In this thesis, we survey some popular pricing methods of average options. They can be classified into three cateogries include approximation, Monte Carlo, and binomial tree approaches. We examine the accuracy of these methods by two cases, exchange rate and stock price. Numerical testing results show the accuracy of approximation and binomial tree are not stable. For the big-size feature of average option, their outputs are doubtful and damaging in pactice. Despite this, they are still valuable. This is because they own the other advantages. For example, the approximation approach can give us a quick formlas to calculate the Greek, and the binomial tree approach can price the American style options.

This dissertation looks at various specific applications of stochastic processes in finance. The motivation for this work has been the work on the valuation of the price of an Asian option by Rogers and Shi (1995). Here, we look at functions of integrals of log - Gaussian processes to obtain approximations to the prices of various financial instruments. We look at pricing of bonds and payments contingent on the interest rate. The interest rate is assumed to be log - Gaussian, thus ensuring that it does not go negative. Obtaining the exact price might not be easy in all cases - hence we use of a combination of a conditioning argument and Jensen's inequality to obtain the lower bound to the prices of the bond as well as payments contingent on interest rates. We look at single driver models as well as multi-driver models. We also look at bonds where default is possible. We try to provide a mathematical justification for the choice of the conditioning factor used throughout the thesis to approximate the price of bonds and options. This is similar to the approach used by Rogers and Shi (1995) to valuing an Asian option; but they had provided no mathematical justification. Another part of this dissertation deals with the problem of pricing European call options on stochastically volatile assets. Further, the price and the volatility processes are in general correlated amongst themselves. Obtaining an exact price is quite involved and computation intensive. Most of the previous work in this field has been based on the solution to a system of partial differential equations. As in the case of pricing bonds, here too, we use a conditioning argument to obtain an approximation to the prices. This method is much faster and less computation intensive. We look at the situations of fixed and stochastic interest rates separately and in each case, we look at the volatility process following a simple Brownian motion and an Ornstein Uhlenbeck process. We also look at the value of stop - loss reinsurance contract for the case of a doubly stochastic Poisson process. Finally, we look at an alternative method of pricing bonds and Asian options. This is done by using a direct expansion and thus avoids the numerical integration that is used in the earlier chapters.

<p>Asian options are an important family of derivative contracts with a wide variety of applications in commodity, currency, energy, interest rate, equity and insurance markets. In this master's thesis, we investigate methods for evaluating the price of the Asian call options with a fixed strike. One of them is the Monte Carlo method. The accurancy of this method can be observed through variance of the price. We will see that the variance with using Monte Carlo method has to be decreased. The Variance Reduction technique is useful for this aim. We will give evidence of the efficiency of one of the Variance Reduction thechniques - Control Variate method - in a mathematical context and a numerical comparison with the ordinary Monte Carlo method.</p>

Asian options are an important family of derivative contracts with a wide variety of applications in commodity, currency, energy, interest rate, equity and insurance markets. In this master's thesis, we investigate methods for evaluating the price of the Asian call options with a fixed strike. One of them is the Monte Carlo method. The accurancy of this method can be observed through variance of the price. We will see that the variance with using Monte Carlo method has to be decreased. The Variance Reduction technique is useful for this aim. We will give evidence of the efficiency of one of the Variance Reduction thechniques - Control Variate method - in a mathematical context and a numerical comparison with the ordinary Monte Carlo method.

An Asian option is a path-dependent option whose payoff depends on the average of the underlying asset price over a certain time interval. It can be European or American. The time interval can be the entire interval of the option's life from the initiation to the expiration, or beginning from some time later than the initiation until the option's expiration. The average can be arithmetic or geometric. This paper derives a closed-form solution for the valuation of European Geometric average fixed strike Asian call option and a closed-form solution for the valuation of a forward-starting Asian call option with arithmetic average floating strike. The valuation formula is obtained by relying upon a slight linear approximation. And the valuation formula of Asian call option with arithmetic average floating strike we have derived is different from that of L. Bouaziz, E. Briys and M. Crouhy (1994). We believe that our argument here is correct, and theirs is wrong.

This thesis empirically evaluates a geometric Brownian motion and a stochastic volatility model for modeling futures prices and hedging Asian call options on the electricity spot price. Estimation of parameters for the models is done based on historical futures prices of futures contracts with a one month delivery period using nonlinear regression and Maximum Likelihood techniques. The models are tested on 2014 data and tracking error for each model is presented. The tracking error is investigated through the median value, the spread between minimum and maximum value along with value at risk at a 95% level. In addition, a third model for modeling spot and futures prices is presented theoretically. It is an exponential additive model with the advantage that it models the future price process from the spot price, instead of modeling the future price process immediately. This bypasses the issue of no information about the future price process during the delivery period, when there is no prices of the futures contracts. The aim of this thesis is to compare the simpler geometric Brownian motion to the more complex stochastic volatility model. It is found that the stochastic volatility model performs better when tested on out-of-sample data. The geometric Brownian motion tends to underestimate the electricity prices, despite that 2014 had low pricest compared to the other years in the data sample. In addition, the approximation of the distribution of the future price process under the geometric Brownian motion model gave a bad fit and led to difficulties when estimating the parameters. The stochastic volatility model produced more stable results and gave a better fit for the distribution.

Arithmetic Asian options are difficult to price and hedge, since, at the present, no closed-form analytical solution exists to price them. This difficulty, moreover, has led to the development of various methods and models used to price these instruments. The purpose of this thesis is two-fold. First, we present an overview of the literature. Secondly, we develop a pseudo-analytical method proposed by Geman and Yor and present an accurate and relatively quick algorithm which can be used to price European-style arithmetic Asian options and their hedge parameters. / Master of Science

An option is a contract between two parties where the holder has the option to buy or sell some underlying asset after a predefined exercise time. Options where the holder only has the right to buy or sell at the exercise time is said to be of European-style, while options that can be exercised any time before the exercise time is said to be of American-style. Asian options are options where the payoff is determined by some average value of the underlying asset, e.g., the arithmetic or the geometric average. For arithmetic Asian options, there are no closed-form pricing formulas, and one must apply numerical methods. Several methods have been proposed and tested for Asian options. For example, the Monte Carlo method isslowforEuropean-styleAsianoptionsandnotapplicableforAmerican-styleAsian options. In contrast, the finite difference method have successfully been applied to price both European- and American-style Asian options. But from a financial point of view, one is also interested in different measures of sensitivity, called the Greeks, which are hard approximate with the finite difference method. For more accurate approximations of the Greeks, researchers have turned to the finite element method with promising results for European-style Asian options. However, the finite element method has never been applied to American-style Asian options, which still lack accurate approximations of the Greeks. Here we present a study of pricing European- and American-style Asian options using the finite element method. For European-style options, we consider two different pricing PDEs. The first equation we consider is a convection-dominated problem, which we solve by applying the so-called streamline-diffusion method. The second equation comes from modelling Asian options as options on a traded account, which we solve by using the so-called cG(1)cG(1) method. For American-style options, the model based on options on a traded account is not applicable. Therefore, we must consider the first convection-dominated problem. To handle American-style options, we study two different methods, a penalty method and the projected successive over-relaxation method. For European-style Asian options, both approaches give good results, but the model based on options on a traded account show more accurate results. For American-style Asian options, the penalty method give accurate results. Meanwhile, the projected successive over-relaxation method does not converge properly for the tested parameters. Our result is a first step towards an accurate and fast method to calculate the price and the Greeks of both European- and American-style Asian options. Because good estimations of the Greeks are crucial when hedging and trading of options, we anticipate that the ideas presented in this work can lead to new ways of trading with Asian options.

In this thesis we investigate Asian oating strike options. We particu-larly focus on options with early exercise - American options. This typeof options are very lucrative to the end-users of commodities or ener-gies who are tend to be exposed to the average prices over time. Asianoptions are also very popular with corporations, who have ongoing cur-rency exposures. The main idea of the pricing is to examine the freeboundary position on which the value of the option is depending. Wefocus on developing a ecient numerical algorithm for this boundary.In the rst Chapter we give an informative description of the nancialderivatives including Asian options. The second Chapter is devoted tothe analytical derivation of the corresponding partial dierential equa-tion coming from the original Black - Scholes equation. The problemis simplied using transformation methods and dimension reduction. Inthe third and fourth Chapter we describe important numerical methodsand discretize the problem. We use the rst order Lie splitting and thesecond order Strang splitting. Finally, in the fth Chapter we makenumerical experiments with the free boundary and compare the resultwith other known methods.

Asian options are exotic financial derivative products which price must be calculated by numerical evaluation. In this thesis, we study certain ways of solving partial differential equations, which are associated with these derivatives. Since standard numerical techniques for Asian options are often incorrect and impractical, we discuss their variations, which are efficiently applicable for handling frequent numerical instabilities reflected in form of oscillatory solutions. We will show that this crucial problem can be treated and eliminated by adopting flux limiting techniques, which are total variation dimishing.