Essays on Derivative Pricing and Risk Management [before doctoral defense]

Kivonat, rövid leírás

This dissertation comprises three main chapters. The first two develop numerical and semi-analytical Hamiltonian methods for pricing single- and multi-asset derivatives, while the third examines sentiment-augmented models for forecasting volatility and Value-at-Risk (VaR). The first study develops a numerical pseudo-Hermitian Hamiltonian framework for pric-ing plain-vanilla and path-dependent options in the Black–Scholes–Merton setting. Using the similarity transformation to a Hermitian operator, Arrow–Debreu state-price densities are re-covered from a stable spectral representation and are then used to price derivatives and corre-sponding risk sensitivities. Two numerical implementations are proposed: a dense Hamiltonian spectral simulation and a shift-invert Lánczos variant that exploits the dominance of low-lying eigenpairs in Euclidean-time propagation. The proposed numerical methods are benchmarked against Monte Carlo simulation, Cox–Ross–Rubinstein binomial trees and Crank–Nicolson fi-nite differences. For the plain-vanilla case, the Hamiltonian spectral method produces the small-est pricing error and the closest approximation to the analytical state-price density. For the double knock-out barrier case, the Hamiltonian spectral and Crank–Nicolson methods generate highly consistent state-price densities and state-price-implied risk measures, while the Lánczos approximation remains nearly indistinguishable from the full spectral solution and exhibits a more favourable empirical runtime profile on sufficiently fine grids. As the method estimates the state-price density directly, option prices, Delta estimates, digital option prices and knock-out probabilities can be obtained within a unified framework. The second study extends the previous pseudo-Hermitian formulation to options that depend on multiple correlated underlying assets. In Hilbert space, the multi-asset Black–Scholes–Merton partial differential equation can be represented as a multi-particle Euclidean-time Schrödinger equation whose evolution is governed by a pseudo-Hermitian Hamiltonian. The corresponding similarity transformation leads to a Hermitian representation. Momentum-space wave functions and superposition are used to price general European multi-asset op-tions, for which the conventional multi-asset Black–Scholes–Merton pricing kernel is recov-ered. The two-asset double knock-out worst-of rainbow option with asset-specific barriers is treated using a multi-particle-in-a-box construction. This provides a computationally tractable semi-analytical approximation. The proposed construction also permits the lower and upper barrier levels to be specified separately for each underlying asset. The third study presents sentiment-augmented GARCH models and a hybrid GARCH–LSTM model that incorporates news and social media sentiment to forecast the volatility and Value-at-Risk (VaR) of individual stocks. Various families of GARCH models and their hy-brid extensions have been developed to improve conditional volatility forecasts. The study investigates whether the performance of these models can be further enhanced by incorporat-ing sentiment indicators from external media platforms. It emphasises the nonlinear predictive relationships between conditional variance and sentiment indices and shows how these rela-tionships can be integrated into volatility forecasting models. The resulting forecasts are then used to examine whether sentiment information can improve the estimation of potential finan-cial losses within the traditional VaR framework. To evaluate the models, an empirical study is conducted on the logarithmic returns of stocks included in the S&P 500 index from 2019 to 2024. In conjunction with standard VaR statistical tests, different loss functions are used to examine potential loss magnitudes. The results indicate that the nonlinear tests detect pre-dictive relationships considerably more often than the linear tests. The sentiment-augmented frameworks perform better than their corresponding benchmarks for a considerable subset of equities. They reduce realised-volatility forecast errors for a majority of the examined stocks under all three reported error measures. The VaR backtesting results are also stronger in five of the six model-by-test comparisons. These findings indicate that incorporating sentiment infor-mation can enhance both the accuracy of volatility forecasts and the reliability of VaR estimates.