We will begin with a review of key financial topics and outline many of the crucial ideas utilized in the latter half of the paper. Formal notation for important variables will also be established. Then, a derivation of the Black-Scholes equation will lead to a discussion of its shortcomings, and the introduction of stochastic volatility models. Chapter 2 will focus on a variation of the CIR Model using stock price in the volatility driving process, and its behavior to a greater degree. The key area of discussion will be to approximate a hedging function for European call option prices by Taylor Expansion. We will apply this estimation to real data, and analyze the behavior of the price correction. Then make conclusions about whether stock price has any positive effects on the model.

In this thesis we will focus on interest rate modelling and related practical aspects. We will explain the significance of generated scenarios of interest rate's movement for economic results of both life and non-life insurance companies. We will analyse presently known ways of approaching this matter and describe the selected models in detail. Taking into consideration the practical focus of this thesis, we will address the applied methods of model's calibration. Furthermore, we will employ these methods in an extensive numerical study, that will aim to reveal the weaknesses and strengths of particular calibration methods while implementing a specific model and to evaluate their potential application in actuarial practice. Central model of this work is CIR (Cox-Ingersoll-Ross) model.

Affine term structure models (ATSMs) are one set of popular models for yield curve modeling. Given that the models forecast yields based on the speed of mean reversion, under what circumstances can we distinguish one ATSM from another? The objective of my dissertation is to quantify the benefit of knowing the “true” model as well as the cost of being wrong when choosing between ATSMs. In particular, I detail the power of out-of-sample forecasts to statistically distinguish one ATSM from another given that we only know the data are generated from an ATSM and are observed without errors. My study analyzes the power and size of affine term structure models (ATSMs) by evaluating their relative out-of-sample performance. Essay one focuses on the study of the oneactor ATSMs. I find that the model’s predictive ability is closely related to the bias of mean reversion estimates no matter what the true model is. The smaller the bias of the estimate of the mean reversion speed, the better the out-of-sample forecasts. In addition, my finding shows that the models' forecasting accuracy can be improved, in contrast, the power to distinguish between different ATSMs will be reduced if the data are simulated from a high mean reversion process with a large sample size and with a high sampling frequency. In the second essay, I extend the question of interest to the multiactor ATSMs. My finding shows that adding more factors in the ATSMs does not improve models' predictive ability. But it increases the models' power to distinguish between each other. The multiactor ATSMs with larger sample size and longer time span will have more predictive ability and stronger power to differentiate between models.

Generating samples from generalized hyperbolic distributions and non-central chi-square distributions by inversion has become an important task for the simulation of recent models in finance in the framework of (quasi-) Monte Carlo. However, their distribution functions are quite expensive to evaluate and thus numerical methods like root finding algorithms are extremely slow. In this paper we demonstrate how our new method based on Newton interpolation and Gauss-Lobatto quadrature can be utilized for financial applications. Its fast marginal generation times make it competitive, even for situations where the parameters are not always constant. / Series: Research Report Series / Department of Statistics and Mathematics

The main goal of this thesis is to suggest an appropriate approach to interest rate risk modeling in the Czech financial market environment in various situations. Three distinct periods are analyzed. These periods, which are the period before the global financial crisis, period during the financial crisis and in the aftermath of the global financial crisis and calming subsequent debt crisis in the eurozone, are characterized by different evaluation of liquidity and credit risk, different relationship between financial variables and market participants and different degree of market regulations. Within this goal, an application of the BGM model in the Czech financial market environment is crucial. Use of the BGM model for the purpose of predicting a dynamics of a yield curve is not very common. This is firstly due to the fact that primary use of this model is a valuation of interest rate derivatives while ensuring the absence of arbitrage and secondly its application is relatively difficult. Nevertheless, I apply the BGM model to obtain predictions of the probability distributions of interest rates in the Czech and eurozone market environment, because its complexity, direct modeling of a yield curve based on market rates and especially a possibility of parameter estimation based on current swaptions volatilities quotations may lead to a significant improvement of predictions. This improvement was also confirmed in this thesis. Use of swaptions volatilities market quotations is especially useful in the period of unprecedented mone- tary easing and increased number of central banks and other regulators interventions into financial markets that occur after the financial crisis, because it reflects current market expectations which also include future interventions. As a consequence of underdevelopment of the Czech financial market there are no market quotations of Czech koruna denominated swaptions volatilities. I suggest their approximations based on quotations of euro denominated swaptions volatilities and also using volatilities of koruna and euro forward rates. Use of this approach ensures that predictions of the Czech yield curve dynamics contain current market expectations. To my knowledge, any other author has not presented similar application of the BGM model in the Czech financial market environment. In this thesis I further predict a Czech and Euro area money market yield curve dynamics using the CIR and the GP models as representatives of various types of interest rates models to compare these predictions with BGM predictions. I suggest a comprehensive system of three criteria, based on comparison of predicti- ons with reality, to describe a predictive power of selected models and an appropria- teness of their use in the Czech market environment during different situations in the market. This analysis shows that predictions of the Czech money market yield curve dynamics based on the BGM model demonstrate high predictive power and the best 8 quality in comparison with other models. GP model also produces relatively good qua- lity predictions. Conversely, predictions based on the CIR model as a representative of short rate model family completely failed when describing reality. In a situation when the economy allows negative rates and there is simultaneously a significant likelihood of their implementation, I recommend to obtain predictions of Czech money market yield curve dynamics using GP model which allows existence of negative interest rates. This analysis also contains a statistical test for validating the predictive power of each model and information on other tests. Berkowitz test rejects a hypothesis of accurate predictions for each model. However, this fact is common in real data testing even when using relatively good model. This fact is especially caused by difficult fulfilment of test conditions in real world. To my knowledge, such an analysis of the predictive power of selected interest rate models moreover in the Czech financial market environment has not been published yet. The last goal of this thesis is to suggest an appropriate approach to obtaining pre- dictions of Czech government bonds risk premium dynamics. I define this risk premium as a difference between government bond yields and fixed rate of CZK IRS with the same length. I apply the GP model to describe the dynamics of this indicator of the Czech Republic credit risk. In order to obtain a time series of the risk premium which are necessary for estimation of GP model parameters I firstly estimate yield curves of Czech government bonds using Svensson model for each trading day since 2005. Resulting si- mulations of risk premium show that the GP model predicts the real development of risk premiums of all maturities relatively well. Hence, the proposed approach is suitable for modeling of Czech Republic credit risk based on the use of information extracted from financial markets. I have not registered proposed approach to risk premium modeling moreover in the Czech financial market environment in other publications.

In this thesis the Black Scholes and the Heston stock prices are investigated and the models are compared. The Black Scholes model assumes that the volatility is constant, while the Heston model allows stochastic volatility which is more flexible and can perform better with empirical data. Both models are analysed and simulated, and the parameters are estimated based on empirical data of S&P 500. Results are based on simulations and characteristic functions which are presented with figures of probability density functions.

<p> In this thesis, we calibrated a one factor CIR model for interest rate and a two factor CIR model for each hazard rate of 21 firms. The time series of the interest rate and each hazard rate for 21 firms are also obtained. Extended Kalman Filter and Quasi-Maximum Likelihood Estimation are used as the numerical scheme. The empirical results suggest that multifactor CIR models are not better than multifactor Hull-White model. Positive correlations between hazard rate and interest rate are discovered, although most hazard rates are found to be negatively correlated with the default-free interest rate. The 21 filtered time series of the hazard rates suggest that there maybe a hidden common factor shared only by the intensities. Monte Carlo Simulation is conducted both for interest rate and hazard rates. The simulation indicate that both the SKF and the EKF work pretty well as a filter tool but may produce bad estimation for the value of the likelihood function. QMLE works fine in linear state space form model, but it does a poor job in the case of non-linear state space form.</p> / Thesis / Master of Science (MSc)

In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR model. We derive the partial differential equations with terminal and boundary conditions which determine the American spread option with stochastic interest rate and formulate the associated free boundary problem. We convert the free boundary problem to the linear complimentarity conditions for the American spread option, so that we can go around the free boundary and compute the option price numerically. Alternatively, we approximate the option price using methods based on the Monte Carlo simulation, including the regression-based method, the Lonstaff and Schwartz method and the dual method. We make the comparisons among the option prices derived by the partial differential equation method and Monte Carlo methods to show the accuracy of the result.

Il sistema delle garanzie finanziarie del mercato assicurativo è, allo stato attuale, oggetto di processi di riforma comunitari (c.d. progetto Solvency II). Le ipotesi fino ad ora elaborate, nonostante siano lontane dal potersi definire conclusive, prevedono l'apprezzamento del margine di solvibilità relativo al rischio di tasso d'interesse (IRR) assumendo che la struttura per scadenza sia oggetto di shift paralleli della curva. Noti i limiti dell'approccio deterministico ed in forza dei principi fino ad ora consolidati, il presente elaborato si propone l'obiettivo di quantificare il requisito di capitale di una compagnia operante nel ramo vita, a fronte dell'IRR, ipotizzando che la dinamica dei tassi sia governata da un processo stocastico nella forma del modello Cox Ingersoll e Ross (CIR). Le simulazioni sono state sviluppate sugli equilibri patrimoniali di una teorica compagnia, al fine di apprezzare in maniera asettica il contributo dell'impostazione promossa. / Nowadays, the financial guarantees system of insurance market is being interested by a Community reform process (Solvency II project). Even if the current hypothesis are far to be definitive, the present guidelines state that the Solvency Capital Requirement (SCR) related to Interest Rate Risk (IRR) has to be quantified assuming deterministic shocks to the yield curve. The aim of the thesis is to improve the assessment of SCR connected to IRR, calculating interest rates according to Cox, Ingersoll and Ross (cir) stochastic model. Simulations are developed on the asset liability equilibria of a theoretical life insurance company, in order to better appreciate the SCR algebra sensitivity to changes in CIR model parameters.

In this paper, we investigate the problem of estimating the autocorrelation of squared returns modeled by diffusion processes with data observed at non-equi-spaced discrete times. Throughout, we will suppose that the stock price processes evolve in continuous time as the Heston-type stochastic volatility processes and the transactions arrive randomly according to a Poisson process. In order to estimate the autocorrelation at a fixed delay, the original non-equispaced data will be synchronized. When imputing missing data, we adopt the previous-tick interpolation scheme. Asymptotic property of the sample autocorrelation of squared returns based on the previous-tick synchronized data will be investigated. Simulation studies are performed and applications to real examples are illustrated.