Search Results for « Volatility »

Black-Scholes option pricing model (1973) assumes that all option prices on the same underlying asset with the same expiration date, but different exercise prices should have the same implied volatility. However, instead of a flat implied volatility structure, implied volatility (inverting the Black-Scholes formula) shows a smile shape across strikes and time to maturity. This paper compares parametric volatility models with stochastic volatility models in capturing this volatility smile. Results show empirical evidence in favor of parametric volatility models. Keywords: smile volatility, parametric, stochastic, Black-Scholes. JEL Classification: C14 C68 G12 G13

We investigate the state dependence of the variance of the instantaneous variance of the S&P 500 index empirically. Time-series analysis of realized variance over a 20-year period shows strong evidence of an elasticity of variance of the variance parameter close to that of a log-normal model, albeit with an empirical autocorrelation function that one-factor diffusion models fail to capture at horizons above a few weeks. When studying option market behavior (in-sample pricing as well as out-of-sample pricing and hedging over the period 2004–2019), messages are mixed, but systematic, model-wise. The log-normal but drift-free SABR (stochastic-alpha-beta-rho) model performs best for short-term options (times-to-expiry of three months and below), the Heston model—in which variance is stationary but not log-normal—is superior for long-term options, and a mixture of the two models does not lead to improvements.