Scholes Pricing Formula Research Articles

The Research on the Black-Scholes Formula

At the beginning of 1970s，Fischer Black and Myron Scholes have made the unprecedented work in the domain of the option pricing theory，and proposed the first complete option pricing model，which named the Black—Scholes pricing formula . It has been accepted by the theoretical and industrial world for its wide application，and becomes the secondary great revolution in the financial domain. However，in the actual financial market，a mass of finance practice has indicated that there is a serious warps between the hypothesis of Black-Scholes model and the actual markets. This dissertation introduces the option elementary knowledge briefly，and then carries out a specific study on the the history of option pricing theory as well as factors that influence option price.

PRICING OPTIONS UNDER STOCHASTIC INTEREST RATE AND THE FRASCA–FARINA PROCESS: A SIMPLE, EXPLICIT FORMULA

Assuming a stochastic interest rate, we introduce a simple formula for pricing European options. In doing so, we provide a complete closed-form formula that does not require any numerical/computational methods. Furthermore, the model and formula are far simpler than the previous models/formulas. Our formula is as simple as the classical Black–Scholes pricing formula. Moreover, it removes the theoretical limitation of the original Black–Scholes model without any added practical complexity.

Pricing options under simultaneous stochastic volatility and jumps: A simple closed-form formula without numerical/computational methods

We overcome the limitations of the previous literature in the European options pricing. In doing so, we provide a closed-form formula that does not require any numerical/computational methods. The formula is as simple as the classical Black–Scholes pricing formula. In addition, we simultaneously include jumps and stochastic volatility. Our approach implies the introduction of a new class of stochastic processes that are based on Clifford algebras. The approach can be easily generalized to higher dimensional problems.

Pricing the American options using the Black–Scholes pricing formula

We develop a simple, exact, explicit, and analytical solution to the American option partial differential equation PDE using the Black–Scholes pricing formula.

A modified Black–Scholes pricing formula for European options with bounded underlying prices

In this paper, a modified Black–Scholes (B–S) model is proposed, based on a revised assumption that the range of the underlying price varies within a finite zone, rather than being allowed to vary in a semi-infinite zone as presented in the classical B–S theory. This is motivated by the fact that the underlying price of any option can never reach infinity in reality; a trader may use our new formula to adjust the option price that he/she is willing to long or short. To develop this modified option pricing formula, we assume that a trader has a view on the realistic price range of a particular asset and the log-returns follow a truncated normal distribution within this price range. After a closed-form pricing formula for European call options has been successfully derived, some numerical experiments are conducted. To further demonstrate the meaning of the proposed model, empirical studies are carried out to compare the pricing performance of our model and that of the B–S model with real market data.

The Performance of Skewness and Kurtosis Adjusted Option Pricing Model in Emerging Markets

In this study, the option pricing performance of the adjusted Black-Scholes model proposed by Corrado and Su (1996) and corrected by Brown and Robinson (2002), is investigated and compared with original Black Scholes pricing model for the Turkish derivatives market. The data consist of the European options written on BIST 30 index extends from January 02, 2015 to April 24, 2015 for given exercise prices with maturity April 30, 2015. In this period, the strike prices are ranging from 86 to 124. To compare the models, the implied parameters are derived by minimizing the sum of squared deviations between the observed and theoretical option prices. The implied distribution of BIST 30 index does not significantly deviate from normal distribution. In addition, pricing performance of Black Scholes model performs better in most of the time. Black Scholes pricing Formula, Carrado-Su pricing Formula, Implied Parameters

Black Scholes Option Pricing Model Brownian Motion Approach

Brownian motion has become one of the fundamental building blocks of modern quantitative finance. The mathematical theory of Brownian motion has been applied in contexts ranging far beyond the movement of particles in fluids. Until recently, stock market researchers have confronted the same problem. While they can chart the path of the market on a minute by minute basis it is very hard for them to observe who buys, who sells and how demand and supply affects price fluctuations. There exist many researchers about how the behavior of different investors makes the option price movement in a stock market. The purpose of this paper is to construct the black Scholes option pricing model in the stock markets by using of Brownian motion approach. The main ambition of this study is fourfold: 1) First we begin our approach to construction of Brownian motion from the simple symmetric random walk. 2) Next we introduce the Black – Scholes option pricing model with stock price movement by using of Geometric Brownian motion. 3) Then we extent this Brownian motion approach in the stock market and 4) Finally we construct the model for the generalization based on the deformation of the standard Brownian motion and Black Scholes pricing formula. And this paper will end with conclusion.

Option pricing: Stock price, stock velocity and the acceleration Lagrangian

The industry standard Black–Scholes option pricing formula is based on the current value of the underlying security and other fixed parameters of the model. The Black–Scholes formula, with a fixed volatility, cannot match the market’s option price; instead, it has come to be used as a formula for generating the option price, once the so called implied volatility of the option is provided as additional input. The implied volatility not only is an entire surface, depending on the strike price and maturity of the option, but also depends on calendar time, changing from day to day. The point of view adopted in this paper is that the instantaneous rate of return of the security carries part of the information that is provided by implied volatility, and with a few (time-independent) parameters required for a complete pricing formula.An option pricing formula is developed that is based on knowing the value of both the current price and rate of return of the underlying security which in physics is called velocity. Using an acceleration Lagrangian model based on the formalism of quantum mathematics, we derive the pricing formula for European call options. The implied volatility of the market can be generated by our pricing formula. Our option price is applied to foreign exchange rates and equities and the accuracy is compared with Black–Scholes pricing formula and with the market price.

ARE REAL OPTIONS "REAL"? ISOLATING UNCERTAINTY FROM RISK IN REAL OPTIONS ANALYSIS

This paper derives an adjusted Black–Scholes pricing formula. In separating risk and uncertainty using the robust control technique, we find that both uncertainty and risk raise management's subjective evaluation of real options. We suggest a simple method to filter the risk of the project and to acquire a more reliable value of real options without the influence of uncertainty. In addition, we propose that an investment opportunity may be postponed inappropriately, as under uncertainty the exercise of investment may be delayed by the project manager. To our knowledge, any similar quantitative methods have not hitherto been mentioned in terms of isolating uncertainty from risk in real options analysis that we consider here.

Solving replication problems in a complete market by orthogonal series expansion

We reconsider the replication problem for contingent claims in a complete market under a general framework. Since there are various limitations in the Black–Scholes pricing formula, we propose a new method to obtain an explicit self-financing trading strategy expression for replications of claims in a general model. The main advantage of our method is that we propose using an orthogonal expansion method to derive a closed-form expression for the self-financing strategy that is associated with some general underlying asset processes. As a consequence, a replication strategy is obtained for a European option. Converse to the traditional Black–Scholes theory, we derive a pricing formula for a European option from the proposed replication strategy that is quite different from the Black–Scholes pricing formula. We provide an implementation procedure and both numerical and empirical examples to show how the proposed trading strategy works in practice and then compare with a replication strategy based on the Black–Scholes theory.

An investor sentiment barometer — Greek Implied Volatility Index (GRIV)

In this paper, we propose a new measure of Greek equity market volatility based on the prices of FTSE/ATHEX-20 index options. Greek Implied Volatility Index is calculated using the model-free methodology that involves option prices summations and is independent from the Black and Scholes pricing formula. The specific method is applied for the first time in a peripheral and illiquid market as the Athens Exchange.The empirical findings of this paper show that the proposed volatility index includes information about future realized volatility beyond that contained in past volatility. In addition, our analysis indicates that there is a statistically significant negative and asymmetric contemporaneous relationship between the returns of the implied volatility index and the underlying equity index. Finally, the volatility transmission effects on the Greek stock exchange from two leading markets, namely the New York Stock Exchange and the Deutsche Börse, are tested and documented.

An Investor Sentiment Barometer - Greek Implied Volatility Index

In this paper a new measure of Greek stock market volatility based on the prices of FTSE/ATHEX-20 index options is proposed. Greek Implied Volatility Index is calculated using the model-free methodology that involves option prices summations and is independent from the Black and Scholes pricing formula. The specific method is applied for the first time in a peripheral and illiquid market as the Athens Exchange. The empirical findings of this paper show that the proposed volatility index includes information about future realized volatility beyond that contained in past volatility and in addition, show that there is a statistically significant negative and asymmetric contemporaneous relationship between the returns of the implied volatility index and the underlying equity index. Finally, the volatility transmission effects on the Greek stock exchange from two leading markets, namely the New York Stock Exchange and the Deutsche Borse, are tested and documented.

The Information Content of VFTSE

The FTSE 100 Volatility Index (VFTSE) reflects the market expectations of the future monthly volatility of the UK benchmark equity index, FTSE100. VFTSE is calculated using the model-free methodology that involves option prices summations and is independent from the Black and Scholes pricing formula. This study tests and documents the information content of VFTSE regarding both the realized volatility and the returns of the underlying equity index. The empirical findings suggest that VFTSE includes information about future volatility beyond that contained in past volatility and in addition show that there is a statistically significant negative and asymmetric contemporaneous relationship between implied volatility changes and the underlying equity index returns.

A cerebellar associative memory approach to option pricing and arbitrage trading

Option pricing is a process to obtain the theoretical fair value of an option based on the factors affecting its price. The classical approaches to option pricing include the Black–Scholes pricing formula and the binomial pricing model. These techniques, however, employ complex and rigid statistical formulations that are not easily comprehensible to novice investors. More recently, non-parametric and computational methods of option valuation that are able to construct a model of the pricing formula from historical data have been proposed in the literature. However, most of these models functioned as black-boxes and may not be able to efficiently and accurately capture the complex market dynamics and characteristics of the option data. This paper proposes a novel brain-inspired cerebellar associative memory model for pricing American-style call options on British pound vs. US dollar currency futures. The proposed model, named PSECMAC, constitutes a local learning model that is inspired by the neurophysiological aspects of the human cerebellum. The PSECMAC-based option-pricing model is subsequently applied in a mis-priced option arbitrage trading system. Simulation results show an encouraging return on investment of 23.1% for some of the traded options.

A methodology using option pricing to determine a suitable discount rate in environmental management

With the present analysis the authors propose an approach for determining an adequate discount rate in environmental management problems, more specifically radioactive-waste management. It is shown that the classical Black–Scholes pricing formula can be used for determining the adequate present funding to be set-aside for the future. The average funding is equal to the net present value (NPV) of the future costs, including technical-scenario uncertainties. For taking into account the financial uncertainties, the NPV is identified with the strike price of a European put option, and the asset value in the managed fund is identified with the current price. The risk-free rate is the expected return rate of the portfolio. The paper shows that the adequate present funding can be determined for given multi-generational risk levels and an asset allocation by fixing the discount rate and adding a premium to the NPV of future costs.

Long memory options: LM evidence and simulations

This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation by simple simulation. Many empirical researchers have observed the non-Fickian degrees of persistence in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black–Scholes’ geometric Brownian motion assumption. Moreover, Elliott and van der Hoek [Elliott, R.J., van der Hoek, J., 2003. A general fractional white noise theory and applications to finance. Math. Finance 13, 301–330] provide a theoretical framework for incorporating these findings into the Black–Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. By using a simple mono-fractal fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the Black–Scholes pricing formula. Long memory options are of considerable importance in corporate remuneration packages, since stock options are written on a company’s own shares for long expiration periods. It makes a significant difference in the valuation when an option is “blue” or when it is “red.” For a proper valuation of such stock options, the degrees of persistence of the companies’ share markets must be precisely measured and properly incorporated in the warrant valuation, otherwise substantial pricing errors may result.

Option Pricing under Stochastic Interest Rates: An Empirical Investigation

Using daily data of the Nikkei 225 index, call option prices and call money rates of the Japanese financial market,a comparison is made of the pricing performance of stock option pricing modelsunder several stochastic interest rate processes proposedby the existing term structure literature.The results show that (1) one option pricing modelunder a specific stochastic interest ratedoes not significantly outperformanother option pricing model under an alternative stochasticinterest rate, and (2) incorporating stochastic interest ratesinto stock option pricing does not contribute to the performanceimprovement of the original Black–Scholes pricing formula.

The Black-Scholes pricing formula in the quantum context.

A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects. The lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects. These characteristics have been noted by traders and economists and appear intrinsic to market dynamics. This explanation is explored here in terms of a corresponding generalization of the Wiener process and its role in the Black-Scholes-Merton theory. The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context. This provides a basis for a corresponding generalization of the Ito theory of stochastic integration. An extension of the Black-Scholes option pricing formula to the quantum context is deduced.

Models for Option Prices

Cox, Ross, and Rubinstein [6] introduced a binomial option price model and derived the seminal Black–Scholes pricing formula. In this paper we characterize all possible stock price models that can be approximated by the binomial models and derive the corresponding approximations for the pricing formulas. We introduce two additional randomizations in the binomial price models seeking more general and more realistic limiting models. The first type of model is based on a randomization of the number of price changes, the second one on a randomization of the ups and downs in the price process.As a result we also obtain price models with fat tails, higher peaks in the center, nonsymmetric etc., which are observed in typical asset return data. Following similar ideas as in [6] we also derive approximating option pricing formulas and discuss several examples.