A Prediction of Option Price via Two Volatility Computations with Application to a 50ETF Option

Abstract

Financial derivative trading is integral to stock markets, leading to high option price volatility due to increased trading volume. Determining a reasonable option price is complex and requires extensive research in fields like Economics, Applied Mathematics, and Finance Engineering. The Black-Scholes (BS) equation provides a scientific pricing tool for options by considering five parameters: stock price (S), option strike price (K), risk-free interest rate (r), time to expiration (τ), and volatility (σ). Notably, all model parameters except volatility σ can be directly observed from market data, necessitating the determination of this parameter from historical data when applying the BS model in practice.

In this report, we examine two widely used computation methods for volatility. The first method involves a simple statistical calculation of historical data, resulting in the historical volatility (HV). The second method utilizes the BS model in a "backward" manner: given any previous option price and four other parameters, we solve the BS equation to obtain the implied volatility (IV). To determine such an IV parameter, we propose a generalized fixed-point iterative solver for solving a complex nonlinear equation. By employing a well-designed initial guess, we demonstrate that this fixed-point solver achieves global and rapid convergence.

The two volatilities lead to different predictions of the option price. This report examines if the BS model accurately predicts the option price using these volatilities. It presents an empirical study on a 50ETF option, where we use t-tests to check the hypothesis and determine the relationship between predicted and actual option prices. The study finds that using implied volatility in the BS model provides significantly more accurate predictions compared to historical volatility.