A closed-form pricing formula of European options is obtained by Fischer Black and Myron Scholes (1973). In such a European option, the payoff depends `linearly' on the underlying asset price at the expiration time. An power option has a payoff which depends nonlinearly on the underlying asset price at the expiration time by raising a certain exponent. In the Black-Scholes model, a closed-form formula of a power option is obtained by Esser (2004). This paper extends Esser's result to the generalized Black- Scholes model. That is, we derive a closed-form pricing formula of a power option in the case when both the interest rate and the stock volatility are time-dependent.

Asian option is an option which payoff depends on the average underlying price over some some specific time period. Although there is no closed form solution of asian option, appropriate change of variable and Num¡¦eraire would reduce some terms of equation satisfies the Asian call price function. This thesis presents asian option¡¦s properties and process of reduction terms.

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)].

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Since small and medium ¡Vsize enterprises (SMEs) have played a fundamental role in the economic development of this country, as a intermediary, the financial assistance of Taiwan SMEG to SMEs, is the greatest contribution to the above, nevertheless, Taiwan SMEG has been immersing in a predicament of financial deficit due to the unbalanced budget for a long time. Except the Package Credit Guarantee has introduced the concept of total risk control, all the others like the Authorized Approach and the Normal Approach have not introduced yet. For its perennial operation, Taiwan SMEG must set up an appropriate system for the rate of guarantee fees to achieve self-contained and self-sufficient condition, also to obtain the best efficiency under its limited resources. This paper uses the market ¡Vbased risk neutral model developed by Kuo (2006) to estimate the probability of default of banking loan assets through the risk premium of each banking loan, further to stimulate the guarantee fees by using the actuarial valuation principles. The purpose of this model is to react the degree of credit risk on the stimulated guarantee fees , making the fees pricing mechanism reasonable and fair . The model also analyzed the cost of subrogation payment under default cases by different industry , making it a reference for banks and Taiwan SMEG. The empirical results show that: 1.The credit guarantee fees stimulated by this model are approximately fair comparative to the actual subrogation payments of default cases reimbursed by Taiwan SMEG. The result provide evidence that the model possess the power of fitness for estimating the default cost of subrogation payment. 2.The degree of credit risk can actually react on the guarantee fees through using this model. It is essential to set up different range for rate of guarantee fees according to the subrogation payment ratio of default cost caused by respective guaranteed industry. The result suggests that Taiwan SMEG should amplify the extreme energy of guarantee through reasonable and fair use of its limited resources. Using the data of Taiwan SMEG guaranteed cases, this paper also analysis the default cost between the cost of funds and earning profits for the banking loans. The banks in Taiwan earn less profits than before under the overbanking environment. The competitive bank¡¦s loan pricing strategy leads to extreme loss while the default cases occur. In the lights of risk management , banks have become a high risk and low return industry. It is essential for banks to emphasize the loan quality and pricing strategy when expanding their loan business. Banks must simultaneously evaluate the RAROC under a perfect risk management system, so that the monetary environment can be improved and banks can take advantage of it by earning reasonable profits.

Investment-linked insurance in Taiwan has been listed for almost a decade since 2001. In 2002, after the big sales of the investment-linked insurance, the domestic insurance companies also joined the market. For the investment-linked insurance, the policyholders retain the protection of the life insurance as well as share the earnings of the investment. Since the main investment instruments of the investment-linked insurance are mutual funds, it is important to study how to optimally allocate the portfolio. This research consider the returns of the mutual funds under tree models assumption. The objective is to find the optimal portfolio which has minimum variance and attained a given expected return level. The problem is also known as mean-variance portfolio problem. In the empirical work, we study eleven daily mutual fund price data from Sep. 2007 to Nov. 2008. Using the data of the first 12 months, we first establish initial tree price models, then update the parameters of the tree model by the EWMAmethod. The optimal trading strategies of the mean-variance portfolio are investigated under this model setting. We class the mutual funds into three categories: equity funds, balanced funds and bond funds. Different combination of these three kinds of funds are considered to find the optimal trading strategy respectively. The results showed that the realized returns using this optimal trading strategy in practice is close to the pre-specified expected return level.

Financial market analysis nowadays constitutes an important pillar in central banks' monetary policy considerations. This is because the inherently forward-looking properties of asset prices can provide policy-makers with valuable information about future macroeconomic prospects, as seen through the eyes of investors. The five essays contained in this thesis elaborate upon three separate but complementary topics within the area of financial market research. First, the price discovery process of asset prices following releases of macroeconomic and monetary policy-related news is investigated. Such analysis can help in improving a central bank's understanding of how market participants update their views about future growth and inflation prospects. Second, an attempt is made to identify the factors which explain the time-varying co-movement of bond and stock prices. This analysis reveals that periods of negative correlation between the two assets tend to coincide with periods of very low investor risk appetite. Third, frequency distributions implied by options prices are often employed by central banks to assess the degree of uncertainty prevailing in markets as well as how the perceived balance of risks concerning future asset price movements is tilted. Various methods have been developed to estimate option-implied frequency distributions and the thesis assesses and compares the robustness of two of the most commonly used methods in central banks. / <p>Diss. Stockholm : Handelshögskolan, 2007 S. 9-16: sammanfattning, s. 17-160: 5 uppsatser</p>

In this thesis, we study the estimation of the nonlinear conditionalrisk-neutral density function (RND) in discrete time. Specifically, weevaluate the extent to which the estimated nonlinear conditional RNDvaluable insights to answer relevant economic questions regarding to optionpricing, the measurement of invertors' preferences and portfolio choice.We make use of large dataset of options contracts written on the S&P 500index from 1996 to 2011, to estimate the parameters of the conditional RNDfunctions by minimizing the squared option pricing errors delivered by thenonlinear models studied in the thesis.In the first essay, we show that a semi-nonparametric option pricing modelwith GARCH variance outperforms several benchmarks models in-sample andout-of-sample. In the second essay, we show that a simple two-state regimeswitching model in volatility is not able to fully account for the pricingkernel and the risk aversion puzzle; however, it provides a reasonablecharacterisation of the time-series properties of the estimated riskaversion.In the third essay, we evaluate linear stochastic discount factormodels using an out-of-sample financial metric. We find that multifactormodels outperform the CAPM when this metric is used, and that modelsproducing the best fit in-sample are also those exhibiting the bestperformance out-of-sample.

In this dissertation, we study the effects of emerging biomass markets on land use changes between alternatives of agricultural production, conventional timber production, and forest woody biomass production for energy use. Along with the uncertainty associated with woody biomass prices and rents, transaction costs incurred to land use play an important role in land allocation decisions and make this study distinct from other work. In Chapter 1, we introduce the background and objectives of our study. In Chapter 2, we analyze the behavior of a risk-neutral private landowner and social planner under uncertainty of woody biomass prices, assuming that there is a market emergence at some unknown time point in the future. Market emergence is characterized by a price jump and a certain timing of the price jump. Six different price jumps and five different timings of bioenergy market emergence are adopted to study their collective effects on land use change between agriculture and forestry. Chapter 3 studies this problem for a risk-averse private landowner. Two measures of relative risk aversion are used to examine how a landowner's preference may affect his or her land use decision. In Chapter 2, we find that, for three different quality categories of land, land rents from forestry increase significantly for higher price jumps and decreases in the length of time until bioenergy market emergence. One of the most important results is concerned with the presence of transaction costs. Here, we find that these costs may require unrealistic market emergence scenarios to lead to bioenergy adoption on any large scale. This result is even more likely with nonlinear transaction costs. Land allocation decisions in Chapter 3 are distinctly different from those in Chapter 2, due to the introduction of landowner risk aversion. In certain market emergence cases, some land units retain in agriculture entirely when the landowner is risk averse . The Chapter 4 studies a stochastic optimization problem of land use, assuming that woody biomass rents follow a stochastic diffusion called geometric Brownian motion that is later discretized by a binomial option pricing approach. The problems in Chapters 2 and 3 assume that the landowner must make all decisions at the beginning of his or her time horizon. This assumption is relaxed in Chapter 4. Now, the landowner is allowed to revise his or her land allocation decision among three alternatives over time as information about market emergence is collected. We observe that the different forms of transaction costs are not as significant as in Chapters 2 and 3. However, different values of volatility of forest biomass rents give rise to different land allocation decisions, especially for the land of high quality. / Ph. D.

Supply and demand in the World oil market are balanced through responses to price movement with considerable complexity in the evolution of underlying supply-demand expectation process. In order to be able to understand the price balancing process, it is important to know the economic forces and the behavior of energy commodity spot price processes. The relationship between the different energy sources and its utility together with uncertainty also play a role in many important energy issues. The qualitative and quantitative behavior of energy commodities in which the trend in price of one commodity coincides with the trend in price of other commodities, have always raised the questions regarding their interactions. Moreover, if there is any interaction, then one would like to know the extent of influence on each other. In this work, we undertake the study to shed a light on the above highlighted processes and issues. The presented study systematically deals with the development of stochastic dynamic models and mathematical, statistical and computational analysis of energy commodity spot price and interaction processes. Below we list the main components of the research carried out in this dissertation. (1) Employing basic economic principles, interconnected deterministic and stochastic models of linear log-spot and expected log-spot price processes coupled with non-linear volatility process are initiated. (2) Closed form solutions of the models are analyzed. (3) Introducing a change of probability measure, a risk-neutral interconnected stochastic model is derived. (4) Furthermore, under the risk-neutral measure, expectation of the square of volatility is reduced to a continuous-time deterministic delay differential equation. (5) The by-product of this exhibits the hereditary effects on the mean-square volatility process. (6) Using a numerical scheme, a time-series model is developed and utilized to estimate the state and parameters of the dynamic model. In fact, the developed time-series model includes the extended GARCH model as special case. (7) Using the Henry Hub natural gas data set, the usefulness of the linear interconnected stochastic models is outlined. (8) Using natural and basic economic ideas, interconnected deterministic and stochastic models in (1) are extended to non-linear log-spot price, expected log-spot price and volatility processes. (9) The presented extended models are validated. (10) Closed form solution and risk-neutral models of (8) are outlined. (11) To exhibit the usefulness of the non-linear interconnected stochastic model, to increase the efficiency and to reduce the magnitude of error, it was essential to develop a modified version of extended Kalman filtering approach. The modified approach exhibits the reduction of magnitude of error. Furthermore, Henry Hub natural gas data set is used to show the advantages of the non-linear interconnected stochastic model. (12) Parameter and state estimation problems of continuous time non-linear stochastic dynamic process is motivated to initiate an alternative innovative approach. This led to introduce the concept of statistic processes, namely, local sample mean and sample variance. (13) Then it led to the development of an interconnected discrete-time dynamic system of local statistic processes and (14) its mathematical model. (15) This paved the way for developing an innovative approach referred as Local Lagged adapted Generalized Method of Moments (LLGMM). This approach exhibits the balance between model specification and model prescription of continuous time dynamic processes. (16) In addition, it motivated to initiate conceptual computational state and parameter estimation and simulation schemes that generates a mean square sub-optimal procedure. (17) The usefulness of this approach is illustrated by applying this technique to four energy commodity data sets, the U. S. Treasury Bill Yield Interest Rate and the U.S. Eurocurrency Exchange Rate data sets for state and parameter estimation problems. (18) Moreover, the forecasting and confidence-interval problems are also investigated. (19) The non-linear interconnected stochastic model (8) was further extended to multivariate interconnected energy commodities and sources with and without external random intervention processes. (20) Moreover, it was essential to extend the interconnected discrete-time dynamic system of local sample mean and variance processes to multivariate discrete-time dynamic system. (21) Extending the LLGMM approach in (15) to a multivariate interconnected stochastic dynamic model under intervention process, the parameters in the multivariate interconnected stochastic model are estimated. These estimated parameters help in analyzing the short term and long term relationship between the energy commodities. These developed results are applied to the Henry Hub natural gas, crude oil and coal data sets.