Monte Carlo path simulations are common in mathematical and computational finance as a way of estimating the expected values of a quantity such as a European put option, which is functional to the solution of a stochastic differential equation (SDE). The computational complexity of the standard Monte Carlo (MC) method grows quite large quickly, so in this thesis we focus on the Multilevel Monte Carlo (MLMC) method by Giles, which uses multigrid ideas to reduce the computational complexity. We use a Euler-Maruyama time discretisation for the approximation of the SDE and investigate how the convergence rate of the MLMC method improves the computational times and cost in comparison with the standard MC method. We perform a numerical analysis on the computational times and costs in order to achieve the desired accuracy and present our findings on the performance of the MLMC method on a European put option compared to the standard MC method.

In biochemically reactive systems with small copy numbers of one or more reactant molecules, stochastic effects dominate the dynamics. In the first part of this thesis, we design novel efficient simulation techniques for a reliable and fast estimation of various statistical quantities for stochastic biological and chemical systems under the framework of Stochastic Reaction Networks. In the first work, we propose a novel hybrid multilevel Monte Carlo (MLMC) estimator, for systems characterized by having simultaneously fast and slow timescales. Our hybrid multilevel estimator uses a novel split-step implicit tau-leap scheme at the coarse levels, where the explicit tau-leap method is not applicable due to numerical instability issues. In a second work, we address another challenge present in this context called the high kurtosis phenomenon, observed at the deep levels of the MLMC estimator. We propose a novel approach that combines the MLMC method with a pathwise-dependent importance sampling technique for simulating the coupled paths. Our theoretical estimates and numerical analysis show that our method improves the robustness and complexity of the multilevel estimator, with a negligible additional cost. In the second part of this thesis, we design novel methods for pricing financial derivatives. Option pricing is usually challenging due to: 1) The high dimensionality of the input space, and 2) The low regularity of the integrand on the input parameters. We address these challenges by developing different techniques for smoothing the integrand to uncover the available regularity. Then, we approximate the resulting integrals using hierarchical quadrature methods combined with Brownian bridge construction and Richardson extrapolation. In the first work, we apply our approach to efficiently price options under the rough Bergomi model. This model exhibits several numerical and theoretical challenges, implying classical numerical methods for pricing being either inapplicable or computationally expensive. In a second work, we design a numerical smoothing technique for cases where analytic smoothing is impossible. Our analysis shows that adaptive sparse grids’ quadrature combined with numerical smoothing outperforms the Monte Carlo approach. Furthermore, our numerical smoothing improves the robustness and the complexity of the MLMC estimator, particularly when estimating density functions.

Dans cette thèse, nous étudions plusieurs problèmes de mathématiques financières liés à la valorisation des produits dérivés. Par différentes approches asymptotiques, nous développons des méthodes pour calculer des approximations précises du prix de certains types d’options dans des cas où il n’existe pas de formule explicite.Dans le premier chapitre, nous nous intéressons à la valorisation des options dont le payoff dépend de la trajectoire du sous-jacent par méthodes de Monte-Carlo, lorsque le sous-jacent est modélisé par un processus affine à volatilité stochastique. Nous prouvons un principe de grandes déviations trajectoriel en temps long, que nous utilisons pour calculer, en utilisant le lemme de Varadhan, un changement de mesure asymptotiquement optimal, permettant de réduire significativement la variance de l’estimateur de Monte-Carlo des prix d’options.Le second chapitre considère la valorisation par méthodes de Monte-Carlo des options dépendant de plusieurs sous-jacents, telles que les options sur panier, dans le modèle à volatilité stochastique de Wishart, qui généralise le modèle Heston. En suivant la même approche que dans le précédent chapitre, nous prouvons que le processus vérifie un principe de grandes déviations en temps long, que nous utilisons pour réduire significativement la variance de l’estimateur de Monte-Carlo des prix d’options, à travers un changement de mesure asymptotiquement optimal. En parallèle, nous utilisons le principe de grandes déviations pour caractériser le comportement en temps long de la volatilité implicite Black-Scholes des options sur panier.Dans le troisième chapitre, nous étudions la valorisation des options sur variance réalisée, lorsque la volatilité spot est modélisée par un processus de diffusion à volatilité constante. Nous utilisons de récents résultats asymptotiques sur les densités des diffusions hypo-elliptiques pour calculer une expansion de la densité de la variance réalisée, que nous intégrons pour obtenir l’expansion du prix des options, puis de leur volatilité implicite Black-Scholes.Le dernier chapitre est consacré à la valorisation des dérivés de taux d’intérêt dans le modèle Lévy de marché Libor qui généralise le modèle de marché Libor classique (log-normal) par l’ajout de sauts. En écrivant le premier comme une perturbation du second et en utilisant la représentation de Feynman-Kac, nous calculons explicitement l’expansion asymptotique du prix des dérivés de taux, en particulier, des caplets et des swaptions. / In this thesis, we study several mathematical finance problems, related to the pricing of derivatives. Using different asymptotic approaches, we develop methods to calculate accurate approximations of the prices of certain types of options in cases where no explicit formulas are available.In the first chapter, we are interested in the pricing of path-dependent options, with Monte-Carlo methods, when the underlying is modelled as an affine stochastic volatility model. We prove a long-time trajectorial large deviations principle. We then combine it with Varadhan’s Lemma to calculate an asymptotically optimal measure change, that allows to reduce significantly the variance of the Monte-Carlo estimator of option prices.The second chapter considers the pricing with Monte-Carlo methods of options that depend on several underlying assets, such as basket options, in the Wishart stochastic volatility model, that generalizes the Heston model. Following the approach of the first chapter, we prove that the process verifies a long-time large deviations principle, that we use to reduce significantly the variance of the Monte-Carlo estimator of option prices, through an asymptotically optimal measure change. In parallel, we use the large deviations property to characterize the long-time behaviour of the Black-Scholes implied volatility of basket options.In the third chapter, we study the pricing of options on realized variance, when the spot volatility is modelled as a diffusion process with constant volatility. We use recent asymptotic results on densities of hypo-elliptic diffusions to calculate an expansion of the density of realized variance, that we integrate to obtain an expansion of option prices and their Black-Scholes implied volatility.The last chapter is dedicated to the pricing of interest rate derivatives in the Levy Libor market model, that generaliszes the classical (log-normal) Libor market model by introducing jumps. Writing the first model as a perturbation of the second and using the Feynman-Kac representation, we calculate explicit expansions of the prices of interest rate derivatives and, in particular, caplets and swaptions

Introduction: This thesis presents a python implementation and evaluation of the Heston-Queue-Hawkes (HQH) model, a recent jump-diffusion model for pricing options. The model is capable of tracking options for a wide range of different underlying assets. The model is expected to perform better on Fourier-based fast pricing algorithms such as the COS Method, however in this thesis we’ll only look at Monte Carlo solvers for the HQH model. The type of option studied in this master’s thesis is European options, however, the implementation could be extended to other types of options.  Methodology: The methodology for evaluating the HQH model (in this paper) involves the use of a custom Monte Carlo simulation implemented in Python. The Monte Carlo method enables simulating multiple scenarios and provides reliable results across a variety of situations, making it an appropriate tool for evaluating the model's performance.  Evaluation: The HQH model is evaluated on ease of implementation in python and it’s general ability to reflect different market phenomena such as volatility in price movements.  Improvement: This thesis also investigates the possibility of improving the model or adding corrections, parameters, readjustments, or the like to the model to improve results. The aim is to enhance the model's usefulness, and this evaluation seeks to identify potential improvements.  Worth noting: The goal of this thesis is to align with the research interests of financial institutions and provide a practical, applied approach to evaluating options pricing models. The research presented in this thesis aims to mirror the type of projects that a company like Visigon may be requested to undertake by a bank (and engineering work in general). Additionally, the findings and methodology developed in this thesis aims to inform and contribute to future research in options pricing models which may help markets perform better.

Spread options are derivative securities, which are written on the difference between the values of two underlying market variables. They are very important tools to hedge the correlation risk. American style spread options allow the holder to exercise the option at any time up to and including maturity. Although they are widely used to hedge and speculate in financial market, the valuation of the American spread option is very challenging. Because even under the classic assumptions that the underlying assets follow the log-normal distribution, the resulting spread doesn't have a distribution with a simple closed formula. In this dissertation, we investigate the American spread option pricing problem. Several approaches for the geometric Brownian motion model and the stochastic volatility model are developed. We also implement the above models and the numerical results are compared among different approaches.

Dealing with uncertainty is at the heart of financial risk management and asset pricing. This cumulative dissertation consists of four independent research papers that study various aspects of uncertainty, from estimation and model risk over the volatility risk premium to the measurement of unobservable variables. In the first paper, a non-parametric estimator of conditional quantiles is proposed that builds on methods from the machine learning literature. The so-called leveraging estimator is discussed in detail and analyzed in an extensive simulation study. Subsequently, the estimator is used to quantify the estimation risk of Value-at-Risk and Expected Shortfall models. The results suggest that there are significant differences in the estimation risk of various GARCH-type models while in general estimation risk for the Expected Shortfall is higher than for the Value-at-Risk. In the second paper, the leveraging estimator is applied to realized and implied volatility estimates of US stock options to empirically test if the volatility risk premium is priced in the cross-section of option returns. A trading strategy that is long (short) in a portfolio with low (high) implied volatility conditional on the realized volatility yields average monthly returns that are economically and statistically significant. The third paper investigates the model risk of multivariate Value-at-Risk and Expected Shortfall models in a comprehensive empirical study on copula GARCH models. The paper finds that model risk is economically significant, especially high during periods of financial turmoil, and mainly due to the choice of the copula. In the fourth paper, the relation between digitalization and the market value of US insurers is analyzed. Therefore, a text-based measure of digitalization building on the Latent Dirichlet Allocation is proposed. It is shown that a rise in digitalization efforts is associated with an increase in market valuations.:1 Introduction 1.1 Motivation 1.2 Conditional quantile estimation via leveraging optimal quantization 1.3 Cross-section of option returns and the volatility risk premium 1.4 Marginals versus copulas: Which account for more model risk in multivariate risk forecasting? 1.5 Estimating the relation between digitalization and the market value of insurers 2 Conditional Quantile Estimation via Leveraging Optimal Quantization 2.1 Introduction 2.2 Optimal quantization 2.3 Conditional quantiles through leveraging optimal quantization 2.4 The hyperparameters N, λ, and γ 2.5 Simulation study 2.6 Empirical application 2.7 Conclusion 3 Cross-Section of Option Returns and the Volatility Risk Premium 3.1 Introduction 3.2 Capturing the volatility risk premium 3.3 Empirical study 3.4 Robustness checks 3.5 Conclusion 4 Marginals Versus Copulas: Which Account for More Model Risk in Multivariate Risk Forecasting? 4.1 Introduction 4.2 Market risk models and model risk 4.3 Data 4.4 Analysis of model risk 4.5 Model risk for models in the model confidence set 4.6 Model risk and backtesting 4.7 Conclusion 5 Estimating the Relation Between Digitalization and the Market Value of Insurers 5.1 Introduction 5.2 Measuring digitalization using LDA 5.3 Financial data & empirical strategy 5.4 Estimation results 5.5 Conclusion

This thesis investigates the application of the Black-Scholes model for pricing long-maturity options, primarily utilizing historical data on S\&P500 options. It compares prices computed with the Black-Scholes formula to actual market prices and critically examines the validity of the Black-Scholes model assumptions over long time frames. The assumptions mainly focused on are the constant volatility assumption, the assumption of normally distributed returns, the constant interest rate assumption and the no transaction cost assumption. The results show that the differences between computed prices and actual prices decrease as options get closer to maturity. They also show that several of the Black-Scholes model assumptions are not entirely realistic over long time frames. The conclusion of the thesis is that there are several limitations to the Black-Scholes model when it comes to pricing long-maturity options.

In this thesis, we consider an optimal stopping problem interpreted as the task of valuating two so called real options written on an underlying asset following the dynamics of an observable geometric Brownian motion with non-observable drift; we have incomplete information. After exercising the first real option, however, the value of the underlying asset becomes observable with reduced noise; we obtain partial information. We then state some theoretical properties of the value function such as convexity and monotonicity. Furthermore, numerical solutions for the value functions are obtained by stating and solving a linear complementary problem. This is done in a Python implementation using the 2nd order backward differentiation formula and summation-by-parts operators for finite differences combined with an operator splitting method.

有鑑於資產報酬常具有自我相關的特性，本文探討當標的資產報酬服從一階移動平均過程之選擇權（MA(1)-type option）評價。研究結果顯示，除了總變異因子（total volatility input）不同外，MA(1)-type option 的評價公式與 Black and Scholes 模型極為相似。而根據數值分析的結果，即使資產報酬間自我相關的程度薄弱，由一階移動平均過程產生之自我相關仍會對選擇權價值造成顯著影韾。 / This paper derives the closed-form formula for a European option on an asset with returns following a continuous-time type of first-order moving average process, which is named as an MA(1)-type option. The pricing formula of these options is similar to that of Black and Scholes except for the total volitility input. Specifically, the total volatility input of MA(1)-type options is the conditional standard deviation of continuous-compounded returns over the option's remaining life, whereas the total volatility input of Black and Scholes is indeed the diffusion coefficient of a geometric Brownian motion times the square root of an option's time to maturity. Based on the result of numerical analyses, the impact of autocorrelation induced by the MA(1)-type process is significant to option values even when the autocorrelation between asset returns is weak.

There has been an on-going debate about choices of the most suitable model amongst a variety of model specifications and parameterizations. The first dissertation essay investigates whether asymmetric leptokurtic return distributions such as Hansen's (1994) skewed tdistribution combined with GARCH specifications can outperform mixed GARCH-jump models such as Maheu and McCurdy's (2004) GARJI model incorporating the autoregressive conditional jump intensity parameterization in the discrete-time framework. I find that the more parsimonious GJR-HT model is superior to mixed GARCH-jump models. Likelihood-ratio (LR) tests, information criteria such as AIC, SC, and HQ and Value-at-Risk (VaR) analysis confirm that GJR-HT is one of the most suitable model specifications which gives us both better fit to the data and parsimony of parameterization. The benefits of estimating GARCH models using asymmetric leptokurtic distributions are more substantial for highly volatile series such as emerging stock markets, which have a higher degree of non-normality. Furthermore, Hansen's skewed t-distribution also provides us with an excellent risk management tool evidenced by VaR analysis. The second dissertation essay provides a variety of empirical evidences to support redundancy of stochastic volatility for SP500 index returns when stochastic volatility is taken into account with infinite activity pure Lévy jumps models and the importance of stochastic volatility to reduce pricing errors for SP500 index options without regard to jumps specifications. This finding is important because recent studies have shown that stochastic volatility in a continuous-time framework provides an excellent fit for financial asset returns when combined with finite-activity Merton's type compound Poisson jump-diffusion models. The second essay also shows that stochastic volatility with jumps (SVJ) and extended variance-gamma with stochastic volatility (EVGSV) models perform almost equally well for option pricing, which strongly imply that the type of Lévy jumps specifications is not important factors to enhance model performances once stochastic volatility is incorporated. In the second essay, I compute option prices via improved Fast Fourier Transform (FFT) algorithm using characteristic functions to match arbitrary log-strike grids with equal intervals with each moneyness and maturity of actual market option prices.