Refined analysis of the no-butterfly-arbitrage domain for SSVI slices

IF 0.8 4区经济学 Q4 BUSINESS, FINANCE

Journal of Computational Finance Pub Date : 2023-01-01 DOI:10.21314/jcf.2023.008

Claude Martini, Arianna Mingone

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引用次数: 0

Abstract

The no-butterfly-arbitrage domain of the Gatheral stochastic-volatility-inspired (SVI) five-parameter formula for the volatility smile has recently been described. It requires in general a numerical minimization of two functions together with a few root-finding procedures. We study here the case of the famous surface SVI (SSVI) model with three parameters, to which we apply the SVI results in order to provide the nobutterfly- arbitrage domain. As side results, we prove that, under simple requirements on parameters, SSVI slices always satisfy Fukasawa’s weak conditions of no arbitrage (ie, the corresponding Black–Scholes functions d1 and d2 are always decreasing), and we find a simple subdomain of no arbitrage for the SSVI model that we compare with the well-known subdomain of Gatheral and Jacquier. We simplify the obtained no-arbitrage domain into a parameterization that requires only one immediate numerical procedure, leading to an easy-to-implement calibration algorithm. Finally, we show that the long-term Heston SVI model is in fact an SSVI model, and we characterize the horizon beyond which it is arbitrage free.

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SSVI切片无蝴蝶套利域的精细化分析

本文描述了随机波动激励(SVI)五参数波动微笑公式的无蝴蝶套利域。它通常需要两个函数的数值最小化和一些寻根程序。本文研究了著名的三参数曲面SVI (SSVI)模型，并将SVI的结果应用于该模型，以提供无蝴蝶套利域。作为副结果，我们证明了在对参数的简单要求下，SSVI片总是满足Fukasawa的弱无套利条件(即对应的Black-Scholes函数d1和d2总是递减)，并且我们找到了SSVI模型的一个简单的无套利子域，并与我们熟知的Gatheral和Jacquier子域进行了比较。我们将得到的无套利域简化为只需要一个直接数值过程的参数化，从而导致易于实现的校准算法。最后，我们证明了长期Heston SVI模型实际上是一个SSVI模型，并且我们描述了超出该模型的无套利范围。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Computational Finance BUSINESS, FINANCE-

CiteScore

0.90

自引率

0.00%

发文量

期刊介绍： The Journal of Computational Finance is an international peer-reviewed journal dedicated to advancing knowledge in the area of financial mathematics. The journal is focused on the measurement, management and analysis of financial risk, and provides detailed insight into numerical and computational techniques in the pricing, hedging and risk management of financial instruments. The journal welcomes papers dealing with innovative computational techniques in the following areas: Numerical solutions of pricing equations: finite differences, finite elements, and spectral techniques in one and multiple dimensions. Simulation approaches in pricing and risk management: advances in Monte Carlo and quasi-Monte Carlo methodologies; new strategies for market factors simulation. Optimization techniques in hedging and risk management. Fundamental numerical analysis relevant to finance: effect of boundary treatments on accuracy; new discretization of time-series analysis. Developments in free-boundary problems in finance: alternative ways and numerical implications in American option pricing.