What is the SABR model?
The SABR model (pronounced as in light saber) is a popular stochastic volatility model. A somewhat more general version than the one we discuss here was introduced by P. Hagan and co-authors in 2002, with applications to interest rate-related derivatives. The model became popular because a (quite difficult) small-time analysis resulted in some easy-to-apply, approximate, “smile” formulas that could fit the authors’ target markets.
While equity applications are rare, the model makes mathematical sense for equities, where the stochastic process pair is . Here is a stock price and is the associated stochastic volatility. Using that equity language, the SABR model, as an SDE (stochastic differential equation) system, reads
In equation (1), is any real number, and is the “volatility of volatility”. The latter can be set equal to one without loss of generality, so we’ll do so for the remainder of this post. Although any real makes mathematical sense, fits to market data will almost always have , so we’ll stick to that range also. Also is a correlation parameter. Finally, is the first time, if any, that the stock price hits 0.
The stock price can hit 0 with a positive probability whenever is strictly less than 1. What happens after that requires a boundary condition. Of course, for equities, to avoid an arbitrage opportunity, once a stock price hits zero, it must stay there (unless the company is recapitalized). So, an additional specification of the equity version of the model is that for all .
For the original fixed income applications, the underlying security is a (forward) interest rate and other boundary conditions are possible.
By a “solution” to equation (1), I mean a solution for the associated transition probability density. Once you have that, option values, for example, are found by an integration. Exact solutions are known for the following special cases:
- Normal SABR model: with
- Lognormal SABR model: with any
The Normal SABR model solution is found by adapting a famous formula for the transition density found by Henry P. McKean in 1970. For that case, it makes more sense to switch gears slightly in the notation and use instead of , now the location of a “particle” which can hit 0 and keep on going to arbitrarily negative values. With that new notation, the Normal SABR model process is:
Now and are two independent Brownian motions. Equation (2) also describes a generalized type of Brownian motion, namely Brownian motion on a two-dimensional hyperbolic space.
Suppose we start the particle at and . Then, the probability of finding the particle at subsequent times is given by Mckean’s formula for . If we let t range from 0.01 to 1, we can plot McKean’s formula to find the following evolution:
If you watch closely, you’ll see that the probability density starts with a spike near and then spreads out in the X direction. In the direction, mass begins to accumulate near That’s due to lack of mean reversion in the volatility process and is an ‘unphysical’ aspect of the model.
Now that we’ve talked about it so much, you may be curious as to what this mysterious formula looks like. Here it is:
To adapt the McKean solution for an equity model, you can use the well-known “method of images”. The method will ensure that the associated equity process , which starts at , is absorbed should it hit
(There’s much more to the model than we have touched upon, and we’ll certainly return to it in future blog posts. This post has been adapted from material in Ch. 8 (“A Closer Look at the SABR Model”) from “Option Valuation under Stochastic Volatility II”).