波动性(金融)
最大值和最小值
高斯分布
有界函数
单调函数
数学
隐含波动率
随机波动
波动微笑
概率密度函数
凸性
正多边形
统计物理学
计量经济学
应用数学
数学分析
经济
几何学
统计
物理
金融经济学
量子力学
作者
Paul Glasserman,Dan Pirjol
标识
DOI:10.1080/14697688.2023.2165448
摘要
The number of crossings of the implied volatility function with a fixed level is bounded above by the number of crossings of the risk-neutral density with the density of a log-normal distribution with the same mean as the forward price. It is bounded below by the number of convex payoffs priced equally by the two densities. We discuss the implications of these bounds for the implied volatility in the N-component Gaussian mixture model, with particular attention to the possibility of W-shaped smiles. We show that the implied volatility in this model crosses any level at most 2(N−1) times. We show that a bimodal density need not produce a W-shaped smile, and a unimodal density can produce an oscillatory smile. We give monotonicity properties of the implied volatility in Gaussian mixtures under stochastic orderings of the location parameters and volatilities of the mixture components. For some of these results we make use of a novel convexity property of the Black-Scholes price at one strike with respect to the price at another strike. The combined constraints from density crossings and extreme strike asymptotics restrict the allowed shapes of the implied volatility. As an application we discuss a symmetric N = 3 Gaussian mixture model which generates three possible smile shapes: U-shaped, W-shaped and an oscillatory shape with two minima and two maxima.
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