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Problem: The standard Black-Scholes framework cannot incorporate the volatility smiles usually observed in the markets. Instead, one must consider alternative stochastic volatility models such as the SABR. Little research about the suitability of the SABR model for Swedish market (swaption) data has been found. Purpose: The purpose of this paper is to account for and to calibrate the SABR model for swaptions trading on the Swedish market. We intend to alter the calibration techniques and parameter values to examine which method is the most consistent with the market. Method: In MATLAB, we investigate the model using two different minimization techniques to estimate the model’s parameters. For both techniques, we also implement refinements of the original SABR model. Results and Conclusion: The quality of the fit relies heavily on the underlying data. For the data used, we find superior fit for many different swaption smiles. In addition, little discrepancy in the quality of the fit between methods employed is found. We conclude that estimating the α parameter from at-the-money volatility produces slightly smaller errors than using minimization techniques to estimate all parameters. Using refinement techniques marginally increase the quality of the fit.

The main aim of my final thesis is to familiar reader with different ways of measuring credit risk by means of particular structural models of credit risk. This issue has been already described by foreign authors. Though, neither Czech nor Slovak economists have been deeply involved in this topic so far. For this reason, I have decided to focus on those models and both describe them as well as put them into the practice. My final thesis gradually focus on individual detailed model description in each chapter in following sequence: Credit Metrics, Black-School model, Merton model, KMV, Credit Grades. Moreover, it also targets model's construction as well as practical application. Regarding practical model's application, Black-School model is applied on IBM and KMV on Kraft Foods Company. Admittedly, that proves the fact that structural models are not only theoretical models, but also practical models applyable on real companies. Finally, I will compare all above mentioned models in selected parameters.

<p>This study examines whether the performance of the Black-Scholes model to price stock index options is influenced by the general conditions of the financial markets. For this purpose we calculated the theoretical values of 5814 options (3366 put option price observations and 2448 call option price observations) under the Black-Scholes assumptions. We compared these theoretical values with the real market prices in order to put the degree of deviations in two different time windows built around the bankruptcy of Lehman Brothers (September 15th 2008) to the test. We find clear evidences to state that the Black-Scholes model performed differently in the period after Lehman Brothers than in the period before; therefore we are able to blame this event for our findings.</p>

This study examines whether the performance of the Black-Scholes model to price stock index options is influenced by the general conditions of the financial markets. For this purpose we calculated the theoretical values of 5814 options (3366 put option price observations and 2448 call option price observations) under the Black-Scholes assumptions. We compared these theoretical values with the real market prices in order to put the degree of deviations in two different time windows built around the bankruptcy of Lehman Brothers (September 15th 2008) to the test. We find clear evidences to state that the Black-Scholes model performed differently in the period after Lehman Brothers than in the period before; therefore we are able to blame this event for our findings.

Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent research in this field.

A num¡¦eraire is the unit of account in which other assets are denominated. One usually takes the num¡¦eraire to be the currency of a country. In some applications one must change the num¡¦eraire due to the finance considerations. And sometimes it is convenient to change the num¡¦eraire because of modeling considerations. A model can be complicated or simple, depending on the choice of thenum¡¦eraire for the method. When change the num¡¦eraire, denominating the asset in some other unit of account, it is no longer a martingale under ˜P . When we change the num¡¦eraire, we need to also change the risk-neutral measure in order to maintain risk neutrality. The details and some applications of this idea developed in this thesis.

The Black-Scholes formula was the widely-used model for option pricing, this formula can be use to calculate the price of option by using current underlying asset prices, strike price, expiration time, volatility and interest rates. The European call option price from the model is a convex and increasing with respect to the initial underlying asset price. Assume underlying asset prices follow a generalized geometric Brownian motion, it is true that option prices increasing with respect to the constant interest rate and volatility, so that the volatility can be a very important factor in pricing option, if the volatility process £m(t) is constant (with £m(t) =£m for any t ) satisfying £m_1 ≤ £m(t) ≤ £m_2 for some constants £m_1 and £m_2 such that 0 ≤ £m_1 ≤ £m_2. Let C_i(t, S_t) be the price of the call at time t corresponding to the constant volatility £m_i (i = 1,2), we will derive that the price of call option at time 0 in the model with varying volatility belongs to the interval [C_1(0, S_0),C_2(0, S_0)].

This article examines the hedging positions derived from the Black-Scholes(B-S) model and the GARCH(1,1) models, respectively, when the log returns of underlying asset exhibits GARCH(1,1) process. The result shows that Black-Scholes and GARCH options deltas, one of the hedging parameters, are similar for near-the-money options, and Black-Scholes options delta is higher then GARCH delta in absolute terms when the options are deep out-of-money, and Black-Scholes options delta is lower then GARCH delta in absolute terms when the options are deep in-the-money. Simulation study of hedging procedure of GARCH(1,1) and B-S models are performed, which also support the above findings.