Projection on a Quadratic Model by Asymptotic Expansion with an Application to LMM Swaption

We develop a technique of parameter averaging and Markovian projection on a quadratic volatility model based on a term-by-term matching of the asymptotic expansions of option prices in volatilities. In doing so, we revisit the procedure of asymptotic expansion and show that the use of the product formula for iterated Ito integrals leads to a considerable simplification in comparison with the approach currently prevalent in the literature.Results are applied to the classic problem of LIBOR Market Model (LMM) swaption pricing. We confirm numerically that the retention of the quadratic term gives a marked improvement over the standard approximation based on the projection on a displaced diffusion