## The SABR Model (Stochastic Alpha Beta Rho)

The SABR model, introduced by Hagen et al, is a stochastic volatility model which is commonly used for valuing interest rate derivatives. It models the time evolution of some forward rate $$F_t$$, of some maturity $$T$$ and tenor $$L$$, as follows:

$dF_t = \alpha_t F_t^\beta dW_t,$

$d\alpha_t = v \alpha_t dZ_t,$

where $$W_t$$ and $$Z_t$$ are two Wiener processes with constant correlation $$\rho$$.

Unlike local volatility models, the model reproduces the correct smile dynamics, so it can be used for hedging. The SABR model is popular because it is simpler than some other stochastic volatility models, and allows an approximate pricing formula, in terms of the SABR model parameters, which gives the price in terms of the implied Black volatility. That is, the implied volatility that when inserted into Black’s model reproduces the price generated by the SABR model. This makes the model much easier to calibrate, and in general it fits the market well. The Hagen formula is straight forward but long, so we don’t reproduce it here.

The SABR model quickly became the market standard for vanilla interest rate derivatives like caps, floors and swaptions.

The first equation above shows that the forward $$F_t$$ will remain positive under the evolution of the equation. Thus, the SABR model needs to be modified if zero and negative rates are considered possible.

## Fitting the model

The SABR model is fitted for a single forward rate, with some expiry T and some tenor L. As mentioned already, a direct formula, in terms of the SABR model parameters, gives you a good approximation for the Black implied vol for any given strike. Therefore, using a set of strikes for which there exist market prices (and therefore implied vols), one can do a least squares fitting which minimises the deviation of the SABR approximate Black vols from the market implied vols. That is, one wishes to find SABR parameters which minimise

$\sum_{i} (\sigma_{K_i}^{SABR} – \sigma_{K_i}^{BS}),$

where the sum is over the set of strikes for a given expiry. The SABR volatilities are computed from the SABR parameters using the Hagen implied volatility formula, avoiding the need to numerically simulate the SABR model. Since $$\beta$$ and $$\rho$$ both affect the slope of the volatility smile, it’s common to fix $$\beta$$ according to some view of the market, and fit only the remaining three parameters using this method.

This procedure can be repeated for all required expiries and tenors for which there are market prices.

Finally, if one wishes to price an option on a forward rate of an expiry and tenor for which there is no analogue in the market, one can construct a set of SABR parameters by interpolating between the SABR parameters of nearby expiries and tenors. Alternatively, one can interpolate the implied volatilities directly.

## Swaptions and caps/floors

For swaptions, one uses the swap rate as the underlying. That is, the rate that makes the swap value equal to zero. For caps and floors, one must first do a bootstrapping process to obtain the individual caplet/floorlet vols from the trading caps and floors.

## The parameters alpha, beta and rho.

The variable $$\alpha_t$$ represents the volatility of the forward process. The first SABR parameter $$\alpha$$ is a non-negative parameter which is defined as the initial value of $$\alpha_t$$ at time $$t=0$$. It’s effect is approximately a parallel shift of the volatility smile vertically.

The parameter $$\beta$$ satisfies $$0 \leq \beta \leq 1$$. As $$\beta$$ increases, the slope of the volatility smile decreases. Note that for $$\beta$$ close to zero, the SABR model is close to a normal, rather than lognormal, model.

As mentioned already, $$\rho$$ represents the correlation between changes in the underlying and changes in it’s volatility. Like $$\beta$$, the effect of $$\rho$$ is to change the slope of the volatility smile.

The parameter $$v$$ is the volatility of the volatility. A higher value for $$v$$ gives a greater curvature to the volatility smile.

## Limitations of the SABR model

The SABR model has some limitations worth noting. Firstly, as it only models a single forward rate, it should not be used to price derivatives whose valuation depends on more than one forward rate. This is because forward rates with nearby maturities and tenors are not independent. However, note that swaptions are nonetheless commonly priced using the SABR model by modelling the swap rate, instead of the constituent swaplets. Secondly, the equation lacks a mean reversion term, limiting its ability to accurately model interest rates which tend to be eventually pulled back to normal values. Finally, the convenient Black implied vol formula can become inaccurate under certain circumstances.

For pricing non-vanilla derivatives which depend on multiple forward rates (of various expiries and tenors), the SABR model must be modified so that forward rates of nearby tenors and expiries are not independent. See the SABR libor market model.