This dissertation studies a collection of problems on trading assets and derivatives over finite and infinite horizons. In the first part, we analyze an optimal switching problem with transaction costs that involves an infinite sequence of trades. The investor's value functions and optimal timing strategies are derived when prices are driven by an exponential Ornstein-Uhlenbeck (XOU) or Cox-Ingersoll-Ross (CIR) process. We compare the findings to the results from the associated optimal double stopping problems and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Our results show that when prices are driven by a CIR process, optimal strategies for the switching problems are of the classic buy-low-sell-high type. On the other hand, under XOU price dynamics, the investor should refrain from entering the market if the current price is very close to zero. As a result, the continuation (waiting) region for entry is disconnected. In both models, we provide numerical examples to illustrate the dependence of timing strategies on model parameters. In the second part, we study the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the OU, CIR or XOU model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor's timing options to enter and exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short. Finally, we analyze the optimal risk-averse timing to sell a risky asset. The investor's risk preference is described by the exponential, power or log utility. Two stochastic models are considered for the asset price -- the geometric Brownian motion (GBM) and XOU models to account for, respectively, the trending and mean-reverting price dynamics. In all cases, we derive the optimal thresholds and certainty equivalents to sell the asset, and compare them across models and utilities, with emphasis on their dependence on asset price, risk aversion, and quantity. We find that the timing option may render the investor's value function and certainty equivalent non-concave in price even though the utility function is concave in wealth. Numerical results are provided to illustrate the investor's optimal strategies and the premia associated with optimally timing to sell with different utilities under different price dynamics.

Submitted by ARIEL TEIXEIRA (ateixeira@fgvmail.br) on 2014-03-14T15:38:35Z No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2014-03-14T16:29:51Z (GMT) No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) / Approved for entry into archive by Maria Almeida (maria.socorro@fgv.br) on 2014-04-09T14:58:17Z (GMT) No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) / Made available in DSpace on 2014-04-09T14:58:29Z (GMT). No. of bitstreams: 1 Pair trading in Bovespa with a quantitative approach cointegration, Ornstein-Uhlenbeck equation and Kelly criterion.pdf: 857529 bytes, checksum: d324d9458357fb950c816b5a8bfd11d2 (MD5) Previous issue date: 2014-02-17 / Pair trading is an old and well-known technique among traders. In this paper, we discuss an important element not commonly debated in Brazil: the cointegration between pairs, which would guarantee the spread stability. We run the Dickey-Fuller test to check cointegration, and then compare the results with non-cointegrated pairs. We found that the Sharpe ratio of cointegrated pairs is greater than the non-cointegrated. We also use the Ornstein-Uhlenbeck equation in order to calculate the half-life of the pairs. Again, this improves their performance. Last, we use the leverage suggested by Kelly Formula, once again improving the results.

M.Sc.

Cette thèse est consacrée au problème d'estimation de la fonction de dérive de certains modèles de processus stochastiques périodiques lorsque la durée d'observation tend vers l'infini. Aucune hypothèse de récurrence n'est posée a priori.Dans un premier temps nous considérons le modèle du type signal plus bruit dζt = f (t, θ)dt + σ(t)dWt,; et puis nous étudions l'estimation du paramètre θ à partir d'une observation continue et puis d'une observation discrète du processus {ζt} sur l'intervalle [0; T]. Les fonctions f (·, ·) et σ(·) sont continues et périodiques en t de même période P > 0, σ(·) > 0 et θ ∈ Θ ⊂R. Nous établissons la convergence en probabilité d'un estimateur du maximum de vraisemblance θˆT , sa normalité asymptotique et son efficacité asymptotique minimax. Lorsque f (t, θ) = θf (t), l'expression de θˆT est explicite et nous obtenons la convergence en moyenne quadratique aussi bien pour le cas d'une observation continue que pour le cas d'une observation discrète. De plus, nous déduisons la convergence presque sûre dans le cas d'une observation continue.Dans la seconde partie nous traitons l'estimation non-paramétrique de la fonction f(_) pour les modèles périodiques du type signal plus bruit et du type Ornstein-Uhlenbeck donnés par dζt = f (t)dt + σ(t)dWt, dξt = f (t)ξtdt + dWt. Pour le premier modèle, un estimateur à noyau périodique est construit, la convergence en moyenne quadratique uniformément sur [0; P] et presque sûre de cet estimateur est établie ainsi que sa normalité asymptotique. Dans le cas du modèle d'Ornstein-Uhlenbeck, la convergence du biais ainsi que la convergence en moyenne quadratique uniformément sur [0; P] sont prouvées, et leurs vitesses de convergence sont étudiées. / In this thesis, we consider a drift estimation problem of a certain class of stochastic periodic processes when the length of observation goes to infinity. Firstly, we deal with the linear periodic signal plus noise model dζt = f (t, θ)dt + σ(t)dWt, ;and we study the parametric estimation from a continuous and discrete observation of the process f_tg throughout the interval [0; T]. Using the maximum likelihood method we show the existence of an estimator θˆT which is consistent, asymptotically normal and asymptotically efficient in the sens minimax. When f(t; _) = _f(t), the expression of ^_T is explicit and we obtain the mean square convergence in the both continuous and discrete observation cases. In addition, we deduce the strong consistency in the case of continuous observation.Secondly, we consider the nonparametric estimation problem of the function f(_) for the next two periodic models of type signal plus noise and Ornstein-Uhlenbeckd_t = f(t)dt + _(t)dWt; d_t = f(t)_tdt + dWt:For the signal plus noise model, we build a kernel estimator, the convergence in mean square uniformly over [0; P] and almost sure convergence are established, as well as the asymptotic normality. For the Ornstein-Uhlenbeck model, we prove the convergence uniformly over [0; P] of the bias and the mean square convergence. Moreover, we study the speed of these convergences.

When an Ornstein-Uhlenbeck (or CAR(1)) process is observed at discrete times 0, h, 2h,··· [T/h]h, the unobserved driving process can be approximated from the ob- served process. Approximated increments of the driving process are used to test the assumption that the process is L\'evy-driven. Asymptotic behavior of the test statis- tic at high sampling frequencies is developed assuming that the model parameters are known. The behavior of the test statistics using an estimated parameter is also studied. If it can be concluded that the driving process is L\'evy, the empirical process of the approximated increments can then be used to carry out more precise tests of goodness-of-fit. For example, one can test whether the driving process can be modeled as a Brownian motion or a gamma process. In each case, performance of the proposed test is illustrated through simulation.

The change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift change point problem for Ornstein-Uhlenbeck processes driven by _-stable Levy motions. In each problem, we establish the consistency of the estimators, determine asymptotic behavior for the changing parameters, and finally, we perform simulation studies to computationally assess the convergence of parameters. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

Weather derivatives are financial tools used to manage the risks related to changes in the weather and are priced considering weather variables such as rainfall, temperature, humidity and wind as the underlying asset. Some recent researches suggest to model the amount of rainfall by considering the mean reverting processes. As an example, the Ornstein Uhlenbeck process was proposed by Allen [3] to model yearly rainfall and by Unami et al. [52] to model the irregularity of rainfall intensity as well as duration of dry spells. By using the Feynman-Kac theorem and the rainfall indexes we derive the partial differential equations (PDEs) that governs the price of an European option. We apply the Lie analysis theory to solve the PDEs, we provide the group classification and use it to find the invariant analytical solutions, particularly the ones compatible with the terminal conditions.

京都大学 / 0048 / 新制・課程博士 / 博士(農学) / 甲第20006号 / 農博第2190号 / 新制||農||1045(附属図書館) / 学位論文||H28||N5015(農学部図書室) / 33102 / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 藤原 正幸, 教授 村上 章, 准教授 宇波 耕一 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...

La motivation principale de cette thèse est l'étude du comportement en temps grand des solutions non-bornées d'équations de Hamilton-Jacobi visqueuses dans RN en présence d'un terme d'Ornstein-Uhlenbeck. Nous considérons la même question dans le cas d'une équation de Hamilton-Jacobi du premier ordre. Dans le premier cas, qui constitue le cœur de la thèse, nous généralisons les résultats de Fujita, Ishii et Loreti (2006) dans plusieurs directions. La première est de considérer des opérateurs de diffusion plus généraux en remplaçant le Laplacien par une matrice de diffusion quelconque. Nous considérons ensuite des opérateurs non-locaux intégro-différentiels de type Laplacien fractionnaire. Le second type d'extension concerne le Hamiltonien qui peut dépendre de x et est seulement supposé sous-linéaire par rapport au gradient. / The main aim of this thesis is to study large time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in RN in presence of an Ornstein-Uhlenbeck drift. We also consider the same issue for a first order Hamilton-Jacobi equation. In the first case, which is the core of the thesis, we generalize the results obtained by Fujita, Ishii and Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a non-local integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear.