SABR volatility model
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.
The SABR model describes a single forward , such as a LIBOR forward rate, a forward swap rate, or a forward stock price. The volatility of the forward is described by a parameter . SABR is a dynamic model in which both and are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations:
with the prescribed time zero (currently observed) values and . Here, and are two correlated Wiener processes with correlation coefficient :
The constant parameters satisfy the conditions .
The above dynamics is a stochastic version of the CEV model with the skewness parameter : in fact, it reduces to the CEV model if The parameter is often referred to as the volvol, and its meaning is that of the lognormal volatility of the volatility parameter .
We consider a European option (say, a call) on the forward struck at , which expires years from now. The value of this option is equal to the suitably discounted expected value of the payoff under the probability distribution of the process .
Except for the special cases of and , no closed form expression for this probability distribution is known. The general case can be solved approximately by means of an asymptotic expansion in the parameter . Under typical market conditions, this parameter is small and the approximate solution is actually quite accurate. Also significantly, this solution has a rather simple functional form, is very easy to implement in computer code, and lends itself well to risk management of large portfolios of options in real time.
It is convenient to express the solution in terms of the implied volatility of the option. Namely, we force the SABR model price of the option into the form of the Black model valuation formula. Then the implied volatility, which is the value of the lognormal volatility parameter in Black's model that forces it to match the SABR price, is approximately given by:
where, for clarity, we have set . The value denotes a conveniently chosen midpoint between and (such as the geometric average or the arithmetic average ). We have also set
The function entering the formula above is given by
Alternatively, one can express the SABR price in terms of the normal Black's model. Then the implied normal volatility can be asymptotically computed by means of the following expression:
It is worth noting that the normal SABR implied volatility is generally somewhat more accurate than the lognormal implied volatility.
SABR for the negative rates
The SABR model can be modified to cover also Negative interest rate:
for and a free boundary condition for . Its exact solution for the zero correlation as well as an efficient approximation for a general case are available.
Another SABR model extension for negative rates that gained popularity in the recent years is the shifted SABR model, where shifted forward rate is assumed to follow a SABR process
for some positive shift . An obvious drawback of this approach is the a priori selection of the shift, and the resulting possibility of needing to adjust this shift further once the rates go still more negative.
- Antonov, Alexandre and Konikov, Michael and Spector, Michael, The Free Boundary SABR: Natural Extension to Negative Rates (January 28, 2015). Available at SSRN: http://ssrn.com/abstract=2557046
- Managing Smile Risk, P. Hagan et al. – The original paper introducing the SABR model.
- Probability Distribution in the SABR Model of Stochastic Volatility, P. Hagan et al. - Introduced the normal SABR model, heat kernel expansion, and asymptotic probability distribution.
- Hedging under SABR Model, B. Bartlett – Refined risk management under the SABR model.
- LIBOR market model with SABR style stochastic volatility, P. Hagan and A. Lesniewski – LMM extension of SABR for term structure modeling.
- Arbitrage Free SABR, P. Hagan et al. - Refined treatment of near zero forwards.
- Fine Tune Your Smile – Correction to Hagan et al.
- A summary of the approaches to the SABR model for equity derivatives smile
- Unifying the BGM and SABR models: a short ride in hyperbolic geometry
- Asymptotic Approximations to CEV and SABR Models
- Test SABR (with calibration) online
- SABR calibration
- Advanced Analytics for the SABR Model - Includes exact formula for zero correlation case
- Small-Strike Implied Volatility Expansion in the SABR Model - Arbitrage-free asymptotic formula for small strikes and for long-dated options
- The Free Boundary SABR: Natural Extension to Negative Rates - SABR for the negative rates