∫0TeσWs+(a−12σ2)sds, which is exact in the limit σ2T→0 at fixed σ2θT2 and aT. This asymptotic result is applied to pricing zero coupon bonds in the Dothan model of stochastic interest rates. The asymptotic result provides an approximation for bond prices which is in good agreement with numerical evaluations in a wide range of model parameters. As a side result we obtain the asymptotics for Asian option prices in the Black-Scholes model, taking into account interest rates and dividend yield contributions in the σ2T→0 limit.

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.

On regime-switching European option pricing

The concern of this article is to derive a regime switching model that can be utilized to price European call options for a financial market that exhibits structural changes with time. The model is formulated based on the fact that the underlying asset process is described by a geometric Brownian motion that is modulated by a continuous-time Markov chain with two regimes. Moreover, by an application of the change of measure technique, an option price is derived under the risk neutral valuation and the model parameter estimates is performed by use of the maximum likelihood estimation. The model implementation is carried out by utilizing the Russell 2000 and Facebook in dices data sets. The model results are compared with that of the Black-Scholes model in order to establish the model with better results in terms of predicting the European call option prices. In general, the data sets have common characteristics of financial time series across the regimes and the volatility process spends longer time in regime 2 than it stays in regime 1. The predicted call option prices from both models are more or less similar across the market indices; however, the results of the Black-Scholes model are a bit closer to the market prices than that of the regime-switching model across the two markets. Therefore, the Black-Scholes model slightly gives better results for the Russell 2000 and Facebook indices data sets as compared with the RS model.

The pricing and static hedging of multi-step double barrier options

As a sequel to Lee et al. (2022b), this paper explores the pricing of multi-step double barrier options with arbitrary European-type payoffs from a static hedging perspective. Using the reflection principle of Brownian motion, we develop how to construct an exact static hedging portfolio consisting of simple discrete barrier options under the Black–Scholes model. This equivalent conversion from continuous monitoring to discrete ones provides an efficient way of evaluating multi-step double barrier options, while overcoming the drawbacks of dynamic hedging. We illustrate our result with numerical examples, and extend it to other asset price dynamics such as jump diffusion.

Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion

In this work, we study the numerical solution for time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator. For simplicity of the analysis, the model problem is converted into a time fractional partial integro-differential equation with a Fredholm integral operator. The L1 discretization is introduced on a graded mesh to approximate the temporal derivative. A second order central difference scheme is used to replace the spatial derivatives and the composite trapezoidal approximation is employed to discretize the integral part. The stability results for the proposed numerical scheme are derived with a sharp error estimation. A rigorous analysis proves that the optimal rate of convergence is obtained for a suitable choice of the grading parameter. Further, we introduce the Adomian decomposition method to find out an analytical approximate solution of the given model and the results are compared with the numerical solutions. The main advantage of the fully discretized numerical method is that it not only resolves the initial singularity occurred due to the presence of the fractional operator, but it also gives a higher rate of convergence compared to the uniform mesh. On the other hand, the Adomian decomposition method gives the analytical solution as well as a numerical approximation of the solution which does not involve any mesh discretization. Furthermore, the method does not require a large amount of computer memory and is free of rounding errors. Some experiments are performed for both methods and it is shown that the results agree well with the theoretical findings. In addition, the proposed schemes are investigated on numerous European option pricing jump-diffusion models such as Merton’s jump-diffusion and Kou’s jump-diffusion for both European call and put options.

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.

Comparison of Black-Scholes and GARCH Option Models on The Jakarta Islamic Index with Collar Strategy

Objective –The revival of the Islamic economy in Indonesia was marked by the establishment of the Jakarta Islamic Index (JII) on July 3, 2000. From 2000 to 2020, two economic crises hit Indonesia in 2008 and 2020. These crises also had an impact on the high price fluctuations of JII. Assets can be protected from significant losses by hedging. One of the hedging strategies was option contracts with a collar strategy. Methodology – The data analysed in this research were the JII closing price data from 2000 to 2020 using the Black Scholes model and the GARCH model with the Collar strategy. Findings – Option with the Collar strategy provided more benefits in crises than without Option. In crisis conditions, the Option with the GARCH method and the collar strategy provided a higher average profit of 3.07% on the one-month Option and 7.01% on the three-month Option than without using the Option with the collar strategy. In non-crisis conditions, the Option using the GARCH method, and the collar strategy provided a 0.16% higher average profit at one-month maturity than without the Option. Meanwhile, the average profit decreased by 1.45% at the three-month maturity. The collar strategy resulted in less volatility. Without a collar strategy, JII volatility was 103%. The collar strategy produced maximum volatility of 12.71%, 15.18%, and 17.14%. The findings also revealed that the GARCH model was better than the Black Scholes based on the AMSE value obtained in crisis conditions with a maturity of one month and three months and in non-crisis conditions with a maturity of one month. Meanwhile, the Black Scholes model showed better results in non-crisis conditions with a maturity of three months. Novelty – Based on the research result, the Option with a collar strategy is effective for hedging in crisis conditions. This research assesses the collar strategy with several different maturity values to get the most effective value. Type of Paper: Empirical JEL Classification: G11, G13. Keywords: Black Scholes; Collar Strategy; GARCH; Jakarta Islamic Index; Option Contracts. Reference to this paper should be made as follows: Hendrawan, R; Arifin, Z. (2023). Comparison of Black-Scholes and GARCH Option Models on The Jakarta Islamic Index with Collar Strategy, J. Fin. Bank. Review, 7(4), 16 – 27. https://doi.org/10.35609/jfbr.2023.7.4(2)

Bridging electricity market and carbon emission market through electric vehicles: Optimal bidding strategy for distribution system operators to explore economic feasibility in China's low-carbon transitions

China, as the world's largest emitter of carbon emissions, has implemented carbon markets to reduce carbon emissions, but the economic feasibility of integrating electric vehicles (EVs) into these markets remains uncertain. To address this challenge, we propose a novel methodology for grid integration of EVs based on the Shenzhen carbon inclusive management measures. Our proposed methodology can optimize EV charging schedules based on carbon emission prices and improve the economic feasibility of low-carbon transitions through EVs in China. We incorporate the Black-Scholes model to measure carbon emission option pricing and develop an optimal bidding strategy that considers spatiotemporal uncertainties of EVs and electricity markets. We use numerical simulations with real-world data from Shenzhen carbon emission market and Guangdong electricity market to demonstrate the effectiveness of our proposed methodology. Results reveal that scheduling 100 EVs' charging/discharging behavior properly can generate a revenue of 2379.4 yuan for the distribution system operator, while the revenue for carbon emission reduction (CER) is only 9.35 yuan. This paper demonstrates that the potential revenue of CER may be substantially based on the future scenario for the carbon emission market. These findings provide valuable insights for policymakers and stakeholders seeking to reduce carbon emissions in China's electricity sector.

A meshless multiquadric quasi-interpolation method for time fractional Black–Scholes model

Based on multiquadric quasi-interpolation, this study presents a meshless numerical method for time fractional Black–Scholes (B–S) model. The method is highly accurate and flexible. The stability and convergence of the proposed scheme are discussed in detail. Numerical results show the accuracy and efficiency of the proposed method.

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

SUBLEADING CORRECTION TO THE ASIAN OPTIONS VOLATILITY IN THE BLACK–SCHOLES MODEL

The short maturity limit [Formula: see text] for the implied volatility of an Asian option in the Black–Scholes model is determined by the large deviations property for the time-average of the geometric Brownian motion. In this note, we derive the subleading [Formula: see text] correction to this implied volatility, using an asymptotic expansion for the Hartman–Watson distribution. The result is used to compute subleading corrections to Asian options prices in a small maturity expansion, sharpening the leading order result obtained using large deviations theory. We demonstrate good numerical agreement with precise benchmarks for Asian options pricing in the Black–Scholes model.

The Black-Scholes Exotic Barrier Option Pricing Formula

The paper considers a specific type of such financial instrument as an option, namely an exotic barrier call option of the European type. Exotic options are gaining popularity among ordinary investors due to the development of information and telecommunication technologies, thanks to which such specific financial instruments as options have become readily available. We investigate the hedging problem for such options with some restrictions on the payment function and the availability of dividend payment on a risky asset in the classical Black-Scholes model. An analogue of the Black-Scholes formula for the mentioned variant of the exotic barrier is proved. In the future, it is planned to generalize the obtained results for put options and for more general payment functions.

Fast Barrier Option Pricing by the COS BEM Method in Heston Model (with Matlab Code)

Abstract In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper.

The Progress of Black-Scholes Model and Black-Scholes-Merton Model

Black-Scholes (BS) model was first proposed in 1973, which has been modified by Robert Merton as the Black-Scholes-Merton (BSM) model subsequently. Contemporarily, these two models have been widely used and praised by financial scholars as well as employees. Plenty of scholars have tried to verify the accuracy of the and expressed their views on the existing defects in above models. Based on the existing literature, this article first introduces and derives the two models step by step and discusses the basic assumptions for these models. Subsequently, the applications of the two models are demonstrated separately. Specifically, the project valuation based on BS model is presented detaily while the applications of BSM model are introduced from four aspects (pricing of intangible assets, risk avoidance, default prediction and employee stock option’s pricing). Afterwards, the limitations and gaps of the models (e.g., volatility smile) ascribed to the ideal assumptions are discussed. In order to tackle the issue, improvement and suggestions are proposed including extending the models with different forms. These results offer a guideline for the option pricing, which can be widely applied in investment strategy.

Implementation of Statistical Tool in Legendary Pokémon Analysis and the Pricing in Stock Market

Statistics are widely applied in wide spread of advanced science researches to give the quantitative crustily, which is now much more well-liked. In this paper, a dataset is analyzed and a pricing method is presented based on two different statistical models. In the random forest model, we first do a simple statistical data analysis to determine the association between attributes and "is legendary" using the Legendary Pokémon dataset. The above analysis results may then be used to determine which characteristics can make a Pokémon legendary. In the Black-Scholes model, we first discuss SDE before using it to provide a stock pricing method. For the former one, the statistical data analysis in the first application may determine that all the characteristics have a positive correlation with "is legendary," and lastly, random forest discovered that the characteristics that can turn a Pokémon legendary are special attack and speed. Through the second application, it can determine how an SDE’s trajectory and coefficients are related. The pricing method is then obtained by applying SDE to the Black-Scholes model. We value a company’s shares using this method and contrast the accuracy of this approach with more conventional approaches. According to data analysis, the first application easily and intuitively demonstrates random forest’s practical application capabilities, as well as its ease and accuracy for us. These results shed light on guiding further exploration of the practice of random forests in data analysis and the application of SDE related models.