A Black–Scholes inequality: applications and generalisations
研究了看涨期权价格曲线的自然非交换半群结构，推导出布莱克-舒尔斯隐含波动率的一个不等式，并证明单参数子半群对应唯一的对数凹概率密度，为构建无套利市场模型提供了新方法。
Abstract The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.