Valuation of financial derivatives through transmutation operator methods

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.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

Trading in Structured Products: Investor Behavior and Pricing Policies

In my work I focus on structured products offering numerous risk-return variations on a broad landscape of underlying assets. I answer the question whether investors put structured products to good use and whether issuers provide a fair and transparent environment

A Method of Pricing European Style Equity Options

The study of option pricing has a very short history, when compared with other elements of economics. Since the publication of a method to price European style equity options by Fischer Black and Myron Scholes in 1973 a vast amount of research on option pricing has occurred. Ultimately, the pricing of equity options depends upon the match of the model and reality. A new method to price option contracts is proposed. It is argued that the distributional assumptions of standard models are uncorrelated with nature. A new model is proposed as a start to a new class of models

Valuation, Empirical Analysis, and Optimal Exercise of Open-End Turbo Certificates

This dissertation analyzes Open-End Turbo Certificates (OETCs), a popular class of retail derivatives. OETCs can be exercised at any time at the investor’s discretion. In order to explain the existence of the certificates jump risk must be considered. We propose and implement an optimal stopping approach to price these securities, which further allows for determining optimal exercise thresholds. They result from the trade-off between benefits from downward jump protection and financing costs. We show that early exercise right has a significant impact on their values. In an empirical analysis pertaining to the years 2007 through 2009 it turns out that certificates which could be rationally held are very rare, although the degree by which the underlying exceeds the optimal exercise thresholds continually declines over the considered period. We suggest three lines of explanation: general market movement, jump risk perception by the market, and increased competition among issuers.Die vorliegende Dissertation behandelt Open-End Turbo Zertifikate (OETCs), eine populäre Klasse von Privatkundenderivaten, die jederzeit durch den Investor ausgeübt werden können. Um ihre Existenz rechtfertigen zu können, muss Sprungrisiko berücksichtigt werden. Zur Preisstellung des Produktes schlagen wir einen Optimal Stopping Ansatz vor und implementieren diesen. Dies erlaubt zudem die Berechnung optimaler Ausübungsschwellen, die aus dem Gegenspiel von Finanzierungskosten einerseits und Schutz gegen Abwärtssprünge andererseits entstehen. Wir zeigen, dass vorzeitige Ausübungsrechte in der Bewertung eine signifikante Rolle spielen. In einer empirischen Analyse für die Jahre 2007 bis 2009 zeigt sich schließlich, dass rationale Investoren nur sehr wenige OETCs halten sollten. Andererseits geht der Grad, um welchen das Underlying die optimale Ausübungsschwelle überschreitet, kontinuierlich zurück. Für diese Beobachtung lassen sich drei Begründungen anführen: allgemeine Marktbewegung, vom Markt wahrgenommenes Sprungrisiko und erhöhter Wettbewerb unter den Anbietern

Risk-mitigation techniques: from (re-)insurance to alternative risk transfer

Insurance risks knowledge is becoming essential for both financial stability and social security purposes, moreover in a country with a very low insurance education like Italy. In insurance industry, Solvency II requirements introduced new issues for actuarial risk management in non-life insurance, challenging the market to have a consciousness of its own risk profile, and also investigating the sensitivity of the solvency ratio depending on the insurance risks and technical results on either a short-term and medium-term perspective. For this aim, in the present thesis, firstly a partial internal model for underwriting risk is developed for multi-line non-life insurers. Specifically, the risk-mitigation and profitability impacts of traditional reinsurance in the underwriting risk model are introduced, and a global framework for a feasible application of this model consistent with a medium-term analysis is provided. Reinsurance have to be considered in the assessment of Non-Life insurers risk profile, with particular regard to the Solvency II Underwriting Risk because of its impact on business and risk strategy. Risk mitigation techniques appear as a key driver of Non-Life insurance business as they can change risk profile over either the short-term or medium-term perspective. They impact the technical result of the year in such a way that it is important to assess how reinsurance strategies decrease the volatility, reducing the capital requirements, but, on the other hand, they also change the mean of distributions in different ways according to the price for risk requested by reinsurers. At the same time, risk mitigation also influences Non-Life insurance management actions as it can improve business strategy and capital allocation (also in potential capital recovery plans). Furthermore, the analysis a medium-term capital requirement would ask insurers to have more capital than in a one-year time horizon, and in this framework risk mitigation effects linked to reinsurance strategies must be assessed on either risk/return perspective trade-off. On the other hand, (re)insurance can play an active role in mitigating physical risks, and in particular natural catastrophe risks. In this context, as well as in natural disasters, Alternative Risk Transfer (ART) is becoming a new significant actuarial and capital management tool for insurers and, potentially, for government measures in recovery actions of economic and social losses in case of natural disasters. Catastrophe Bonds are insurance-linked securities that have been increasingly used as an alternative to traditional reinsurance for two decades. In exchange for a Spread over to the risk-free rate, protection is provided against stated perils that could impact the insured portfolio. A broad literature has flourished to investigate what are the features that significantly influence the Spread, in addition to the portfolio’s expected loss. Almost all proposed models are based on multivariate linear regression, that has provided satisfactory predictive performance as well as easily interpretability. This thesis also explores the use of Machine Learning models in modeling the determinant at issuances, contrasting both their predictive performance and their interpretability with respect to traditional models. An overview of the economics of CAT bonds, on current literature and on the statistical methodologies will be provided also. Aim of this Thesis is to provide a solid framework of insurance risk transfer for both pure underwriting and catastrophe risks, investigating risk transfer practices from traditional to alternative and most innovative technique. In these fields, firstly a suitable risk model is used in order to describe main impacts on insurance business model. Then, the main innovative alternative risk transfer for catastrophe risks are illustrated and CAT Bond will be adequately described, investigating main pricing models using a machine learning approach. Finally, a possible Italian CAT Bond issuance is provided in order to investigate an integrated solution with a traditional reinsurance underlying an alternative risk transfer in order to achieve a public-private partnership to natural catastrophe

Mathematical methods for valuation and risk assessment of investment projects and real options

In this thesis, we study the problems of risk measurement, valuation and hedging of financial positions in incomplete markets when an insufficient number of assets are available for investment (real options). We work closely with three measures of risk: Worst-Case Scenario (WCS) (the supremum of expected values over a set of given probability measures), Value-at-Risk (VaR) and Average Value-at-Risk (AVaR), and analyse the problem of hedging derivative securities depending on a non-traded asset, defined in terms of the risk measures via their acceptance sets. The hedging problem associated to VaR is the problem of minimising the expected shortfall. For WCS, the hedging problem turns out to be a robust version of minimising the expected shortfall; and as AVaR can be seen as a particular case of WCS, its hedging problem is also related to the minimisation of expected shortfall. Under some sufficient conditions, we solve explicitly the minimal expected shortfall problem in a discrete-time setting of two assets driven by correlated binomial models. In the continuous-time case, we analyse the problem of measuring risk by WCS, VaR and AVaR on positions modelled as Markov diffusion processes and develop some results on transformations of Markov processes to apply to the risk measurement of derivative securities. In all cases, we characterise the risk of a position as the solution of a partial differential equation of second order with boundary conditions. In relation to the valuation and hedging of derivative securities, and in the search for explicit solutions, we analyse a variant of the robust version of the expected shortfall hedging problem. Instead of taking the loss function $l(x) = [x]^+$ we work with the strictly increasing, strictly convex function $L_{\epsilon}(x) = \epsilon \log \left( \frac{1+exp\{−x/\epsilon\} }{ exp\{−x/\epsilon\} } \right)$. Clearly $lim_{\epsilon \rightarrow 0} L_{\epsilon}(x) = l(x)$. The reformulation to the problem for L_{\epsilon}(x) also allow us to use directly the dual theory under robust preferences recently developed in [82]. Due to the fact that the function $L_{\epsilon}(x)$ is not separable in its variables, we are not able to solve explicitly, but instead, we use a power series approximation in the dual variables. It turns out that the approximated solution corresponds to the robust version of a utility maximisation problem with exponential preferences $(U(x) = −\frac{1}{\gamma}e^{-\gamma x})$ for a preferenes parameter $\gamma = 1/\epsilon$. For the approximated problem, we analyse the cases with and without random endowment, and obtain an expression for the utility indifference bid price of a derivative security which depends only on the non-traded asset