Maturity cycles in implied volatility. (English) Zbl 1063.91066

Fouque, Jean-Pierre; Papanicolaou, George; Sircar, Ronnie; Solna, Knut

Finance Stoch. 8, No. 4, 451-477 (2004).

The purpose of the paper is to study the performance of the observed implied volatility surface \(I\) that is reduced to an affine function of the log-moneyness-to-maturity ratio (LMMR) of the form \[ I = a\times \text{LMMR} + b,\qquad \text{LMMR} = \frac{\log(K/S)}{T-t}, \] where \(K\) is the strike price, \(T\) is the expiration date, and \(S\) is the asset price, or index price, at time \(t\). The aim is to fit the model to real data and to introduce an extended theory with time-dependent periodic parameters that picks up a significant feature of the data. The criteria used here are goodness of the in-sample fit and stability of the fitted parameters overtime. The authors look at the daily slope and intercept the estimates from S&P 500 index option implied volatilities. The adjusted asymptotic theory in the general case of time-dependent volatility parameters is presented and then specialized to only the speed to be time-dependent. The time variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. It is shown that this behaviour can be explained in a manner consistent with the large model class for the underlying price dynamics with time-periodic volatility coefficients.

Reviewer: Yuliya Mishura

MSC:

91B70	Stochastic models in economics
60F05	Central limit and other weak theorems
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

fast mean-reverting stochastic volatility; asymptotic expansions