SABR Smile Dynamics and Hedging for Interest Rate Swaptions

# Sabr vs Heston for IR swaptions


# Sabr vs Heston for IR swaptions












Why is SABR considered the model of choice for swaptions? Is the Heston model not suitable? Does Heston produce unrealistic dynamics with respect to the swaption market?

## Answer by LePiddu (score 4)

https://quant.stackexchange.com/a/65876

TL:DR With SABR you can still use your favorite Black-76 / Bachelier formula and improve your hedging strategy pretty much for free. And when trading options there's no good price without a good hedging.

More in details, interest rates derivative desks don't price vanilla instruments using the SABR model only in part due to the absence of analytical formulae, and that's because finite differences solutions to the PDE are pretty fast and precise.

When pricing european vanilla options (swaptions, caps/floors), they use Black-76 Formula or Bachelier Formula. SABR is (mostly) used as a model-backed interpolation of the implied volatility curve (the so-called smile or skew). What you gain is smile-coherent hedging ratios, which is what is actually missing from those "simple" models.

Let me be a bit more formal, let $C(\tau, F, K, \sigma)$ be the option price on a forward (rate or price, doesn't matter) $F$ with strike $K$, instantaneous volatility $\sigma$ and time to expiry $\tau$.

To hedge option's market-risks you have to set up your portfolio buying/selling a "delta amount" of the underlying

$$\Delta_{blk} = \frac{\delta C(F)}{\delta F}$$

If you admit any kind of dependency of the instantaneous volatility on the underlying, that is $\sigma(F)$, the actual delta amount becomes

$$\Delta_{SABR} = \frac{\delta C(F,\sigma(F))}{\delta F} = \underbrace{\frac{\delta C(F)}{\delta F}}_{\Delta_{blk}} + \underbrace{\frac{\delta C(\sigma(F))}{\delta \sigma}\frac{\delta \sigma(F)}{\delta F}}_{\text{Shadow Delta}}$$

Notice how in the basic setup (Black-75 / Bachelier) $\frac{\delta \sigma(F)}{\delta F}=0$: you end up with the "wrong amount" of underlying in your hedging portfolio and therefore increasing the variance of your profit and loss distribution.

With the SABR model you have a clear parametrization of $\sigma(F)$ and therefore you can obtain $\frac{\delta \sigma(F)}{\delta F}$, correcting you hedging amount.

In the same way, you can compute the correct "vega" amount.

Moreover, SABR parameters are highly interpretable in terms of movements of the implied volatility curve, therefore you can compute the exposure against tilting / bending of such curve and improve the hedging even more.

Finally SABR calibration is a matter of instants and you get an extra-feature: you can follow in real-time the changes in the level of implied volatility by calibrating the alpha parameter to the At-the-money quotes. Such quotes are tipically very liquid and fast moving with respect to out-of-the-money smile wings (especially in swaptions market) and this is as fast as computing the root of a 3rd degree polynomial (or even faster in some approximations of the SABR implied volatility).

References

Oblój, J. (2007). Fine-tune your smile: Correction to Hagan et al. arXiv preprint arXiv:0708.0998.

Bartlett, B. (2006). Hedging under SABR model. Wilmott magazine, 4, 2-4.

Hagan, P. S., Kumar, D., Lesniewski, A. S., & Woodward, D. E. (2002). Managing smile risk. The Best of Wilmott, 1, 249-296.