Black-Scholes Research Articles

APPRAISAL OF STOCHASTIC PRICING FORMULA FOR HYBRID EQUITY WARRANTS

An evaluator of warrant models is essential for enhancing the current procedures used in the country's stock market and economy. This would provide additional insight for investors to make informed investment decisions and help to better understand the dynamics of warrant pricing in Malaysia. It has been established that the conventional process for valuing equity warrants using call option pricing models, like the Black-Scholes model, has numerous flaws, including the presumption of constant volatility and interest rates. As a matter of fact, existing alternative models were more concerned with providing techniques for pricing than with conducting empirical tests of warrant pricing models. Consequently, it is crucial to have a mathematical model that takes into account stochastic volatility and stochastic interest rates when pricing and analyzing equity warrants. Recent research by Roslan, Ibrahim, Jameel, and Ibrahim (2020) and Roslan, Ibrahim, Karim (2020) examined the pricing of hybrid equity warrants in the presence of stochastic factors. There was no comparative analysis conducted in these studies to ascertain the relative efficacy of the two methods used. This study's main goal is to empirically investigate how well equity warrant pricing models perform in the Malaysian market. This evaluation will take into account the impact of stochastic interest rates, using the Cox-Ingersoll-Ross (CIR) stochastic interest rate model, and stochastic volatility, using the Heston stochastic volatility model. To perform the analysis, the study will make use of a number of mathematical techniques, such as partial differential equations, Fourier transform approaches, and the change of measure method. The research methodology is divided into three main phases. It is anticipated that the research's findings will provide new perspectives that go beyond the domain of warrant pricing, creating chances for more widespread applications in fields like foreign exchange, insurance, and mortgage contracts.

Penerapan Model Harga Opsi Black Scholes dalam Penentuan Premi Asuransi Jiwa Dwiguna Unit Link

Unit-linked life insurance is one of the most popular insurance products. This product connects the element of protection with investment assets in a product. Regarding option pricing, the Black Scholes model is one method that can be used. The advantage of this model is that it is a call option valuation model and is in great demand among financial associations because the option rate obtained from the calculation of this model is an accurate value. This research aims to determine the premium value of unit-linked endowment life insurance by applying the Black Scholes option pricing model. This research begins by selecting the stocks used and completing the information needed, including the insured's age, gender, insurance period, and life expectancy based on TMI in 2011. The return value and stock volatility can be calculated based on the stock data used. Furthermore, applying the Black Scholes option pricing model on customer data with male gender, age 25 years, a selected interest rate of 5.75%, and an insurance period of 5 years obtained a unit link insurance premium value of Rp123,058,412. The results showed that the Black Scholes option price model is determined by stock prices, interest rates, insurance periods, and volatility. When the longer period of insurance taken causes the premium value to be higher, and for different ages, with increasing age, the value of premiums paid will also increase.

The Benefits and Potential Risks of Using Option Pricing for Investment in Chinas OTC Derivatives Market under the Black-Scholes Model

In this paper, based on the qualitative research and analysis of Chinas financial market, both the benefits and potential risks of using option pricing for investment in Chinas OTC derivatives market under the Black-Scholes model are obtained. In China, OTC derivatives mainly refer to the one-to-one contract trading between institutions, non-on-exchange financial derivatives, naturally, its environment is called OTC derivatives market; Black-Scholes model is a mathematical model for option pricing. Based on the assumption that stock price obeys geometric Brownian motion, it measures the relationship between option price and underlying asset price, term, interest rate, volatility and exercise price. Based on the above variables and their overall two aspects, the gains and losses of Chinas OTC derivatives market under the Black-Scholes pricing model are obtained. Some predictions and expectations are also expressed for the future of this field.

Exploring Volatility: Evolution, Advancements, Trends, and Applications

Volatility is a fundamental notion in financial markets, influencing investment decisions, risk management techniques, and market dynamics. This paper provides a thorough overview of the historical evolution and practical implications of volatility, focusing on important works and key advancements in the field. The overview begins with early conceptions of volatility and the necessity for measurement prompted by market collapses, then progresses to advanced quantitative models and computer tools. The study includes key innovations such as the Black-Scholes model, which revolutionized options pricing and pioneered the concept of implied volatility. The Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models introduced frameworks for modeling time-varying volatility, paving the way for greater forecasting accuracy. Advancements in computing techniques have made it easier to analyze high-frequency data and estimate realized volatility, providing timely insights into market trends. The review also investigates contemporary trends, such as the use of machine learning algorithms and the issues provided by cryptocurrency marketplaces. Furthermore, the article examines the various characteristics and metrics of volatility, emphasizing its multidimensional nature and diverse uses in risk management, portfolio optimization, derivative pricing, and market analysis. Practical examples show how investors, traders, and financial professionals may use volatility to navigate complex market settings and make sound judgments. Finally, the study highlights the enduring significance of volatility in financial markets and highlights the need for continuing research and analysis to improve our understanding of market behavior. Acknowledging the complexities of volatility prepares market participants with valuable understandings to manage risks effectively and capitalize on market opportunities, thus contributing to financial stability and optimal portfolio performance.

A deep-genetic algorithm (deep-GA) approach for high-dimensional nonlinear parabolic partial differential equations

We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their corresponding backward stochastic differential equations (BSDEs). Recognizing the sensitivity of the solver to the initial guess selection, we embed a genetic algorithm (GA) into the solver to optimize the selection. We aim to achieve faster convergence for the nonlinear PDEs on a broader interval than deep-BSDE. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes (BS) equation with default risk and the Hamilton-Jacobi-Bellman (HJB) equation. We compare the results of our method with those of the deep-BSDE and show that our method provides comparable accuracy with significantly improved computational efficiency.

Research on the Pricing Model of DerivativesBased on the Black-Scholes Model and Binomial Model

The pricing of financial derivatives plays a significant role in risk management and financial transactions. The models of pricing are mainly divided into Black-Scholes Model and Binomial Model. The two models have different characteristics and distinguishing the models is helpful for price makers. The paper aims to analyse the distinction between the two models on the Huaxia Stock Exchange 50ETF by using Python. This paper selected the stock price, the volatility and the option price from the Eastmoney database. First, edit the formulas of two models and then introduce the parameters into the formula by Python and get the model calculated price. Comparison between the calculated prices helps to compare the two pricing models. The result is that Black-Scholes Model is more accurate and Binomial Model is more simplified which is suitable for new officers.

The Valuation of Internet Companies Based on Adjusted Black -Scholes Model

With the continuous breakthrough of Internet technology, we can clearly see the huge growth space and potential profitability of related companies. However, due to their unique instant change and uncertainty, high returns and high risks exist simultaneously and finding new approaches different from traditional ones to evaluate internet companies is necessary and significant. Based on adjusted Black-Scholes Model, this study discusses three approaches: Metcalfe approach values internet companies that relies on user numbers. DCF approach and Fuzzy real option approach estimate internet companies with strong cash flow and long-period potential growth, focusing on probability and fuzzy theory separately.

Comparing Black-Scholes and GARCH Models in Long Strangle Option Strategies for LQ45 Index

This study compared the Black-Scholes and GARCH models in a long strangle strategy applied to the LQ45 index using closing price data from 1998 to 2021. It aimed to assess the benefits, calculate returns during crises and non-crisis periods, and evaluate performance through Average Mean Square Error (AMSE). The Black-Scholes model consistently outperformed GARCH in one- and three-month options. One-month options had an average return of 28.64%, and three-month options, 43.31%. In crises, Black-Scholes delivered average profits of 43.36% for one-month and 45.14% for three-month options. In non-crisis conditions, profits averaged 26% for one-month and 42.84% for three-month options. Model performance varied by option type and market context. Black-Scholes excelled in one-month call options (1.268% error), while GARCH performed better in one-month put options (1.0981% error). For three-month options, GARCH outperformed in call options (1.270% error), and Black-Scholes dominated put options (3.117% error). In summary, the choice between models should consider market conditions, favoring GARCH during crises and Black-Scholes in non-crisis scenarios.

인공신경망(ANN)을 이용한 옵션가격결정 및 딥헤징 (DH)에 대한 SNN 및 RNN 결과의 비교·분석

The purpose of this study is to present alternative methods to overcome market limitations such as the assumption of fixing the dynamics of underlying assets and market friction for the traditional Black-Scholes option pricing model. As for the research method of this paper, Adam is used as the gradient descent optimizer for the hedging model of the call option short portfolio using artificial neural network models SNN (simple neural network) and RNN (recurrent neural network) as analysis models, and deep neural network (deep neural network) is used as the hedging model. neural network) methodology was applied.
 The research results of this paper are as follows. Regarding the ATM call option price, the BS model showed 10.245, the risk neutral model showed 10.268, the SNN-DH model showed 11.834, and the RNN-DH model showed 11.882. Therefore, it appears that there is a slight difference in the call option price according to each analysis model. At this time, the DH analysis data sample is 100,000 (1×) training samples generated by Monte Carlo Simulation, and the number of testing samples is the training sample. 20% (20,000 pieces) was used, and the option payoff function is lambda(), which is -max(, 0). In addition, the option price and P&L (P&L) of SNN-DH and RNN-DH appear linearly on the basis, and the longer the remaining maturity, the closer the delta value of SNN-DH and BS to the distribution of the S-curve, and the closer the expiration date, the closer the two models are. Delta is densely distributed between 0 and 1. As for the total loss of delta hedging of the short call option position, SNN-DH was -0.0027 and RNN-DH was -0.0061, indicating that the total hedged profit and loss was close to 0.
 The implications of this study are to present DL techniques based on AI methods as an alternative way to overcome limitations such as fixing underlying asset dynamics and market friction of the traditional Black-Scholes model. In addition, it is an independent analysis model that considers valid robust tools by applying deep neural network algorithms for portfolio hedging problems. One limitation of the research is that it analyzed the model under the assumption of zero transaction costs, so future studies should consider this aspect.

Pricing Asian and Barrier Options Using a Combined Heston Model and Monte Carlo Simulation Approach with Artificial Intelligence

The computation of fair values for exotic options often necessitates complex pricing techniques, which remain sparsely addressed in academic literature. Predominantly, the assessment of fair value for vanilla options relies on methodologies such as the Black-Scholes model or Monte Carlo simulations. This study proposes an innovative, dynamic approach to pricing, leveraging artificial intelligence in conjunction with the Heston model and a Monte Carlo simulation engine. This approach aims to furnish estimates of the prices for Barrier and Asian options. To enhance the accuracy of the model, calibration was performed employing a supervised machine learning algorithm, a continuous risk-free curve, and a dynamic implied volatility surface, derived from the current market data of vanilla options on S&P 500 futures. The amalgamation of these models yields instantaneous pricing for exotic option derivatives, contingent on the investor's determination of time to maturity and barrier levels. The efficacy of the model was evaluated by comparing the output prices to theoretical model predictions and a selection of over-the-counter traded options. Our findings indicate that the proposed dynamic, integrated approach substantially reduces the disparity between the theoretical models and current market prices. The prices calculated by our model demonstrate a marginal error of merely 0.33% in comparison to market prices, a significant improvement over the considerably larger error of 3.12% exhibited by traditional models.

The modified homotopy perturbation method and its application to the dynamics of price evolution in Caputo-fractional order Black Scholes model

BackgroundFollowing a financial loss in trades due to lack of risk management in previous models from market practitioners, Fisher Black and Myron Scholes visited the academic setting and were able to mathematically develop an option pricing equation named the Black–Scholes model. In this study, we address the solution of a Caputo fractional-order Black–Scholes model using an analytic method named the modified initial guess homotopy perturbation method.MethodologyForemost, the classical Black Scholes model relaxed for European option style is generalized to be of Caputo derivative. The introduced method is established by coupling a power series function of arbitrary order with the renown He’s homotopy perturbation method. The convergence of the method is demonstrated using the fixed point theorem, and its methodology is illustrated by solving a generalized theoretical form of the fractional order Black Scholes model. The applicability of the method is proven by solving three different fractional order Black–Scholes equations derived from different market scenarios and its effectiveness is confirmed as feasible series of arbitrary orders that accelerate fast to the exact solution at an integer order were obtained. The computation of these results was carried out using Mathematica 12 software. Subsequently, the obtained outcomes were utilized in Maple 18 software to conduct a series of numerical simulations. These simulations aimed to analyze the influence of the fractional order on the dynamics of payoff functions regarding the share value as the option approached its expiration date under varying market constraints. In all three scenarios, the results showed that option values decrease as the expiration date approaches the integer order. Furthermore, the comparative outcomes reveal that Caputo fractional order derivatives control the flexibility of the classical Black–Scholes model because its payoff curve exhibits more sensitivity to changes associated with market characteristic parameters, such as volatility and interest rates.RecommendationsWe propose that the results of this work should be examined and implemented by mathematicians and economists to better comprehend the influence of Caputo-fractional order derivatives in understanding the dynamics of option price evolution of financial assets.

The kind of silence: managing a reputation for voluntary disclosure in financial markets

We create a continuous-time setting in which to investigate how the management of a firm controls a dynamic choice between two generic voluntary disclosure decision rules (strategies) in the period between two consecutive mandatory disclosure dates: one with full and transparent disclosure termed candid, the other, termed sparing, under which values only above a dynamic threshold are disclosed. We show how parameters of the model such as news intensity, pay-for-performance and time-to-mandatory-disclosure determine the optimal choice of candid versus sparing strategies and the optimal times for management to switch between the two. The model presented develops a number of insights, based on a very simple ordinary differential equation characterizing equilibrium in a piecewise-deterministic model, derivable from the background Black–Scholes model and Poisson arrival of signals of firm value. It is shown that in equilibrium when news intensity is low a firm may employ a candid disclosure strategy throughout, but will otherwise switch (alternate) between periods of being candid and periods of being sparing with the truth (or the other way about). Significantly, with constant pay-for-performance parameters, at most one switching can occur.

Valuation of deposit insurance Black–Scholes model using Banach contraction principle

Deposit insurance is a mechanism by which financial institutions are stabilized. The danger of a bank’s inability to meet its consumer commitments due to its suspended license is insured through deposit insurance practices. A flat-rate insurance scheme would contribute to moral hazard and a financial panic when banks indulge in dangerous practices. Hence, a reliable model with an explicit solution is required. This paper considers a risk rate model for deposit insurance engendered by the classical Black Scholes Option Pricing Model. The solutions are obtained via the application of Banach Contraction Mapping or Method. The procedures involved are straightforward, easy, and flexible, even without giving up accuracy. The desired explicit solutions are obtained with less computational time.

Pricing formula for a Barrier call option based on stochastic delay differential equation

We derive new explicit pricing formulae for a type of Barrier call option, down and in call option when underlying asset price processes are represented by a stochastic delay differential equation (hereafter “SDDE”). We note the conditional normality of a stochastic integral with respect to a Wiener process to find the joint distribution of the stochastic integral and their minimum. On the basis of this result, we obtain pricing formulae for the Barrier call option which extends ones in the classical Black-Scholes models without delay. Finally, through Monte-Carlo simulations, we demonstrate that our theoretical prices for a Barrier option are correct.

Optionality in Australian Football League draftee contracts.

Though player drafts have commonly been utilised to equitably disperse amateur talent and avoid bidding wars, often they have also been accused of creating a monopsony labour market which restricts player movement. Within the Australian Football League (AFL) some have called for the increase of the initial draftee contract from two to three seasons, which further pushes the envelope on monopsony power. Instead of increasing the contract length, this paper suggests a call option to be purchased by the teams allowing them to add a further season to the draftee contract at a predetermined compensation package should they choose to do so at the end of the initial contract. The call prices per pick were calculated using the Black-Scholes model and were valued between 1% and 1.5% of the pick value. However, it failed to follow a monotonic function similar to pick value, owing to managerial overconfidence and sunk investment plays. Overall, the findings allow teams to procure the option of increasing initial draftee contracts and not impede further on a player's ability to move.

Review and Practice of Option Pricing Research

This paper reviews the past research on option pricing, including the Bachelier option pricing formula, Black-Scholes model, and the Cox-Ross-Rubinstein (CRR) binomial option pricing model. Then some practical calculations are performed on the Black-Scholes model and CRR binomial model and the differences between the results of these two models are analyzed for investors to choose the appropriate model for option prices. From the practical calculation results, it is verified that the Black-Scholes model can give out the analytical solutions of option prices and decrease the calculation of solution, while it can only be used for European options with stock prices regarded as a continuous process. For the CRR binomial model, although it can only give numerical solutions and requires lots of calculation because of too many branches, but it can solve the American option prices with early exercises and the European option prices with stock prices regarded as a discrete process.

A Method for Teaching the Binomial Option Pricing Model in Investments Courses

One of the most important concepts in modern finance in general, and in teaching an investments course in particular, is option pricing. While the Black-Scholes model is the most well known option pricing method, we have found that the binomial option pricing model is sometimes better for illustrating how different inputs affect the value of the option. The binomial option pricing model provides students with an intuitive understanding of the mechanics of option pricing. The binomial option pricing model is also widely used in practice. This paper presents an implementation of the one-period binomial model in Excel. By using spin buttons, students can directly observe the effects of changes in volatility, the underlying asset price, and the risk free rate, on the price of a call option. The model also allows students to compare prices of options with different strike prices. In addition to calculating binomial option prices and providing students with improved intuition, this spreadsheet provides a graphical representation of option value relative to the value of the underlying asset.

A Method for Teaching the Black-Scholes Option Pricing Model Using Excel

One of the most important concepts in modern finance practice and education is option pricing. The Black-Scholes model of option pricing is possibly the most commonly-used model. This paper presents an implementation of the Black-Scholes model of option pricing that includes graphs of the option value, the intrinsic value, and the time value. These graphs are dynamic, allowing the user to change the value of the volatility of underlying asset returns, time to maturity, and the risk-free rate. The user can also see how price of an option changes with movements in the underlying asset price. By illustrating both option prices and the components of option prices (intrinsic and time value) over a range of underlying asset prices, students can more easily visualize the effects of the drivers of option prices.

Funding startups using contingent option of value appreciation: theory and formula

PurposeThis paper provides a structural model to value startup companies and determine the optimal level of research and development (R&D) spending by these companies.Design/methodology/approachThis paper describes a new variant of float-the-money options, which can act as a financial instrument for financing R&D expenses for a specific time horizon or development stage, allowing the investor to share in the startup's value appreciation over that duration. Another innovation of this paper is that it develops a structural model for evaluating optimal level of R&D spending over a given time horizon. The paper deploys the Gompertz-Cox model for the R&D project outcomes, which facilitates investigation of how increased level of R&D input can enhance the company's value growth.FindingsThe author first introduces a time-varying drift term into standard Black-Scholes model to account for the varying growth rates of the startup at different stages, and the author interprets venture capital's investment in the startup as a “float-the-money” option. The author then incorporates the probabilities of startup failures at multiple stages into their financial valuation. The author gets a closed-form pricing formula for the contingent option of value appreciation. Finally, the author utilizes Cox proportional hazards model to analyze the optimal level of R&D input that maximizes the return on investment.Research limitations/implicationsThe integrated contingent claims model links the change in the financial valuation of startups with the incremental R&D spending. The Gompertz-Cox contingency model for R&D success rate is used to quantify the optimal level of R&D input. This model assumption may be simplistic, but nevertheless illustrative.Practical implicationsOnce supplemented with actual transaction data, the model can serve as a reference benchmark valuation of new project deals and previously invested projects seeking exit.Social implicationsThe integrated structural model can potentially have much wider applications beyond valuation of startup companies. For instance, in valuing a company's risk management, the level of R&D spending in the model can be replaced by the company's budget for risk management. As another promising application, in evaluating a country's economic growth rate in the face of rising climate risks, the level of R&D spending in this paper can be replaced by a country's investment in addressing climate risks.Originality/valueThis paper is the first to develop an integrated valuation model for startups by combining the real-world R&D project contingencies with risk-neutral valuation of the potential payoffs.

A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black–Scholes model

The Black–Scholes (B–S) equation has been recently extended as a kind of tempered time-fractional B–S equations, which becomes an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B–S model. To achieve high-order accuracy in space and overcome the weak initial singularity of exact solution, we combine the compact difference operator with L1-type approximation under nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods.