I am currently doing a thesis on a class of SDE parameter inference methods and using the SABR model as an example for inference. I want to extend the application to market data. My question is does it make sense to use a forward curve (e.g 3m LIBOR) as the input to my method and infer the SABR parameters (excluding beta) this way and then price derivatives? I only ask as usually it seems derivative prices are the input to the parameter fitting not the forward itself. Thank you in advance.
The SABR model is really Black Scholes in the volatility dimension. This means that volatility is not constant but a stochastic process itself. Hence, σ itself is governed by an SDE, just like the forward rate (as assumed in Black Scholes). That will be the problem with your approach in my opinion. If you look at the forward rate, you miss out on the entire information of the volatility component itself.
Option implied Vols frequently exhibit very pronounced smiles (IVOL for far OTM options are significantly larger compared to ATM), meaning they imply different vol processes. Quoting from Just What You Need To Know About Variance Swaps - JP Morgan Equity Derivatives
For each strike and maturity there is a different implied volatility which can be interpreted as the market’s expectation of future volatility between today and the maturity date in the scenario implied by the strike. For instance, out-of-the money puts are natural hedges against a market dislocation (such as caused by the 9/11 attacks on the World Trade Center) which entail a spike in volatility; the implied volatility of out-of-the money puts is thus higher than in-the-money puts.
Implied vol is also not directly related to historical vol (the vol of the underlying) for at least two reasons:
1 ) Empirically, IV tends to overestimate RV, commonly referred to as Volatility Risk Premium
2 ) IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk.
For example, if you look at GBPUSD, you have Brexit as a major event. Uncertainty meant that IVOL not only anticipated the higher realized / historical vol, but also meant that it was heavily skewed towards OTM Puts (on GBP). Below is a screenshot of the smile on the day of Brexit and during normal times.
Given β, the SABR model parameters describe the shape of the surface and the SABR price correction is much stronger away from the money, resulting in a volatility smile:
Once you have $\beta$,
- $\alpha$ mainly controls the overall height (like CEV),
- $\rho$ (correlation) controls the skew (for set beta) and
- $\nu$ (vol of vol) controls the smile
The resulting vol surface looks like this (taken from this answer):
Nonetheless, it is an interesting question and research idea. You can get market vols from Bloomberg for example (many universities and libraries like the New York Public Library have Windows PCs running Bloomberg Terminal software). You can also look at this answer for a working quantlib code that you can use to fit the SABR model to market quotes. Here is a PySABR code.