Abstract not available

published_or_final_version / Economics and Finance / Master / Master of Philosophy

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

Research conducted in mathematical finance focuses on the quantitative modeling of financial markets. It allows one to solve financial problems by using mathematical methods and provides understanding and prediction of the complicated financial behaviors. In this thesis, efforts are devoted to derive and extend stochastic optimization models in financial economics and establish practical algorithms for representing and solving problems in mathematical finance. An option gives the holder the right, but not the obligation, to buy or sell an underlying asset at a specified strike price on or before a specified date. In this thesis, a valuation model for a perpetual convertible bond is developed when the price dynamics of the underlying share are governed by Markovian regime-switching models. By making use of the relationship between the convertible bond and an American option, the valuation of a perpetual convertible bond can be transformed into an optimal stopping problem. A novel approach is also proposed to discuss an optimal inventory level of a retail product from a real option perspective in this thesis. The expected present value of the net profit from selling the product which is the objective function of the optimal inventory problem can be given by the actuarial value of a real option. Hence, option pricing techniques are adopted to solve the optimal inventory problem in this thesis. The goal of risk management is to eliminate or minimize the level of risk associated with a business operation. In the risk measurement literature, there is relatively little amount of work focusing on the risk measurement and management of interest rate instruments. This thesis concerns about building a risk measurement framework based on some modern risk measures, such as Value-at-Risk (VaR) and Expected Shortfall (ES), for describing and quantifying the risk of interest rate sensitive instruments. From the lessons of the recent financial turmoils, it is understood that maximizing profits is not the only objective that needs to be taken into account. The consideration for risk control is of primal importance. Hence, an optimal submission problem of bid and ask quotes in the presence of risk constraints is studied in this thesis. The optimal submission problem of bid and ask quotes is formulated as a stochastic optimal control problem. Portfolio management is a professional management of various securities and assets in order to match investment objectives and balance risk against performance. Different choices of time series models for asset price may lead to different portfolio management strategies. In this thesis, a discrete-time dynamic programming approach which is flexible enough to deal with the optimal asset allocation problem under a general stochastic dynamical system is explored. It’s also interesting to analyze the implications of the heteroscedastic effect described by a continuous-time stochastic volatility model for evaluating risk of a cash management problem. In this thesis, a continuous-time dynamic programming approach is employed to investigate the cash management problem under stochastic volatility model and constant volatility model respectively. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Decreasing block pricing refers to the practice of selling a product at successively lower marginal prices as the amount purchased in any one time period increases. In more familiar terms, this practice can be thought of as any quantity discount scheme as long as marginal price does not vary continuously with quantity. Decreasing block pricing results in a faceted, non-convex budget set, and under standard assumptions concerning consumer preferences, yields several nonstandard theoretical implications. The central goal of this paper is to formulate an estimation technique which is consistent with these implications. When the budget set is not convex, the uniqueness of consumer equilibrium is no longer guaranteed. It also follows that discontinuities in demand occur whenever consumer equilibrium shifts from one facet of the budget constraint to another. Prior empirical studies have not made use of demand functions consistent with these results. In Chapter 2, a utility-maximizing algorithm was developed to determine consumer equilibrium given the declining block pricing schedule and income for a Cobb-Douglas utility function. In developing this algorithm, it was made clear that the proper approach for estimating individual demand was through the use of a block-dependent independent variable. The coefficient of this block-department independent variable provided an estimate of a utility function parameter which completely specified the Cobb-Douglas form. Incorporating this utility function estimate into the utility-maximation algorithm made it possible to obtain estimates of consumption given changes in any or all of the rate schedule components. While the use of a block-dependent independent variable is the theoretically correct method for estimating demand, it poses an inescapable problem of errors-in-variables. A Monte Carlo study was performed in Chapter 2 to investigate, among other things, the seriousness of the errors-in-variables bias. The results were quite encouraging. When using data incorporating extremely large error variances, amazingly precise estimates were obtained. Another encouraging Monte Carlo result was when comparing samples not containing a discontinuity with those with one, it was found that the latter produced estimates with statistically significant superiority. Chapter 3 generalized the estimation technique of the previous chapter to allow the estimation of demand using cross-sectional data. The data base recorded monthly electricity consumption for households from a number of cities whose utilities had decreasing block rates. Seven of these cities were selected for analysis. The data also included various demographic characteristics and electric appliance stock information. The generalization was accomplished by assuming that all households had a Stone-Geary utility function. Also, the utility function parameter representing the minimum required quantity of electricity was assumed to depend linearly on the household's appliance stock and demographic characteristics. This allowed demand to vary across households on the basis of this parameter and income. The results of applying this regression technique to the cross-sectional data were then compared with results from a conventional, non-theoretically based demand specification. The data were used in pooled and individual month form with the former yielding much better statistical results. The Stone-Geary form provided a greater number of significant coefficients for price and income variables than the conventional version. The predominant failure of the conventional version was that the coefficient of marginal price was rarely significant and when significant, frequently of the wrong sign. For the same samples, the Stone-Geary results were quite acceptable except for the regressions involving one of the cities. Thus, it was demonstrated that a method consistent with the theoretical implications of decreasing block pricing is easily applied to cross-sectional data and produces better results than conventional techniques.

We present a tree algorithm, called the willow tree, for financial derivative pricing. The setup of the tree uses a fixed number of spatial nodes at each time step. The transition probabilities are determine by solving linear programming problems. The willow tree method is radically superior in numerical performance when compared to the binomial tree method.

The present thesis provides a definite answer to the question of how comparisons of certain aggregate quantities and price levels should be made across two or more geographic regions. It does so from the viewpoint of both economic theory and the “test” (or “axiomatic”) approach to index-number theory. Chapter 1 gives an overview of the problem of multilateral interspatial comparisons and introduces the rest of the thesis. Chapter 2 focuses on a particular domain of comparison involving consumer goods and services, countries and households in developing a theory of international comparisons in terms of the the (Kontis-type) cost-of-living index. To this end, two new classes of purchasing power parity measures are set out and the relationship between them is explored. The first is the many-household analogue of the (single-household) cost-of-living index and, as such, is rooted in the theory of group cost-of-living indexes. The second Consists of sets of (nominal) expenditure-share deflators, each corresponding to a system of (real) consumption shares for a group of countries. Using this framework, a rigorous exact index- number interpretation for Diewert’s “own-share” system of multilateral quantity indexes is provided. Chapter 3 develops a novel multilateral test approach to the problem at hand by generalizing Eichhorn and Voeller’s bilateral counterpart in a sensible manner. The equivalence of this approach to an extended version of Diewert’s multilateral test approach is exploited in an assessment of the relative merits of several alternative multilateral comparison formulae motivated outside the test-approach framework. Chapter 4 undertakes an empirical comparison of the formulae examined on theoretical grounds in Chapter 3 using an appropriate cross-sectional data set constructed by the Eurostat—OECD Purchasing Power Parity Programme. The principal aim of this comparison is to ascertain the magnitude of the effect of choosing one formula over another. In aid of this, a new indicator is proposed which facilitates the measurement of the difference between two sets of purchasing power parities, each computed using a different multilateral index-number formula.

This dissertation is in the form of three essays on the topic of component and long memory GARCH models. The unifying feature of the thesis is the focus on investigating European index option evaluation using these models. / The first essay presents a new model for the valuation of European options. In this model, the volatility of returns consists of two components. One of these components is a long-run component that can be modeled as fully persistent. The other component is short-run and has zero mean. The model can be viewed as an affine version of Engle and Lee (1999), allowing for easy valuation of European options. The model substantially outperforms a benchmark single-component volatility model that is well established in the literature. It also fits options better than a model that combines conditional heteroskedasticity and Poisson normal jumps. While the improvement in the component model's performance is partly due to its improved ability to capture the structure of the smirk and the path of spot volatility, its most distinctive feature is its ability to model the term structure. This feature enables the component model to jointly model long-maturity and short-maturity options. / The second essay derives two new GARCH variance component models with non-normal innovations. One of these models has an affine structure and leads to a closed-form option valuation formula. The other model has a non-affine structure and hence, option valuation is carried out using Monte Carlo simulation. We provide an empirical comparison of these two new component models and the respective special cases with normal innovations. We also compare the four component models against GARCH(1,1) models which they nest. All eight models are estimated using MLE on S&P500 returns. The likelihood criterion strongly favors the component models as well as non-normal innovations. The properties of the non-affine models differ significantly from those of the affine models. Evaluating the performance of component variance specifications for option valuation using parameter estimates from returns data also provides strong support for component models. However, support for non-normal innovations and non-affine structure is less convincing for option valuation. / The third essay aims to investigate the impact of long memory in volatility on European option valuation. We mainly compare two groups of GARCH models that allow for long memory in volatility. They are the component Heston-Nandi GARCH model developed in the first essay, in which the volatility of returns consists of a long-run and a short-run component, and a fractionally integrated Heston-Nandi GARCH (FIHNGARCH) model based on Bollerslev and Mikkelsen (1999). We investigate the performance of the models using S&P500 index returns and cross-sections of European options data. The component GARCH model slightly outperforms the FIGARCH in fitting return data but significantly dominates the FIHNGARCH in capturing option prices. The findings are mainly due to the shorter memory of the FIHNGARCH model, which may be attributed to an artificially prolonged leverage effect that results from fractional integration and the limitations of the affine structure.

Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closed-form solutions for option prices. However, relatively little is known about the empirical shortcomings of this model. In the first essay, we investigate alternatives to the SQR model, by comparing its empirical performance with that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources. We first use realized volatilities to assess the properties of the SQR model and to guide us in the search for alternative specifications. We then estimate the models using maximum likelihood on a long sample of S& P500 returns. Finally, we employ nonlinear least squares on a time series of cross sections of option data. In the estimations on returns and options data, we use the particle filtering technique to retrieve the spot volatility path. The three sources of data we employ all point to the same conclusion: the SQR model is misspecified. Overall, the best of alternative volatility specifications is a model we refer to as the VAR model, which is of the GARCH diffusion type. / In the second essay, we estimate the Constant Elasticity of Variance (CEV) model in order to study the level of nonlinearity in the volatility dynamic. We also estimate a CEV process combined with a jump process (CEVJ) and analyze the effects of the jump component on the nonlinearity coefficient. Estimation is performed using the particle filtering technique on a long series of S&P500 returns and on options data. We find that both returns data and returns-and-options data favor nonlinear specifications for the volatility dynamic, suggesting that the extensive use of linear models is not supported empirically. We also find that the inclusion of jumps does not affect the level of nonlinearity and does not improve the CEV model fit. / The third essay provides an empirical comparison of two classes of option valuation models: continuous-time models and discrete-time models. The literature provides some theoretical limit results for these types of dynamics, and researchers have used these limit results to argue that the performance of certain discrete-time and continuous-time models ought to be very similar. This interpretation is somewhat contentious, because a given discrete-time model can have several continuous-time limits, and a given continuous-time model can be the limit for more than one discrete-time model. Therefore, it is imperative to investigate whether there exist similarities between these specifications from an empirical perspective. Using data on S&P500 returns and call options, we find that the discrete-time models investigated in this paper have the same performance in fitting the data as selected continuous-time models both in and out-of-sample.

This research studies normative pricing strategies for information technology (IT) used by suppliers to supplement an underlying primary good. Transactions with consumers and customer firms are considered. Characteristics of IT are divided into IT impacts on customers, and IT impacts on suppliers. IT impacts on customers include vertical differentiation or reduced turnover costs for the primary good, and positive IT adoption costs. IT impacts on suppliers include reduced production costs for the primary good, and the costs of IT. Optimal pricing for the IT and the primary good is modelled for monopoly, and Bertrand competition based on IT and the primary good is modelled for oligopoly. Two part tariffs are used for the IT and IT enhanced primary good. Results of pricing to consumers show that the fixed component of an optimal (or equilibrium) two part tariff can either be a net tax or a net subsidy, confirming the possibility of taxed or subsidized IT adoption. For the monopolist offering the IT and IT enhanced primary good only, the consumer's adoption/switching cost limits the possible subsidy. Consistent with previous economics research, in a duopoly where one supplier has IT, the IT supplier abandons the original primary good. Two suppliers with identical IT cannot attain a positive profit equilibrium. Analogous results obtain for a special case of pricing to customer firms. Empirical results support differential (premium) pricing for an IT enhanced primary good over an original good. / Business, Sauder School of / Operations and Logistics (OPLOG), Division of / Graduate