Option pricing under the normal stochastic alpha–beta–rho model with Gaussian quadratures

Abstract

The stochastic alpha–beta–rho (SABR) model has been widely adopted in options trading. In particular, the normal (ˇ D 0) SABR model is a popular model choice for interest rates because it allows negative asset values. The option price and delta under the SABR model are typically obtained via asymptotic implied volatility approximation, but the results are often inaccurate and arbitrageable. Using a recently discovered price transition law, we propose a Gaussian quadrature integration scheme to price options under the normal SABR model. The compound Gaussian quadrature sum over only 49 points can calculate a very accurate price and delta that are arbitrage-free.