Delta hedging is a risk management strategy used to neutralize the impact of price movements in the underlying asset of an option. It involves adjusting the position in the underlying asset to offset the sensitivity of the option’s value, measured by its “delta.” Delta represents the rate of change in an option’s price relative to changes in the price of the underlying asset. As the price of the underlying asset fluctuates, the delta also changes, requiring frequent rebalancing of the hedge.
Equity index option traders often use delta to hedge vega risks. This approach is feasible due to the strong negative correlation between the equity index and its implied volatility. Reference [1] formalized this practice by developing a so-called minimum-variance (MV) delta. Essentially, the minimum-variance delta is the Black-Scholes delta with an additional adjustment term. The authors pointed out,
This paper has investigated empirically the difference between the practitioner Black-Scholes delta and the minimum variance delta. The negative relation between price and volatility for equities means that the minimum variance delta is always less than the practitioner Black-Scholes delta. Traders should under-hedge equity call options and over-hedge equity put options relative to the practitioner Black-Scholes delta.
The main contribution of this paper is to show that a good estimate of the minimum variance delta can be obtained from the practitioner Black-Scholes delta and an empirical estimate of the historical relationship between implied volatilities and asset prices. We show that the expected movement in implied volatility for an option on a stock index can be approximated as a quadratic function in the option’s practitioner Black-Scholes delta divided by the square root of time. This leads to a formula for converting the practitioner Black-Scholes delta to the minimum variance delta. When the formula is tested out of sample, we obtain good results for both European and American call options on stock indices. For options on the S&P 500 we find that our model gives better results that either a stochastic volatility model or a model based on the slope of the smile.
In summary, the new delta hedging scheme using the MV delta performs well for certain underlyings, even outperforming stochastic volatility models like SABR and local volatility models.
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References
[1] John Hull and Alan White, Optimal Delta Hedging for Options, Journal of Banking and Finance, Vol. 82, Sept 2017: 180-190
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