Time Series Analysis and Calibration to Option Data: A Study of Various Asset Pricing Models

In this chapter, we study three asset pricing models for valuing financial derivatives; namely, the constant elasticity of variance (CEV) model, the Bessel-K model, derived from the squared Bessel (SQB) process, and the unbounded Ornstein–Uhlenbeck (UOU) model, derived from the standard OU process. All three models are diffusion processes with linear drift and nonlinear diffusion coefficient functions. Specifically, the Bessel-K and UOU models are constructed based on a so-called diffusion canonical transformation methodology (Campolieti and Makarov, Int J Theor Appl Financ 10:1–38, 2007; Solvable Nonlinear Volatility Diffusion Models with Affine Drift, 2009; Math Finance 22:488–518, 2012). The models are calibrated to market prices of European options on the S&P500 index. It follows from the calibration analysis that the Bessel-K, UOU, and CEV models provide the best fit for pricing options that mature in 1 month, 3 months, and 1 year, respectively. The UOU model captures option data with a pronounced smile and hence it can be better calibrated to option data with short maturities. The CEV model provides a skewed local volatility and hence it works best for options with longer maturities. Moreover, we demonstrate that the CEV model is reasonably consistent through recalibration analysis on time series data in comparison with the Black–Scholes implied volatility.