scholarly journals Entropic Dynamics of Stocks and European Options

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 765 ◽  
Author(s):  
Mohammad Abedi ◽  
Daniel Bartolomeo
Keyword(s):  
Stock Price ◽  
Inductive Inference ◽  
Planck Equation ◽  
Derivative Securities ◽  
European Options ◽  
Scale Invariant ◽  
No Arbitrage ◽  
Black Scholes ◽  
Risk Neutral ◽  
Log Price

We develop an entropic framework to model the dynamics of stocks and European Options. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available. The objective of the paper is to lay down an alternative framework for modeling dynamics. An important information about the dynamics of a stock’s price is scale invariance. By imposing the scale invariant symmetry, we arrive at choosing the logarithm of the stock’s price as the proper variable to model. The dynamics of stock log price is derived using two pieces of information, the continuity of motion and the directionality constraint. The resulting model is the same as the Geometric Brownian Motion, GBM, of the stock price which is manifestly scale invariant. Furthermore, we come up with the dynamics of probability density function, which is a Fokker–Planck equation. Next, we extend the model to value the European Options on a stock. Derivative securities ought to be prices such that there is no arbitrage. To ensure the no-arbitrage pricing, we derive the risk-neutral measure by incorporating the risk-neutral information. Consequently, the Black–Scholes model and the Black–Scholes-Merton differential equation are derived.

scholarly journals Entropic Dynamics of Exchange Rates and Options

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 586 ◽  
Author(s):  
Mohammad Abedi ◽  
Daniel Bartolomeo
Keyword(s):  
Exchange Rate ◽  
Exchange Rates ◽  
Inductive Inference ◽  
Relevant Information ◽  
Scale Transformation ◽  
European Options ◽  
No Arbitrage ◽  
Model Dynamics ◽  
The Exchange Rate ◽  
Entropic Dynamics

An Entropic Dynamics of exchange rates is laid down to model the dynamics of foreign exchange rates, FX, and European Options on FX. The main objective is to represent an alternative framework to model dynamics. Entropic inference is an inductive inference framework equipped with proper tools to handle situations where incomplete information is available. Entropic Dynamics is an application of entropic inference, which is equipped with the entropic notion of time to model dynamics. The scale invariance is a symmetry of the dynamics of exchange rates, which is manifested in our formalism. To make the formalism manifestly invariant under this symmetry, we arrive at choosing the logarithm of the exchange rate as the proper variable to model. By taking into account the relevant information about the exchange rates, we derive the Geometric Brownian Motion, GBM, of the exchange rate, which is manifestly invariant under the scale transformation. Securities should be valued such that there is no arbitrage opportunity. To this end, we derive a risk-neutral measure to value European Options on FX. The resulting model is the celebrated Garman–Kohlhagen model.

scholarly journals PENERAPAN METODE BINOMIAL TREE DALAM MENGESTIMASI HARGA KONTRAK OPSI TIPE AMERIKA

2016 ◽  
Vol 5 (4) ◽  
pp. 156
Author(s):  
I GUSTI AYU MITA ERMIA SARI ◽  
KOMANG DHARMAWAN ◽  
TJOKORDA BAGUS OKA
Keyword(s):  
Stock Price ◽  
Price Movement ◽  
Binomial Tree ◽  
Option Contracts ◽  
Black Scholes ◽  
Risk Neutral ◽  
Price Option ◽  
Tree Method

Binomial tree is a method that can be used to determine price option contracts. In this method, the stock price movement is presented in the form of a  tree with each branch representing the probability of the stock price to move up or move down. The purpose of this paper was to determine the price of the options contracts with the American type on Binomial Tree method and compare the three methods that is variance matching, proportional , and risk neutral of determining the value of price option contracts used in Binomial Tree method with Black-Schole method. The result of this research was the value of the options contract using the variance matching more similar with the value of the Black-Scholes contract.

BLACK–SCHOLES–MERTON IN RANDOM TIME: A NEW STOCHASTIC VOLATILITY MODEL WITH PATH DEPENDENCE

2007 ◽  
Vol 10 (05) ◽  
pp. 847-872 ◽  
Author(s):  
DMITRY OSTROVSKY
Keyword(s):  
Brownian Motion ◽  
Implied Volatility ◽  
Time Derivative ◽  
Sufficient Conditions ◽  
Random Time ◽  
Average Price ◽  
European Options ◽  
No Arbitrage ◽  
Volatility Model ◽  
Black Scholes

A generalized Black–Scholes–Merton economy is introduced. The economy is driven by Brownian motion in random time that is taken to be continuous and independent of Brownian motion. European options are priced by the no-arbitrage principle as conditional averages of their classical values over the random time to maturity. The prices are path dependent in general unless the time derivative of the random time is Markovian. An explicit self-financing hedging strategy is shown to replicate all European options by dynamically trading in stock, money market, and digital calls on realized variance. The notion of the average price is introduced, and the average price of the call option is shown to be greater than the corresponding Black–Scholes price for all deep in- and out-of-the-money options under appropriate sufficient conditions. The model is implemented in limit lognormal random time. The significance of its multiscaling law is explained theoretically and verified numerically to be a determining factor of the term structure of implied volatility.

scholarly journals European Option Pricing Formula in Risk-Aversive Markets

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Shujin Wu ◽  
Shiyu Wang
Keyword(s):  
Risk Measure ◽  
European Option ◽  
Underlying Asset ◽  
No Arbitrage ◽  
European Option Pricing ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Risk Neutral ◽  
Arbitrage Opportunity ◽  
Weaker Conditions

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

A Risk-Neutral Stochastic Volatility Model

1998 ◽  
Vol 01 (02) ◽  
pp. 289-310 ◽  
Author(s):  
Yingzi Zhu ◽  
Marco Avellaneda
Keyword(s):  
Stochastic Volatility ◽  
Implied Volatility ◽  
Monte Carlo Study ◽  
Stochastic Volatility Model ◽  
No Arbitrage ◽  
Volatility Model ◽  
Black Scholes ◽  
Risk Neutral ◽  
The Difference ◽  
Log Normal

We construct a risk-neutral stochastic volatility model using no-arbitrage pricing principles. We then study the behavior of the implied volatility of options that are deep in and out of the money according to this model. The motivation of this study is to show the difference in the asymptotic behavior of the distribution tails between the usual Black–Scholes log-normal distribution and the risk-neutral stochastic volatility distribution. In the second part of the paper, we further explore this risk-neutral stochastic volatility model by a Monte-Carlo study on the implied volatility curve (implied volatility as a function of the option strikes) for near-the-money options. We study the behavior of this "smile" curve under different choices of parameter for the model, and determine how the shape and skewness of the "smile" curve is affected by the volatility of volatility ("V-vol") and the correlation between the underlying asset and its volatility.

scholarly journals PENGARUH PEMBAGIAN DIVIDEN MELALUI MODEL BLACK-SCHOLES

2021 ◽  
Vol 2 (3) ◽  
pp. 351-354
Author(s):  
Diana Purwandari
Keyword(s):  
Stock Prices ◽  
Stock Price ◽  
European Options ◽  
Financial Contracts ◽  
Stock Trading ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Dividend Distribution ◽  
Option Pricing Model ◽  
The Right

Stock trading has a risk that can be said to be quite large due to fluctuations in stock prices. In stock trading, one alternative to reduce the amount of risk is options. The focus of this research is on European options which are financial contracts by giving the holder the right, not the obligation, to sell or buy the principal asset from the writer when it expires at a predetermined price. The Black-Scholes model is an option pricing model commonly used in the financial sector. This study aims to determine the effect of dividend distribution through the Black-Scholes model on stock prices. The effect of dividend distribution through the Black-Scholes model on stock prices results in the stock price immediately after the dividend distribution being lower than the stock price shortly before the dividend distribution

THE FORWARD PDE FOR EUROPEAN OPTIONS ON STOCKS WITH FIXED FRACTIONAL JUMPS

2005 ◽  
Vol 08 (02) ◽  
pp. 239-253 ◽  
Author(s):  
PETER CARR ◽  
ALIREZA JAVAHERI
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Stock Price ◽  
Arrival Rate ◽  
European Options ◽  
Standard Assumption ◽  
Integro Differential Equation ◽  
Local Variance ◽  
Calendar Dates

We derive a partial integro differential equation (PIDE) which relates the price of a calendar spread to the prices of butterfly spreads and the functions describing the evolution of the process. These evolution functions are the forward local variance rate and a new concept called the forward local default arrival rate. We then specialize to the case where the only jump which can occur reduces the underlying stock price by a fixed fraction of its pre-jump value. This is a standard assumption when valuing an option written on a stock which can default. We discuss novel strategies for calibrating to a term and strike structure of European options prices. In particular using a few calendar dates, we derive closed form expressions for both the local variance and the local default arrival rate.

scholarly journals OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS

2004 ◽  
Vol 07 (07) ◽  
pp. 901-907
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Strike Price ◽  
Call Options ◽  
Alternative Approach ◽  
Black Scholes ◽  
Market Practice ◽  
The Difference ◽  
Risk Free Rate

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

Approach to the Delta Greek of nonlinear Black–Scholes equation governing European options

2022 ◽  
Vol 402 ◽  
pp. 113790
Author(s):  
Rui M.P. Almeida ◽  
Teófilo D. Chihaluca ◽  
José C.M. Duque
Keyword(s):  
European Options ◽  
Black Scholes
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