The Small and Large Time Implied Volatilities in the Minimal Market Model

TL;DR

This paper derives explicit formulas for the asymptotic behavior of implied volatility in the minimal market model, highlighting the influence of interest rates over long time horizons.

Contribution

It provides explicit formulas for small and large time implied volatility limits, revealing interest rate effects in the long-term behavior.

Findings

Interest rates affect long-term implied volatility.

Explicit formulas for small and large time limits.

Interest rates are negligible in short-term implied volatility.

Abstract

This paper derives explicit formulas for both the small and large time limits of the implied volatility in the minimal market model. It is shown that interest rates do impact on the implied volatility in the long run even though they are negligible in the short time limit.