Дисертації з теми "Valuation equation"

<p>The famous Black-Scholes (BS) model used in the option pricing theory</p><p>contains two parameters - a volatility and an interest rate. Both</p><p>parameters should be determined before the price evaluation procedure</p><p>starts. Usually one use the historical data to guess the value of these</p><p>parameters. For short lifetime options the interest rate can be estimated</p><p>in proper way, but the volatility estimation is, as well in this case,</p><p>more demanding. It turns out that the volatility should be considered</p><p>as a function of the asset prices and time to make the valuation self</p><p>consistent. One of the approaches to this problem is the method of</p><p>uncertain volatility and the static hedging. In this case the envelopes</p><p>for the maximal and minimal estimated option price will be introduced.</p><p>The envelopes will be described by the Black - Scholes - Barenblatt</p><p>(BSB) equations. The existence of the upper and lower bounds for the</p><p>option price makes it possible to develop the worse and the best cases</p><p>scenario for the given portfolio. These estimations will be financially</p><p>relevant if the upper and lower envelopes lie relatively narrow to each</p><p>other. One of the ideas to converge envelopes to an unknown solution</p><p>is the possibility to introduce an optimal static hedged portfolio.</p>

Nous entamons ce rapport de thèse par l’évaluation de l’espérance espérée qui représente une des composantes majeures des ajustements XVA. Sous l’hypothèse d’indépendance entre l’exposition et les coûts de financement et de crédit, nous dérivons dans le chapitre 3 une représentation nouvelle de l’exposition espérée comme la solution d’une équation différentielle ordinaire par rapport au temps d’observation du défaut. Nous nous basons, pour le cas unidimensionnel, sur des arguments similaires à ceux de la volatilité locale de Dupire. Et pour le cas multidimensionnel, nous nous référons à la formule de la Co-aire. Cette représentation permet d’expliciter l’impact de la volatilité sur l’exposition espérée : Cette valeur temps fait intervenir la volatilité des sous-jacents ainsi que la sensibilité au premier ordre du prix, évalués sur un ensemble fini de points. Malgré des limitations numériques, cette méthode est une approche précise et rapide pour la valorisation de la XVA unitaire en dimension 1 et 2.Les chapitres suivants sont dédiés aux aspects du risque de corrélations entre les enveloppes d’expositions et les coûts XVA. Nous présentons une modélisation du risque général de corrélation à travers une diffusion stochastique multivariée, comprenant à la fois les sous-jacents des dérivés et les intensités de défaut. Dans ce cadre, nous exposons une nouvelle approche de valorisation par développements asymptotiques, telle que le prix d’un ajustement XVA correspond au prix de l’ajustement à corrélation nulle, auquel s’ajoute une somme explicite de termes correctifs. Le chapitre 4 est consacré à la dérivation technique et à l’étude de l’erreur numérique dans le cadre de la valorisation de dérivés contingents au défaut. La qualité des approximations numériques dépend uniquement de la régularité du processus de diffusion de l’intensité de crédit, et elle est indépendante de la régularité de la fonction payoff. Les formules de valorisation pour CVA et FVA sont présentées dans le chapitre 5. Une généralisation des développements asymptotiques pour le cadre bilatéral de défaut est adressée dans le chapitre 6.Nous terminons ce mémoire en abordant un cas du risque spécifique de corrélation lié aux contrats de migration de rating. Au-delà des formules de valorisation, notre contribution consiste à présenter une approche robuste pour la construction et la calibration d’un modèle de transition de ratings consistant avec les probabilités de défaut implicites de marché<br>The point of departure of this thesis is the valuation of the expected exposure which represents one of the major components of XVA adjustments. Under independence assumptions with credit and funding costs, we derive in Chapter 3 a new representation of the expected exposure as the solution of an ordinary differential equation w.r.t the default time variable. We rely on PDE arguments in the spirit of Dupire’s local volatility equation for the one dimensional problem. The multidimensional extension is addressed using the co-area formula. This forward representation gives an explicit expression of the exposure’s time value, involving the local volatility of the underlying diffusion process and the first order Greek delta, both evaluated only on finite set of points. From a numerical perspective, dimensionality is the main limitation of this approach. Though, we highlight high accuracy and time efficiency for standalone calculations in dimensions 1 and 2.The remaining chapters are dedicated to aspects of the correlation risk between the exposure and XVA costs. We start with the general correlation risk which is classically modeled in a joint diffusion process for market variables and the credit/funding spreads. We present a novel approach based on asymptotic expansions in a way that the price of an XVA adjustment with correlation risk is given by the classical correlation-free adjustment to which is added a sum of explicit correction terms depending on the exposure Greeks. Chapter 4 is consecrated to the technical derivation and error analysis of the expansion formulas in the context of pricing credit contingent derivatives. The accuracy of the valuation approach is independent of the smoothness of the payoff function, but it is related to the regularity of the credit intensity model. This finding is of special interest for pricing in a real financial context. Pricing formulas for CVA and FVA adjustments are derived in Chapter 5, along with numerical experiments. A generalization of the asymptotic expansions to a bilateral default risk setting is addressed in Chapter 6.Our thesis ends by tackling the problem of modeling the specific Right-Way Risk induced by rating trigger events within the collateral agreements. Our major contribution is the calibration of a rating transition model to market implied default probabilities

<p>e will evaluate forward contracts in the electricity market. A thorough presentation of stochastic analysis for processes with discontinuous paths are provided, and some results concerning these from mathematical finance are stated. Using a Feynman-Kac-type theorem by Pham we derive a partial integro-differential equation giving the forward price from the spot dynamics taken from Geman and Roncoroni. This spot model is regime switching, so we get two equations. These equations are then attempted solved numerically. We suggest the following approach: When implementing boundary-conditions numerically we use values obtained from a Monte Carlo simulation of the spot dynamics to calibrate the boundary.</p>

In this thesis we are concerned with valuing options on traded accounts using both continuous and discrete time models. An option on a traded account is a zero strike call on the balance of a trading account which consists of a position of size $\theta$ in a risky asset (which we refer to as a stock) and the remaining wealth in a risk-free account. The choice of trading positions throughout the life of the option are made by the buyer, subject to constraints specified in the contract at the time of purchase. The specification of these trading constraints gives rise to some of the more well known examples including passport options and vacation options. At maturity, the option buyer is entitled to any positive wealth accumulated in the trading account whilst any losses are covered by the option seller. First, we examine the problem of valuing these options in continuous time. A review of some existing methods is presented, including a complete derivation of the pricing formula for the passport option and the option on a traded account following the methods proposed by Hyer et al. (1997) and Shreve and Vecer (2000), though we often use different techniques to those authors. We also present an alternative derivation for the value of a passport option using our own methodology which we believe is simpler than those currently available. Secondly, we consider the valuation problem in a discrete time setting by looking at one specific discrete time model, the binomial tree. This is a new contribution to the literature as binomial models for these options have not been previously examined. Using this approach, the greatest difficulty is the determination of an optimal trading strategy which is required to price this class of option. We show that in general, binomial models and continuous time models do not have the same trading strategy, and in fact that the analytic determination of the trading strategy for an option on a traded account may in fact be impossible to obtain. We then turn to passport options, where we are able to derive an analytic optimal strategy which in this case is identical to that used in the continuous time models, thus the problem of valuing passport options is reduced to the same computational burdens as a binomial valuation without recombining branches. Lastly, we examine some numerical methods which could be used to value options on traded accounts with binomial models. Our problem is shown to be an NP-hard convex maximisation which we convert into both an l1-norm convex maximisation and an indefinite quadratic program. Whilst we present algorithms which are guaranteed to obtain the optimal solution, they are also known to be inefficient and thus inappropriate for any likely application beyond a few time steps. We conclude by summarising our results and give directions for future research in this area.

This thesis explores the price determinants of Bitcoin using a macroeconomic model based on the economic equation of exchange presented by Joseph Wang (2014). The thesis provides a concise and structured introduction to Bitcoin and a comprehensive literature review on Bitcoin. The analysis begins with the application of the functions of money to Bitcoin, arguing that while Bitcoin does fulfill the three classical functions of money to a certain extent, its use remains mainly as a speculative instrument. Wang's model is criticized and amended to reflect the realities of empirically analyzing the Bitcoin market. Using the daily number of transactions and Bitcoin days destroyed as proxies for economic activity and inactivity - to measure Bitcoin's velocity on the block chain - vector autoregression modelling is used to determine if there is Granger causality between the price of bitcoin and the two proxies. The results demonstrate that there is a bidirectional Granger-causal relationship between Bitcoin days destroyed and the price of bitcoin and that there is none between the daily number of transactions and the price of bitcoin; proving Wang's two main assumptions. Impulse- response functions are provided to illustrate and discuss this bidirectional relationship. The results are in line with the...

Nowadays there is a fast development of the methods based on transmutation operators (TO) theory for solving differential equations. The possibility to construct the images of solutions for TO in certain cases allowed the construction of accurate numerical solutions to several problems that appear in different applied fields. In the present work, for the first time, it is shown that these methods can be effectively applied to the optimal stopping problems that appear in mathematical finance. The first part of the thesis (Chapter 2) consists of an application of the method to the valuation of European-style double-barrier knock-out options (DBKO). This is done by using the efficient computation of eigenvalues for the Shrodinger equation and a representation of solutions in terms of Neumann series of Bessel functions. This knowledge was used in the construction of a novel analytically tractable method for pricing (and hedging) DBKO, which can be applied to the whole class of one-dimensional timehomogeneous diffusions even for the cases where the corresponding transition density is not known. The proposed numerical method is shown to be efficient and simple to implement. To illustrate the flexibility and computational power of the algorithm it is constructed an extended jump to default model that is able to capture several empirical regularities commonly observed in the literature. The second part of the thesis (Chapters 3 and 4) is dedicated to the study of the more complicated problems: the free boundary problems. For this purpose, the method was first (in some certain sense) generalized and tested on the Stefan-like problem. The method consists in efficiently constructing a complete system of solutions for parabolic equation from known solutions for the heat equation, heat polynomials (HP). This way it was possible to extend the numerical method that existed only for the heat equation to the large class of parabolic equations. However, for the selected financial problem, Russian option with finite horizon (ROFH), the numerical computation from the method based on HP revealed to be non-efficient. This is due to the more complex structure of the problem, specifically the non-consistent boundary conditions. Hence, it was developed another variation of the method that uses different systems of solutions for the heat equation: the generalized exponential basis. The constructed method proved to be accurate, relatively easy to implement and can it can be applied to the large class of the free boundary problems. The value of the ROFH has been an important theme of discussion in the last decades. The application of the method to this problem confirmed several results that have appeared recently in the literature and shred some light on the differences that were present.The constructed methods have a large scope of applications not only in financial field, but also in other disciplines. Both studies also open a variety of future research and applications that are discussed in the text.<br>Actualmente estamos a assistir a um rápido desenvolvimento de métodos baseados nos operadores de transmutação (OT) para a resolução de equações diferenciais. Em certos casos, é possível calcular as imagens de soluções para OT, o que permite construir soluções numéricas com um elevado grau de precisão para diversos problemas aplicados. No presente trabalho, pela primeira vez, e desenvolvida e ilustrada uma aplicação eficiente destes métodos aos problemas de paragem óptima que surgem na matemática financeira. A primeira parte da tese (Capítulo 2) consiste na aplicação do método ao problema de avaliação de opção com dupla barreira "knock-out" (DBKO) de estilo europeu. A construção do método passa por um apurado cálculo de valores próprios do respectivo problema de Schrodinger e a representação de soluções em termos de séries de Neumann de funções de Bessel. Esse conhecimento foi utilizado para construir um novo método de expressão analítica para definição de preço (e cobertura) de DBKO. O método pode ser aplicado a toda uma classe de difusões uni-dimensionais homogéneas no tempo, mesmo para os casos em que não é conhecida a função de densidade de transição. Neste capítulo é demonstrado que o método proposto é eficiente e simples de implementar. Para ilustrar a flexibilidade e a robustez computacional do respectivo algoritmo é construído um modelo estendido de salto para o incumprimento que oferece a possibilidade de captar certos efeitos empíricos presentes na literatura. A segunda parte da tese (Capítulos 3 e 4) e dedicada ao estudo de problemas mais complexos: problemas de fronteira livre. Para esse propósito, o método foi (em certo sentido) generalizado e testado no problema do tipo de Stefan. O método consiste numa construção eficiente de um sistema completo de soluções para uma equação diferencial parabólica a partir de um sistema completo de soluções para a equação de calor, os polinómios de calor (PC). Deste modo, foi possível estender o método numérico que existia apenas para equação de calor para uma larga classe de equações parabólicas. No entanto, para o problema financeiro seleccionado, a opção russa com horizonte finito (ORHF), o método baseado nos PCs revelou-se computacionalmente ineficiente. Isso deve-se a uma estrutura mais complicada do problema, nomeadamente as não-consistentes condições de fronteira. Como tal, foi desenvolvida uma outra variação do método que usa um sistema de soluções diferente de PCs: uma base exponencial generalizada. O método construído provou ser preciso, de relativamente fácil implementação e pode ser aplicado a uma larga classe de problemas de fronteira livre. O valor de ORHF foi e continua a ser um importante tema de discussão nas últimas décadas. A aplicação do método a esse problema confirmou vários resultados que surgiram recentemente na literatura e revelou o porque de algumas diferenças. Os métodos construídos têm uma larga gama de aplicações, tanto no âmbito de matemática financeira como em outras disciplinas. Ambos os estudos abrem várias possibilidades para futuras investigações e aplicações, as discussões das quais se encontram no texto.

The discretionality and the appraisers’ subjectivity that characterize traditional real estate valuation are still allowed to take part in the formation of the asset price even when respecting international standards (EVS, IVS) or Appraisal Institution´s regulations (TEGOVA, RICS, etc.). The application of econometric and statistical methods to real estate valuation aims at the elimination of subjectivity on the appraisal process. But the unanswered question underneath this subject is the following: How important is the subjective component on real estate appraisal value formation? On this study Structural Equation Models (SEM) are used to determine the importance of the objective and subjective components on real estate valuation value formation as well as the weight of economic factors and the current economic context on real estate appraisal for mortgage purposes price formation. There were used two latent variables, Objective Component and Subjective Component, witch aggregate objective observed variables and subjective observed and unobserved variables, respectively. Factorial Exploratory Analysis is the statistical technique used in order to link the observed variables extracted from the valuation appraisal reports to the latent constructs derived from the theoretical model. SEM models were used to refine the model, eliminate non‐significant variables and to determine the weight of Objective and Subjective latent variables. These techniques were applied to a sample of over 11.000 real estate assets appraisal reports throughout the time period between November of 2006 and April of 2012. The real assets used on this study are located on Lisbon’s Metropolitan Area – “Grande Lisboa” –, Portugal. From this study, we conclude that Subjective Component has a considerable weight on real estate appraisal value formation and that the external factor Economic Situation has a very small impact on real estate appraisal value formation.

A robust computational framework is presented for the risk-neutral valuation of capital guarantees written on discretely-reallocated portfolios following the Constant Proportion Portfolio Insurance (CPPI) strategy. Aiming to address the (arguably more realistic) cases where analytical results are unavailable, this framework accommodates risky-asset jumps, volatility surfaces, borrowing restrictions, nonuniform reallocation schedules and autonomous CPPI floor trajectories. The two-asset state space representation developed herein facilitates visualising the CPPI strategy, which in turn provides insight into grid design and interpolation. It is demonstrated that given a deterministic process for the risk-free rate, the pricing problem can be cast as solving cascading systems of 1D partial integro-differential equations (PIDEs). This formulation’s stability and monotonicity are studied. In addition to making more sense financially, the limited borrowing variant of the CPPI strategy is found to be better suited than the classical (unlimited borrowing) counterpart for bounded-domain calculations. Consequently, it is demonstrated how the unlimited borrowing problem can be approximated by imposing an artificial borrowing limit. For implementation validation, analytical solutions to special cases are derived. Numerical tests are presented to demonstrate the versatility of this framework.