Path-dependent implied Volatility Surfaces

In insurance, among financial risks, equity volatility surfaces are a key input for risk assessment, strategic asset allocation, or economic capital calculation, among other applications. However, to date, simple methods are often considered to model this risk, as well as the dependency with the underlying asset returns. In this context, our aim is to enhance the modelling catalog of insurance practitioners with more realistic implied volatility models. Following the work of Guyon and Lekeufack (2023), we study how implied volatility can be predicted using the past trajectory of the underlying asset price. Our empirical study reveals that a large part of the movements of the at the money implied volatility for up to two years maturities can be explained using the past returns and their squares. Moreover, we show that this feedback effect gets weaker when the maturity increases and that up to four years of the past evolution of the underlying price should be used for the prediction. Building on this new stylized fact, we fit to historical data a parsimonious version of the SSVI parameterization (Gatheral and Jacquier, 2014) of the implied volatility surface relying on only four parameters and show that the two parameters ruling the at the money implied volatility as a function of the maturity exhibit a path-dependent behavior with respect to the underlying asset price. By adding this feedback effect to the path-dependent volatility model of Guyon and Lekeufack for the underlying asset price and by specifying a regime-switching diffusion model for the residuals of these two parameters, and the two other parameters, we are able to simulate highly realistic paths of implied volatility surfaces that are arbitrage-free. Based on this stochastic model for implied volatilities, an application to the hedging of RILA products will be presented.