Black-Scholes Research Articles

Multivariate Option Pricing with Gaussian Mixture Distributions and Mixed Copulas

Recently, it has been reported that the hypothesis proposed by the classical black scholes model to price multivariate options in finance were unrealistic, as such, several other methods have been introduced over the last decades including the copulas methods which uses copulas functions to model the dependence structure of underlying assets. However, the previous work did not take into account the use of mixed copulas to assess the underlying assets' dependence structure. The approach we propose consists of selecting the appropriate mixed copula’s structure which captures as much information as possible about the asset’s dependence structure and apply a copulas-based martingale strategy to price multivariate equity options using monte Carlo simulation. A mixture of normal distributions estimated with the standard EM algorithm is also considered for modeling the marginal distribution of financial asset returns. Moreover, the Monte Carlo simulation is performed to compute the values of exotic and up and out barrier options such as worst of, spread, and rainbow options, which shows that the clayton gumble and clayton gaussian have relatively large values for all the options. Our results further indicate that the mixed copula-based approach can be used efficiently to capture heterogeneous dependence structure existing in multivariate assets, price exotic options and generalize the existing results.

Predictable forward performance processes in complete markets

We establish existence of Predictable Forward Performance Processes (PFPPs) in conditionally complete markets, which has been previously shown only in the binomial setting. Our market model can be a discrete-time or a continuous-time model, and the investment horizon can be finite or infinite. We show that the main step in construction of PFPPs is solving a one-period problem involving an integral equation, which is the counterpart of the functional equation found in the binomial case. Although this integral equation has been partially studied in the existing literature, we provide a new solution method using the Fourier transform for tempered distributions. We also provide closed-form solutions for PFPPs with inverse marginal functions that are completely monotonic and establish uniqueness of PFPPs within this class. We apply our results to two special cases. The first one is the binomial market and is included to relate our work to the existing literature. The second example considers a generalized Black–Scholes model which, to the best of our knowledge, is a new result.

Option pricing using deep learning approach based on LSTM-GRU neural networks: Case of London stock exchange

<abstract> <p>This study is a review of literature on machine learning to examine the potential of deep learning (DL) techniques in improving the accuracy of option pricing models versus the Black-Scholes model and capturingcomplex features in financial data.</p> <p>Neural networks and other machine learning models have been proposed for option pricing and have improved accuracy compared withtraditional models. However, such use of machine learning also presents practical challenges such as data availability and quality, computational resources, model selection and validation, interpretability and overfitting. This study discusses several of these challenges and highlights the need for careful evaluation and validation of machine learning models in London option pricing during the Coronavirus disease 2019 pandemic. Moreover, to investigate the quality of the models used, we compare the performances of these algorithms in option pricing through the application of significance statistical tests.</p> </abstract>

A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models

<abstract><p>The numerical solution of the time-fractional Black-Scholes model for European and American options is presented using a local meshless collocation approach based on hybrid Gaussian-cubic radial basis functions with polynomials is presented. The approach is then expanded to a nonlinear time-fractional model for an option with transaction costs in a market with low liquidity. The spatial derivatives of the models are discretized using the proposed meshless technique. Numerical experiments are carried out for the American option, European option, and nonlinear transaction cost option models. In order to evaluate the effectiveness and precision of the suggested meshless approach, $ L_{\infty} $ and $ L_{rel} $ error norms are utilized. Both call and put option volatility is explored. A non-uniform grid customized around the strike price region is also used to determine the prices of European call and American put options. The methods described in literature are compared with the numerical results.</p></abstract>

A Study on Value Assessment of E-Commerce Enterprises Based on the Model of Real Options

With the rapid development of network technology, global e-commerce enterprises are mushrooming, and corresponding economic behaviors such as investment, financing and mergers and acquisitions of e-commerce enterprises are constantly appearing. Therefore, it is very important to construct the value assessment system of e-commerce enterprises. This paper takes the characteristics and value of e-commerce enterprises as a starting point, and explores the value assessment of e-commerce enterprises based on the real option theory of Black-Scholes (B-S), which is a model with good applicability in market transactions. Not only can the model reduce the computational complexity of enterprise value, it can also improve the accuracy of value assessment, which is now an important reference for the value assessment of e-commerce enterprises in the era of big data.

Extrapolation of Crude Oil Futures Implied Return by Black Scholes Model and Monte Carlo Simulation

Since the global COVID-19 outbreak in 2020, central banks around the world have adopted loose monetary policies to hedge against the downward pressure on the economy through quantitative easing. With the gradual increase in the number of currencies, from the beginning of 2022, global commodity prices began to rise continuously. Among them, the price of fossil fuels such as oil has hit record highs due to the intensification of geopolitical conflict between Russia and Ukraine. During this period, as the divergence in oil prices intensified, speculation opportunities in the derivatives market also emerged one after another. [1] Therefore, in this article, the principles of two pricing models, the Black Scholes Model and the Monte Carlo Model, are used to predict the price trend of Crude oil in the future. The implied volatility calculated according to the Black Scholes model will be used as the volatility of the oil futures price trend in the Monte Carlo model, and the current option premium will be used to reverse the implied return in the Monte Carlo simulation. It serves as a forecast of future yields for oil futures. By using the data from March to September 2022, the correlation coefficient between the predicted return and the actual return is 0.67, which is a relatively convincing prediction accuracy. However, the volatility of the predicted return will be higher than the actual return due to factors such as the volatility smile.

A Study on the Applicability of Black-Scholes Option Pricing Model in Chinese Stock Index Option Market

Compared with the developed countries, Chinese stock index option market developed late and accumulated less historical data. At the same time, it has the characteristics that many developed countries' stock option markets do not have. Black-Scholes Model is an important model in the field of option pricing. It is not enough for a financial model to be reasonable in theory, but it needs to be tested by the actual market. Therefore, based on the main transaction data of the CSI 300 stock index option contracts from 2010 to 2020, this paper analyzes the compliance of the CSI 300 stock index option market to the basic assumptions of the B-S option pricing model, the main parameters of the CSI 300 stock index options in the B-S option pricing model, and the fitting degree of the B-S model's prediction and actual prices. The result shows that there is a certain deviation between the pricing conclusions of the B-S model and the actual option prices. This paper analyzes the reasons for the deviation from many aspects, and puts forward the corresponding improvement strategies that the users of the model should adopt, so as to improve the applicability of the B-S option pricing model in Chinese stock index option market.

Testing Black Scholes and Garch Model Options on Gold Price Index With Long Strangle Strategy Using 1985-2020 Data

Objective - This research examines the implementation of options contracts on gold prices using GARCH and Black Scholes models accompanied by a long strangle strategy. Methodology – This research used 36 years of secondary data for daily gold prices from www.gold.org obtained from the year 1985 to 2020. The implementation of options contracts on gold prices using GARCH and Black Scholes models with long strangle strategy through three types of maturity dates (1-month, 2-months, and 3-months). These results were tested by comparing the average percentage value of the actual options premium price and the options calculated using the AMSE (Average Mean Square Error) methodology, where the smallest percentage value is a more precise reflection of the model. Findings – The research indicates that the results of the error percentage for 1-month, 2-months, and 3-months maturities increased linearly along with the size of the maturity period of the option contract also profit percentage of long strangle strategy for gold price option contract in average 17-27% or below 30%. Novelty – Based on the research result, the long strangle strategy is not the best strategy regarding the gold price option contract, and this study can contribute to current practices for the investor. Also, this research is unique because no research used gold price data for 36 years (daily basis) for the options contracts. Type of Paper - Empirical Keywords: Gold Price; Options Contract; Black Scholes; GARCH; Long Strangle; AMSE JEL Classification: G11, G13. Reference to this paper should be made as follows: Hendrawan, R; Maharsasi, D.E. (2022). Testing Black Scholes and Garch Model Options on Gold Price Index With Long Strangle Strategy Using 1985-2020 Data, J. Fin. Bank. Review, 7(3), 160 – 174. https://doi.org/10.35609/jfbr.2022.7.3(3)

Option pricing in emerging markets using pure jump processes: explicit calibration of IST30 European option

Purpose- This study aims to illustrate the efficiency of pure jump processes, more specifically Variance Gamma (VG) and Normal Inverse Gaussian models (NIG), in option pricing by comparing with the Black Scholes (BS) option pricing model for emerging markets. Methodology- This study presents an alternative derivation of option pricing formulas for VG and NIG models. Then, it investigates the VG and NIG models' option pricing performance with the help of new derivation by comparing them with the BS option pricing model for emerging markets for an emerging country, Turkey. The data consists of the BIST30 index daily price and European options written on this index extend from 05 May 2018 to 05 May 2020 for given exercise prices with a maturity of 90 days. In this period, the European call options' strike prices range from 1200 to 1650, and the European put options' strike prices range from 1000 to 1400. To compare the models' efficiency, first, we calibrate the models by minimizing the sum of squared deviations between the observed and theoretical option prices. Second, we compute the option prices and compare the results with the observed option prices. Findings- The significant contribution to the literature is the calibration of the pure jump processes (VG and NIG processes) using the characteristic functions, the continuous BS prices for an emerging market, and the computation of European options prices in BIST. We find that while the NIG process performs better than VG and BS models, the BS model is the worst in option pricing. Conclusion- The pure jump processes (VG and NIG processes) can be calibrated using the characteristic functions, and option price estimations with them are better than the continuous BS prices for an emerging market. Thus, the pure jump processes are more efficient in market modeling than the BS model.

Testing Black Scholes and Garch Option Models on Pharmaceutical State-Owned Enterprises Holding

This study aims at testing the implementation of option contracts using Black Scholes and GARCH Option Models on the pharmaceutical State-Owned Enterprises (BUMN, Badan Usaha Milik Negara) holding using Long Straddle Strategy. The data were the closing stock price from 2002 to 2021 of two companies holding: INAF and KAEF. Results of this study were calculated by comparing percentage of average mean squared error of the Black Scholes and GARCH model with the implementation of Long Straddle Strategy, in which the smaller the percentage the better the model. The result showed that for one-month due date option contract, Black Scholes model was better than GARCH with error value on call option of 6.28% and put option of 4.279% for INAF and error value on call option of 5.24% and put option of 3.29%. With three-month due date option contract, Black Scholes model continued to show better results for call option with error value of 20.38% for INAF and 14.59% for KAEF. Conversely, GARCH model was better on the put option with 14.69% error value for INAF and 9.50% for KAEF. Keywords: Black Scholes, GARCH, option contract, Long Straddle

The Collar Strategy as a Hedgıng Effort in the Jakarta Islamıc Index

The revivals of the Islamic economy in Indonesia was marked by the establishment of the Jakarta Islamic Index (JII) on July 3, 2000. The crises in 2008 and 2020 caused JII to experience relatively high fluctuations. Hedging was necessary to protect assets from sharp declines. One of the hedging strategies was to use option contracts with a collar strategy. This research focuses on testing and comparing various types of collar strategy option using the Black Scholes method with historical variance and GARCH (1,1) variance on JII in the period of one and three months. According to Sinclair (2010), the collar strategy is a combination of a covered call option and a protective put option. The collar strategy provides limited profits but the collar provides hedging with little or no cost if the buying price of the put option is the same as the selling price of the call option. Based on the above phenomenon and previous research, this research aimed at comparing the Black Scholes and GARCH models using the Collar Strategy as a Hedging Effort in the Jakarta Islamic Index (JII). Keywords: Black Scholes, GARCH, Jakarta Islamic Index, Option Contracts

Optimal investment strategies and profit shares distributions: A stochastic control approach

With the advent of Solvency II regulations in 2016, European insurance companies are tasked with calculating technical provisions by either using a standard formula given by regulators or developing their own internal model. This paper first reviews a model adapted to Solvency II regulations for an insurance company’s participating contracts, which was proposed by Hainaut [1]. The objective of the insurer is to optimize profit shares distributions and asset allocation strategies. In this setting, a stochastic control approach is used to formulate a dynamic portfolio optimization problem. The insurer invests in an account or bond that is risk-free, she also invests in a risky asset that is based on the Black-Scholes model. A closed form solution to the problem under a power utility function on future profit shares and the insurer’s future economic wealth is calculated. The paper ends with a numerical example, where Tesla and Apple share prices are used as the insurer’s risky asset, in order to test the usability of the model in industry.

Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter

Black-Scholes (BS) equations, which are in the form of stochastic partial differential equations, are fundamental equations in mathematical finance, especially in option pricing. Even though there exists an analytical solution to the standard form, the equations are not straightforward to be solved numerically. The effective and efficient numerical method will be useful to solve advanced and non-standard forms of BS equations in the future. In this paper, we propose a method to solve BS equations using an approach of optimization problems, where a metaheuristic optimization algorithm is utilized to find the best-approximated solutions of the equations. Here we use the Adaptive Differential Evolution with Learning Parameter (ADELP) algorithm. The BS equations being solved are meant to find values of European option pricing that is equipped with Barrier option pricing. The result of our approximation method fits well to the analytical approximation solutions.

Psychological barriers and option pricing in a local volatility model

In this paper, we investigate the possible existence of psychological barriers in the SSE 50 Index and find that the volatility of the index returns and the implied volatility around psychological barriers are significantly different. To capture this special behavior, a local volatility model (LVM) is introduced, and the price of the European call option is derived under this framework with the help of the Laplace transform approach. We also report the LVM’s empirical and delta hedging performance relative to the Black–Scholes (BS) model, constant elasticity of variance (CEV) model and Jang et al.’s threshold model and find that for 50ETF and S&P 500 option pricing, the LVM is superior to the other three models.

IS THE BLACK–SCHOLES MODEL GOOD ENOUGH FOR RETAIL INVESTORS IN CHINA?

This study answers a simple question for Chinese investors, especially Chinese retail investors: Is the Black–Scholes model good enough for them to make investment decisions? Using the absolute out-of-sample error as a measure of model efficiency, I find that the volume-weighted mean absolute out-of-sample error is 12.03% of the option premium and investors have to tolerate more than 1% absolute error in almost all subsample groups. The significant out-of-sample error indicates that using the Black–Scholes model solely in the decision-making process may have a negative impact on the investments’ performance.

Hahn hybrid functions for solving distributed order fractional Black–Scholes European option pricing problem arising in financial market

The main purpose of this work is to present a new numerical method based on Hahn hybrid functions (HHFs) for solving of Black–Scholes option pricing distributed order time‐fractional partial differential equation. To this end, HHFs are introduced and their fractional integral operator with some properties of HHFs is calculated. In the next, with the help of fractional integral operator of HHFs, Gauss–Legendre quadrature formula and collocation method, distributed order time‐fractional Black–Scholes model is reduced to a system of algebraic equations. Furthermore, convergence analysis of the mentioned scheme is discussed. Finally, some test problems have been included to confirm the validity and efficiency of the mentioned numerical scheme. Moreover, Black–Scholes equations are studied through a bibliometric viewpoint.

Nonparametric estimation for i.i.d. paths of a martingale-driven model with application to non-autonomous financial models

This paper deals with a projection least squares estimator of the function $J_{0}$ computed from multiple independent observations on $[0,T]$ of the process $Z$ defined by $dZ_{t} = J_{0}(t)d\langle M\rangle _{t} + dM_{t}$ , where $M$ is a continuous square-integrable martingale vanishing at 0. Risk bounds are established for this estimator, an associated adaptive estimator and an associated discrete-time version used in practice. An appropriate transformation allows us to rewrite the differential equation $dX_{t} = V(X_{t})(b_{0}(t)dt +\sigma (t)dB_{t})$ , where $B$ is a fractional Brownian motion with Hurst parameter $H\in [1/2,1)$ , as a model of the previous type. The second part of the paper deals with risk bounds for a nonparametric estimator of $b_{0}$ derived from the results on the projection least squares estimator of $J_{0}$ . In particular, our results apply to the estimation of the drift function in a non-autonomous Black–Scholes model and to nonparametric estimation in a non-autonomous fractional stochastic volatility model.

Numerical Approaches of Pricing European Options in the Cox-Ross-Rubinstein Models

The Cox-Ross-Rubinstein (CRR) market model is one of the simplest and easiest ways to analyze the option pricing model. CRR has been employed to evaluate a European Option Pricing (call options) model without complex elements, including dividends, stocks, and stock indexes. Instead, it considers only a continuous dividend yield, futures, and currency options. The CRR model is simple but strong enough to describe the general economic intuition behind option pricing and its principal techniques. Also, it gives us basic ideas on how to develop financial products based on current deviations and volatilities. The paper investigates the CRR model using numerical approaches with python code. It provides a practical event using the mathematical model to demonstrate the application of the model in the financial market. First, the paper provides a simple example to figure out the basic concept of the model. Only a two-period binomial model based on the introductory definitions of the call options makes us understand the concept more easily and quickly. Next, we used actual data on Tesla stock fluctuations from the Nasdaq website (See section 3). We developed the python code to make it easier to understand figures, including tables and graphs. The code allows us to visualize and simplify the model and output data. The code analyzes the stock data to evaluate the probability of the stock’s price increasing or decreasing. Then, it used the CRR model to estimate all possible cases for the stock’s prices and investigate the call and put option pricing. The code was based on the introductory code of binomial option pricing, but we improved it to get more information and provide more detailed results from the data. The detailed codes are provided in section 3 of the paper. As a result, we believe the CRR model is a fundamental formula, but the improved python code can suggest a new direction for evaluating the probability and investigating the value of the stocks. Also, we expect to develop the code to extend the Black Scholes Pricing model, increasing the number of periods.

A Study on the Impact of Nonlinear Source Term in Black-Scholes Option Pricing Model

In this work, we study the effect of nonlinear source term in Black-Scholes model by finding the solution of it. We use the mathematical concepts of existence and uniqueness to arrive the conclusion. The transformation of the nonlinear equation into heat equation leads to the existence of solution through fixed-point theorems, semigroup theory, and certain regularity conditions imposed on variables.

Arbitrage in the Hermite Binomial Market

Much attention has been paid to the arbitrage opportunities in the Black–Scholes model when it is driven by fractional Brownian motions. It is natural to ask whether there exists arbitrage or not when we focus on other fractional processes, such as the Hermite process. We set forth an approximation of the Hermite Black–Scholes model by random walks in the Skorokhod topology, and apply the Donsker type approximation to the Hermite process as the Hurst index is greater than 12. We find that the binary model approximation of the Black–Scholes model driven by Hermite processes also admits arbitrage opportunities. Several numerical examples of the Hermite binomial model are presented as demonstration. Moreover, we provide an option pricing model when the geometric Hermite processes drives the price fluctuations of the underlying asset.