In this paper we investigate the potential of the analysis of noisy non-stationary time series by quantizing it into streams of discrete symbols and applying finite-memory symbolic predictors. The main argument is that careful quantization can reduce the noise in the time series to make model estimation more amenable given limited numbers of samples that can be drawn due to the non-stationarity in the time series. As a main application area we study the use of such an analysis in a realistic setting involving financial forecasting and trading. In particular, using historical data, we simulate the trading of straddles on the financial indexes DAX and FTSE 100 on a daily basis, based on predictions of the daily volatility differences in the underlying indexes. We propose a parametric, data-driven quantization scheme which transforms temporal patterns in the series of daily volatility changes into grammatical and statistical patterns in the corresponding symbolic streams. As symbolic predictors operating on the quantized streams we use the classical fixed-order Markov models, variable memory length Markov models and a novel variation of fractal-based predictors introduced in its original form in (Tino, 2000b). The fractal-based predictors are designed to efficiently use deep memory. We compare the symbolic models with continuous techniques such as time-delay neural networks with continuous and categorical outputs, and GARCH models. Our experiments strongly suggest that the robust information reduction achieved by quantizing the real-valued time series is highly beneficial. To deal with non-stationarity in financial daily time series, we propose two techniques that combine ``sophisticated" models fitted on the training data with a fixed set of simple-minded symbolic predictors not using older (and potentially misleading) data in the training set. Experimental results show that by quantizing the volatility differences and then using symbolic predictive models, market makers can generate a statistically significant excess profit. However, with respect to our prediction and trading techniques, the option market on the DAX does seem to be efficient for traders and non-members of the stock exchange. There is a potential for traders to make an excess profit on the FTSE 100. We also mention some interesting observations regarding the memory structure in the studied series of daily volatility differences. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

This thesis explores numerical methods for solving nonlinear partial differential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and efficient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep. Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions. In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived. Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems. In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques. In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems. Finally, two applications are briefly explored. The first application involves static hedging to reduce the bid-ask spread implied by uncertain volatility pricing. While static hedging has been carried out previously for one factor models, examples for two factor models are provided. The second application uses passport option theory to examine trader compensation strategies. By changing the payoff, it is shown how the expected distribution of trading account balances can be modified to reflect trader or bank preferences.

This thesis addresses two main issues: first, forecasting short-term electricity market prices; and second, the application of short-term electricity market price forecasts to operation planning of demand-side Bulk Electricity Market Customers (BEMCs). The Ontario electricity market is selected as the primary case market and its structure is studied in detail. A set of explanatory variable candidates is then selected accordingly, which may explain price behavior in this market. In the process of selecting the explanatory variable candidates, some important issues, such as direct or indirect effects of the variables on price behavior, availability of the variables before real-time, choice of appropriate forecasting horizon and market time-line, are taken into account. Price and demand in three neighboring electricity markets, namely, the New York, New England, and PJM electricity markets, are also considered among the explanatory variable candidates. Electricity market clearing prices in Ontario are calculated every five minutes. However, the hourly average of these 5-minute prices, referred to as the Hourly Ontario Energy Price (HOEP), applies to most Ontario market participants for financial settlements. Therefore, this thesis concentrates on forecasting the HOEP by employing various linear and non-linear modeling approaches. The multivariate Transfer Function (TF), the multivariate Dynamic Regression (DR), and the univariate Auto Regressive Integrated Moving Average (ARIMA) are the linear time series models examined. The non-linear approaches comprise the Multivariate Adaptive Regression Splines (MARS), and the Multi-Layer Perceptron (MLP) neural networks. Multivariate HOEP models are developed considering two forecasting horizons, i.e. 3 hours and 24 hours, taking into account the case market time-line and the ability of market participants to react to the generated forecasts. Univariate ARIMA models are also developed for day-ahead market prices in the three neighboring electricity markets. The developed models are used to generate price forecasts for low-demand, summer peak-demand, and winter peak-demand periods. The HOEP forecasts generated in this work are significantly more accurate than any other available forecast. However, the accuracy of the generated HOEP forecasts is relatively lower than those of the price forecasts for Ontario's neighboring electricity markets. The low accuracy of the HOEP forecasts is explained by conducting a price volatility analysis across the studied electricity markets. This volatility analysis reveals that the Ontario electricity market has the most volatile prices compared to the neighboring electricity markets. The high price volatility of the Ontario electricity market is argued to be the direct result of the real-time nature of this market. It is further observed that the inclusion of the just-in-time publicly available data in multivariate HOEP models does not improve the HOEP forecast accuracy significantly. This lack of significant improvement is attributed to the information content of the market data which are available just-in-time. The generated HOEP forecasts are used to plan the short-term operation of two typical demand-side case-study BEMCs. The first case-study BEMC is a process industry load with access to on-site generation facilities, and the second one is a municipal water plant with controllable electric demand. Optimization models are developed for the next-day operation of these BEMCs in order to minimize their total energy costs. The optimization problems are solved when considering market price forecasts as the expected future prices for electricity. The economic impact of price forecast inaccuracy on both the case study is analyzed by introducing the novel Forecast Inaccuracy Economic Impact (FIEI) index. The findings of this analysis show that electricity market price forecasts can effectively be used for short-term scheduling of demand-side BEMCs. However, sensitivity to price forecast inaccuracy significantly varies across market participants. In other words, a set of price forecasts may be considered ``accurate enough'' for a customer, while leading to significant economic losses for another.

In this thesis I discuss the method of time-change and its applications in quantitative finance. I mainly consider the time change by writing a continuous diffusion process as a Brownian motion subordinated by a subordinator process. I divide the time change method into two cases: deterministic time change and stochastic time change. The difference lies in whether the subordinator process is a deterministic function of time or a stochastic process of time. Time-changed Brownian motion with deterministic time change provides a new viewpoint to deal with option pricing under stochastic interest rates and I utilize this idea in pricing various exotic options under stochastic interest rates. Time-changed Brownian motion with stochastic time change is more complicated and I give the equivalence in law relation governing the ``original time" and the ``new stochastic time" under different clocks. This is readily applicable in pricing a new product called ``timer option". It can also be used in pricing barrier options under the Heston stochastic volatility model. Conclusion and further research directions in exploring the ideas of time change method in other areas of quantitative finance are in the last chapter.

This study seeks to assess the correlation which exists between exchange rate of Nigerian naira and Unites States dollar and oil price on the basis of monthly data from 1999-2009. The research employ the fundamental variables which were assumed to be the monthly spot crude oil price, monthly exchange rate of Nigeria naira and monthly exchange rate of United States dollar. The empirical result adopted the ordinary least square using regression analysis and also the correlation model which shows that there is a weak/negative relationship between exchange rate and oil price as there are other factors that brings about changes in oil price other than the exchange rate. The activities of cartel pricing policy and oil speculators too have come to greatly affect the price of crude oil, and it will be interesting to examine the impact speculators have on the change in price of crude oil against the normal drivers of crude oil price.

The relationship between exchange rate volatility and trade flow has been examined in a number of previous researches. The paper mainly focuses on investigating the impact of exchange rate volatility on export values from Sweden to Germany during 2000:01 and 2011:06. The Auto Regressive Distributed Lag (ARDL) model is employed to obtain the estimates of the long run equilibrium and the short run dynamics, simultaneously. The results indicate that the exchange rate volatility has significant short run effects on export value in majority of estimated industries while its meaningful long run impacts do not appear in any cases. However, applying the “bounds test” approach, the co-integration is also found in more than half cases due to long run impacts of other factors such as foreign income on export earnings.

This article presents a new approach for calculating sensitivities of European swaptions. The sensitivities are found by applying an adjoint method to a stochastic volatility model, namely the SABR/LIBOR market model. This market model predicts the volatility smile and follows the market fluctuations more accurately than earlier used deterministic volatility market models for complex derivatives. The new adjoint method involves not only sensitivity calculations, it also presents a way of estimating the time discretization error using an a posteriori approach. The error calculation is described in this document but not investigated further. The first step in order to calculate the sensitivities is to calibrate the SABR/LIBOR market model to some market data. In our calculations we used data from June 15 2011 with 6 month intervals between the maturity times. When this calibration is complete all of the parameters in the SABR/LIBOR market model are specified and we can continue with the sensitivity calculations using the new adjoint method. The results from these calculations show that the method is a good choice for estimating sensitivities if we consider a complex financial derivative like the European swaption. The method is quite computational so we recommend that it is only used on a small number of securities with respect to a large number of parameters. The method provides more market-driven price and sensitivity estimations than earlier used methods and can benefit hedging of portfolios.

The aim of the thesis is to identify an appropriate model in forecasting Value-at-Risk on a morevolatile period than that one from which the model is estimated. We estimate 1-day-ahead and10-days-ahead Value-at-Risk on a number of exchange rates. The Value-at-Risk estimates arebased on three models combined with three distributional assumptions of the innovations, andthe evaluations are made with Kupiec's (1995) test for unconditional coverage. The data rangesfrom January 1st 2006 through June 30th 2011. The results suggest that the GARCH(1,1) andGJR-GARCH(1,1) with normally distributed innovations are models adequately capturing theconditional variance in the series.

This dissertation consists of three related chapters that study financial market volatility, jumps and the economic factors behind them. Each of the chapters analyzes a different aspect of this problem. The first chapter examines tests for jumps based on recent asymptotic results. Monte Carlo evidence suggests that the daily ratio z-statistic has appropriate size, good power, and good jump detection capabilities revealed by the confusion matrix comprised of jump classification probabilities. Theoretical and Monte Carlo analysis indicate that microstructure noise biases the tests against detecting jumps, and that a simple lagging strategy corrects the bias. Empirical work documents evidence for jumps that account for seven percent of stock market price variance. Building on realized variance and bi-power variation measures constructed from high-frequency financial prices, the second chapter proposes a simple reduced form framework for modelling and forecasting daily return volatility. The chapter first decomposes the total daily return variance into three components, and proposes different models for the different variance components: an approximate long-memory HAR-GARCH model for the daytime continuous variance, an ACH model for the jump occurrence hazard rate, a log-linear structure for the conditional jump size, and an augmented GARCH model for the overnight variance. Then the chapter combines the different models to generate an overall forecasting framework, which improves the volatility forecasts for the daily, weekly and monthly horizons. The third chapter studies the economic factors that generate financial market volatility and jumps. It extends the recent literature by separating market responses into continuous variance and discontinuous jumps, and differentiating the market’s disagreement and uncertainty. The chapter finds that there are more large jumps on news days than on no-news days, with the fixed-income market being more responsive than the equity market, and non-farm payroll employment being the most influential news. Surprises in forecasts impact volatility and jumps in the fixed-income market more than the equity market, while disagreement and uncertainty influence both markets with different effects on volatility and jumps. JEL classification: C1, C2, C5, C51, C52, F3, F4, G1, G14 / Dissertation

In this dissertation we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form and present a convergence analysis for the two-dimensional Black-Scholes equation arising in the Hull-White model for pricing European options with stochastic volatility. We formulate a non-conforming Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. We show that the bilinear form of the finite element method is coercive and continuous and establish an upper bound of order O(h) on the discretization error of method, where h denotes the mesh parameter of the discretization. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presentd.