Price Of Call Option Research Articles

Two-factor Rough Bergomi Model: American Call Option Pricing and Calibration by Interior Point Optimization Algorithm

Two-factor Rough Bergomi Model: American Call Option Pricing and Calibration by Interior Point Optimization Algorithm

Analysis of Put and Call Option Pricing on BCA Stock Using the Black-Scholes Model: Financial and Risk Management Perspective

Analysis of Put and Call Option Pricing on BCA Stock Using the Black-Scholes Model: Financial and Risk Management Perspective

RISK ANALYSIS OF GOOGL & AMZN STOCK CALL OPTIONS USING DELTA GAMMA THETA NORMAL APPROACH

Stocks, as investment products, tend to carry risks due to fluctuations. The tendency of stock prices to rise over time leads investors to opt for call options, which are one of the derivative investment products. However, call options are influenced by several factors that can pose risks and have nonlinear dependence on market risk factors. Therefore, methods are needed to measure the risk of call options, such as Delta Normal Value at Risk and Delta Gamma Normal Value at Risk. Delta and Gamma are part of Option Greeks, parameters that measure the sensitivity of options to various factors used in determining option prices with the Black-Scholes model. This study uses an approach with the addition of Theta, which can measure the sensitivity of options to time. This study aims to analyze Value at Risk with the Delta Gamma Theta Normal approach for call options on Google (GOOGL) and Amazon (AMZN) stocks. The analysis uses closing stock price data from September 7, 2022, to September 7, 2023, and three in-the-money and out-of-the-money call option prices. The study begins by collecting closing stock prices and call option contract components, testing the normality of stock returns, calculating volatility, , Delta, Gamma, and Theta, then calculating the Value at Risk. Based on the analysis, it is found that GOOGL and AMZN call options have a Value at Risk of $0.89588 and $0.92760, respectively, at a 99% confidence level with a strike price of $120. Furthermore, based on the comparison of Value at Risk between in-the-money and out-of-the-money call options, it can be concluded that out-of-the-money call options tend to have larger estimated losses.

European Call Option under Stochastic Interest Rate in a Fractional Brownian Motion with Transaction Cost

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price.

Option pricing for Barndorff–Nielsen and Shephard model by supervised deep learning

This paper aims to develop a supervised deep learning scheme to compute call option prices for the Barndorff-Nielsen and Shephard model with a non-martingale asset price process having infinite active jumps. In our deep learning scheme, teaching data are generated through the Monte Carlo method developed by Arai and Imai [(2024). Monte Carlo simulation for Barndorff-Nielsen and Shephard model under change of measure, Mathematics and Computers in Simulation, 218, 223–234]. Moreover, the BNS model includes many parameters, which makes the deep learning accuracy worse. Therefore, we will create another input parameter using the Black–Scholes formula. As a result, the accuracy is improved dramatically.

The Analysis of Black-Scholes Model of Option Pricing with Time-Varying Parameters on Share Prices for Capital Market

This paper studied the perception of European option which is geared towards valuation of financial assets at a prescribed time in the future. In particular, the analytic formula of Black-Scholes model was considered for share prices of Fidelity and Access banks which gave closed form prices of call options. The explicit price on the variations of maturity days is found accordingly. The closed form prices of both banks were compared. The simulation results show that: the share price determines the value of call option prices. An increase on the maturity days dominantly increases the value of call option for both Fidelity Bank, Access Bank and their future merger Bank. Merging of the two banks improved the value of call option prices. Fidelity bank has a good maximum value of call option prices during the period of investments, Kolmogorov ?Smirnov (KS) test shows that the two call option prices (Fidelity and Access) do not come from a common distribution, the normality test of both banks are not significant. The results presents to the Banks management, a basis for taking vital decisions, depending on the levels of their investments.

Optimizing Investment Strategies: A Case Study on JPMorgan Chase & Co. Stock Options Using the Black-Scholes Model and What-If Analysis in Excel

This research focuses on applying the Black-Scholes Model to evaluate European options on JPMorgan Chase & Co. stocks. This model has been a critical foundation in evaluating financial instruments, especially options, since its development in 1973 by Fisher Black, Myron Scholes, and Robert Merton. The study utilizes secondary data from some sources to obtain current information regarding stock prices, strike prices, expiration time, volatility, and relevant risk-free interest rates for option valuation as of December 19, 2023. Through this approach, our aim is to gain a better understanding of how the Black-Scholes Model is used as a framework in determining option prices for these stocks. The research methodology involves What-If analysis, exploring variations in key variables such as current stock price, strike price, expiration time, stock price volatility, and risk-free interest rates to assess how these changes affect the prices of both call and put options. Additionally, the study presents graphs representing stock prices, strike prices, interest rates, time, and volatility to visually support the research findings. The analysis results reveal that the prices of both call and put options are highly responsive to changing market conditions. An increase in the current stock price tends to raise the call option price while reducing the put option price. Conversely, an increase in the strike price has the opposite effect. Moreover, variations in the risk-free interest rates influence the option prices, with rising rates increasing the call option price and decreasing the put option price. Furthermore, as the expiration time approaches or stock price volatility increases, both call and put option prices tend to rise. These findings provide a comprehensive understanding of the dynamics of JPMorgan Chase & Co.'s stock option pricing, offering a foundation for investors to make informed and adaptable investment decisions amid constantly evolving financial markets. Sensitivity to changes in key variables is an essential aspect to consider in designing effective investment strategies in the face of ever-changing financial markets.

Research on the Impact of COVID-19 on Netflix Call Option Prices based on the Black-Scholes Model

Research on the Impact of COVID-19 on Netflix Call Option Prices based on the Black-Scholes Model

Research on the option pricing of Amazon and Google based on B-S model and Monte Carlo simulation

The emergence of financial derivatives has helped people to effectively hedge the risks present in the financial markets while giving them the ability to make leveraged investments in the financial markets. Although derivatives did not originate in the financial market, it still has a huge role in the financial market so far. In recent years, science and technology innovation is known as a growing concern for governments due to factors such as environmental issues. In this paper, we choose to study the financial derivative, European call option pricing problem of technology-based companies AMZN and GOOGL. In this paper, the European call options of these two companies are priced by using the BSM model, and the results predicted by the BSM model are improved by using Monte Carlo Simulation. Ultimately, the paper finds that Monte Carlo Simulation prices European call options on technology-based firms nearly three times higher than those priced by the BSM model. The results suggest that using a single model to price European call options on technology-based firms is inaccurate. The purpose of this paper is to help investors in pricing European call options on technology-based firms and to warn investors about the inaccuracy of a single option pricing model.

Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps

We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes.

European Option Pricing under Sub-Fractional Brownian Motion Regime in Discrete Time

In this paper, the approximate stationarity of the second-order moment increments of the sub-fractional Brownian motion is given. Based on this, the pricing model for European options under the sub-fractional Brownian regime in discrete time is established. Pricing formulas for European options are given under the delta and mixed hedging strategies, respectively. Furthermore, European call option pricing under delta hedging is shown to be larger than under mixed hedging. The hedging error ratio of mixed hedging is shown to be smaller than that of delta hedging via numerical experiments.

PRICING OF CALL OPTIONS USING THE QUASI MONTE CARLO METHOD

A call option is a type of option that grants the option holder the right to buy an asset at a specified price within a specified period of time. Determining the option price period of time within a certain period of time is the most important part of determining an investment strategy. Various methods can be employed to determine the prices of options, such as Quasi-Monte Carlo and Monte Carlo simulations. The purpose of this research is to determine the price of European-type call options using the Quasi-Monte Carlo method. The data used is daily stock closing price data on the Apple Inc. for the period October 1, 2021, to September 30, 2022. Apple Inc. stock options in this study were chosen because it is the largest technology company in the world in 2022. The steps taken in this study are to determine the parameters obtained from historical data such as the initial risk-free interest rate (r), stock price (, volatility , maturity time (T), and strike price (K). Next is to generate Halton’s quasi-randomized sequence and simulate the stock price by substituting the parameters by substituting the parameters. Then proceed to calculate the call option payoff and estimate the call option price by averaging the call option payoff values. The results showed that the call option price of the company Apple Inc. using the Quasi-Monte Carlo with Halton’s quasi-randomized sequence on the 1000000th simulation with a standard error of 0,045 is $90,163. The call option price obtained can be used as a reference for investors in purchasing options to minimize losses from call option investments in that period.

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.

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.

A Multilayer Perceptron Learning Architecture for Black-Scholes Call Options Pricing

This work explores whether an artificial neural network trained with synthetic call option pricing data can perform well on semi-real market data. The approach that is followed is based on a multilayer perceptron using a large synthetic dataset for training and a semi-real dataset from market data simulating top ten S&P 100 stocks. The baseline for errors and price estimations is the Black-Sholes model. Results, demonstrate the overall capability of the network to learn the Black-Scholes function using synthetic data, and estimate call option prices with high level of accuracy, competitively to Black-Sholes model. This is a valuable result for practitioners who might want to use machine learning paradigms for option pricing in various assets and markets. Future work will include feature selection and advanced sampling for training, plus alternative network architectures and further experimentation with hyperparameters. The most important outcome from the work is that machine learning models can be used from practitioners as an alternative method for option pricing.

Pricing of European call options using generalized Wishart processes

<p>This study explores a multiple-security, high-risk pricing model where the implied volatility has been portrayed through Generalized Wishart affine processes. The presence of dual dependency matrices distinctively characterizes this multifaceted model. These matrices encapsulate the relationship between the generalized Wishart processes and the evolving dynamics of several securities. The adaptability of the proposed model makes it a perfect fit for high-frequency market data, whether dealing with either long or short-term maturities of calls. The main objective paper is on its derivation and addressing the call option pricing problem within the context of the volatility mode using generalized Wishart stochastic. A combination of Fourier transforms techniques and perturbation methods are utilized, mainly focusing on pricing European call options. The model proposed in this study is theoretical and practical, showcasing the strong potential for real-world applications within the financial derivative market.</p>

Limiting Cases of the Black-Scholes Type Asymptotics of Call Option Pricing in the Generalised CRR Model

The article concerns the generalised Cox‑Ross‑Rubinstein (CRR) option pricing model with new formulas for changes in upper and lower stock prices. The formula for option pricing in this model, which is the Black‑Scholes type formula, and its asymptotics are presented. The aim of the paper is to analyse limiting cases of the obtained asymptotics using probability theory and later data from the Warsaw Stock Exchange. Empirical analyses of option pricing in the generalised CRR model confirm the calculated limits.

Simulasi Harga Opsi Call Asia Dengan Suku Bunga Tidak Konstan

In the financial world, the thing that is interesting for investors today is to predict a reasonable Asian Call Option price that does not cause the seller or buyer of the option to experience losses. The constant interest assumption is more often used by previous research as an assumption in predicting the price of Asian Call Options. This is not in line with the fact that interest rate movements are not constant. The use of incorrect assumptions will affect the accuracy of prediction results. In overcoming this problem, this study aims to predict the price of Asian Call Options with non-constant interest rates using Monte Carlo simulation. The non-constant interest rate model used is Cox-Ingersoll-Ross (CIR) because this model produces positive interest rates. This study begins with a screening of Normally distributed interest rate data. The data is used to determine the parameters of the CIR interest rate model. Next, an Asian Call Option price simulation with non-constant interest rates was carried out and ended by analyzing a graph of the Asian Call Option price simulation results with non-constant interest rates. The simulated price result of the Asian Call Option with non-constant interest rates converged at the value of $170.82. Based on the price chart of Asian Call Options with different strike prices, the larger the strike price causes, the larger the Asian Call Option price, which corresponds to the price characteristics of the Asian Call Option. In addition, by extending the maturity period, the price of Asian Call Options with non-constant interest rates is greater according to the price characteristics of Asian Call Options.

An Optimization Approach for pricing of Discrete European Call options Based on the Preference of Investors

An Optimization Approach for pricing of Discrete European Call options Based on the Preference of Investors

On regime-switching European option pricing

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