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Stochastic Volatility Jump
In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates. This model fits the observed volatility surface, implied volatility surface well. The model is a Heston model, Heston process for stochastic volatility with an added Jump_diffusion#In_economics_and_finance, Merton log-normal jump. It assumes the following correlated processes: : dS=\mu S\,dt+\sqrt S\,dZ_1+(e^ -1)S \, dq : d\nu =\lambda (\nu - \overline) \, dt+\eta \sqrt \, dZ_2 : \operatorname(dZ_1, dZ_2) =\rho : \operatorname(dq=1) =\lambda dt where ''S'' is the price of security, ''μ'' is the constant drift (i.e. expected return), ''t'' represents time, ''Z''1 is a standard Brownian motion, ''q'' is a Poisson counter with density ''λ''. References Mathematical finance Financial models {{econometrics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volatility Surface
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices (and thus implied volatilities) than what is suggested by standard option pricing models. These options are said to be either deep in-the-money or out-of-the-money. Graphing implied volatilities against strike prices for a given expiry produces a skewed "smile" instead of the expected flat surface. The pattern differs across various markets. Equity options traded in American markets did not show a volatility smile before the Crash of 1987 but began showing one afterwards. It is believed that investor reassessments of the probabilities of fat-tail have led to higher prices for out-of-the-money options. This anomaly implies de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heston Model
In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process. Basic Heston model The basic Heston model assumes that ''St'', the price of the asset, is determined by a stochastic process, : dS_t = \mu S_t\,dt + \sqrt S_t\,dW^S_t, where \nu_t, the instantaneous variance, is given by a Feller square-root or CIR process, : d\nu_t = \kappa(\theta - \nu_t)\,dt + \xi \sqrt\,dW^_t, and W^S_t, W^_t are Wiener processes (i.e., continuous random walks) with correlation ρ. The model has five parameters: * \nu_0, the initial variance. * \theta, the long variance, or long-run average variance of the price; as ''t'' tends to infinity, the expected value of ν''t'' tends to θ. * \rho, the correlation of the two Wiener processes. * \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Volatility
In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others. Stochastic volatility models are one approach to resolve a shortcoming of the Black–Scholes model. In particular, models based on Black-Scholes assume that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying security. However, these models cannot explain long-observed features of the implied volatility surface such as volatility smile and skew, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jump Diffusion
Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, option pricing, and pattern theory and computational vision. In physics In crystals, atomic diffusion typically consists of jumps between vacant lattice sites. On time and length scales that average over many single jumps, the net motion of the jumping atoms can be described as regular diffusion. Jump diffusion can be studied on a microscopic scale by inelastic neutron scattering and by Mößbauer spectroscopy. Closed expressions for the autocorrelation function have been derived for several jump(-diffusion) models: * Singwi, Sjölander 1960: alternation between oscillatory motion and directed motion * Chudley, Elliott 1961: jumps on a lattice * Sears 1966, 1967: jump diffusion of rotational degrees of freedom * Hall, Ross 1981: jump diffusion within a restricted volume In economics and finance I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |