<p>Condfidence in the future is a signicant factor for business development. However frequently, accurate and specific purposes are spread over the market environment influence.Thus,it is necessary to make an appropriate consideration of instability, which is peculiar to the dynamic development. Volatility, variance and standard deviation are used to</p><p>characterize the deviation of the investigated quantity from mean value.</p><p>Volatility is one of the main instruments to measure the risk of the asset.</p><p>The increasing availability of financial market data has enlarged volatility research potential but has also encouraged research into longer horizon volatility forecasts.</p><p>In this paper we investigate stock index volatility with respect to capitalization with help of GARCH-modelling.</p><p>There are chosen three indexes of OMX Nordic Exchange for our research. The Nordic list segment indexes comprising Nordic Large Cap,</p><p>Mid Cap and Small Cap are based on the three market capitalization groups.</p><p>We implement GARCH-modeling for considering indexes and compare our results in order to conclude which ones of the indexes is more volatile.</p><p>The OMX Nordic list indexis quiet new(2002)and reorganized as late as October 2006. The current value is now about 300 and no options do exist. In current work we are also interested in estimation of the Heston</p><p>model(SVmodel), which is popular in financial world and can be used in option pricing in the future.</p><p>The results of our investigations show that Large Cap Index is more volatile then Middle and Small Cap Indexes.</p>

<p>In this paper we investigate pricing of variance swaps contracts. The</p><p>literature is mostly dedicated to the pricing using replication with</p><p>portfolio of vanilla options. In some papers the valuation with stochastic</p><p>volatility models is discussed as well. Stochastic volatility is becoming</p><p>more and more interesting to the investors. Therefore we decided to</p><p>perform valuation with the Heston stochastic volatility model, as well</p><p>as by using replication strategy.</p><p>The thesis was done at SunGard Front Arena, so for testing the replica-</p><p>tion strategy Front Arena software was used. For calibration and testing</p><p>of the Heston model we used MatLab.</p>

Condfidence in the future is a signicant factor for business development. However frequently, accurate and specific purposes are spread over the market environment influence.Thus,it is necessary to make an appropriate consideration of instability, which is peculiar to the dynamic development. Volatility, variance and standard deviation are used to characterize the deviation of the investigated quantity from mean value. Volatility is one of the main instruments to measure the risk of the asset. The increasing availability of financial market data has enlarged volatility research potential but has also encouraged research into longer horizon volatility forecasts. In this paper we investigate stock index volatility with respect to capitalization with help of GARCH-modelling. There are chosen three indexes of OMX Nordic Exchange for our research. The Nordic list segment indexes comprising Nordic Large Cap, Mid Cap and Small Cap are based on the three market capitalization groups. We implement GARCH-modeling for considering indexes and compare our results in order to conclude which ones of the indexes is more volatile. The OMX Nordic list indexis quiet new(2002)and reorganized as late as October 2006. The current value is now about 300 and no options do exist. In current work we are also interested in estimation of the Heston model(SVmodel), which is popular in financial world and can be used in option pricing in the future. The results of our investigations show that Large Cap Index is more volatile then Middle and Small Cap Indexes.

In this paper we investigate pricing of variance swaps contracts. The literature is mostly dedicated to the pricing using replication with portfolio of vanilla options. In some papers the valuation with stochastic volatility models is discussed as well. Stochastic volatility is becoming more and more interesting to the investors. Therefore we decided to perform valuation with the Heston stochastic volatility model, as well as by using replication strategy. The thesis was done at SunGard Front Arena, so for testing the replica- tion strategy Front Arena software was used. For calibration and testing of the Heston model we used MatLab.

Thesis (doctoral)--Erasmus Universiteit Rotterdam, 2008.

In this thesis, we study the quadratic rough Heston model and the corresponding simulation methods. We calibrate the model using real-world market data. We compare and implement the three commonly used schemes (Hybrid, Multifactor, and Multifactor hybrid). We calibrate the model using real-world market SPX data. To speed up calibration, we apply quasi-Monte Carlo methods. We study the effect of the various calibration parameters on the volatility smile.

Stochastic volatility models on option pricing have received much study following the discovery of the non-at implied surface following the crash of the stock markets in 1987. The most widely used stochastic volatility model is introduced by Heston (1993) because of its ability to generate volatility satisfying the market observations, being non-negative and mean-reverting, and also providing a closed-form solution for the European options. However, little research has been done on Heston model used to price early-exercise options. This presumably is largely due to the absence of a closed-form solution and the increase in computational requirement that complicates the required calibration exercise. This thesis examines the performance of the Heston model versus the Black-Scholes model for the American Style equity option of Microsoft and the index option of S&P 100 index. We employ a finite difference method combined with a Projected Successive Over-relaxation method for pricing an American put option under the Black-Scholes model, while an Alternating Direction Implicit method is utilized to decompose a multi-dimensional partial differential equation into several one dimensional steps under the Heston model. For the calibration of the Heston model, we apply a two step procedure where in the first step we apply an indirect inference method to historical stock prices to estimate diffusion parameters under a probability measure and then use a least squares method to estimate the instantaneous volatility and the market risk premium which are used to switch from working under the probability measure to working under the risk-neutral measure. We find that option price is positively related with the value of the mean reverting speed and the long-term variance. It is not sensitive to the market price of risk and it is negatively related with the risk free rate and the volatility of volatility. By comparing the European put option and the American put option under the Heston model, we observe that their implied volatility generally follow similar patterns. However, there are still some interesting observations that can be made from the comparison of the two put options. First, for the out-of-the-money category, the American and European options have rather comparable implied volatilities with the American options' implied volatility being slightly bigger than the European options. While for the in-the-money category, the implied volatility of the European options is notably higher than the American options and its value exceeds the implied volatility of the American options. We also assess the performance of the Heston model by comparing its result with the result from the Black-Scholes model. We observe that overall the Heston model performs better than the Black-Scholes model. In particular, the Heston model has tendency of underpricing the in-the-money option and overpricing the out-of-the-money option. Whereas, the Black-Scholes model is inclined to underprice both the in-the-money option and the out-of-the-money option.b

Asset pricing models are well established and have been used extensively by practitioners both for pricing options as well as for hedging them. Though Black-Scholes is the original and most commonly communicated asset pricing model, alternative asset pricing models which incorporate additional features have since been developed. We present three asset pricing models here - the Black-Scholes model, the Heston model and the Merton (1976) model. For each asset pricing model we test the hedge effectiveness of delta hedging, minimum variance hedging and static hedging, where appropriate. The options hedged under the aforementioned techniques and asset pricing models are down-and-out call options, lookback options and cliquet options. The hedges are performed over three strikes, which represent At-the-money, Out-the-money and In-the-money options. Stock prices are simulated under the stochastic-volatility double jump diffusion (SVJJ) model, which incorporates stochastic volatility as well as jumps in the stock and volatility process. Simulation is performed under two ’Worlds’. World 1 is set under normal market conditions, whereas World 2 represents stressed market conditions. Calibrating each asset pricing model to observed option prices is performed via the use of a least squares optimisation routine. We find that there is not an asset pricing model which consistently provides a better hedge in World 1. In World 2, however, the Heston model marginally outperforms the Black-Scholes model overall. This can be explained through the higher volatility under World 2, which the Heston model can more accurately describe given the stochastic volatility component. Calibration difficulties are experienced with the Merton model. These difficulties lead to larger errors when minimum variance hedging and alternative calibration techniques should be considered for future users of the optimiser.

Black and Scholes have proposed a model for pricing European options where the underlying asset follows a so-called geometric Brownian motion which assumes constant volatility. The proposed Black-Scholes model has an exact solution. However, it has been shown that such an assumption of constant volatility is not realistic, and numerous extensions have been developed. In addition, models usually do not have a closed-form solution which makes pricing a challenging task. The thesis focuses on pricing Bermudan options under two stochastic volatility Heston-type models using an Almost-Exact scheme for simulation. Namely, we focus on deriving the Almost-Exact scheme for Heston and Double Heston model and numerically study the behaviour of the scheme. We show that the AES works well when the number of simulated steps is equal to the number of exercise dates which makes it efficient.

MASTER THESIS ABSTRACT TITLE: Trading Strategy in Incomplete Market AUTHOR: Tomáš Bunčák DEPARTMENT: Department of Probability and Mathematical Statistics, Charles University in Prague SUPERVISOR: Andrea Karlová We focus on the problem of finding optimal trading strategies (in a meaning corresponding to hedging of a contingent claim) in the realm of incomplete markets mainly. Although various ways of hedging and pricing of contingent claims are outlined, main subject of our study is the so-called mean-variance hedging (MVH). Sundry techniques used to treat this problem can be categorized into two approaches, namely a projection approach (PA) and a stochastic control approach (SCA). We review the methodologies used within PA in diversely general market models. In our research concerning SCA, we examine the possibility of using the methods of optimal stochastic control in MVH, and we study the problem of our interest in several settings of market models; involving cases of pure diffusion models and a jump- diffusion case. In order to reach an exemplary comparison, we provide solutions of the MVH problem in the setting of the Heston model via techniques of both of the approaches. Some parts of the thesis are accompanied with numerical illustrations.