Multiscale stochastic elasticity of variance for options and equity linked annuity; A Mellin transform approach

Abstract

It is important to impose a persistent stochastic factor on the underlying asset model to obtain the fair value of financial derivatives with long time-to-maturities. Our empirical study, including the Covid-19 pandemic crisis period, indicates the presence of both fast and slow-scale in the elasticity of variance of S&P 500. This paper extends the elasticity in terms of multiscale stochastic process and obtains a closed form analytic pricing formula for European options and then derive the fair value of Equity-Linked-Annuity (ELA). The Mellin transform method for solving the relevant partial differential equations provides a computationally-efficient pricing formula for the options and the ELA. The prices can be easily calculated simply by taking derivatives of the Black–Scholes option price. Our results reveal the sensitivity of the ELA term structure to the fast-scale or slow-scale related group parameters.