On the Stochastic Alpha Beta Rho Model and Hamiltonian Monte Carlo Techniques

A popular model for capturing the volatility skew in the interest-rate market is the Stochastic Alpha Beta Rho (SABR) model. This model was developed to address the deficiencies observed with the dynamics of the Dupire local volatility model. In this chapter, we apply a first-in-literature Bayesian inference framework to infer the interest-rate volatility skew using simulated and ZAR market swaption prices modeled using the SABR model. With the Bayesian framework, one can obtain confidence bands for any predictions made using the model and uncertainties in the parameters of the calibrated SABR model. Using Markov Chain Monte Carlo (MCMC) techniques, being the Metropolis-Adjusted Langevin Algorithm, Hamiltonian Monte Carlo, and the Separable Shadow Hamiltonian Hybrid Monte Carlo, we calibrate the SABR model using swaption implied normal volatility data. The SABR model parameters are produced as a distribution using our method. The empirical findings demonstrate that our method can precisely price the interest-rate volatility skew on both simulated and ZAR swaption data, with the added advantage of producing a distribution of the model predictions and parameters. We find that the MCMC algorithms are fairly confident (as measured by the width of the confidence bands) of the skew around the at-the-money point but get less confident as we move toward the deep-in and deep-out of the money options. Furthermore, the results show that the methods are more confident in the skew’s right-wing predictions than the skew’s left wing.