Analyzing South African Equity Option Prices Using Normalizing Flows

Black and Scholes’ foundational research on European option pricing launched contemporary derivative pricing in the markets. Since then, much research has been devoted to developing option pricing models that more closely fit actual market data, with the recent focus being on utilizing data-driven and machine learning techniques. Machine learning-based pricing approaches, such as deep neural networks, typically require a custom loss function to prevent the possibility of arbitrage in the predicted option prices. In this chapter, we learn the risk-neutral density implied by South African equity options on the Johannesburg Stock Exchange All Share Index using a mixture of normalizing flows framework recently introduced in Yang and Hospedales (2023). Learning the risk-neutral density is ideal since it naturally incorporates the no-arbitrage conditions, requiring no custom loss function, and the option prices can then be easily generated from this density. Our analysis shows that the mixture of normalizing flows approach can reproduce the option prices in a developing economy for various option maturities and moneyless levels. We further find that the options close to expiry require more mixture components than those far from expiry, which raises the question of how to select the optional number of normalizing flow mixture components. We find that the largest errors are typically associated with low strikes, with the error generally reducing with increasing strikes.