Utilizing Parametric and Non-Parametric methods in Black-Scholes Process with Application to Finance (original) (raw)
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Geometric Brownian motion is used as a process for stock prices in the Black-Scholes model. Empirically, stock prices often show jumps caused by unpredictable events or news, also distribution of the log returns of asset exhibits the excess kurtosis. This leads to consideration of an alternative asset price model based on levy process instead of Wienner process. A three parameter stochastic process termed the variance gamma process that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time change given by gamma process. The two additional parameters are the drift of the Brownian motion and volatility of the time change. These additional parameters provide control over skewness and kurtosis of return distribution. Closed forms are obtained for return density and the prices of European call options.
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In this paper, we propose a new study of the Black and Scholes Process (BSP). The main objective is to add a threshold parameter to the Black and Scholes Process. Using Kolmogorov equations, we obtain the probability density function and the moments of the process. Estimators of the parameters are studied by considering discrete sampling of the sample trajectories of the model and then using the maximum likelihood method and the Wicksell method.
An Empirical Investigation of the Black-Scholes Model: Evidence from the Australian Stock Exchange
This paper evaluates the probability of an exchange traded European call option being exercised on the ASX200 Options Index. Using single-parameter estimates of factors within the Black-Scholes model, this paper utilises qualitative regression and a maximum likelihood approach. Results indicate that the Black-Scholes model is statistically significant at the 1% level. The results also provide evidence that the use of implied volatility and a jump-diffusion approach, which increases the tail properties of the underlying lognormal distribution, improves the statistical significance of the Black-Scholes model.
Black-Scholes and Extended Black-Scholes Models: A Comparative Statistical Analysis
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Much research has been done on options pricing. Black and Scholes [12] set the benchmark in 1973 with their model for arbitrage-free, risk-neutral options valuation. Arbitrage-free refers to a market environment where prices are such that trading opportunities with no risk do not exist and risk-neutral commodities earn a risk free interest rate. Since then the literature has seen a multitude of models improving the fit of the traditional Black -Scholes (BS) model. A brief overview of options and these models is given. A derivation and discussion of BS is followed by a derivation and discussion of the Extended Black-Scholes (EBS) model by Modisett and Powell [39] which augments BS with the addition of a small drift parameter. A solution to both BS and EBS is the n given. A bootstrap method is used to test whether EBS is a significant improvement over BS using S&P500 options data. It is concluded that not only does EBS significantly improve BS for the given data, but that EBS is argua...
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In this paper, we discuss the estimation of the diusion coecient of an Itô process from high-frequency data using a nonparametric approach by Nadayara-Watson estimator. The principal purpose is to estimate the diusion coecient using selective estimators of spot volatility proposed by several authors, which are based on the observed trajectories. In general, statistical and econometrical criteria are used for comparing spot volatility estimators used in nonparametric estimators. We want to resort to merely nancial metrics to achieve the same task. More precisely, the accuracy of dierent spot volatility estimates is measured in terms of how accurately they can reproduce market option prices. The model is implemented using S&P 500 data, and successively, we used it to estimate european call option prices written on the S&P 500 index. The estimation results are compared to well-known parametric alternative available in the literature. Empirical results not only provide strong evidence that most traditional pricing model are mispecied, but also conrm that the nonparametric model generates signicantly dierent prices of common derivatives.
The Black-Scholes Model: A Comprehensive Analysis
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The Black-Scholes Model is a cornerstone of financial economics, revolutionizing options pricing and modern finance. Developed by Fischer Black and Myron Scholes in 1973, it provides a mathematical framework for valuing options and derivatives. This research thoroughly examines the Black-Scholes Model, encompassing its historical context, theoretical foundations, practical applications, limitations, and critically reviews the existing literature on the proposed exact as well as the numerical solutions to the Black-Scholes model and recent advancements. By analyzing its multifaceted aspects, this paper aims to deepen understanding and shed light on its significance in contemporary financial markets.
Different models of pricing stock options are tested systematically. The most useful is Black-Scholes model. Using this model to price the behavior of stock options, it is found that pricing errors and implied volatility estimates differ across exercise price and time to maturity. FTSE/ASE-20 were first introduced in August 2000 and nowadays they represent the 91,3% of the Greek options' market. This paper examines the pricing of FTSE/ASE-20 stock options using the Black-Scholes model. It is found that observed prices and predicted prices by the model differ systematically because the model assumes the market to be frictionless. The model overprices both call and put options. Furthermore, implied volatility reduces as contracts become out-of-the-money.
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In this paper we discuss parameter estimation in black scholes model. A non-parametric estimation method and well known maximum likelihood estimator are considered. Our aim is to estimate the unknown parameters for stochastic differential equation with discrete time observation data. In simulation study we compare the non-parametric method with maximum likelihood method using stochastic numerical scheme named with Euler Maruyama.
A Stochastic Model of the Dynamics of Change in Stock Price
NIGERIAN ANNALS OF PURE AND APPLIED SCIENCES
The solutions of many mathematical models resulting in stochastic differential equations are based on the assumption that the drift and the volatility coefficients were linear functions of the solutions. We formulated a model whose basic parameters could be derived from observations over discretized time intervals rather than the assumption that the drift and the volatility coefficients were linear functions of the solutions. We took into consideration the possibility of an asset gaining, losing or stable in a small interval of time instead of the assumption of the Binomial Asset pricing models that posited that the price could appreciate by a factor p or depreciate by a factor 1-p. A multi-dimensional stochastic differential equation was obtained whose drift is the expectation vector and the volatility the covariance of the stocks with respect to each other. The resulting system of stochastic differential equations was solved numerically using the Euler Maruyama Scheme for multi-di...