Principle:Avhz RustQuant SABR Volatility

Knowledge Sources	Hagan, Kumar, Lesniewski, Woodward - Managing Smile Risk (2002)
Domains	Volatility_Modeling, Derivatives_Pricing
Last Updated	2026-02-07 21:00 GMT

Overview

The SABR (Stochastic Alpha Beta Rho) volatility model for computing implied volatilities that capture the market's volatility smile and skew.

Description

The SABR Volatility model in RustQuant implements the Hagan et al. (2002) approximation formula for implied volatility. The model is represented by the Sabr02 struct containing five parameters:

• f -- The forward price of the underlying asset.
• alpha -- The initial volatility level ($\alpha$).
• beta -- The CEV (constant elasticity of variance) exponent ($\beta \in [0,1]$), controlling the backbone of the smile.
• rho -- The correlation between the asset and volatility processes ($\rho \in [-1,1]$).
• nu -- The volatility of volatility ($\nu$).

Special cases of $\beta$:

• $\beta =0$ corresponds to the stochastic normal model.
• $\beta =0.5$ corresponds to the stochastic CIR model.
• $\beta =1$ corresponds to the stochastic lognormal model.

The library provides two main capabilities:

• Volatility computation -- The volatility(k, t) method computes the SABR implied volatility for a given strike k and time to expiry t, suitable for input to the Black (1976) model.
• Model calibration -- The fit(volatilities, strikes, t) method calibrates the model parameters ($\alpha$, $\rho$, $\nu$) to market data using particle swarm optimization via the argmin crate. The $\beta$ parameter is held fixed during calibration.

Additionally, a SABR stochastic process struct is available in the stochastics crate, defining the model parameters as ModelParameter types for potential use in Monte Carlo simulation.

Usage

Use the SABR Volatility model when pricing options across a range of strikes where the Black-Scholes assumption of constant volatility is insufficient. The model is the industry standard for interest rate derivatives, foreign exchange options, and equity index options where capturing the volatility smile and skew is essential for accurate pricing and risk management.

Theoretical Basis

The SABR model describes the joint dynamics of the forward price $F$ and its volatility $\sigma$:

$d{F}_t={\sigma }_t{F}_t^{\beta }d{W}_1$

$d{\sigma }_t=\nu {\sigma }_{t}d{W}_2$

with $d{W}_{1}d{W}_2=\rho dt$.

The Hagan et al. (2002) approximation for the implied Black volatility is:

${\sigma }_B(K,T)=\frac{z}{\chi (z)}\cdot \frac{\alpha [1+(\frac{(1-\beta {)}^2{\alpha }^2}{24(FK{)}^{1-\beta }}+\frac{\rho \beta \alpha \nu }{4(FK{)}^{(1-\beta )/2}}+\frac{(2-3{\rho }^2){\nu }^2}{24})T]}{(FK{)}^{(1-\beta )/2}[1+\frac{(1-\beta {)}^2}{24}{\ln }^2\frac{F}{K}+\frac{(1-\beta {)}^4}{1920}{\ln }^4\frac{F}{K}]}$

where:

$z=\frac{(FK{)}^{(1-\beta )/2}\ln (F/K)}{\alpha }$ (absorbed into the coefficient/chi ratio in the implementation)

$\chi (z)=\frac{\ln (\sqrt{1-2\rho z+z^2}+z-\rho )}{1-\rho }$

Calibration minimizes the root-mean-square error between model and market volatilities using particle swarm optimization with bounds $\alpha \in (\epsilon ,\mathrm{\infty })$, $\rho \in [-1,1]$, and $\nu \in [-1,\mathrm{\infty })$.

Related Pages

Implemented By

• Implementation:Avhz_RustQuant_SABR_Model