Option Hedging in the Presence of Measurement Errors

June 2006

This paper examines how small, realistic measurement errors in the inputs to option pricing distort the hedge ratios used to manage option risk, and how better estimation reduces the resulting hedging errors. To hedge an option, a risk manager computes partial derivatives of the option price, such as delta and gamma, from observed inputs like the underlying asset price and an estimate of volatility. In practice, those inputs are observed with error, from bid-ask spreads, tick sizes, and non-synchronous quotes. The paper shows that these small input errors can be amplified in some hedge ratios, producing substantial noise and, in turn, large hedging errors. It traces the error in option deltas back to its dominant sources, develops efficient statistical estimators of the hedge parameters that account for measurement error, and shows through simulation how much these estimators reduce hedging errors relative to standard practice. It is written for a technical audience of derivatives researchers, risk managers, and quantitatively minded investors.

What This Paper Examines

• How measurement errors in option inputs propagate into hedge ratios like delta, gamma, and theta.
• Which inputs are the dominant sources of error in computed option deltas.
• How to construct efficient estimates of hedge parameters that account for measurement error.
• How much efficient estimation reduces hedging errors across different hedging strategies.
• Why unnecessary flexibility in volatility modeling can make hedging errors worse.

Key Findings

• Small input errors are amplified into large hedge-ratio errors. Realistic measurement errors in observed prices and volatilities translate into substantial noise in some hedge ratios, which produces materially larger hedging errors than the inputs alone would suggest.
• The underlying price and implied volatility dominate the error in deltas. A variance decomposition shows that, of all the inputs, errors in the underlying asset price and in implied volatility are by far the most important sources of error in computed deltas.
• Efficient estimation of hedge parameters materially reduces hedging errors. Estimators that explicitly account for measurement error, built on a generalized method of moments and generalized least squares approach, cut hedging errors substantially relative to methods that ignore it.
• Unnecessary flexibility in the volatility specification backfires. Allowing a richer volatility smile than the data warrant sharply increases hedging errors, even when noisy observations are down-weighted, so disciplined specification matters.
• Delta-gamma hedging with efficient estimates delivers the largest gains. Neutralizing both delta and gamma, using efficient hedge-parameter estimates, produces hedged payoffs an order of magnitude smaller than delta-only hedging, and these measurement-error costs are incurred regardless of rebalancing frequency.

The Authors

This paper is part of the long lineage of quantitative research that Versor’s founders began earlier in their careers and continue to build on at the firm today.

Ludger Hentschel, Founding Partner, Investment Advisor

Ludger Hentschel joined Versor Investments as a Founding Partner and is based in New York. Ludger has over 20 years of experience in quantitative research and investing.

Disclaimer: Past performance is not necessarily indicative of future results. Not an offer to sell or a solicitation of any type with respect to any securities or financial products.

Methodology: The paper first derives, within the Black-Scholes-Merton framework, how measurement errors in each input propagate into the option hedge parameters delta, gamma, and theta, and decomposes the variance of the resulting hedge-ratio errors into its sources. It then develops efficient estimators of the hedge parameters that exploit the cross-sectional restrictions the model places on options that share an underlying and expiration, using a generalized method of moments procedure together with generalized least squares estimates of implied volatility.

Performance is evaluated through simulation, with ten thousand runs per experiment, on index call options with twenty days to maturity and a one-day holding period. Three hedging strategies are compared: hedging an option with the underlying asset, hedging with an at-the-money option, and delta-gamma hedging. Throughout, simulated prices are constrained to respect no-arbitrage bounds so that the measurement-error structure is realistic.