Option pricing using local volatility function: how to specify its knots?

Abstract

European options of an asset are priced following a continuous 1-factor diffusion model and the generalized Black-Scholes equation. Volatility knots are determined by many specifications that are the number and the location of knots at any stock price and time to expiration. Considering the volatility knots as a set of decision variables, we can approximate the local volatility function with a bicubic spline function to compute option values fitted to a finite set of market data by solving a constrained nonlinear optimization problem. With the real market data of SET50 Index options, pricing accuracy of different knots’ specifications are demonstrated. The experiment in using a set of volatility knots to price options that are out of the calibrated dataset is also provided. The patterns of volatility surface that approximates option values sufficiently close to the market are illustrated.