# The SABR Model

It was introduced by Hagan and al in 2002.

If the ß = 1 version of the model is used, it suffers from none of the problems of Heston, but it comes with some of its own.

These though are associated with how the model tends to be used rather than any fundamental flaw.

It is easier to write down the model in terms of the forward to a fixed expiry rather than in terms of a spot level. In terms of F, the BS process becomes as a simple application of Ito shows.

SABR is practically the simplest extension of BS to a stochastic volatility model. It has SDEs: The model is named after three of its parameters: it is the stochastic- (SABR) model.

Like the Heston model, the SABR model has a vol-of-vol parameter v that controls the convexity of the IV smile, and a correlation parameter that controls the skew.

There is no mean reversion, and this is a drawback compared to Heston as it means that when instantaneous volatility follows a path in which it becomes very large, it is likely to stay large.

Ordinary, say in equity and FX markets, we choose ß = 1 so that the process is approximately log-normal  in the limit of small vol-of-vol.

On the other hand, IR traders are fond of SABR and do make use of the ß parameter (usually ß = 0,5).

If we use ß not equal to 1, we could rewrite the process as: so that the instantaneous "log-normal vol" is In this way we add some dependence between the forward level to the volatility, and therefore ß also impacts the skew.

We can to some extent play off ß and against one another.

However, ß will also impact the convexity of the IV smile.

Caution is required in understanding the process when ß is different from 1. In the special case of v = 0, for which the volatility is no longer stochastic, the process is known as constant elasticity of variance (CEV)

This process is well studied and the following results are provided by Andersen and Andreasen in 1998:

1) For ß ≥ 0,5 --> the SDE has a unique solution

2) For ß between 0 and 1 --> the process can reach Ft = 0 but never go negative.

3) For ß ≥ 1 --> the process can never reach  Ft = 0.

4) For ß = 0 --> the process is normal, and therefore Ft can go negative.

5) For ß between 0 and 0,5 --> the SDE only has a unique solution if one adds a boundary condition at  Ft = 0. For the process to be arbitrage free, the boundary condition must be that when Ft hits zero it stays there.

SABR is popular because Hagan et al. (2002) were able to provide an approximate solution that is valid in the limit of small time to expiry T. We provide their result in the case ß = 1.

Given the price of a call with strike K, the IV is given by the SABR formula: where For our purpose, the ß = 1 version of the model is perfectly good. We will add later a local component into the volatility that would allow for dynamics in which ß is different from 1. For now, we will continue our discussion of the merits of SABR to ß = 1, and call it log-normal SABR.

Log-normal SABR has a volatility process that cannot hit zero and therefore it does not suffer from the numerical difficulties associated with Heston.

Unlike the ß < 1 version of SABR, log-normal SABR also has a spot that stays positive. Therefore the only drawback with the process itself is the lack of mean reversion, allowing paths in which instantaneous volatilities become very large.

This is not such a problem. We can simply add in a mean reversion term to obtain the  By including mean reversion, we lose the analytic approximation for the IV smile in terms of our stochastic volatility parameters.

We have seen that the SABR or itself is attractive.

It is the asymptotic IV formula that causes problems. The approximation is only valid at short expiries, and furthermore, it can imply negative probability densities at strikes that are far away from ATM.

These are problems for people who want to use the SABR formula as a method to define or interpolate the IV smile.

Forrtunately, in this book, we are concerned with smile modelling, and these issues will not cause us any problems.