Black-Scholes Research Articles

Continuity correction: on the pricing of discrete double barrier options

This article deals with the pricing of double-barrier options monitored discretely. A continuity correction method is established to provide an analytical approximation for the price of such discrete options under the Black–Scholes model. We achieve this by applying the smooth-fit principle simultaneously to the two flat boundaries (barriers) associated. The resulting correction form still involves adjustments in the levels of barriers, but the amounts adjusted can be different for different boundaries. More interestingly, the shift for each boundary can also be in different directions, which depends largely on the position of the current level relative to the two boundaries. Numerical examples are provided as well which support our theoretical achievements.

Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach

AbstractIn this paper, we propose a new explicit series expansion formula for the price of an arithmetic Asian option under the Black–Scholes model and Merton's jump‐diffusion model. The method is based on an equivalence in law relation together with the diffusion operator integral method proposed by Heath and Platen. The method yields explicit series expansion formula for the Asian options' prices. The theoretical convergence of the expansion to the true value is established. We also consider the American Asian option (i.e., Amerasian option) and derive the corresponding expansion formula through the early exercise premium representation. Numerical results illustrate the accuracy and efficiency of the method as compared with benchmarks in the literature.

Options Pricing Comparison between the Black-Scholes Model and the Binomial Tree Model: A Case Study of American Equity Option and European-style Index Option

In recent years, quantitative researchers used a wide range of models to price options, from the Black-Scholes model to more complex models such as the Heston model. This paper aims to analyze the effectiveness of the Black-Scholes model and the Binomial Tree model by using them to price Berkshire Hathaway’s equity options and European-style S&P 100 index options. The method used in this paper is gathering the market data of the options first. Second, using the data gathered to price the options by applying the Black-Scholes and Binomial Tree models. Third, comparing the derived theoretical price with the market price by getting the Sum of Square Errors. Lastly, determining the best model for each type of option. Through this research, the author found that comparing the two models, the Binomial Tree model derives a smaller Sum of Square Errors when pricing European-style index options, and the Black-Scholes model derives a smaller Sum of Square Errors when pricing American equity options. Thus, the Binomial Tree model is a better model to price European-style index options, and the Black-Scholes model is more effective when pricing American equity options.

Comparing the Payoff Differences Between the Barrier and European Options Based on the Black-sholes Model

As more people want to invest in the options market, there are basic traditional options in the futures market, such as European options. Still, there are also some exotic options like the barrier option that is conditional on the stock price before expiration and is called a barrier (1). This paper analyzes the payoffs of European and barrier options based on the Black Scholes model by calculating them and comparing them. This study analysis of the results shows that the price of the barrier option is lower than that of the European option. The barrier option is essentially close to the European option under conditions. This article helps newcomers to the options and those unfamiliar with exotic options to understand the differences between barrier options and European options in terms of payoff.

Economic Significance of Arbitrage Transactions in the Tesla Industry

Arbitrage transactions have played a significant role in allowing Tesla to exploit the minor differences in price between its similar and identical assets in two or more markets. In this case, as an arbitrage trader, Tesla undertakes the responsibility of purchasing assets in one market and selling them out at the same period to pocket any differences between the two prices. In this paper, there are two main parts of arbitrage transactions. The first part is three methods that can be utilized to analyze the options prices. These methods are linear regression and random Walk Theory, and Black Scholes. Their functionalities, influences, and theory are demonstrated in this paper. When using Black Scholes to calculate desired options prices and figure it in linear regression, assuming time to the expiration date, volatility, and risk-free rate are the same. The estimated trend and outcome are stated and described in this paper. The second part is about arbitrage transactions based on option prices. The circumstances of arbitrage transactions used by Tesla are analyzed and the utilized method called fundraising is demonstrated, and its functionality is analyzed with data and facts to prove the significance of arbitrage transactions. The paper also points out it is possible to use black scholes to estimate option prices in linear regression graphs and random walk theory. Call and put options prices are related to stock prices. They all shared a similar trend when time flows. The true value of this research is that people do not need to simply use a single method to estimate values, but to combine them, use them together to better understand the estimation of prices.

Pricing of Chooser Option Based on Black-Scholes Model: Case of AAPL Company

Financial derivatives traded in the markets are becoming more complex and investors' complicated situations also should be responsible for this trend. This study examines the Black-Scholes Model theoretically and practically. By using the information of AAPL company, the simulation of stock price is calculated with related parameter, in the comparison of the straddle, the chooser option has a relatively high potential profit. The paper also gives sensitivity analysis and the conclusion of variables, the result is that the volatility has the largest impact on the stock price, followed by maturity period and date chosen period. Finally, this paper introduces wider application scenarios of our study, it can be used to estimate whether the future is bullish or bearish to obtain higher returns, to predict stock volatility, and to evaluate the value of growing companies, which fill the vacancy in past researches of financial market and gives suggestions for investors.

Chooser Option Pricing based on Black-Scholes Model and Monte-Carlo Simulations

Since February 2022, the intensification of the Ukrainian conflict has triggered significant impacts on financial market. Huge changes in supply and demand have led to a spike in stock volatility, which resulted in the growth in risk aversion among investors. The gold is regarded as a safe haven asset and has hedge function in the stock markets. In this study, a Monte-Carlo pricing simulation based on Black-Scholes model for chooser option was investigated using the underlying asset of gold. The gold price and volatility index were sourced from Yahoo Finance website and the Chicago Board Options Exchange (CBOE). According to the analysis, the present value of the average return for one-month chooser option is simulated as $58.91, using the initial gold price of $1965.10 and the strike price of $1977.3. The present value of the average return for one-month chooser option is significantly higher than traditional call or put options. According to the sensitivity analysis, the return of chooser option under the pricing model adopted is not sensitive to the changes in volatility and time, which indicates that this pricing model is able to resist to risks and play a better role in a high-volatility market. These results shed light on the optimization of investment portfolio under complex situations. The utilization of a new pricing method paved a path for the research of exotic options and the state-of-art applications of them in the financial innovation procedures.

Black-Scholes Model’s application in rainbow option pricing

In this paper, we use excel as a tool to explore the pricing of rainbow options and their advantages based on the Black-Scholes Model. Two-color rainbow options are mainly explored in the paper, in which the underlying stocks are Apple and ExxonMobil. Simulating the price of two stocks is performed through Excel. Return on the corresponding European options and rainbow options is obtained after that. Next, the differences between the return on rainbow options and European options and pricing on rainbow option are analyzed. Finally, sensitivity analysis is carried out to further explore rainbow option pricing.

A Mellin transform approach to pricing barrier options under stochastic elasticity of variance

AbstractThis article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi‐closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black–Scholes price for a barrier option with respect to the underlying price and computing the one‐dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte‐Carlo simulation and find the SEV effect on the Black–Scholes barrier option prices.

Improvized implied volatility function and nonparametric approach to unbiased estimation

The purpose of this paper is to assess unbiased options pricing predictions via ad hoc Black–Scholes model approaches. This paper investigates a number of technical issues when fitted values of implied volatility from linear regression are plugged into the Black–Scholes model, which leads to biased estimation. First, the study observes that the implied volatility linear regression can yield a negative outcome, which is meaningless. Therefore, a logarithmic transformation is applied to the linear function to ensure that the forecast is positive. Second, the retransformation from log to original metric to fitted values of implied volatility and the nonlinearity of Black–Scholes to implied volatility yields biased forecasts. A smearing technique has been applied in this study to correct this bias. Finally, the smearing estimation method also provides biased results if there is heteroscedasticity in the OLS estimation residuals. This study applies the weighted least square regression technique in order to avoid heteroscedasticity. According to the performance measures such as mean bias (MB), mean absolute error (MAE) and mean absolute relative error, the study concludes that the smearing method is the most effective to correct the bias in ad hoc Black–Scholes approaches as well as that an absolute smile approach is better than a relative smile approach without the smearing technique, but with smearing methods, relative smile performs superior to absolute.

An Analysis of the Fractional-Order Option Pricing Problem for Two Assets by the Generalized Laplace Variational Iteration Approach

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations.

Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations

We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion (BtH) and a Poisson process (Nt) associated with a stochastic process (At) such that: dXt=μXtdt+σXtdBtH+AtXt−dNt,X0=x0>0. The solution of this SDE is analyzed and properties of its trajectories are presented. Estimators of the model parameters are proposed when the observations are carried out in discrete time. Some convergence properties of these estimators are provided according to conditions concerning the value of the Hurst index and the nonequidistance of the observation dates.

The Quantitative Assessment of the Model Risk of Option Pricing Models When Volatility is Stochastic

Using Monte Carlo simulation experiments, we examine the model risk of writing call option contracts when the underlying process driving security prices is characterized by stochastic volatility and trading occurs discretely. Simulations in this study reveal that discrete rebalacing risk is the main risk factor of total model risk while model misspecification risk and parameter estimation risk are relatively unimportant when single instrument hedge is utilized. These simulation results suggest that hedging based on the misspecified Black-Scholes model yields the hedging performance as good as that of the fully sprecified stochastic volatility model even when the true process driving security price movement is characterized by stochastic volatility. Hedging based on the Black-Scholes model is the simplest to implement. This may explain why the Black-Scholes formula is one of the best choices of option professionals even though it is widely accepted that the constant volatility assumption of the Black-Scholes model is violated in real world financial market. The results of this paper also suggest that for risk managers of financial institutions, the Black-Scholes option formular can be used as a viable risk management tool even under the financial markets characterized by stochastic volatilities.

The Best Option Pricing Model for KOSPI 200 Weekly Options

This study finds the best option pricing model for KOSPI 200 weekly options. It examines the in-sample pricing, out-of-sample pricing and hedging performances of the short-term options with a maximum maturity of seven days or less, which have not been analyzed in previous studies. The Black and Scholes (1973) model, Ad Hoc Black-Scholes model, and stochastic volatility and jumps models are compared. As a result, one of the Ad Hoc BlackScholes models, the absolute smile model using the strike price as an independent variable shows the best performance. However, its performance is not significantly different from that of the Black and Scholes (1973) model. In the early days of the KOSPI 200 weekly options market, it is confirmed that the Black and Scholes (1973) model continues to show the excellent performance in pricing and hedging options.

Fourier변환을 갖는 Tempered Stable Lévy 프로세스(6모수 CGMY 모형)에 의한 옵션가격결정

In the real world, products in the stock market do not generally follow a normal distribution, but are known to follow a fat-tail distribution with high kurtosis and thick tails of skewness. Therefore, the purpose of this study is to analyze option pricing using a distribution in which random variables follow a stable and independent distribution and well represent the properties of the actual distribution. At this time, for the empirical analysis data, 252 daily KOSPI200 closing price indexes were used from November 15, 2019 to November 20, 2020. Call Options(Put options) were set as OTM(ITM) and ITM (OTM) and analyzed.
 In the research methodology of this paper, option pricing was performed by the CTS (or CGMY) model assuming a Tempered Stable Distribution that is capable of Furier transformation. In addition, the Black-Scholes model assuming a normal distribution and Fourier transformation, and the Fourier transformation were performed. The option pricing model using the Merton Diffusion Process and Variance-Gamma process was compared and analyzed.

Numerical Techniques for Determining Implied Volatility in Option Pricing

One of the assumptions of the Black–Scholes model is that volatility is constant in the market. However, in reality, volatility cannot be constant. In this paper we examined an inverse problem of determining the time-dependent non-constant volatility from the observed market price of a European put/call option with a strike price.These unknown non-constant volatilities are from the time the option contract was signed to the time it was exercised (strike price) where the fluctuations of option price is known but the non-constant volatilities that was generated by option price within that time period is unknown.Although many studies have used the portfolio stocks to reconstruct volatility, no study have retrieved volatility from the option market. The aim is to recover unknown non-constant volatilities in vector form from one option contract period using manufacturer observed market data, that is, data from agreed contract period using Black–Scholes Partial Differential Equation (BSPDE). This would be done by using the chain and product rule to take the derivative with respect to volatility in the theoretical(Black-Schoes PDE) model to obtain the non-constant volatilities. By keeping the fluctuations data from observed option prices constant,we differentiate the volatility in the BSPDE to obtain a gradient equation and discretize the gradient equation with the finite difference method to obtain model simulated data. It is expected that non-constant volatility that would be recovered from the market using the gradient descent method would be matched with the non-constant market volatility from the model simulated data and if they are the same or approximately same, the volatility would have deem recovered. All these would be done in the least square sense.

Application of Lie symmetries to solving modified Black-Scholes equation

We perform Lie symmetry analysis on the modified Black-Scholes model described as a partial differential equation (PDE). As a result, a new complete Lie symmetry group and infinitesimal generators of the one-dimensional modified fractional Black-Scholes model are derived. Furthermore, we compute a family of exact invariant solutions that constitute the modified fractional Black-Scholes model using the associated infinitesimal generators and the corresponding similarity reduction equations. Using known solutions, more solutions are generated via group point transformations.

Inverse Multiquadric Function to Price Financial Options under the Fractional Black–Scholes Model

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order 2−α is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers.

Numerical solution of the three-asset Black–Scholes option pricing model using an efficient hybrid method

In this paper, a numerical method for solving three-asset Black–Scholes partial differential equation is presented. The model is based on the Crank–Nicolson and the radial basis function methods. Also, the convergency of the proposed method is proved. Implementation of the proposed method is specially studied on cash-or-nothing option. The numerical results show the efficiency and high accuracy of the proposed method.

Regime recovery using implied volatility in Markov modulated market model

AbstractIn the regime switching extension of Black–Scholes–Merton model of asset price dynamics, one assumes that the volatility coefficient evolves as a hidden pure jump process. Under the assumption of Markov regime switching, we have considered the locally risk minimizing theoretical price of European vanilla options. By pretending these prices or their numerical approximations as traded prices, we have first computed the implied volatility (IV) of the underlying asset. Then by performing several numerical experiments on ternary regime models we have investigated the dependence of IV on the time to maturity (TTM) and strike price of the vanilla options. We have observed a clear dependence that is at par with the empirically observed stylized facts. Furthermore, we have experimentally validated that IV time series, obtained from contracts with moneyness and TTM varying in a particular narrow range, can recover the transition instances of the hidden Markov chain having three states. Such regime recovery for any arbitrary state‐space and transition parameters has also been established in a theoretical setting. Moreover, the novel scheme for computing option price is shown to be stable.