Optimizing the A-Share ETF Option Pricing Model and Empirical Analysis Based on Partial Differential Equation Tools

Abstract

Since the listing of the CSI 300 ETF options in 2015, the A-share ETF options market has developed a multi-tiered product system covering broad-based, industry, and thematic sectors. By the end of 2023, the average daily trading volume of ETF options across the market exceeded 1.2 million contracts, with open interest exceeding 4 million, making them a core tool for investors to hedge risk and optimize returns. However, the traditional Black-Scholes model assumes "constant volatility, a fixed risk-free rate, and no liquidity differences." This is inconsistent with the reality of A-shares, where volatility clusters, contract liquidity differentiation, and short-term fluctuations in the risk-free rate occur. This results in high pricing errors and makes it difficult to adapt to practical applications. This paper focuses on partial differential equations (PDEs) to optimize the pricing pain points of A-share ETF options. First, we analyze market characteristics and pricing issues using trading data from the CSI 300 and CSI 500 ETF options. Then, based on the classic Black-Scholes PDE, we introduce time-varying volatility fitted by GARCH (1, 1), a liquidity adjustment factor constructed using the Amihud indicator, and a dynamic risk-free rate represented by the one-year Treasury bond yield to form an optimized PDE model. Finally, we use daily data from the 2023 CSI 300 (510300) and CSI 500 (510500) ETF options to solve the PDE using the finite difference method. The PDE is then compared with the classic model using the AE, RE, and RMSE. Results show that the optimized model reduces the RMSE for CSI 300 ETF options to 0.082 and for CSI 500 ETF options to 0.091, with the optimization effect being more pronounced during periods of high volatility and low liquidity. Research has confirmed that PDE models incorporating the characteristics of A-shares can more accurately capture option pricing patterns, providing theoretical and practical support for investor decision-making, market maker pricing, and regulatory risk monitoring.