Stock Option Pricing Research Articles

Numerical Solution of European Put Option for Black-Scholes Model Using Keller Box Method

In this study, we propose to determine option pricing by using Black-Scholes model numerically. The Keller box method, a numerical method with a box-shaped implicit scheme, is chosen to solve the problem of pricing stock options, especially European-put option. This option pricing involves several parameters such as stock price volatility, risk-free interest rate and strike price. The numerical stability of the method is checked using Von Neumann stability before the simulation is conducted. The influence of interest rates, volatility, and strike price on the option price state that the higher the value of the interest rate parameter, the lower the option price value, while the greater the value of stock price volatility and strike price, the higher the option price.

The Brownian motion, Wiener process, and Pu migration as examples of random walk

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

PERHITUNGAN HARGA OPSI EROPADENGAN METODE TRINOMIAL PADA PERUSAHAAN MITSUBISHI

Investment is an important instrument in the financial market. The investment objective is to obtain large profits with minimum capital. The trend that is currently developing is that investors are not only interested in investing in real assets, but they have great attention to investing in financial assets such as shares. However, to gain profits, investors must face risk and uncertainty. Among the solutions to reduce these risks are derivatives. Stock options are one of the widely used derivative products. An important instrument in stock options is the method of determining option prices according to the type of option. European option prices can be determined using several methods, namely the Black-Scholes method, binomial method, and trinomial method. This research aims to implement the trinomial method in determining European option prices, both Call options and Put options on Mitsubishi UFJ Financial Group (MUFG) shares. The results of determining option prices using the trinomial method are then compared with the results of the Black-Scholes method which aims to analyze the convergence of the two. The research results show that the trinomial method can be used to determine stock option prices for the Mitsubishi company more flexibly than the Black-Scholes method. In addition, the results show that the trinomial model converges to the Black-Scholes method.

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.

Volatility, Uncertainty, and Option Pricing

Amidst global and Chinese uncertainties, this paper delves into stock market volatility and option pricing within the current economic policy context. Focusing on China's stock market, it calculates returns and volatility changes in key indices, analyzing differences among indices and funds, along with temporal variations. Shared data characteristics and reflections on new market trends emerge. Employing methods such as the GARCH model, Black-Scholes formula, and Monte Carlo algorithm, it analyzes volatility and option pricing using extensive annual and monthly data. Visualizations showcase market change patterns. The study defines volatility as a measure of financial asset price fluctuation extent, reflecting asset risk. Higher volatility indicates pronounced price fluctuations and uncertainty, while lower volatility signifies smoother fluctuations and greater certainty. Merging data with volatility's significance, the study probes China's securities market uncertainty, investigating the link between option pricing and volatility. It concludes by identifying the connection between volatility, uncertainty, and option pricing, pointing to future research directions and challenges. Future work will track the latest market trends to enrich understanding.

Rank and File Employees' Stock Option and Audit Pricing: Evidence From S&P 1500 firms

Rank and File Employees' Stock Option and Audit Pricing: Evidence From S&P 1500 firms

Sensitivity Analysis Based on Portfolio Option Models Using Tesla and Nio as Examples

The electric vehicle (EV) industry is undergoing a significant transformation, driven by environmental concerns, advancements in battery technology, and changing consumer preferences. Companies like Tesla and China's NIO are at the forefront of this revolution, showcasing the global potential of EVs. This study undertook a sensitivity analysis of these two leading EV manufacturers, Tesla and NIO, to understand the dynamics of their stock prices and implications for investment strategies. Data was meticulously collected from various financial databases to capture daily stock prices, which were then used to compute monthly returns. The study also delved into options, focusing on the strike prices and volatility of Tesla and NIO options. Using a combination of stock prices, strike prices, and volatility, the intrinsic and time value of these options were analyzed. The research employed a composite option model to describe the price variation of an option, considering factors like asset price volatility, risk-free rate, and dividend yield. The findings revealed that stock prices and option prices exhibited certain sensitivities to market variables. For instance, Tesla's option prices showed an upward trend with rising stock prices, while NIO's option prices displayed varied responses to stock price changes. The study underscores the importance of understanding these sensitivities for informed investment decisions in the evolving EV market.

Pricing of Arithmetic Average Asian Option by Combining Variance Reduction and Quasi-Monte Carlo Method

Financial derivatives have developed rapidly over the past few decades due to their risk-averse nature, with options being the preferred financial derivatives due to their flexible contractual mechanisms, particularly Asian options. The Black–Scholes stock option pricing model is often used in conjunction with Monte Carlo simulations for option pricing. However, the Black–Scholes model assumes that the volatility of asset returns is constant, which does not square with practical financial markets. Additionally, Monte Carlo simulation suffers from slow error convergence. To address these issues, we first correct the asset volatility in the Black–Scholes model using a GARCH model. Then, the low error convergence rate of the Monte Carlo method is improved using variance reduction techniques. Meanwhile, the quasi-Monte Carlo approach based on low discrepancy sequences is used to refine the error convergence rate. We also provide a simulation experiment and result analysis to validate the effectiveness of our proposed method.

Forecasting stock options prices via the solution of an ill-posed problem for the Black–Scholes equation

In the previous paper (2016 Inverse Problems 32 015010), a new heuristic mathematical model was proposed for accurate forecasting of prices of stock options for 1–2 trading days ahead of the present one. This new technique uses the Black–Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black–Scholes equation was solved in the positive direction of the time variable, this ill-posed initial boundary value problem was solved by the so-called quasi-reversibility method (QRM). This approach with an added trading strategy was tested on the market data for 368 stock options and good forecasting results were demonstrated. In the current paper, we use the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate.

Stock Option Pricing by Augmented Monte-Carlo Simulation models

Studying stock options is still a pristine area of research in the Iranian capital market. This is due to the lack of data as well as the complexity of valuation methodologies. In the present paper, using the Monte-Carlo simulation, we have estimated the value of stock options traded on Tehran Stock Exchange and examined whether the use of a control variate or antithetic variate augmented methods can lower the standard error of estimates. Furthermore, the estimated values of the three models under consideration, including crude Monte-Carlo, control variates augmented Monte-Carlo, and antithetic variates augmented Monte-Carlo are compared with each other and with options market prices. The results show that the standard error of the antithetic variate method is less than the crude method and control variate method. However, the control variate augmented Monte-Carlo model is more powerful than the crude Monte-Carlo and antithetic variate augmented Monte-Carlo method. Therefore, we can conclude that the control variate augmented Monte-Carlo model has a better performance in estimating the value of trading stock options and its estimated values are closer to the market prices.

Rational repricing of risk during COVID‐19: Evidence from Indian single stock options market

Could the COVID‐19 related market crash and subsequent rebound be explained as a rational response to evolving conditions? Our results using multiple forward‐looking measures of uncertainty implied from stock option prices suggest so. First, we find a gradual build‐up of volatility during the month preceding the spike at the start of the pandemic. Second, while tail risk declined after government interventions, the level of uncertainty remained elevated for stocks across industries. Third, the dynamics of decline in tail risk in stocks was industry‐dependent, suggesting that the market performed a fine‐grained analysis of each stock's uncertainty through the pandemic.

LINEAR STOCHASTIC DIVIDEND MODEL

In this paper, we propose a new model for pricing stock and dividend derivatives. We jointly specify dynamics for the stock price and the dividend rate such that the stock price is positive and the dividend rate nonnegative. In its simplest form, the model features a dividend rate that is mean-reverting around a constant fraction of the stock price. The advantage of directly specifying dynamics for the dividend rate, as opposed to the more common approach of modeling the dividend yield, is that it is easier to keep the distribution of cumulative dividends tractable. The model is nonaffine but does belong to the more general class of polynomial processes, which allows us to compute all conditional moments of the stock price and the cumulative dividends explicitly. In particular, we have closed-form expressions for the prices of stock and dividend futures. Prices of stock and dividend options are accurately approximated using a moment matching technique based on the principle of maximal entropy.

Black-Scholes Model of European Call Option Pricing in Constant Market Condition

Investment is a saving activity with the aim of overcoming price increases or often called inflation. Investments can be in the form of gold, property, silver or stock investments. Stock investment is considered more profitable than just saving at a bank. Currency values are declining due to inflation. This results in a tendency to invest in shares. Stock investment carries a great risk. Therefore, in 2004 stock options began to trade. Stock options are contracts that give the holder the right to buy / sell shares at the agreed time, at a certain price. Stock option prices tend to be cheaper than stock prices. Therefore, determining the right price of stock options is needed. In this study, we will focus on the European type of buying options, the right to buy shares at an agreed price at maturity. The purpose of this study is the completion of the Black-Scholes model of European type option prices at a constant market, assuming stock movements meet the stochastic differential equation, fixed risk-free interest rates, companies distributing dividends, no taxes, no transaction costs, and free market arbitration. The results of this research are in the form of differential equations and the settlement of the Black-Scholes model of European type call option prices, and a case study used by stock option contracts with a maturity of January 4, 2010, PT Aqua Golden Mississippi Tbk.

Pricing individual stock options using both stock and market index information

Abstract When it comes to individual stock option pricing, most applications consider a univariate framework. From a theoretical point of view this is unsatisfactory as we know that the expected return of any asset is closely related to the exposure to the market risk factors. To address this, we model the evolution of the individual stock returns together with the market index returns in a flexible bivariate model in line with theory. The model parameters are estimated using both historical returns and aggregated option data from the index and the individual stocks. We assess the model performance by pricing a large set of individual stock options on 26 major US stocks over a long time period including the global financial crisis. Our results show that the losses from using a univariate formulation amounts to 11% on average when compared to our preferred bivariate specification.

Convergence of binomial tree methods to Black Scholes model on determining stock option prices

There are many alternative investment tools that can be used to be the choice of investors. One of them is a derivative product. Derivative products that are more widely known and traded on financial markets are options. Option is a contract or agreement between two parties to buy or sell an instrument. To minimize risk, it is necessary to determine the option price by determining the fair price of the option. This option pricing can be done by using the Binomial Tree method and the Black Scholes method. Some factors that influence options are stock prices, strike prices, maturity, volatility, and interest rates. This paper discusses the European option pricing call on the shares of Bank Central Asia (BCA) with the Binomial Tree method and the Black Scholes Method. From the results of the research, it is found that the Binomial Tree method will converge to the Black Scholes method if the time partition increases.

What Is the Expected Return on a Stock?

ABSTRACTWe derive a formula for the expected return on a stock in terms of the risk‐neutral variance of the market and the stock's excess risk‐neutral variance relative to that of the average stock. These quantities can be computed from index and stock option prices; the formula has no free parameters. The theory performs well empirically both in and out of sample. Our results suggest that there is considerably more variation in expected returns, over time and across stocks, than has previously been acknowledged.

Linear Stochastic Dividend Model

In this paper we propose a new model for pricing stock and dividend derivatives. We jointly specify dynamics for the stock price and the dividend rate such that the stock price is positive and the dividend rate non-negative. In its simplest form, the model features a dividend rate that is mean-reverting around a constant fraction of the stock price. The advantage of directly specifying dynamics for the dividend rate, as opposed to the more common approach of modeling the dividend yield, is that it is easier to keep the distribution of cumulative dividends tractable. The model is non-affine but does belong to the more general class of polynomial processes, which allows us to compute all conditional moments of the stock price and the cumulative dividends explicitly. In particular, we have closed-form expressions for the prices of stock and dividend futures. Prices of stock and dividend options are accurately approximated using a moment matching technique based on the principle of maximal entropy.

Pricing of stochastic volatility stock index option based on Feynman path integral

<sec>Under the background that stock index options urgently need launching in China, the research on stock option pricing model has important theoretical and practical significance. In the traditional B-S-M model it is assumed that the volatility remains unchanged, which differs tremendously from the market’s reality. When the market fluctuates drastically, it is difficult to realize the risk management function of stock index options. Although in the Heston model, as one of the traditional stochastic volatility option pricing models, the correlation risk between the volatility and underlying price is taken into consideration, its pricing accuracy is still to be improved. From the quantum finance perspective, in this paper we use the Feynman path integral method to explore a more practical stock index option pricing model.</sec><sec>In this paper, we construct a Feynman path integral pricing model of stock index options with stochastic volatility by taking Hang Seng index option as the research object and Heston model as the control group. It is found that the Feynman path integral pricing model is significantly superior to the Heston model either at different strike prices on the same expiration date or at different expiration dates for the same strike price. The stock index option pricing model constructed in this paper can give the numerical solution of Feynman's pricing kernel, and directly realizes the forecast of stock index option price. The pricing accuracy is significantly improved compared with the pricing accuracy given by the Heston model through using the characteristic function method.</sec><sec>The remarkable advantages of Feynman path integral stock index option pricing model are as follows. Firstly, the path integral has advantages in solving multivariate problems: the Feynman pricing kernel represents all the information about pricing and can be easily expanded from one-dimensional to multidimensional case, so the change of closing price of stock index and volatility of underlying index can be taken into account simultaneously. Secondly, based on the relationship between the Feynman path generation principle and the law of large number, the mean values of pricing kernel obtained by MATLAB not only optimizes the calculation process, but also significantly improves the pricing accuracy. Feynman path integral is the main method in quantum finance, and the research in this paper will provide reference for its further application in the pricing of financial derivatives.</sec>

Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions

Equity index volatility variation and its interaction with the index return can come from three distinct channels. First, index volatility increases with the market’s aggregate financial leverage. Second, positive shocks to systematic risk increase the cost of capital and reduce the valuation of future cash flows, generating a negative correlation between the index return and its volatility, regardless of financial leverage. Finally, large negative market disruptions show self-exciting behaviors. This article proposes a model that incorporates all three channels and examines their relative contribution to index option pricing and stock option pricing for different types of companies.

Why do employees like to be paid with Options?: A multi-period prospect theory approach

The use of options as compensation for non-executive employees is a puzzle. Standard, rational, valuation models show that the cost of issuing options is larger than the value placed on the options by employees. Existing explanations for this puzzle are based upon static models that ignore the considerable dynamic aspects of employee stock option pricing and exercise behavior. We develop dynamic, multiperiod models of employee preferences considering risk aversion, loss aversion, overconfidence and probability weighting to test possible explanations of the use of employee stock options. We find that a cumulative prospect theory model generates scenarios where employees would prefer options to either cash or equity payments, and also optimally exercise their options early. This is the only model where options are preferred and also optimally exercised early.