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Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior. Some of the arguments for using GBM to model stock prices are: The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.
The Black–Scholes / ˌblæk ˈʃoʊlz / [1] or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives ...
Black–Scholes formula. Itô's lemma can be used to derive the Black–Scholes equation for an option. Suppose a stock price follows a geometric Brownian motion given by the stochastic differential equation dS = S(σdB + μ dt). Then, if the value of an option at time t is f(t, S t), Itô's lemma gives
Risk-neutral measure. In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to ...
Stock market prediction. Stock market prediction is the act of trying to determine the future value of a company stock or other financial instrument traded on an exchange. The successful prediction of a stock's future price could yield significant profit. The efficient market hypothesis suggests that stock prices reflect all currently available ...
The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William F. Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time ...
Definition of Greeks as the sensitivity of an option's price and risk (in the first row) to the underlying parameter (in the first column). First-order Greeks are in blue, second-order Greeks are in green, and third-order Greeks are in yellow. Vanna, charm and veta appear twice, since partial cross derivatives are equal by Schwarz's theorem.
Stock valuation is the method of calculating theoretical values of companies and their stocks.The main use of these methods is to predict future market prices, or more generally, potential market prices, and thus to profit from price movement – stocks that are judged undervalued (with respect to their theoretical value) are bought, while stocks that are judged overvalued are sold, in the ...