G13 - Contingent Pricing; Futures Pricing; option pricingReturn

The main goal of this paper is to compare the classical MCMC estimation method with a universal Neural Network (NN) approach to estimate unknown parameters of the Heston stochastic volatility model given a series of observable asset returns. The main idea of the NN approach is to generate a large training synthetic dataset with sampled parameter vectors and the return series conditional on the Heston model. The NN can then be trained reverting the input and output, i.e. setting the return series, or rather a set of derived generalized moments as the input features and the parameters as the target. Once the NN has been trained, the estimation of parameters given observed return series becomes very efficient compared to the MCMC algorithm. Our empirical study implements the MCMC estimation algorithm and demonstrates that the trained NN provides more precise and substantially faster estimations of the Heston model parameters. We discuss some other advantages and disadvantages of the two methods, and hypothesize that the universal NN approach can in general give better results compared to the classical statistical estimation methods for a wide class of models.

Recently, there has been a considerable interest in machine learning (ML) applications to valuation of options. The main motivation is the speed of calibration or, for example, calculation of the credit valuation adjustments (CVA). It is usually assumed that there is a relatively liquid market with plain vanilla option quotations that can be used to calibrate (using an ML model) the volatility surface, or to estimate parameters of an advanced stochastic model. In the second stage the calibrated volatility surface (or the model parameters) are used to value given exotic options, again using a trained NN (or another ML model). The NNs are typically trained “off-line” by sampling many model and market parameters´ combinations and calculating the options´ market values. In our research, we focus on the quite common situation of a non-liquid option market where we lack sufficiently many plain vanilla option quotations to calibrate the volatility surface, but we still need to value an exotic option or just a plain vanilla option subject to a more advanced stochastic model as it is typical on energy and carbon derivative markets. We show that it is possible to use selected moments of the underlying historical price return series complemented with a volatility risk premium estimate to value such options using the ML approach.

Aim of this paper is to use Variance Gamma process in the option pricing model and compare it with the well-known and the most widely used option pricing model, the Black-Scholes model. The Variance Gamma model is, in contrast to the one-parameter Black-Scholes model, a three-parameter model. In addition, these two parameters, which are included in the Variance Gamma model, serve to model the skewness and kurtosis of the empirical distribution of the logarithmic returns of the underlying asset. An important part of this work is also a comparison of suitable valuation algorithms for calculation of the option price using the Variance Gamma model. The comparison of both models will be performed primarily on historical empirical distributions of logarithmic returns of selected stocks. Then, performance and pricing error of both models will be tested when estimating implied coefficients based on market data of the option. The performance of both models will be measured by traditional statistical-econometric methods such as RMSE, Likelihood ratio, Akaike information criterion and last but not least by the Natural spline regression model, which estimates the effect of the variable "Moneyness" (distance between the strike price and the current asset value) on the pricing error. All tests performed in this work suggest that the Variance Gamma model is a more accurate model for calculating the price of options.