Negative Prices: The beta-Black Formula

Abstract

A simple idea is put forth to handle negative prices. The context for the need for such formula is negative settlements observed on the black gold (Crude Oil) May 2020 contract on the two days leading to expiry and the subsequent large market reaction. A novel formula, the beta-Black (BB) is proposed. This defines new BB-implied volatility coordinates which allow smooth transition around zero. An added benefit is that when beta goes to zero, the formula degenerates to Black. Indeed it would be hazardous in those troubled times to completely change the way we think about volatility.