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 requir ...

, the CEV or constant elasticity of

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

model is a

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 ...

model that attempts to capture stochastic volatility and the leverage effect. The model is widely used by practitioners in the financial industry, especially for modelling

equities In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company ...

and

commodities In economics, a commodity is an economic good, usually a resource, that has full or substantial fungibility: that is, the market treats instances of the good as equivalent or nearly so with no regard to who produced them. The price of a co ...

. It was developed by John Cox in 1975.

Dynamic

The CEV model describes a process which evolves according to the following

stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock ...

: :

\mathrmS_t = \mu S_t \mathrmt + \sigma S_t ^ \gamma \mathrmW_t

in which ''S'' is the spot price, ''t'' is time, and ''μ'' is a parameter characterising the drift, ''σ'' and ''γ'' are other parameters, and ''W'' is a Brownian motion. And so we have :

\sigma(S_t, t)=\sigma S_t^

The constant parameters

\sigma,\;\gamma

satisfy the conditions

\sigma\geq 0,\;\gamma\geq 0

. The parameter

\gamma

controls the relationship between volatility and price, and is the central feature of the model. When

\gamma < 1

we see an effect, commonly observed in equity markets, where the volatility of a stock increases as its price falls and the leverage ratio increases. Conversely, in commodity markets, we often observe

\gamma > 1

,Geman, H, and Shih, YF. 2009. "Modeling Commodity Prices under the CEV Model." The Journal of Alternative Investments 11 (3): 65–84. whereby the volatility of the price of a commodity tends to increase as its price increases and leverage ratio decreases. If we observe

\gamma = 0

this model is considered the model which was proposed by

Louis Bachelier Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part ...

in his PhD Thesis "The Theory of Speculation".

References

External links

Asymptotic Approximations to CEV and SABR ModelsPrice and implied volatility under CEV model with closed formulas, Monte-Carlo and Finite Difference MethodPrice and implied volatility of European options in CEV Model
delamotte-b.fr {{Stochastic processes Options (finance) Derivatives (finance) Financial models

Dynamic

See also

References

External links