Individual Stock Versus Index: A Comparison of Black-Scholes Calibration and Delta Hedging on TSLA and SPY (2019–2022)

Abstract

The Black–Scholes option pricing model assumes that the volatility of the underlying asset is constant, yet observed market option prices imply a volatility that changes with the strike price. This study investigates whether adjusting the volatility input to match market prices improves option pricing and delta hedging, and whether the benefit differs between a high-volatility individual stock and a broad market index. Daily option data for Tesla and the SPDR S&P 500 exchange-traded fund from 2019 to 2022 are used, covering calm markets, the 2020 market crash, and the subsequent monetary tightening. Two settings are compared: one using historical volatility estimated from past returns, and one using an implied volatility obtained by fitting the Black–Scholes price to market option prices through least squares. Pricing accuracy is assessed by the pricing residuals across strike prices, and hedging effectiveness by the variability of the hedging profit and loss over forty-eight monthly positions. The results show that a single fitted volatility cannot match option prices at all strike prices, confirming that the constant-volatility assumption does not hold. Using the fitted volatility reduces the variability of the hedging profit and loss by about twelve percent for the individual stock and twenty-three percent for the index. The larger improvement for the index indicates that the value of calibration depends on the volatility structure of the underlying asset.