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.

中文翻译：

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Black-Scholes不等式：应用和推广

通话价格曲线的空间具有自然的非交换半群结构，具有对合。一个基本的例子是Black-Scholes的叫价面，由此得出Black-Scholes隐含波动率的有趣不等式。二元运算与凸序兼容，因此一个参数的半分组产生了无套利的市场模型。结果表明，每个这样的单参数半群都对应于唯一的对数-凹面概率密度，从而提供了集Gatheral-Jacquier SVI曲面的精神为特点的一系列易于处理的调用价格曲面参数化。给出一个明确的例子来说明这个想法。关键观察结果是同构关系，该同构关系将初始看涨价曲线与标的资产的终端价格的升空区域链接起来。