>Magister Scientiae - MSc / In Monte Carlo path simulations, which are used extensively in computational -finance, one is interested in the expected value of a quantity which is a functional of the solution to a stochastic differential equation [M.B. Giles, Multilevel Monte Carlo Path Simulation: Operations Research, 56(3) (2008) 607-617] where we have a scalar function with a uniform Lipschitz bound. Normally, we discretise the stochastic differential equation numerically. The simplest estimate for this expected value is the mean of the payoff (the value of an option at the terminal period) values from N independent path simulations. The multilevel Monte Carlo path simulation method recently introduced by Giles exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. This new method improves on the computational complexity of the standard Monte Carlo approach by considering Monte Carlo simulations with a geometric sequence of different time steps following the approach of Kebaier [A. Kebaier, Statistical Romberg extrapolation: A new variance reduction method and applications to options pricing. Annals of Applied Probability 14(4) (2005) 2681- 2705]. The multilevel method makes computation easy as it estimates each of the terms of the estimate independently (as opposed to the Monte Carlo method) such that the computational complexity of Monte Carlo path simulations is minimised. In this thesis, we investigate this method in pricing path-dependent options and the computation of option price sensitivities also known as Greeks.

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The aim of the thesis is to present a robust conceptual framework for risk management of non-financial companies in order to improve decision making in the area of hedging with derivative instruments. Application of modern quantitative methods.

The thesis focuses on methods of option prices calculations using two different pricing models which are Heston and Black-Scholes models. The first part describes theory of these two models and conlcudes with a comparison of the risk-neutral measures of these two models. In the second part, the relations between input parameters and the option price generated by these models are clarified. This part ends up with an analysis of the market data and it answers the question which model predicts better.

This thesis studies interest rates (even negative), interest rate derivatives and term structure of interest rates. We review the different types of interest rates and go through the evaluation of a derivative using risk-neutral and forward-neutral methods. Moreover, the construction of interest rate models (term-structure models), pricing of bonds and interest rate derivatives, using both equilibrium and no-arbitrage approaches are discussed, compared and contrasted. Further, we look at the HJM framework and the LMM model to evaluate and simulate forward curves and find the forward rates as the discount factors. Finally, the new framework (after financial crisis in 2008), under the collateral agreement (CSA) has been taken into consideration.