Delta hedging is a method used to reduce or eliminate the directional risk of an options position. In most delta hedging schemes, delta is calculated using the Black-Scholes-Merton (BSM) model. However, the BSM delta is not always accurate due to the assumptions embedded in the model. For a more accurate hedge, adjustments need to be made. We have discussed such an adjustment proposed by Hull and White.
Following this line of research, Reference [1] examined the use of skew-adjusted delta for hedging. The paper retested the method developed by Vähämaa [2] and applied it to S&P 500 index options during the COVID-19 pandemic. Specifically, the author modified the BSM delta as follows,
The author pointed out,
Considering the second hypothesis (H2), the performance of the smile-adjusted delta was retested with data from the S&P 500 index. Both models’ quantitative fit is documented to be approximately 90%, but SAD explains daily option price movements slightly better than the BS delta, regardless of the moneyness and maturity of the option. Indicating, it is reliable to finally compare the delta hedging performance of the two deltas.
The overall conclusion is that SAD outperforms the BS delta in delta hedging, meaning that the second hypothesis (H2) is also true. To contradict Vähämaa’s (2004) results, the empirical finding from delta hedging indicates that SAD outperforms the BS delta most distinctly among ITM options. When the hedging horizon is longer than a day, SAD outperforms the BS delta regardless of the moneyness and maturity of the option. That out-performance becomes even more notable as the hedging horizon lengthens. This out-performance increase is even more distinct than what Vähämaa (2004) documented. This may be due to the relatively steep volatility smile making the smile-adjustment term, or specifically considering the slope of the smile, more crucial.
In short, the article shows that skew-adjusted delta (SAD) performs better than BSM delta, especially for in-the-money options. When the hedge lasts more than one day, SAD outperforms BSM delta across all moneyness and maturities, with the advantage growing as the hedging period gets longer.
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References
[1] Berg, Wille, From the inaccurate Black-Scholes model to more efficient delta hedging with smile-adjusted extension, University of VAASA, 2025
[2] Vähämaa, S., Delta hedging with the smile, Financial Markets and Portfolio Management, 18(3), 241-255, 2004
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