SABR volatility model

Overview

In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick S. Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.

Dynamics

The SABR model describes a single forward

F

{\displaystyle F}

, such as a LIBOR forward rate, a forward swap rate, or a forward stock price. This is one of the standards in market used by market participants to quote volatilities. The volatility of the forward

F

{\displaystyle F}

is described by a parameter

σ

{\displaystyle \sigma }

. SABR is a dynamic model in which both

F

{\displaystyle F}

and

σ

{\displaystyle \sigma }

are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations:

d

F

t

=

Asymptotic solution

We consider a European option (say, a call) on the forward

F

{\displaystyle F}

struck at

K

{\displaystyle K}

, which expires

T

{\displaystyle T}

years from now. The value of this option is equal to the suitably discounted expected value of the payoff

max (

F

T

− K ,

0 )

{\displaystyle \max(F_{T}-K,\;0)}

under the probability distribution of the process

F

t

{\displaystyle F_{t}}

SABR for the negative rates

A SABR model extension for negative interest rates that has gained popularity in recent years is the shifted SABR model, where the shifted forward rate is assumed to follow a SABR process

d

F

t

=

σ

t

(

F

t

+ s

)

β

d

W

t

,

{\displaystyle dF_{t}=\sigma _{t}(F_{t}+s)^{\beta }\,dW_{t},}

d

σ

Arbitrage problem in the implied volatility formula

Although the asymptotic solution is very easy to implement, the density implied by the approximation is not always arbitrage-free, especially not for very low strikes (it becomes negative or the density does not integrate to one). One possibility to "fix" the formula is use the stochastic collocation method and to project the corresponding implied, ill-posed, model on a polynomial of an arbitrage-free variables, e.g. normal. This will guarantee equality in probability at the collocation points while the generated density is arbitrage-free. Using the projection method analytic European option prices are available and the implied volatilities stay very close to those initially obtained by the asymptotic formula. Another possibility is to rely on a fast and robust PDE solver on an equivalent expansion of the forward PDE, that preserves numerically the zero-th and first moment, thus guaranteeing the absence of arbitrage.

Extensions

The SABR model can be extended by assuming its parameters to be time-dependent. This however complicates the calibration procedure. An advanced calibration method of the time-dependent SABR model is based on so-called "effective parameters". Alternatively, Guerrero and Orlando show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Explicit solutions obtained by said techniques are comparable to traditional Monte Carlo simulations allowing for shorter time in numerical computations.

Simulation

As the stochastic volatility process follows a geometric Brownian motion, its exact simulation is straightforward. However, the simulation of the forward asset process is not a trivial task. Taylor-based simulation schemes are typically considered, like Euler–Maruyama or Milstein. Recently, novel methods have been proposed for the almost exact Monte Carlo simulation of the SABR model. Extensive studies for SABR model have recently been considered. For the normal SABR model (

β = 0

{\displaystyle \beta =0}

with no boundary condition at

F = 0

{\displaystyle F=0}

), a closed-form simulation method is known.

See also

Volatility (finance) Stochastic volatility Risk-neutral measure

Further reading

Hagan, Patrick; Lesniewski, Andrew; Woodward, Diana (2005-03-22). "Probability Distribution in the SABR Model of Stochastic Volatility" (PDF). Archived from the original (PDF) on 2021-03-08. Retrieved 2022-04-30. Bartlett, Bruce (February 2006). "Hedging under SABR Model" (PDF). Wilmott. Archived from the original (PDF) on 2020-12-30. Retrieved 2022-04-30. Hagan, Patrick; Lesniewski, Andrew (2008-04-30). "LIBOR market model with SABR style stochastic volatility" (PDF). Archived from the original (PDF) on 2022-03-03. Retrieved 2022-04-30. Hagan, Patrick S.; Kumar, Deep; Lesniewski, Andrew S.; Woodward, Diana E. (2014-01-29). "Arbitrage Free SABR". Wilmott. Vol. 2014, no. 69. pp. 60–75. doi:10.1002/wilm.10290. Retrieved 2022-04-30. Obloj, Jan (2008-03-18). "Fine Tune Your Smile – Correction to Hagan et al". arXiv:0708.0998 [q-fin.CP]. West, Graeme. "A summary of the approaches to the SABR model for equity derivatives smile". Riskworx. Archived from the original on 2015-09-14. Retrieved 2022-04-30. Henry-Labordere, Pierre (2005-02-15). "Unifying the BGM and SABR models: a short ride in hyperbolic geometry". arXiv:physics/0602102. Jordan, Richard; Tier, Charles (2011-05-17). "Asymptotic Approximations to CEV and SABR Models". SSRN 1850709. "SABR calibration". 2012-12-26. Archived from the original on 2016-05-27. Retrieved 2022-04-30. Antonov, Alexandre; Spector, Michael (2012-03-23). "Advanced Analytics for the SABR Model". SSRN 2026350. Antonov, Alexandre; Konikov, Michael; Spector, Michael (2019-05-02). Modern SABR Analytics: Formulas and Insights for Quants, Former Physicists and Mathematicians (Springer Briefs in Quantitative Finance) 1st ed. doi:10.1007/978-3-030-10656-0. ISBN 978-3-030-10655-3. ISSN 2192-7014. S2CID 182484805. / Press Release, New York, NY – June 24, 2019 Gulisashvili, Archil; Horvath, Blanka; Jacquier, Antoine (Jack) (2016-11-22). "Mass at Zero in the Uncorrelated SABR Model and Implied Volatility Asymptotics". SSRN 2563510.