Constant Elasticity of Variance (CEV) Lab

Q: What is the Constant Elasticity of Variance (CEV) model?

The Constant Elasticity of Variance (CEV) model, introduced by John C. Cox in 1975, is a continuous-time local volatility diffusion model: dS_t = (r - q) S_t dt + sigma S_t^beta dW_t. The instantaneous local volatility sigma(S) = sigma * S^(beta - 1) depends directly on the asset price level, allowing the model to generate realistic implied volatility skews.

Q: How does the elasticity parameter beta generate an implied volatility skew?

When beta is less than 1.0, the local volatility function sigma(S) = sigma * S^(beta - 1) increases as the asset price drops (d sigma / dS < 0). This models Fischer Black's (1976) leverage effect—as firm equity falls, the debt-to-equity ratio increases, amplifying equity volatility. This generates a downward-sloping implied volatility skew where out-of-the-money puts trade at higher implied volatilities than calls.

Q: What happens when beta equals 1.0, 0.5, or 0.0 in the CEV model?

When beta = 1.0, the diffusion term is sigma * S_t dW_t, which is Black-Scholes geometric Brownian motion with constant volatility. When beta = 0.5, the diffusion term is sigma * sqrt(S_t) dW_t, matching the Cox-Ingersoll-Ross (CIR 1985) square-root process. When beta = 0.0, the diffusion is sigma dW_t, representing Louis Bachelier's (1900) arithmetic Brownian motion.

Q: Why does the CEV model have an absorbing boundary at zero and default probability?

For beta < 1.0, Feller's boundary classification demonstrates that the origin S = 0 acts as an absorbing boundary that can be reached in finite time with strictly positive probability. When the asset price hits zero, the firm enters bankruptcy and the stock remains at zero forever. The cumulative default probability by expiration T is evaluated analytically via the regularized incomplete gamma function.

Q: How does the CEV model serve as the foundation for the SABR model?

The celebrated SABR model (Hagan et al. 2002) adopts the CEV diffusion backbone dF_t = sigma_t F_t^beta dW_t as its local volatility driver while introducing a second stochastic differential equation for the volatility coefficient sigma_t. The CEV beta parameter governs the slope of the implied volatility backbone across different forward rate regimes.

Model John C. Cox's (1975) local volatility diffusion framework. Price European options using analytical non-central chi-square distributions, evaluate endogenous credit default probability at the absorbing boundary, and analyze leverage skews.

CEV Call Price

$0.00

European Call

CEV Put Price

$0.00

European Put

vs Black-Scholes

+$0.00

Model Spread

ATM Implied Vol

0.00%

Inverted Black-Scholes

Default Risk

0.000%

Absorption at S=0

Variance Elasticity

-1.20

2(β − 1)

Option & Underlying Parameters

Spot Price S₀ ($)

Strike Price K ($)

Expiry T (Yrs)

Rate r (%)

Div Yield q (%)

CEV Diffusion Parameters

Normalized ATM Volatility σ_ATM (%)

Instantaneous annualized volatility when asset price equals S₀.

Elasticity Exponent β (Beta)

• β < 1.0: Leverage effect (volatility rises as stock falls)
• β = 1.0: Lognormal Black-Scholes (flat volatility)
• β = 0.5: Cox-Ingersoll-Ross square-root diffusion
• β > 1.0: Inverse leverage (commodity upside spikes)

CEV Local Volatility Visualizer

Implied Volatility Smile / Skew: Blue curve charts CEV implied volatility across strikes K ∈ [0.7 S₀, 1.3 S₀]. Dashed gray line indicates flat Black-Scholes benchmark volatility.

Sankaran non-central χ² CDF & regularized upper incomplete Γ default absorption.

Analytical Model Decomposition

Component	Value	Notes
CEV European Call	$0.00	Non-central χ² analytical formula
CEV European Put	$0.00	Put-call parity with absorption
Black-Scholes Call (β = 1)	$0.00	Constant volatility baseline
Black-Scholes Put (β = 1)	$0.00	Constant volatility baseline
Model Spread (ΔC)	+$0.00	CEV Call − BS Call
Local Vol Scale (σ)	0.0000	σ_ATM × S₀^1−β
Default Probability (P(S_T=0))	0.0000%	Absorption at origin barrier
χ² Degrees of Freedom	0.00	2 + 1/(1−β) parameter
Put-Call Parity Error	< $0.000001	Exact Martingale Match

Risk-Neutral Sensitivity Greeks

Delta (Δ) 0.0000

∂C / ∂S₀ — Share hedge ratio

Gamma (Γ) 0.00000

∂²C / ∂S₀² — Convexity curvature

Vega (Ν) $0.00

∂C / ∂σ_ATM — Dollar per 1% vol

Theta (Θ) $0.00

∂C / ∂t — 1-day calendar decay

Rho (Ρ) $0.00

∂C / ∂r — Dollar change per 1% interest rate shift

Moneyness & Implied Volatility Smile Table

Strike (Moneyness)	CEV Call	CEV Put	CEV IV (%)	Black-Scholes IV (%)	Skew Spread

Sensitivity 1: Elasticity β vs. ATM Volatility

Evaluates 90% OTM Put Implied Volatility (%). Demonstrates how lower elasticity β steepens downside skew across volatility regimes.

Sensitivity 2: Strike Price K vs. Expiry Tenor T

Evaluates CEV Call Option Price ($) across strike moneyness and maturity horizons.

Theoretical Foundations: The Constant Elasticity of Variance Model

The Constant Elasticity of Variance (CEV) model was introduced by John C. Cox (1975) to resolve the empirical observation that equity volatility is not constant, but instead exhibits an inverse relationship with the underlying stock price. Under the risk-neutral pricing measure $\mathbb{Q}$, the asset price $S_t$ follows the stochastic differential equation:

$$dS_t = (r - q) S_t dt + \sigma S_t^\beta dW_t$$

Dividing both sides by $S_t$ reveals the local volatility function:

$$\frac{dS_t}{S_t} = (r - q) dt + \sigma(S_t) dW_t, \quad \sigma(S_t) = \sigma S_t^{\beta - 1}$$

The sensitivity of volatility to the asset price is captured by the elasticity of variance:

$$\frac{d \ln(\sigma(S)^2)}{d \ln S} = 2(\beta - 1)$$

This elasticity parameter $\beta$ dictates the structural behavior of the asset and its derivatives:

$\beta < 1$ (Leverage Effect): As the stock price drops, the instantaneous local volatility increases ($\partial \sigma(S) / \partial S < 0$). This directly models the economic leverage effect documented by Fischer Black (1976)—as equity value declines relative to debt, financial leverage and equity volatility rise. This creates a downward-sloping implied volatility skew characteristic of equity indices.
$\beta = 1$ (Black-Scholes Baseline): The diffusion term simplifies to $\sigma S_t dW_t$, recovering classical geometric Brownian motion with a flat implied volatility surface.
$\beta = 0.5$ (Square-Root Diffusion): The diffusion term is $\sigma \sqrt{S_t} dW_t$, identical to the Cox-Ingersoll-Ross (CIR 1985) short-rate process.
$\beta > 1$ (Inverse Leverage): Volatility increases as the asset price rises ($\partial \sigma(S) / \partial S > 0$), generating upward-sloping implied volatility smiles common in agricultural and energy commodities.

Absorbing Boundary & Endogenous Default Risk

A unique feature of the CEV model with $\beta < 1$ is that the origin $S=0$ is an absorbing boundary. Unlike geometric Brownian motion (which never reaches zero because $\ln S_t \to -\infty$), the CEV asset price can hit zero in finite time with strictly positive probability. When $S=0$, the firm defaults, equity becomes worthless, and the price remains at zero forever. The cumulative default probability by expiration $T$ is evaluated analytically:

$$P(\text{Default by } T) = P(S_T = 0) = Q\left(\frac{1}{2(1 - \beta)}, \frac{x}{2}\right) = \frac{\Gamma\left(\frac{1}{2(1-\beta)}, \frac{x}{2}\right)}{\Gamma\left(\frac{1}{2(1-\beta)}\right)}$$

where $Q(a, z)$ is the regularized upper incomplete gamma function.

Analytical Option Pricing via Non-Central Chi-Square Distributions

Cox (1975) and Mark Schroder (1989) derived the exact closed-form solution for European Call options under CEV using the cumulative distribution function of the non-central chi-square distribution $\chi^2(z; \nu, \lambda)$:

$$C(S_0, K, T) = S_0 e^{-q T} \left[ 1 - \chi^2\left(y; 2 + \frac{1}{1 - \beta}, x\right) \right] - K e^{-r T} \chi^2\left(x; \frac{1}{1 - \beta}, y\right)$$ $$a = \frac{r - q}{\sigma^2 (1 - \beta) (e^{2(r - q)(1 - \beta) T} - 1)}, \quad x = 2 a S_0^{2(1 - \beta)} e^{2(r - q)(1 - \beta) T}, \quad y = 2 a K^{2(1 - \beta)}$$

CEV Model Mastery Quiz

Test your quantitative derivatives understanding of local volatility dynamics and elasticity.

1. Why does the CEV model with elasticity parameter β < 1 generate an implied volatility skew?

2. What well-known stochastic process corresponds to the CEV diffusion when β = 0.50?

3. What is the economic significance of the absorbing boundary at S = 0 when β < 1?

4. How does the CEV model relate to the widely traded SABR model in quantitative finance?

Frequently Asked Questions