Black–Scholes Pricing and Market Mispricing of XSP European Index Options

Abstract

This study compares Black–Scholes prices with 6,414 XSP European option quotes from August 7, 2026. The calculation uses the XSP index level, a three-month U.S. Treasury yield, an annualized dividend yield, and 20-day historical volatility from S&P 500 log returns. Signed, absolute, and absolute relative pricing errors (ARPE) are examined by option type, moneyness, maturity, and liquidity. Model prices broadly follow market midquotes, but calls have a larger negative bias and larger absolute errors than puts. Errors vary most clearly with moneyness, with near-at-the-money calls showing the largest absolute deviations. Maturity patterns depend on the error measure, while spreads and volume show no simple monotonic relationship with errors. Excluding low-priced contracts reduces ARPE substantially and weakens its maturity pattern, but the main absolute-error and moneyness findings remain. Black–Scholes provides a useful benchmark for this sample, although one historical-volatility estimate cannot reproduce price differences across all contracts. The evidence is descriptive and comes from a single trading day, so its generality requires further testing.