Simulations of Implied Volatility and Option Pricing using Neural Networks and Finite Difference Methods for Heston Model

Abstract

The theory of option pricing made a dramatic step forward when Black and Scholes published a centennial paper with a solution for the prices of European call and put options. However, their solution only deals with the perfect markets. In the realworld, the markets are not perfect and the predictions from their formula deviate significantly from the market prices. This is because of the assumptions on which the model is based. Heston model is one of the newer models which extends the classical Black-Scholes model and can produce better estimates of option prices with non-constant variable volatility. However, this variable volatility needs to be accurately predicted itself, but the price which is often taken as input for finding implied volatility is not always given. In this thesis, two neural network models for learning historical volatility were constructed and compared in order to predict the implied volatility for the option without knowing its price. The results of prediction were tested and evaluated. The better performing model was used to approximate implied volatility and the result was incorporated into the solution grid and finite difference scheme for the Heston model was applied to produce the option price. The generated prices were compared to the prices generated by the classical Black-Scholes model in several different scenarios.