Uygulamalı Matematik Enstitüsü

Faize dayalı türevlerde opsiyon fiyatlaması

The valuation of interest rate derivatives and embedded options in fixed-income securities is crucial for market practitioners. Although there have been many models to price interest rate derivatives, the inconsistency across the assumptions of the models creates difficulty in aggregating interest rate exposures. Besides, the models tend to be applicable to specific cases. In this regard, adaptation of a general methodology to price all interest rate derivatives without making additional assumptions has critical importance. This study is expected to contribute to the literature by providing a general approach that can be applied to any fixed-income security with regular or irregular cash flows using the Vasicek model. The methodology involves four main steps: (i) deriving the closed-form solution for the interest rate derivatives traded in the market, (ii) estimating the Vasicek model parameters, (iii) deriving the exhibit solution for the interest rate derivatives and (iv) plugging the estimated Vasicek model parameters to price the security. This methodology provides a general solution that is applicable to all interest rate derivatives with regular or irregular cash flows. Additionally, it allows aggregation of exposures to different interest rate derivatives and allows the derivation of sensitivities of the option values to the changes in model parameters. Although the study provides empirical evidence for European type of options, it also can be applied to price American or Bermudan type of options as well. Besides, the methodology can be implemented using other interest rate models with desirable properties.

Market mikro yapısının faktörlerinin analizi: Fiyat etkisi, likidite ve oynaklık

First chapter of this thesis is an attempt to model the price impact. It is assumed that the market is not perfectly efficient so that it takes to time to adjust new equilibrium price. Thus, in order to model the price impact, two new concepts are introduced which are market resiliency and speed of price informativeness. It is showed that market resiliency and price impact tend to raise as speed of price information increases which emphasizes the fact that speed of information matters in financial markets and market resiliency is not a phenomenon that can be neglected. In the second chapter, it is tried to stress the importance of the liquidity which is considered as the neglected dimension of the financial risk. To do that, a new approach called Liquidity Augmented Stochastic Volatility with Jump (LASVJ) model is introduced and it is compared with the Stochastic Volatility with Jump (SVJ) model in terms of stability and performance. This analysis includes both simulation and calibration analysis. The simulation results suggest that LASVJ model outperforms SVJ as it has lower bias and Root Mean Square Error. In the calibration part, ten companies listed in Dow-Jones 30 are used and it is found that the estimated probability of default and credit spread with LASVJ model are higher than those with SVJ model. In the third chapter, it is aimed to improve the volatility prediction which included in the financial risk management. As a well-performing volatility prediction sheds light on the uncertainty in the financial market, it is an important task to model it. To this end, GARCH-type models as well as SVR-GARCH. To this end, GARCH-type models as well as SVR-GARCH model are used to model the volatility and the results are compared based on the performance metrics. In part of empirical analysis, finding indicates that SVR-GARCH outperforms the traditional models in predicting volatility and also produce more reliable result in Value-at-Risk estimation.

Gerçekleme ataklarına karsı matematıksel tedbırler

Parola tabanlı kimliği doğrulanmış anahtar değişim protokolleri üzerine

Authentication and key agreement protocols play an important role in today's digital world. Key agreement methods mostly mimic Diffie-Hellman key exchange protocol, but unfortunately they are susceptible to man-in-the-middle attacks. Password based authenticated key exchange (PAKE) protocols promise to handle these key agreement and authentication without requiring existence of certificate authorities or trusted third parties. More importantly, PAKE protocols enable agreement on low-entropy passwords rather than high-entropy cryptographic keys shared by only involved parties. Even if PAKE protocols are not widely used in practice, they are already included in IEFT (RFC), ISO security standards and TLS cryptographic suite. In this thesis, by following these recent developments, we first present these PAKE protocols in three forms, namely balanced PAKE protocols, augmented PAKE protocols and password authenticated key retrieval (PAKR) protocols and within both single and multi server settings. Particularly, we revisit EKE, SPEKE, PAK, PPK, J-PAKE, SPAKE, SESPAKE balanced, and SRP, AugPAKE, OPAQUE, B-SPEKE augmented protocols. Then, we summarize security attacks to these protocols. Afterwards, detailed explanations of the attacks against these protocols are given. We further present current state of the art for PAKE protocols. Finally, we draw attention to possible extensions for PAKE protocols and state currently open questions about the subject.

Tekil Markov olmayan son değerlerli geriye doğru stokastik diferansiyel denklemlerin deterministik vadelerde çözümlerinin süreklilikleri

In this thesis we study a class of Backward Stochastic Differential Equations (BSDE) with superlinear driver process f adapted to a filtration F = fFt; t 2 [0; T]g supporting at least a d dimensional Brownian motion and a Poisson random measure on Rm n f0g in a deterministic time interval [0; T]. The superlinearity of f allows terminal conditions that can take the value +1 with positive probability. Such terminal conditions are called "singular." A terminal condition is said to be Markovian if it is a deterministic function of a Markov process.The first goal of the present thesis is to construct solutions to the class of BSDE we work with when they are coupled with singular non-Markovian terminal conditions. We consider the following class of terminal conditions: 1 = 1 1f 1 Tg + A 1f 1>Tg where 1 is any stopping time with a bounded density in a neighborhood of T and 2 = 1 1AT +A 1Ac T where At, t 2 [0; T] is a decreasing sequence of events adapted to the filtration F that is continuous in probability at T (equivalently, AT = f 2 > Tg where 2 is any stopping time such that P( 2 = T) = 0). In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they are continuous at time T and attain almost surely their terminal values. Let X be a d-dimensional diffusion process driven by the Brownian motion and with strongly elliptic covariance matrix. The second goal of the present thesis is to derive density formulas for the first exit time of X from a time varying domain. The existence of these densities show that such exit times can be used as 1 and 2 to define the terminal conditions 1 and 2: We also discuss the implications of our results in stochastic optimal control.

Simetrik olmayan matrisler için spektral bölümlemeyi yeniden değerlendirme

Parallel solutions to scientific problems having graph representation require efficient tasks and partitioning data. In this thesis, various parallel graph partitioning algorithms are studied. While these algorithms are applicable to both directed and undirected graphs, we focus on the directed case whose matrix representations are sparse and unsymmetric arising in linear system of equations representing various application domains such as computational fluid dynamics and thermal problems. Strategies inspected in this study are ParMETIS with the Multilevel Kernighan-Lin algorithm and the spectral partitioning algorithm with k-means clustering (SPEC) as well as the recursive spectral partitioning algorithm in CHACO. We have implemented SPEC in C programming language using PETSc and SLEPc libraries, whereas CHACO and ParMETIS are called from PETSc. Weighted partitioning is done under the consideration of the edge weights of the graph. SPEC is compared with the libraries only when the unweighted partitioning is made due to the limitations of the libraries for weighted partitioning. Hence, for weighted partitioning, only various eigensolver tolerances in SLEPc are studied in terms of the edge-cut and partitioning time. Another study is performed for the spectral partitioning algorithm based on eigensolver tolerance used with the k-means algorithm in MATLAB. The comparison is based on the quality of the partitioning (edge-cut and partition imbalance) and the number of iterations. The quality of partitioning is determined by the edge-cut and the load imbalance, which could be based on the edge and vertex imbalance ratios of partitions depending on the application. Since the adjacency matrix of a graph is structurally symmetric, the eigenvalue problem can only be solved approximately when the matrix is unsymmetric. Thus, only approximate results are provided in this study. It is deduced that using SPEC performs better than the existing software libraries when the number of cut edges is compared in unweighted partitioning of unsymmetric matrices.

Kimlik ve özellik tabanlı eşleme tabanlı kimlik doğrulama protokolleri

Authentication is one of the most important goals in cryptography. It provides sharing information with only authorized people and protecting data from being modified. Authentication can be achieved in various ways such as password-based, symmetric-key and public-key. The public-key authentication is the most preferred one among these options. It provides construction of key pairs and verification with based on hard mathematical problems. Public-key authentication is used as a basis for two important idea: Identity-Based Authentication (IBA) and Attribute-Based Authentication (ABA). The IBA systems are actually specialized publickey encryption systems where the public key is generated using an identity information of the user. ABA systems, which are the other important idea, are the generalizations of the IBA systems. While IBA systems cover only one attribute about users, ABA systems cover more than one attribute. In this thesis, identity-based and attribute-based authentication protocols are analyzed and a new attribute based authentication protocol is proposed. First, it is given the details and comparison of ID-based authentication protocols Shim,Yuan and Li, Tseng 2017 and Tseng 2015 which are based on elliptic curve. Shim,Yuan and Li and Tseng 2017 use pairings for authentication, while Tseng 2015 uses hash functions. Their securities and performances are analyzed. They provide the security properties such as known-key security, forward secrecy, key-compromise impersonation and unknown-key share. They also resist the passive, man-in-the-middle and reveal attacks. The protocols are more efficient than the protocols which are based on public-key by virtue of elliptic curve. Tseng 2015 is the most efficient one among them since it uses only hash functions. Also, the Zhang, Mu and Zhang attribute-based authentication protocol is studied. Then a new protocol, which is inspired from it, is designed. While Zhang et al. is based on public-key, the new protocol is constructed on elliptic curve basis. Moreover, controlling of attributes is simplified. It decreases the number of operations to determine necessary attributes. In this way, computational cost is reduced. The security analysis of the new protocol is presented and showed that the protocol is resistant to the following attack scenarios; adaptive chosen ciphertext, keycompromise impersonation, probing resistance, indistinguishable to eavesdroppers, forward secrecy and unknown key-share.

İkinci mertebeden market hacimli almgren-chriss modelinin özel fonksiyonlar yardımıyla çözümü

One of the current topics of research in mathematical finance is the scheduling of buy or sell orders to liquidate a position. A well-known framework for this problem is the one proposed by Almgren and Chriss that poses the problem as the maximization of the expected utility of the final terminal wealth. In the simplest formulation of the problem the market trading volume is taken as a constant. Under this and other assumptions an optimal trading curve can be computed in terms of the sinh function. This study aims to consider the same model under the assumptions that the market trading volume is an affine function of time, i.e, V_t = at+b and a quadratic function of time i.e, V_t = at^2+bt+c. A solution of the differential equations arising from the Almgren Chriss model under these assumptions is given in terms of the Modified Bessel function (for the affine volume curve) and the confluent hypergeometric function (for the quadratic volume curve). We also provide numerical examples on how the optimal trading curve varies with the model parameters a, b and c.

Kafes tabanlı algoritmaların verimli gerçeklenmesi

Quantum computing and quantum computers have been discussed for almost three decades. However, they remain mainly in theory. Almost all big companies like Google, IBM, and Microsoft have put their effort to build the most scalable quantum computers in recent years. These computers can change the game in cryptography since the known hard problems such as integer factorization and discrete logarithms can be broken with a large-scale quantum computer. These computers would seriously jeopardize the confidentiality and integrity of all digital communications. The question of when large-scale quantum computers will be built is still an open question. There are some educated predictions that sufficiently large quantum computers will be built to break all public-key schemes currently in use within the next ten or so years. That is why the National Institute of Standards and Technology (NIST) has announced a standardization process in 2016 to prepare our information security systems to be able to resist in quantum computing. The first two rounds of this process have been completed, and there are seven on-going candidates and eight alternate candidates. Among these 15 schemes, seven of them are based on lattices. In this thesis, we study the efficient implementation of lattice-based schemes. We first propose an efficient and compact variant of NewHope, one of the most efficient second-round candidates of the NIST post-quantum standardization project. The proposed algorithm NewHope-Compact heavily uses recent advances on Number Theoretic Transform (NTT), so that transformation from one polynomial to another is easy. To make it possible, we changed the definition of a component in component-wise multiplication during polynomial multiplication and show that changing the security level only requires to change the size of the polynomial and the definition of a component. Then, we present various optimizations for lattice-based KEMs using the NTT on the popular ARM Cortex-M4 microcontroller. Improvements come in the form of a faster code using more efficient modular reductions, optimized small-degree polynomial multiplications, and more aggressive layer merging in the NTT, but also in the form of reduced stack usage. We test our optimizations in software implementations of Kyber, one of the round three candidates in the NIST post-quantum project, NewHope, and NewHope-Compact.

FIPS 140 ve ISO/IEC 15408'e göre şifreleme modülleridoğrulama süreci

With the advancement of technology, questions have arisen regarding the reliability of information technology products. Some standards have emerged to ensure the reliability of these products and to validate this reliability internationally. One of the standards issued to meet this need is ISO/IEC 15408 Common Criteria Standard. Thanks to this standard, the relevant institutions authorized to evaluate information technology products are evaluated and the reliability of this product is provided by the certificates issued by the competent authorities. In addition to this standard, the FIPS 140 standard has been created, which specifies the requirements of cryptographic modules specifically and is used for the approval and verification of these modules. This standard is used to determine the reliability of cryptographic modules. In this thesis, we first define these standards and how the evaluation process in accordance with them takes place in the world and how they should be implemented in Turkey. Then, we discuss the vulnerabilities of the AES-GCM algorithm in order to prevent the vulnerabilities of the cryptographic algorithms used in the cryptographic algorithm verification process, which is a part of the cryptographic module verification process, and talk about alternative AES modes. Finally, we complete the thesis by talking about test vectors that help us detect these vulnerabilities.