Constant elasticity of variance model

In mathematical finance, the CEV or constant elasticity of variance model is a stochastic volatility model, although technically it would be classed more precisely as a local 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:

\mathrm {d} S_{t}=\mu S_{t}\mathrm {d} t+\sigma S_{t}^{\gamma }\mathrm {d} W_{t}

in which S is the spot price, t is time, and μ is a parameter characterising the drift, σ and γ are volatility parameters, and W is a Brownian motion.^[2] In terms of general notation for a local volatility model, written as

\mathrm {d} S_{t}=\mu S_{t}\mathrm {d} t+v(t,S_{t})S_{t}\mathrm {d} W_{t}

we can write the price return volatility as

v(t,S_{t})=\sigma S_{t}^{\gamma -1}

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.^[3] Conversely, in commodity markets, we often observe $\gamma >1$ ,^[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 $\gamma =1$ this model becomes a geometric Brownian motion as in the Black-Scholes model, whereas if $\gamma =0$ and either $\mu =0$ or the drift $\mu S$ is replaced by $\mu$ , this model becomes an arithmetic Brownian motion, the model which was proposed by Louis Bachelier in his PhD Thesis "The Theory of Speculation", known as Bachelier model.

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 Unpublished draft, Stanford University, 1975.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

Derivatives market

Derivative (finance)

Options

Terms	Delta neutral Exercise Expiration Moneyness Open interest Pin risk Risk-free interest rate Strike price Synthetic position the Greeks Volatility
Vanillas	American Bond option Call Employee stock option European Fixed income FX Option styles Put Warrants
Exotics	Asian Barrier Basket Binary Callable bull/bear contract Chooser Cliquet Commodore Compound Forward start Interest rate Lookback Mountain range Rainbow Spread Swaption
Strategies	Backspread Box spread Butterfly Calendar spread Collar Condor Covered option Credit spread Debit spread Diagonal spread Fence Intermarket spread Iron butterfly Iron condor Jelly roll Ladder Naked option Straddle Strangle Protective option Ratio spread Risk reversal Vertical spread (Bear, Bull)
Valuation	Bachelier Binomial Black Black–Scholes (equation) Finite difference Garman–Kohlhagen Heston Lattices Margrabe Put–call parity MC Simulation Real options Trinomial Vanna–Volga

Swaps

Exotic derivatives

Other derivatives

Market issues

Business portal

v t e Volatility
Modelling volatility	Implied volatility Volatility smile Volatility clustering Local volatility Stochastic volatility Jump-diffusion models ARCH and GARCH
Trading volatility	Volatility arbitrage Straddle Volatility swap IVX VIX

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category