论文标题
前向方差曲线模型中的弱近似值和VIX期权价格上涨
Weak approximations and VIX option price expansions in forward variance curve models
论文作者
论文摘要
We provide explicit approximation formulas for VIX futures and options in forward variance models, with particular emphasis on the family of so-called Bergomi models: the one-factor Bergomi model [Bergomi, Smile dynamics II, Risk, 2005], the rough Bergomi model [Bayer, Friz, and Gatheral, Pricing under rough volatility, Quantitative Finance, 16(6):887-904, 2016年]，以及一种增强的粗糙模型版本，可以为Vix微笑产生逼真的积极偏斜 - 由de Marco [Bachelier World Congress，2018年]和Guyon [Bachelier World Congress，2018]同时推出[Bergomi，Smile smile smile smile smile smile smile dynamisics iii，2008年，2008年]，我们参考了我们的'comp of bergomi模型。遵循[Gobet和Miri中设置的方法，平均扩散过程的弱近似值。随机过程。\ appl。，124（1）：475-504，2014]，我们得出了VIX定律的弱近似值，从而在黑色 - choles价格和希腊人的明确组合形式下导致期权价格近似。作为新的贡献，我们应对在粗糙模型中出现的分数集成内核，并处理非平滑收益的情况，以包含Vix Futures，呼叫和放置选项。我们强调的是，我们的方法不依赖于小型渐近学，也不依赖于小参数（例如少量波动性）渐近性，因此可以应用于任何选项成熟度和广泛的参数配置。通过几个数值实验和VIX市场数据的校准测试来说明我们的结果。
We provide explicit approximation formulas for VIX futures and options in forward variance models, with particular emphasis on the family of so-called Bergomi models: the one-factor Bergomi model [Bergomi, Smile dynamics II, Risk, 2005], the rough Bergomi model [Bayer, Friz, and Gatheral, Pricing under rough volatility, Quantitative Finance, 16(6):887-904, 2016], and an enhanced version of the rough model that can generate realistic positive skew for VIX smiles -- introduced simultaneously by De Marco [Bachelier World Congress, 2018] and Guyon [Bachelier World Congress, 2018] on the lines of [Bergomi, Smile dynamics III, Risk, 2008], that we refer to as 'mixed rough Bergomi model'. Following the methodology set up in [Gobet and Miri, Weak approximation of averaged diffusion processes. Stochastic Process.\ Appl., 124(1):475-504, 2014], we derive weak approximations for the law of the VIX, leading to option price approximations under the form of explicit combinations of Black-Scholes prices and greeks. As new contributions, we cope with the fractional integration kernel appearing in rough models and treat the case of non-smooth payoffs, so to encompass VIX futures, call and put options. We stress that our approach does not rely on small-time asymptotics nor small-parameter (such as small volatility-of-volatility) asymptotics, and can therefore be applied to any option maturity and a wide range of parameter configurations. Our results are illustrated by several numerical experiments and calibration tests to VIX market data.
