A GARCH Parameterization of the Volatility Surface

Slicing the Swaption Cube

“…where F(t, T n−1 , T n ) is the forward rate for a deposit between T n−1 and T n (possibly forward LIBOR), as seen from time t. The shorter the tenor, the less terms are contained in this sum, and the greater the influence of the term structure of the particular forward rates F(t, T n−1 , T n ). Nevertheless, even the six-parameter model, which is effectively the stock market version, achieves a precision comparable to the original approach in Mazzoni (2015). The full model enhances the fit considerably.…”

“…As detailed in Mazzoni (2015), the class of asymmetric GARCH (AGARCH) models introduced by Engle (1990),…”

“…(23) would be a simple linear recursion formula. In the original paper (Mazzoni, 2015), E[h t (T)] was computed in terms of a perturbation series, where the unperturbed solution was the one neglecting the square expectation. A subsequent analysis revealed that the relative error of the unperturbed solution was already smaller than 1 percent.…”

“…A parametrization of the implied volatility surface has to accommodate these stylized features in order to be applicable in equity and fixed-income scenarios. A framework suitable for equity markets, based on a GARCH vehicle, was introduced in Mazzoni (2015). To derive a model framework that extends to the swaption market and generates sufficiently precise implied volatility surfaces, this approach is generalized in two ways, Firstly, the natural model period of one trading day is replaced by an arbitrary time interval.…”

Asymptotic Expansion of Risk-Neutral Pricing Density

“…The key idea in the approach suggested here is to generate modified dynamics of the arbitrage-free pricing density by asymptotic expansion around the classical Black-Scholes dynamics of complete markets. Asymptotic analysis has proven a very potent tool in deriving new results over the last fifteen years (see for example Basu and Ghosh 2009;Hagan et al 2002;Kim 2002;Mazzoni 2015;Medvedev and Scaillet 2003;Uchida and Yoshida 2004;Whalley and Willmot 1997) whenever certain parts of a problem can be assumed as small. The first step in this approach is to express the complete market dynamics of the risk-neutral pricing density in a new coordinate frame, where it looks stationary.…”