Implied Volatility Surface Construction

Sammanfattning: Implied volatility surfaces are central tools used for pricing options. This thesis treats the topic of their construction. The main purpose is to uncover the most appropriate methodology for constructing implied volatility surfaces from discrete data and evaluate how well it performs. First some methods and techniques in use for such surface constructing are presented. Then the most attractive approach, chosen to contain 4 interesting models is studied. The models’ performances are tested on two price grids from the EURO STOXX 50 and Nikkei 225 indices. The found implied volatility surfaces give good and decent fits to the data, respectively. The surfaces are evaluated in terms of presence of static arbitrage and are found to have it, although mostly for strike price and time to maturity combinations which are somewhat abnormal and rarely traded. Arbitrage is found to be more prevalent in surfaces when the input data is of lower quality. The volatility surfaces’ shapes and absolute values are compared in between models but also within models for some of the best-fit producing parameter sets. The surfaces are found to differ in both cases for some strike and maturity combinations - sometimes with relative differences of more than 10%. This suggests that surfaces with good fits to the input data still can produce distinctly differing prices for some options. Calibrating the models with the chosen approach involves calculations with complex numbers in ways which potentially introduce problematic discontinuities due to branch crossings. This is investigated numerically as well as theoretically for the 4 models and found to be a significant problem in one of them. The three other models are found to avoid these problems under all valid parameter sets.