Smile‐implied hedging with volatility risk

Abstract

Options can be dynamically replicated using model-free Greeks extracted from the volatility smile. However, smile-implied delta and delta-gamma hedging do not achieve minimum variance in the presence of price-volatility correlation, and these strategies have shown poor performance relative to the Black-Scholes (BS) benchmark. We propose a way to extend smile-implied option replication with volatility risk management. Large-scale evidence on S&P 500 index options indicates that smile-implied delta-gamma-vega hedging strategies outperform the BS approach as well as more sophisticated option hedging frameworks, including stochastic volatility and jumps.

Delta Hedging Bitcoin Options with a Smile

“…† There are several previous empirical studies of smile-implied and/or smile-adjusted delta hedging, but all of them study equity index options. Not all of the results are consistent: Vähämaa (2004) shows that some smile-adjusted deltas out-perform the BS delta for FTSE 100 index options, but only during excessively volatile periods; Crépey (2004) confirms these findings for DAX 30 options; Attie (2017) claims that smile-implied deltas consistently out-perform the BS delta for hedging S&P 500 options; Alexander et al (2012) extend the Derman (1999) framework to a Markov-switching setting which reflects the correct smile-adjusted delta for the prevalent market regime, showing that, for S&P 500 options, it is only possible to improve on the BS delta by using this Markov-switching extension; and François and Stentoft (2021) also examine S&P 500 options and confirm that standard adjustments cannot out-perform the BS delta or delta-gamma hedges, but their new smileimplied delta-gamma-vega hedge substantially improves on the BS model. Much less is known about the success of smile-adjusted delta hedges for other types of options.…”

Harvesting the volatility smile in a large emerging market: A Dynamic Nelson–Siegel approach

“…Given our focus on implementing our models for trading strategies, only near-month options data are used, and to avoid issues arising from expiration day effects (Bollen & Whaley, 1999) we remove all data with less than 5 days before the expiry date. Following François and Stentoft (2021a), we also remove all observations which violate the option bounds, and following Mixon (2009), we only consider those stock option days where at least five unique options have been traded. Overall, this leaves us with a final sample consisting of a total of 867,165 daily observations.…”

“…These effectively constitute statistical methods that describe the shape of the smile parametrically or nonparametrically, often using unobservable latent factors (Dumas et al, 1998). The different approaches to modeling volatility smile this way in the more modern literature range from the use of polynomials (Choi et al, 2015; Kim, 2021; Le & Zurbruegg, 2014; Sui et al, 2020; Yue et al, 2021; Zhang & Xiang, 2008) to use of stochastic volatility models (François & Stentoft, 2021a) and semiparametric methods (Fengler & Hin, 2015, and references therein). In a recent study with a detailed review of the related literature, Kim (2021) concludes that such atheoretical models (the author calls them ad hoc Black–Scholes models) are “… the best alternative to mathematically sophisticated option‐pricing models given its simplicity of implementation.” A study which is methodologically closest to ours, Chen et al (2018), also acknowledges that “… structured parametric forecasting models achieve superior out‐of‐sample results.”…”

“…There is a growing literature on understanding the behavior of volatility smile in the options markets, the name given to the commonly observed empirical relationship between implied volatility (IV) and option strike. In the last decade, using data from economies with liquid option markets, there have been many articles published on the modeling of volatility smile, as well as on its applications in the larger literature on option pricing, financial engineering, and risk management (Carr & Wu, 2003; Christoffersen et al, 2009; Cont & Fonseca, 2002; François & Stentoft, 2021a; García‐Machado & Rybczyński, 2017; GrØborg & Lunde, 2016; Hagan et al, 2002; Jain et al, 2019; Kim et al, 2020; Kim, 2021; Taylor et al, 2010; Wong & Heaney, 2017).…”

Venturing into uncharted territory: An extensible implied volatility surface model

“…Third, the specification for the IV surface proposed in this paper can be applied to the dynamic hedging of options. Delta and gamma are derived analytically and are shown to be smile‐implied Greeks (Alexander & Nogueira, 2007; Bates, 2005; François & Stentoft, 2021). That is, these Greeks are consistent with the observed shape of the volatility smile and they do not depend on any assumption regarding the underlying asset dynamics.…”