Black-Scholes Research Articles

On the Implied Volatility of Asian Options Under Stochastic Volatility Models

In this paper, we study the short-time behaviour of the at-the-money implied volatility for arithmetic Asian options with fixed strike price. The asset price is assumed to follow the Black–Scholes model with a general stochastic volatility process. Using techniques of the Malliavin calculus developed in Alòs, García-Lorite, and Muguruza [2022. On Smile Properties of Volatility Derivatives: Understanding the VIX Skew. SIAM Journal on Financial Mathematics. 13(1): 32–69. https://doi.org/10.1137/19M1269981], we give sufficient conditions on the stochastic volatility in order to compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew slope of the implied volatility that depends on the correlation between prices and volatilities and the Hurst parameter of the volatility model. We apply our general results to the SABR and fractional Bergomi models, and provide numerical simulations that confirm the accurateness of the asymptotic formulas.

An Empirical Study of Stock Price Returns Using Black-Scholes Option Pricing Model

Abstract: This study examined empirically the predictive ability of Black-Scholes Option pricing model in Nigerian Stock Market by testing whether there is any significant difference between the market (underlying) price of the stock and the theoretical prices. The data used for this study was Coca-Cola Stock prices from 2018 to 2022 obtained from http://www.investing.com and were analyzed using Statistical packages such as Microsoft Excel and Minitab to obtain the result. The results showed that there is a significant difference between the underlying and the theoretical price (Black-Scholes Option Pricing Model). Based on the findings of this study, we conclude that the Black- Scholes model is not accurate in its price predictive ability on the Coca-Cola Stock prices over the years under study.

Using Simpson’s Rule to Evaluate the Normal Cumulative Distribution Function in an Algorithmic Option Pricing Problem

Algorithmic problems are frequently used to help ensure academic honesty during assessments in quantitative courses, asthey allow instructors to easily generate multiple versions of a test, quiz, or assignment with questions that are similar but whoseinput variables and correct answers differ. In general, instructors who create an algorithmic problem must program its solutionby entering an algebraic expression using the input variables. While this is usually straightforward enough, it can be problematicwhen the solution requires a more advanced function that has no algebraic representation. One such challenge occurs withfinancial option pricing problems based on the Black-Scholes model, which relies on the normal cumulative distribution function(CDF). Although students can easily obtain normal CDF values from a calculator or spreadsheet function, no such function isavailable within testing software, presenting an obstacle for instructors attempting to author algorithmic problems on optionpricing. I demonstrate how to overcome this difficulty by using Simpson’s Rule to numerically integrate the underlying normalprobability density function, thereby generating a highly accurate approximation of the CDF that can be programmed into thesoftware and used in the Black-Scholes formula.

Rescaling the double-mean-reverting 4/2 stochastic volatility model for derivative pricing

The popular 4/2 stochastic volatility model reveals important features of volatility but a closed-form formula for derivative prices is still lacking. This paper proposes a modified form of the 4/2 model into which two scale double-mean-reverting stochastic volatility is incorporated in order to remedy its shortcomings. We obtain a closed-form formula for the prices of European derivatives. The formula can be explicitly calculated by taking derivatives of the Black–Scholes prices and thus faster calibration of the 4/2 model becomes available. We also show that the rescaled 4/2 model is flexible enough to capture essential features (skew or smile) of market implied volatilities.

PENENTUAN KONTRAK OPSI TIPE EROPA MENGGUNAKAN MODEL SIMULASI VARIANCE GAMMA (VG)

Options are used as a hedge against stock price uncertainty brought on by unstable stock prices fluctuation. The price of an option contract can be determined using a variety of approaches, one of which is the Variance Gamma. The purpose of this study is to compare the Black Scholes method with the Variance Gamma simulation model to determine the European call option contract price. The first thing that needs to be done is to figure out the moment variance gamma method. These parameters were used as initial values to get an idea of what the parameters that will be used in the simulation will be like. The European call option contract's price is calculated using the simulation results, which are then compared to the Variance Gamma simulation model and the Black Scholes model for the European call option contract. This study shows that the European call option contract's price, which was calculated using the Variance Gamma simulation, is less expensive than the Black Scholes contract's price.

Analytical solution of time fractional Black-Scholes equation with two assets through new Sumudu Transform iterative method

There is a scopious rise in the study of financial derivatives over the past two or three decades. Mathematical model proposed by Black and Scholes expounds financial derivatives in a more momentous way. The Black-Scholes model on a single asset is a partial differential equation characterizing the behavior of European options. In this article, we introduce the new Sumudu transform iterative method (NSTIM) as a new technique to obtain the analytical solution of time fractional Black-Scholes model involving European options with two assets. The proposed model is the advanced version of the regular Black-Scholes model. Explicit solution of the problem has been obtained with the help of generalized Mittag-Leffer function. The numerical analysis prove that this method is efficacious in solving various problems of financial theory.

Determination of Agricultural Insurance Premium Prices Based on Rainfall Index with the Black-Scholes Model

This article discusses the use of the European put option formula to calculate agricultural insurance premium prices based on the rainfall index. The data used in this study is data on rainfall and rice production in Banjarnegara Regency from 2014 to 2019 which are arranged in quarterly form. The research was completed by literature study and online secondary data searching. From the results of the research, rainfall data that has a strong correlation is rainfall in the tenth month (October) so that rainfall data on October is used as a rainfall index. From the calculation results, at the 5th percentile, the October rainfall is 2.2 mm. For this rainfall, a premium price of IDR 2,515,549.00 is obtained, if the trigger data for the last rainfall (2.2 mm) is used. When used as a reference, the average overall rainfall data (9.92 mm) obtained a premium of IDR 562,664.00. These results show that the selection of references gives very different results. The calculation results also show that the high and low percentiles affect the price of agricultural insurance premiums in Banjarnegara Regency. The higher the percentile value, the more rainfall will increase, and the premium price will increase.

PRICING AMERICAN OPTION USING A MODIFIED FRACTIONAL BLACK–SCHOLES MODEL UNDER MULTI-STATE REGIME SWITCHING

The American option pricing problem is examined in this work using a regime switching finite moment log-stable model. The option prices under this model are governed by a coupled system of fractional partial differential equations. Combination of the coupled system and the spatial-fractional derivative makes it extremely difficult to find an analytic solution. We have constructed a numerical algorithm to numerically solve such problems. The developed predictor-corrector type method is highly efficient and reliable in solving coupled system in each regime having different volatility and interest rates. Two-sided Riesz space fractional diffusion term is approximated using fractional finite difference scheme whereas the classical space derivative term is approximated using central difference formula. Splitting technique is utilized to construct a highly efficient scheme which can also be implemented on parallel processors. Stability and error analysis of the scheme is proved analytically and demonstrated through numerical experiments. Effect of the order of the fractional derivative (also called tail index) on the option prices is shown through graphs by performing numerical experiments for different values of the tail index.

Analytical and Computational Analysis of Fractional Stochastic Models Using Iterated Itô Integrals

Biological and financial models are examples of dynamical systems where both stochastic and historical behavior are important to be considered. The fractional Brownian motion (fBM) is commonly used, sometimes with fractional-order derivatives, to model the combined stochastic and fractional effects. Recently, spectral techniques are used to analyze models with fBM using, e.g., iterated Itô fractional integrals such as the fractional Wiener-Hermite (FWHE). In the current work, FWHE is generalized and adapted to be consistent with the Malliavin calculus approach. The conditions for existence and uniqueness are outlined in addition to the proof of convergence. The solution algorithm is described in detail. Using FWHE, the stochastic fractional model is replaced by a deterministic fractional-order system that can be handled using well-known mathematical tools to evaluate the solution statistics. Analytical solutions can be obtained for many important models such as the fractional stochastic Black–Scholes model. The convergence is studied and compared with the exact solution and high convergence is noticed compared with other techniques. A general numerical algorithm is described to analyze the resultant deterministic system in the case of no feasible analytical solutions. The algorithm is applied to study and simulate the population model with nonlinear losses for different values of the Hurst parameter. The results show the efficiency of FWHE in analyzing practical linear and nonlinear models.

An option-value framework for valuing mining streams

This paper introduces an option-value framework for valuing mining streams. A stream has the same economic characteristics as a portfolio of European call options. Since a stream can be replicated with a portfolio of European call options, the option-value approach to valuing streams is correct. At the same time, there are reasons to use a simpler alternative to the Black-Scholes Model. This paper provides a simple option-value equation for valuing streams.

Generalized finite integration method with Volterra Operator for pricing multi-asset barrier option

We investigate in this paper the pricing of European-style barrier options under the Black–Scholes model. Based on the recently developed Generalized Finite Integration Method with Volterra operator (GFIM-V), we apply the Crank–Nicolson scheme to treat the time variable in the governing Black–Scholes equation for pricing multi-asset barrier options. For verification on the accuracy and efficiency of the proposed approach, we construct several numerical experiments for the solutions of multi-asset barrier option prices with various time step sizes and number of spatial nodal points. Comparisons with available exact solution and existing spectral convergent method indicate the advantages of the GFIM-V method in superior accuracy and unconditional stability.

Physical wave propagation and dynamics of the Ivancevic option pricing model

In this study, the Ivancevic option pricing model (IOPM), a nonlinear wave alternative to the traditional Black–Scholes option pricing model, is investigated. The Ivancevic option pricing model is formalized by adaptive nonlinear Schrodinger equations that characterize the option-pricing wave function in terms of stock price and time. This model exhibits the characteristic regulated Brownian behavior observed in financial markets. We use the unified auxiliary equation method, which is one of the most powerful methods to explore analytical solutions of the Ivancevic option pricing model. Via this method, we obtain several new exact solutions such as singular, periodic, bright, dark, hyperbolic, trigonometric, exponential and Jacobi elliptic functions solutions. In addition, we simulate 3D surface 2D graphs and counter plots graphics for some obtained solutions by giving specific values for the involved parameters. To further analyze the obtained solutions, we conduct simulations and generate visual representations in the form of 3D surface plots, 2D graphs, and counter plots. By assigning specific values to the parameters involved in the model, we are able to visualize and examine the dynamics of the solutions in a more tangible and intuitive manner. These graphical representations provide valuable insights into the behavior and patterns exhibited by the solutions, enhancing our understanding of the model’s implications and potential applications Furthermore, we present the option price wave functions of the dependent variable in 2D graphs under the effect of the time variable t.

Currency change and capital structure decisions: evidence from the birth of the Euro area

PurposeThe paper investigates if the process that led to the birth of the Euro Area had a significant impact in homogenizing the capital structure decisions of European firms since the first introduction of the common currency.Design/methodology/approachA large sample of firms was constructed, and a Tobit-censored regression model was utilized to investigate the determinants of firms' observed capital structures. The Black–Scholes–Merton model was used to infer market values of assets, as well as the volatility of those values, from the observed market values of equity and the corresponding volatility. The existing differences in national tax rules were considered for estimating firm-specific marginal tax rates.FindingsIt was found that, despite the currency union and the institutional harmonization process, certain factors still play a different role. In particular, the impact of profitability is consistent with the pecking order view in some countries, and with the trade-off theory in others. Assets risk, measured as the annualized volatility of the market enterprise value, is the best predictor of observed leverage ratios. The sector of activity is significant in determining leverage decisions even when assets' risk is taken into account. Despite the monetary union and the increased financial and institutional integration in the Euro Area, the country of origin still plays a significant role in capital structure decisions, suggesting that other country-level factors may affect firms' financing behaviour.Practical implicationsThe paper indicates that, despite the long harmonization process of institutions, regulations and public budget required to join the Euro, firms' financing decisions are still affected by country-specific factors once the common currency is introduced. Therefore, new entrant countries in the Euro area should not expect their companies to immediately conform with those located in other countries within the common currency area.Originality/valueThis article investigated the impact of the currency change from national currencies to the Euro on the determinants of capital structure choices. It was shown that, despite the long harmonization process that led to the birth of the Euro Area, national factors still affect firms' financing decisions. This provides guidance for policymakers in countries that are planning to join the Euro about the impact this will have on firms' financing decisions in the entrant country.

Romancing with Black-Scholes Model: An Overview

This paper is an overview of our previous and recent results associated with the modifications of Black-Scholes model and formula. There were and are attempts to extend the classical Black-Scholes model by introducing the randomness into the coefficients r, �, s (interest rate, drift and volatility, respectfully) or to describe them by SDEs, and so on, and then to find explicit or close-form formulas for the hedging strategies and option prices. If coefficients r, �, s depend on some parameter x ?X, where X is some state space, i.e., r(x), �(x), s(x) then we call it (B, S, X)-security markets or (B, S)-security markets in random environment X. We consider different (B, S, X)-securities markets, including Markov, semi-Markov cases, jumps, and analogues/generalizations of Black-Scholes formulas for them. Black-Scholes formula with delay is also presented. Then we consider two types of telegraph process, classical/symmetric and asymmetric, and give two applications of those telegraph processes in finance by presenting European call and put option prices. Moreover, We introduce a new type of Hawkes process, namely, the exponential one-dimensional general compound Hawkes process (E1DGCHP), consider its limit theorems (LLN and FCLT) and present analogue of Black-Scholes formula in this case. Numerical examples are presented for both telegraph processes and E1DGCHP.

Hedging using reinforcement learning: Contextual k-armed bandit versus Q-learning

The construction of replication strategies for contingent claims in the presence of risk and market friction is a key problem of financial engineering. In real markets, continuous replication, such as in the model of Black, Scholes and Merton (BSM), is not only unrealistic but is also undesirable due to high transaction costs. A variety of methods have been proposed to balance between effective replication and losses in the incomplete market setting. With the rise of Artificial Intelligence (AI), AI-based hedgers have attracted considerable interest, where particular attention is given to Recurrent Neural Network systems and variations of the Q-learning algorithm. From a practical point of view, sufficient samples for training such an AI can only be obtained from a simulator of the market environment. Yet if an agent is trained solely on simulated data, the run-time performance will primarily reflect the accuracy of the simulation, which leads to the classical problem of model choice and calibration. In this article, the hedging problem is viewed as an instance of a risk-averse contextual k-armed bandit problem, which is motivated by the simplicity and sample-efficiency of the architecture, which allows for realistic online model updates from real-world data. We find that the k-armed bandit model naturally fits to the Profit and Loss formulation of hedging, providing for a more accurate and sample efficient approach than Q-learning and reducing to the Black-Scholes model in the absence of transaction costs and risks.

Pricing Technique for European Option and Application

In financial mathematics, the pricing technique for derivatives is constantly debated. In this paper, the pricing technique of the European Option is mainly discussed, and the binomial tree (BN) model is first applied to the pricing process of European options. The previous results show that carbon credit index can be traded as an option, and BN model can correctly simulate the future price of call option constructed by consuming the carbon credit index. Secondly, the Black-Scholes (BN) model is also a crucial technique for pricing European options, and it is successfully applied to predicting the future three months' CSI 300 index option price. Finally, BN model is compared with BS model, and the result reflects that BN model can perform as well as BS model for pricing European Option When the step reaches 2000. However, the efficiency of the BN model is stable under low volatility. Under higher volatility, such as 1.5 sigmas, the required steps will increase to achieve the same accuracy level. For American options, the BN simulator of a put option is close to the actual value, but the call option simulator will fluctuate. For the stock-pricing process, both models estimate far above Monte-Carlo method. The result of this paper is to provide some clues for pricing European options with different methods.

Female CEOs and default risk in listed family firms

PurposeThe purpose of this paper is to examine the effect of female CEO board members on listed family firms’ corporate default risk, integrating upper echelons theory with social role theory and the socio-emotional wealth approach and proxying default risk with the Black–Scholes–Merton model. It also searches for possible differences attributable to the type of female CEO.Design/methodology/approachThis study is applied to a longitudinal sample of listed US family firms. After a preliminary analysis of the main descriptive, several models are estimated with the system GMM estimator, which is a panel data estimator. The models are dynamic, including the lagged value of the dependent variable. In addition, the model estimation is repeated with a different measure of default risk, for robustness.FindingsThis research findings show that default risk diminishes in the presence of a female CEO, whose reduction is even greater if she is a family member. The results are proven to be robust to the measure for proxying default risk.Originality/valueThis study primarily contributes to the existing literature by exploring a possible link between female CEOs, particularly those with a family affiliation, and a lower level of default risk in family firms. It also provides practical implications for policymakers, who would be advised to promote conditions enabling women to contribute towards family business viability. In addition, this study offers encouragement for family business owners to value the potential of their female family members in company succession processes.

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).