1,721,010 research outputs found
Sıçramalı stokastik volatilie modellerinin BIST opsiyonlarına uygulaması.
No full textThis thesis gives a derivation of call and put option pricing formulas under stochastic volatility models with jumps; the precise model is a combination of Merton and Heston models. The derivation is based on the computation of the characteristic function of the underlying process. We use the derived formulas to fit the model to options written on two stocks in the BIST30 index covering the first two months of 2017. The fit is done by minimizing a weighted distance between the observed prices and the model prices. M.S. - Master of Scienc
Heston modelinin BIST30 varantları üzerindeki uygulamaları: üretme ve fiyatlama.
No full textThe Heston model is one of the first and known stochastic volatility models. The aim of this work is to study the performance of the Heston Model on pricing and hedging the warrants written on BIST30 and the compatibility between the observation of the Heston Model in the literature and BIST30 data.M.S. - Master of Scienc
Markov modülasyonlu kısıtlı rastgele yürüyüşlerin çıkış olasılıkları.
No full textLet X be the constrained random walk on Z2+ with increments (0, 0), (1, 0), (−1, 1), (0, −1) whose jump probabilities are determined by the state of a finite state Markov chain M. X represents the lengths of two queues of customers (or packets, tasks, etc.) waiting for service from two servers working in tandem; the arrival of customers occur with rate λ(Mk), service takes place at rates μ1(Mk), and μ2(Mk) where Mk denotes the current state of the Markov chain M. We assume that the average arrival rate is less than the average service rates, i.e., X is assumed stable. Stability implies that X moves in cycles that restart each time it hits the origin. Let τn be the first time X hits the line ∂An={x:x(1)+x(2)=n}, i.e., when the sum of the queue lengths equals n for the first time ; if the queues share a common buffer, τn represents the time of a buffer overflow and pn = P(x,m)(τn 0. We then construct a class of harmonic functions for (Y,M) and use their linear combinations to develop approximate formulas for P(y,m)(τ < ∞). The construction is based on points on a characteristic surface associated with Y defined through the eigenvalues of a matrix whose components depend on the transition matrix of the modulating chain and the jump probabilities of Y . We indicate possible applications of our results and approach in finance and insurance.Ph.D. - Doctoral Progra
Enflasyona endeksli̇ tahvi̇lleri̇n ve gömülü deflasyon koruma opsi̇yonlarının fi̇yatlanması: Türk tahvi̇l pi̇yasasi üzeri̇ne anali̇z
No full textFixed income securities are financial contracts that provide a stream of cash flows to the investors. Such cash flows are exposed to inflation risk. From the perspective of the lender, the purchasing power of provided cash flows might be subject to erosion in inflationary pressures. Inflation indexed bonds issued in a way that to hedge this risk and provide real return to the bond holder. UK issued first inflation indexed bonds as sovereign in 1981. Then, developed countries such as US, Canada, France, Germany, and emerging market countries such as Turkey, Brazil, Mexico had high interest in issuing these bonds. The principal and coupon payments of these bonds are linked to the changes in the reference price index. Inflation indexed bonds might be issued as plain, which excludes any protection against deflation. On the other hand, bonds might be issued with deflation protection (put) option embedded since bond cash flows will be less than the nominal value in a deflationary economic environment. Deflation protection might cover only principal value as in TIPS issued by US or both principal and coupon payments as inflation indexed bonds issued by Turkish Treasury. This thesis aims to price deflation protection option premium in Turkish bond market by decomposing bond structure into plain and option components. First, we review the Jarrow-Yildirim model under the HJM framework and the analytical formulas for derivative prices available in that model. Then, we use historical bond market data to estimate model parameters and the price of the embedded deflation protection option. Finally, we examine historical course of this premium for bonds with different characteristics and inflation expectations.Sabit getirili menkul kıymetler, yatırımcılara düzenli nakit akımı saglayan finansal sözleşmelerdir. Bu kıymetlerin sagladığı nakit akımları enflasyon riskine maruzdur. Enflasyonist ortamda, saglanan nakit akımlarının satın alma gücü borç veren açısın dan azalmaktadır. Enflasyona endeksli tahviller yatırımcıları bu riskten korumak ve reel getiri saglamak üzere ihraç edilmektedir. Enflasyona endeksli devlet tahvilleri ilk olarak İngiltere tarafından 1981 yılında ihraç edilmiştir. Daha sonra, ABD, Kanada, Fransa, Almanya gibi gelişmiş ülkelerin yanı sıra Türkiye, Brezilya, Meksika gibi gelişmekte olan ülkeler de bu tahvillere yogun ilgi göstermiştir. Tahvillerin anapara ve kupon demeleri referans fiyat endeksindeki degişime bağlı dır. Enflasyona endeksli tahviller deflasyona karşı herhangi bir koruma saglamayan, yapılandırılmamış, şekilde ihraç edilebilir. Diger yandan deflasyonist ekonomik or tamda nakit akımının nominal degerinin altına düşmesini engellemek amacıyla def lasyon koruma (satım) opsiyonu gömülü şekilde de ihraç edilebilir. Deflasyon koruması ABD tarafından ihraç edilen tahvillerde oldugu şekilde yalnızca anaparayı kap sayabilecegi gibi Türkiye’de Hazine tarafından ihraç edilen tahvillerde olduğu şekilde hem anapara hem de kupon ödemesini kapsayabilmektedir. Bu çalışma, tahvilleri yapılandırmamış ve opsiyon bileşenlerine ayırarak Türk tah vil piyasasında deflasyon koruma primini hesaplamaktadır. İlk olarak, HJM çerçevesinde Jarrow-Yildirim modeli gözden geçirilmekte ve analitik formüller türetilmetedir. Daha sonra, tahvil piyasası verileri kullanılarak parametre tahmini gerçekleştirilmekte ve deflasyon koruma opsiyonu fiyatlanmaktadır. Son olarak, deflasyon priminin tarihse seyri incelenmekte ve enflasyon beklentileri ile olan ilişkisi incelenmektedir.M.S. - Master of Scienc
Kısıtlı basit rastgele yürüyüşlerin çıkış olasılıkları.
No full textConsider a nearest neighbor stable two dimensional random walk X constrained to remain on the positive orthant. X is assumed stable, i.e., its average increment points toward the origin. X represents the lengths of two queues (or two stacks in computer science applications) working in parallel. The probability pn that the sum of the components of this random walk reaches a high level n before the random walk returns to the origin is a natural performance measure, representing the probability of a buffer overflow in a busy cycle. The stability of the walk implies that pn decays exponentially in n. Let Y be the same constrained random walk as X, but constrained only on its second component and the jump probabilities on its first component reversed. The present thesis shows that one can approximate pn with the probability that components of Y ever equal each other, with exponentially decaying relative error, if X starts from an initial point with nonzero first component. We further construct a class of Y -harmonic functions from single and conjugate points on a characteristic surface, with which the latter probability can be either computed perfectly in some cases, or approximated with bounded relative error in general. We provide numerical examples showing the effectiveness of the computed approximations and indicate possible applications of our results in finance and insurance.Ph.D. - Doctoral Progra
Avrupa tipi satın alma opsiyonu fayda kayıtsızlığı fiyatlamasının korunma performansı.
No full textHedging performance of the Utility Indifference Pricing model presented by Davis et. al [European option pricing with transaction costs, SIAM J. Con. & Opt., 31(2), 1993] is studied in this thesis. Their indifference pricing approach is based on utility indifference of an investor towards portfolios with and without a short position in the European call option contract. The option price is defined as a difference of the minimum amount of initial endowments that make the maximum utilities from these portfolios equal to zero. Furthermore, Davis et al. considered an incomplete market where transaction costs are included. They worked with an exponential utility function which eliminates the dependence of investments in stocks to total wealth. This framework is adopted and hedging strategy is defined as a difference of two control variables that solves the utility maximization problems for the portfolios. Thus, finding the call option price embedded in utility maximization problems is studied via Optimal Control Theory. Markov Chain Approximation is utilized to compute the problem numerically and the option price is derived. Ending wealth from the portfolio consisting of short position in the option is measured. Hedging error is defined as the losses incurred in this portfolio. Furthermore, hedging performance is measured by computing the conditional expected value of losses as a percentage of the option price. Hedging performance is evaluated against different levels of transaction costs, degree of risk aversion, volatility and option moneyness. Our findings suggest that hedging performance measure is large when volatility and risk aversion rates are low, and when transaction costs are high. We also find that moneyness of option has a decreasing effect on the hedging performance measure.Thesis (M.S.) -- Graduate School of Applied Mathematics. Financial Mathematics
Geriye doğru stokastik diferansiyel denklemler üzerine iki çalışma.
No full textBackward stochastic differential equations appear in many areas of research including mathematical finance, nonlinear partial differential equations, financial economics and stochastic control. The first existence and uniqueness result for nonlinear backward stochastic differential equations was given by Pardoux and Peng (Adapted solution of a backward stochastic differential equation. System and Control Letters, 1990). They looked for an adapted pair of processes {x(t); y(t)}; t is in [0; 1]} with values in Rd and Rd×k respectively, which solves an equation of the form: x(t) + int_t^1 f(s,x(s),y(s))ds + int_t^1 [g(s,x(s)) + y(s)]dWs = X. This dissertation studies this paper in detail and provides all the steps of the proofs that appear in this seminal paper. In addition, we review (Cvitanic and Karatzas, Hedging contingent claims with constrained portfolios. The annals of applied probability, 1993). In this paper, Cvitanic and Karatzas studied the following problem: the hedging of contingent claims with portfolios constrained to take values in a given closed, convex set K. Processes intimately linked to BSDEs naturally appear in the formulation of the constrained hedging problem. The analysis of Cvitanic and Karatzas is based on a dual control problem. One of the contributions of this thesis is an algorithm that numerically solves this control problem in the case of constant volatility. The algorithm is based on discretization of time. The convergence proof is also provided.M.S. - Master of Scienc
ÜÇ BOYUTTA SINIRLI BASİT RASTGELE YÜRÜYÜŞLERİN ÇARPMA OLASILIKLARI
No full textWe study the constrained simple random walk in three dimensions modeling the state of a queueing system with three nodes working in parallel. The process is assumed to be stable, i.e., the service rate at each node is greater than the arrival rate. The stability assumption implies that the process follows a repeating cycle, starting anew each time the process hits the origin. Consider the probability pn that the sum of the components of the process equals n before the process hits the origin, which can be thought of as the probability of a buffer overflow in a cycle. The stability assumption implies that pn decays exponentially in n. The goal of the present thesis is to develop approximation formulas for pn. In the literature, this problem is treated for two dimensional simple walks using an affine transformation of the problem. We extend this analysis to three dimensions. As in two dimensions, the affine transformation yields a limit process and a limit hitting probability. We show, for the case of the three dimensional stable constrained simple random walk, the limit probability approximates pn with an exponentially diminishing relative error, assuming that the first component of the initial point of the process is nonzero. We further approximate the limit probability by harmonic functions of the limit process constructed from solutions of harmonic systems associated with the problem. We provide a numerical example and discuss a possible application to finance.Çalışmada, üç ağın paralel olarak çalıştığı bir kuyruk sisteminin durumunu modelleyen üç boyutlu sınırlı basit rastgele yürüyüş incelenmektedir. Sürecin dengeli olduğu varsayılmaktadır, diğer bir deyişle her ağdaki servis oranı varış oranından daha büyüktür. Dengelilik varsayımı, sürecin orijine her ulaştığında yeniden başlayarak tekrarlayan bir döngüyü takip ettiği anlamına gelmektedir. Sürecin başlangıç noktasına ulaşmadan önce bileşenlerinin toplamının n’ye eşit olma olasılığı pn olsun. Bu olasılık, bir döngüde bir arabellek aşım olasılığı olarak düşünülebilir. Sürecin dengeli olması varsayımı, pn’nin n arttıkça üssel olarak azaldığını ima etmektedir. Bu tezin amacı, pn için yaklaşık hesaplama formülleri geliştirmektir. Literatürde bu problem, problemin afin dönüşümü kullanılarak iki boyutlu basit rastgele yürüyüşler için ele alınmaktadır. Bu analiz, mevcut çalışmada üç boyuta genişletilmektedir. İki boyutta olduğu gibi, afin dönüşüm sonrasında bir limit süreci ve bir limit çarpma olasılığı elde edilmektedir. Üç boyutlu dengeli kısıtlı basit rastgele yürüyüş için, elde edilen limit olasılığının, sürecin başlangıç noktasının ilk bileşeninin sıfır olmadığı varsayılarak, üstel olarak azalan bir göreli hata ile pn’ye yaklaştığı gösterilmektedir. Ayrıca, problemle ilişkili olan harmonik sistemin çözümlerinden elde edilerek oluşturulan harmonik fonksiyonlar ile limit olasılığı yaklaşık olarak hesaplanmaktadır. Sayısal bir örnek sağlanmış ve finans sisteminde olası bir uygulamadan bahsedilmiştir.Ph.D. - Doctoral Progra
PİYASA YAPICILIK MODELİNDE RİSK LİKİDİTE PRİMİNİN STOK SÜRECİNE ETKİSİ
No full textA basic market making model in mathematical finance is the one proposed by Avellaneda and Stoikov (AS), which is formulated as a stochastic optimal control problem. This model includes a risk liquidity premium function ℓ which penalizes the remaining inventory at terminal time. To the best of our understanding, in the currently available literature, at least in the context of the AS model, this function is usually assumed zero or is ignored. One explanation given for this assumption is that the dynamics of the inventory process is mean reverting and the ℓ function has little impact on this. In this thesis, we study numerically whether this assumption holds by computing the dynamics of the inventory process for non-zero ℓ functions. We see that depending on model parameters this function can have a nontrivial impact on inventory process dynamics. We present a numerical study of how this impact depends on model parameters.Finansal matematikteki temel piyasa yapıcılık modeli Avellaneda ve Stoikov (AS) tarafından önerilen ve stokastik optimal kontrol problemi olarak formüle edilen modeldir. Bu model terminal zamanda kalan envanteri cezalandıran risk likidite primi fonksiyonu ℓ’i içermektedir. Anlayışımız dahilinde halihazırda mevcut literatürde, özellikle AS modelinin bağlamında, bu fonksiyon genellikle sıfır olarak varsayılmıştır veya göz ardı edilmiştir. Bu varsayım için yapılan bir açıklama envanter süreci dinamiklerinin ortalamaya dönüş olması ve ℓ fonksiyonunun bu durum üzerinde az etkisinin olmasıdır. Bu tezde, ilgili varsayımın doğru olup olmadığı sıfırdan farklı ℓ fonksiyonları için envanter süreci dinamiklerini hesaplayarak sayısal olarak incelenmiştir. Model parametrelerine bağlı olarak bu fonksiyonun envanter süreci dinamiklerine önemsiz olmayan etkisi olabileceği görülmüştür. Bu etkinin model parametrelerine nasıl bağlı olduğuna ilişkin sayısal çalışma sunulmuştur.M.S. - Master of Scienc
Merton’un portföy probleminin, sabit volatile ile stokastik volatile olduğu durumlarda karşılaştırılması.
No full textMerton's Portfolio Problem is a dynamic portfolio choice problem, which assumes asset returns and covariances are constant. There is well documented evidence that, stock returns and volatilities are correlated. Therefore, stochastic volatility models in dynamic portfolio problems can give better results. The work [J. Liu, Portfolio selection in stochastic environments, Review of Financial Studies, 20(1), 2007] developed a general dynamic portfolio model that allows the parameters of the model to depend on an external process X; this general model includes Merton's portfolio problem with Heston stochastic volatility (Merton H) and constant volatility as special cases. Liu's solution involves substituting solutions of a specific form into the Hamilton Jacobi Bellman (HJB) equation associated with the problem and reducing it first to a simpler Partial Differential Equation (PDE), and then reducing this PDE into a sequence of Ordinary Differential Equations (ODE). In this thesis we give the details of these reductions. We then use the explicit solutions provided by Liu for the Merton H model to see the effect of replacing stochastic volatility with constant volatility in Merton's problem. We find that, a ratio(sensitivity to stochastic volatility ratio) depending on mean reversion rate, risk aversion and Sharpe ratio is the most important parameter in this respect. When the value of this ratio is small, incorporating stochastic volatility into the model has little effect on the optimal portfolio. When it is large (when Sharpe ratio is high and the investor has low risk aversion) taking stochastic volatility into consideration is meaningful.M.S. - Master of Scienc
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