No Arbitrage Conditions and Characters of Implied Volatility Surface: A Review for Implied Volatility Modelers

Abstract

Since the market model based approach gained more popularity in implied volatility modeling literature, it is very important to understand the restriction on implied volatility surface imposed by no arbitrage conditions. There are many papers offering conclusions in different forms under kinds of assumptions. Thus a review of these conditions and a discussion of the relationship among these conditions will be helpful for implied volatility modeler as they can avoid wrong models. We will first introduce some results in the literature about no arbitrage conditions and general properties on implied volatility surface. Then we will present the sufficient and close to necessary conditions for no static arbitrage and talk about no dynamic arbitrage. Finally, we will show that when we are building models, check the last group of conditions, which can imply other necessary conditions, is quite enough.