Calibrating S&P 500 index options under alternative formulations of the Heston (1993) model

• Main Author: Silva, João Frederico Freitas
• Publication Date: 2024
• Format: Master thesis
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10071/33603
• topic: Modelo Heston -- Heston model; Stochastic volatility; Option pricing; Volatilidade estocástica; Avaliação de opções

The Heston model, renowned for incorporating stochastic volatility, is crucial for accurate pricing of financial derivatives. However, its calibration poses significant challenges, particularly in computational efficiency and numerical stability. The model’s inherent complexity arises from its use of stochastic differential equations to describe both the stock price and its volatility, requiring sophisticated numerical techniques for parameter estimation. Ensuring numerical stability while maintaining accuracy is a delicate balance. The purpose of this thesis is to efficiently calibrate S&P 500 index options using various calibration techniques under alternative formulations of the Heston (1993) model. The calibration techniques examined include: Mean Squared Error (MSE); Relative Mean Squared Error (RMSE); and Christoffersen (2009) (IVRMSE). The formulations examined include: the original Heston model where we address the "little Heston trap"; consolidated single integral formulation; Attari (2004) representation; fast Fourier transform (FFT) formulation by Carr and Madan (1999); and, Chourdakis (2005) fractional fast Fourier transform (FRFT).