In this thesis, a new method based on characteristic functions is proposed to estimate the jump component in a finite-activity Levy process, which includes the jump frequency and the jump size distribution. Properties of the estimators are investigated, which show that this method does not require high frequency data. The implementation of the method is discussed, and examples are provided. We also perform a comparison which shows that our method has advantages over an existing threshold method. Finally, two applications are included: one is the classification of the increments of the model, and the other is the testing for a change of jump frequency.

In this thesis, we suggest using nano-relays in a network using molecular com- munication in free space to improve the performance of the system in terms of delay. An approximation method for jump diffusion processes, which is based on Markov chains, is used to model molecular propagation in such scenarios. The model is validated through comparing analytic results with simulation results. The results illustrate the advantage of using nano-relays over diffusion in terms of delay. The proposed model is then used to inves- tigate the effect of different parameters, such as filaments’ length and the number of filaments attached to each nano-relay, on the delay performance of the communication technique. We used transient solution of the model in the first set of results. How- ever, stationary solution of the model can generate useful results, too. In the second set of results, the model is extended for an unbounded scenario. Con- sidering the propagation as a one-sided skip free process and using matrix analytic methods, we find the final distribution for the position of informa- tion molecules. It will be shown that it is possible to keep molecules in a desired region. The effect of different parameters on the final distribution for the position of information molecules is investigated, too. This analysis can be useful in drug delivery applications. / February 2016

>Magister Scientiae - MSc / We discuss efficient Monte Carlo methods for pricing of electricity derivatives. Electricity derivatives are risk management tools used in deregulated electricity markets. In the past,research in electricity derivatives has been dedicated in the modelling of the behaviour of electricity spot prices. Some researchers have used the geometric Brownian motion and the Black Scholes formula to offer a closed-form solution. Electricity spot prices however have unique characteristics such as mean-reverting, non-storability and spikes that render the use of geometric Brownian motion inadequate. Geometric Brownian motion assumes that changes of the underlying asset are continuous and electricity spikes are far from being continuous. Recently there is a greater consensus on the use of Mean-Reverting Jump-Diffusion (MRJD) process to describe the evolution of electricity spot prices. In this thesis,we use Mean-Reverting Jump-Diffusion process to model the evolution of electricity spot prices. Since there is no closed-form technique to price these derivatives when the underlying electricity spot price is assumed to follow MRJD, we use Monte Carlo methods to value electricity forward contracts. We present variance reduction techniques that improve the accuracy of the Monte Carlo Method for pricing electricity derivatives.

In this note, we consider an inverse problem in a jump diffusion model. Using characteristic functions we prove the injectivity of the forward operator mapping the five parameters determining the model to the density function of the return distribution.

博士（経営） / 85 p. / 一橋大学

In this dissertation, we introduce novel exponential jump-diffusion models for pricing options. Firstly, the normal convolution gamma mixture jump-diffusion model is presented. This model generalizes Merton's jump-diffusion and Kou's double exponential jump-diffusion. We show that the normal convolution gamma mixture jump-diffusion model captures some economically important features of the asset price, and that it exhibits heavier tails than both Merton jump-diffusion and double exponential jump-diffusion models. Secondly, the normal convolution double gamma jump-diffusion model for pricing options is presented. We show that under certain configurations of both the normal convolution gamma mixture and the normal convolution double gamma jump-diffusion models, the latter exhibits a heavier left or right tail than the former. For both models, the maximum likelihood procedure for estimating the model parameters under the physical measure is fairly straightforward; moreover, the likelihood function is given in closed form thereby eliminating the need to embed a probability density function recovery procedure such as the fast Fourier transform or the Fourier-cosine expansion methods in the parameter estimation procedure. In addition, both models can reproduce the implied volatility surface observed in the options data and provide a good fit to the market-quoted European option prices.

The jump phenomenons of many financial assets prices have been observed in many empirical papers. In this paper we extend the Heath-Jarrow-Morton model to include the jump component to derive the European-style pricing formula of the interest rate futures options. We use numerical method to simulate the options prices and analyze how each component of HJM model under jump-diffusion processes affects the interest rate futures options. Finally, we utilize LSM method which are presented by Longstaff and Schwartz to derive American options prices and compare it with European options.

In this note, we consider an inverse problem in a jump diffusion model. Using characteristic functions we prove the injectivity of the forward operator mapping the five parameters determining the model to the density function of the return distribution.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.