Dynamic options pricing with variance reduction for limited and unlimited price stochastic procesess

Innovation is in the heart of business. An improvement and development process is rapidly evolving on every area which includes financial markets. The most innovative product among a number of different assets in financial markets is options. Options are fully featured deriative products which can serve as a tool for many targets. First of all options make financial markets deeper. Investors would be convinced to come and make investments on risky assets by existence of a related option. This helps to enlarge daily trade amount in financial markets. Secondly options are dedicated to a wide scope of investors attitude. Options allow the holder to have protection against market risk while offering higher profit. Market risk refers to a lower investment value on behalf of unstationary price movements. Options could be exercised immediately in case of any unexpected market condition to refresh the present value of portfolios. Mainly options are purchased in order to fix the underlying price for a reasonable time period which gives a safeguard to the owner of the option. Options provide to control the risk level of investment portfolios when traded for risk aversion purposes. Moreover option portfolios may result in superior returns when expectations are realized. Leverage property of options allow to have higher profit. These aspects make options one of the most traded assets in financial markets. Trade volume of options accelerated in global markets and passed the stock exchange and forex market volumes one by one. An option is simply a contract between two sides about trading another asset at a future time. One side is the writer that fixes the terms of the option contract and sells the option. The other side is the holder who purchases the option for the market price namely option premium. At the beginning option premium is a kind of cost for the holder and it is a kind of profit for the writer. Whereas final score could be seen at the exercise date or at the expiry date. If market conditions grow up in accordance with the holder's expectations, the holder can exercise the option and get some payoff. At the exercise date writer has to pay the payoff to the holder. Whilst there is no obligation for the holder to exercise the option and holder does not need to pay any more to the writer at the exercise date. Consequently the balance may change at the exercise date. Only one side of option contract can get profit. That's why all possible scenarios should be taken into consideration while determining the fair value of option premium. Fair value of the option price can be calculated by some mathematical and numerical methods. It is impossible to specify the probability distribution of risky asset returns. Options pricing is based on a different probability space that calculations are done with respect to arbitrage free principle and risk neutral pricing. In this probability space asset price is simply an expectation of the discounted measure of its terminal value. The expectation is taken under an equivalent (semi) martingale measure which is a mapping of original probability distribution of the asset returns. Arbitrage free principle provides existence of an equivalent martingale measure which is not unique if the market is incomplete. The market price and the fair value of the option may be different. This is an indicator for investors about the market price is whether cheap or expensive. Trade decisions are considered in this respect. Therefore fair value should be calculated accurately to have a correct decision. Options pricing includes highly complicated mathematical and statistical calculations. Black-Scholes model gives a closed form solution for standart options. It is assumed that logarithmic returns have an equivalent normal distribution in Black-Scholes model. Hence, distribution of the asset price becomes log-normal. Other options pricing methods are Binomial lattices, finite difference methods and Monte Carlo Integration technique. Especially Monte Carlo Integration technique appears to be the unique way to price some exotic options like Barrier options and Quanto options. Barrier options may be in somehow unlimited number of different kinds like up and knock in, up and knock out or down and knock in etc. It is possible to have two or three or more barriers. Quanto options allow to pay in a different currency at the exercise date. Monte Carlo Integration technique is a common way used to price exotic options. Nevertheless most of the options traded in global markets are exotic styles. Options are traded in huge amounts. Little differences in options premium may result in bigger transactions. The error term in Monte Carlo calculations should be reduced in order to have more accurate results. The error term is measured with the Variance or Mean Squared Error. Any variance reduction acchieved in Monte Carlo applications constitutes smaller confidence intervals. Importance Sampling technique can be occupied in an attempt to reduce variance in financial applications. Moreover Importance Sampling method can be applied together with any other variance reduction techniques like Antithetic Variates. In developed financial markets firms and individuals seek new methods to minimize the risk arises from their transactions. At the same time there is a challenging competition to increase their market share. Research and Development facilities goes on continuosly to generate any kinds of competetive assets like derivatives. As a result whether new generation derivatives are frequently introduced to the markets or some regular rules are implemented in financial markets to feed the competitive environment. One of the regulations is price margins which is used to improve investors interest. Price margin is seen in some markets that asset prices are allowed to have some limited values during one day regular sessions. Namely asset price can increase up to a maximum level in a session and fall to a minimum level. Ceil and floor prices are calculated according to a base price architecture. The base price is simply weighted average value of the previous session. Margins are generally in ten percent intervals of the base price. It is expected that the market gets charming to the risk aversion investors. Although there are more than one reasons to apply such a rule in a market, mainly increase in the trade amount is desired. It is possible to prevent huge losses in the market with the floor price. Nevertheless unreasonable price movements are reduced to minimal with the ceil price. Those two prevent big manipulative attacks to the market which shows some regulations are needed. Price margin enables regular market processing and casuality in the market. Price movements should be caused by daily data and news from the market according to Efficient Market Hypothesis. Moreover trade decisions should be given after balance sheet analyses. Options allow investors to control the risk level when included to the portfolios. Huge amount of transactions in the options market makes options pricing one of the most important topics. Price margins should be implemented to the existing options pricing models with bounded stochastic price procesess. Very famous Black-Scholes model lightened options pricing process. Their assumptions have been analyzed enormously. By this PhD thesis it is provided to calculate option prices for the markets which have price margins. To the literature bounded stochastic price process is included into options pricing model. There are a number of options pricing models. First selected options price is calculated with Monte Carlo integration. Confidence interval for the price is reduced with the Importance Sampling technique. The results are collected for the two cases of price margin is in use and not in use. The different points of bounded and unbounded stochastic price proceses are visualized. Specially bounded log-normal process is examined in terms of variance reduction capability. Last but not the least volatility is taken into consideration. Low and high volatility situations are tested in order to see how the option price is effected with the stochastic price processes. As a result there is no sensible difference in unbounded and bounded stochastic price processes for low volatility. However it is observed that high volatility makes options prices clearly different for unbounded and bounded stochastic price processes. High volatility makes call option prices lower for bounded stochastic price processes whereas it makes put option prices higher. Financial options gain more importance day by day as high volatility is seen more frequently in financial markets. With this study it is aimed to contribute the options market which is developing level.