Constant elasticity of variance model: Difference between revisions
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The '''CEV''' model describes a process which evolves according to the following [[stochastic differential equation]]: |
The '''CEV''' model describes a process which evolves according to the following [[stochastic differential equation]]: |
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:<math>\mathrm{d}S_t = \mu S_t \mathrm{d}t + \sigma S_t ^ \gamma \mathrm{d}W_t</math> |
:<math>\mathrm{d}S_t = \mu S_t \mathrm{d}t + \sigma S_t ^ \gamma-1 \mathrm{d}W_t</math> |
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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.<ref>[https://web.archive.org/web/20180220152049/http://rafaelmendoza.org/wp-content/uploads/2014/01/CEVchapter.pdf <nowiki>Vadim Linetsky & Rafael Mendozaz, 'The Constant Elasticity of Variance Model', 13 |
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.<ref>[https://web.archive.org/web/20180220152049/http://rafaelmendoza.org/wp-content/uploads/2014/01/CEVchapter.pdf <nowiki>Vadim Linetsky & Rafael Mendozaz, 'The Constant Elasticity of Variance Model', 13 |
Revision as of 19:31, 26 February 2024
In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility 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 and commodities. It was developed by John Cox in 1975.[1]
Dynamic
The CEV model describes a process which evolves according to the following stochastic differential equation:
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.[2] And so we have [citation needed]
The constant parameters satisfy the conditions .
The parameter controls the relationship between volatility and price, and is the central feature of the model. When 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.[3] Conversely, in commodity markets, we often observe ,[4][5] whereby the volatility of the price of a commodity tends to increase as its price increases and leverage ratio decreases. If we observe this model is considered the model which was proposed by Louis Bachelier in his PhD Thesis "The Theory of Speculation".
See also
References
- ^ Cox, J. "Notes on Option Pricing I: Constant Elasticity of Diffusions." Unpublished draft, Stanford University, 1975.
- ^ Vadim Linetsky & Rafael Mendozaz, 'The Constant Elasticity of Variance Model', 13 July 2009. (Accessed 2018-02-20.)
- ^ Yu, J., 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127, 165–178.
- ^ Emanuel, D.C., and J.D. MacBeth, 1982. "Further Results of the Constant Elasticity of Variance Call Option Pricing Model." Journal of Financial and Quantitative Analysis, 4 : 533–553
- ^ Geman, H, and Shih, YF. 2009. "Modeling Commodity Prices under the CEV Model." The Journal of Alternative Investments 11 (3): 65–84. doi:10.3905/JAI.2009.11.3.065
External links
- Asymptotic Approximations to CEV and SABR Models
- Price and implied volatility under CEV model with closed formulas, Monte-Carlo and Finite Difference Method
- Price and implied volatility of European options in CEV Model delamotte-b.fr