An efficient valuation of participating life insurance contracts under Lévy process.

Wong, Shiu Fung."July 2010."Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (leaves 36-38).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Participating policy --- p.4Chapter 3 --- Levy Process and its use in financial modelling --- p.8Chapter 3.1 --- Levy process in asset modelling --- p.8Chapter 3.2 --- Levy process in derivative pricing --- p.11Chapter 3.2.1 --- Review of FFT methods in option pricing --- p.12Chapter 3.2.2 --- Expectation using FFT --- p.13Chapter 4 --- Network methodology --- p.17Chapter 4.1 --- Asset dynamic: Network Approach --- p.17Chapter 4.1.1 --- Transition probability by FFT --- p.18Chapter 4.1.2 --- Example in American option pricing --- p.19Chapter 4.2 --- Extended Network for Participating Contract --- p.20Chapter 4.3 --- Practical network construction --- p.22Chapter 4.3.1 --- Modified network-drift offsetting --- p.23Chapter 4.3.2 --- Logarithmic scale network --- p.25Chapter 4.4 --- Incorporating surrender rights and mortality --- p.26Chapter 4.4.1 --- Surrender right --- p.26Chapter 4.4.2 --- Mortality --- p.27Chapter 4.5 --- Proof of convergence --- p.28Chapter 5 --- Numerical Results --- p.32Chapter 5.1 --- The Black and Scholes model --- p.33Chapter 5.2 --- The Merton's Jump diffusion model --- p.33Chapter 5.3 --- Variance gamma model --- p.34Chapter 6 --- Conclusion --- p.35Bibliography --- p.3

Hedging error in Lévy models with a Fast Fourier Transform approach

We measure, in terms of expectation and variance, the cost of hedging a contingent claim when the hedging portfolio is re-balanced at a discrete set of dates. The basic point of the methodology is to have an integral representation of the payoff of the claim, in other words to be able to write the payoff as an inverse Laplace transform. The models under consideration belong to the class of Lévy models, like NIG, VG and Merton models. The methodology is implemented through the popular FFT algorithm, used by many financial institutions for pricing and calibration purposes. As applications, we analyze the effect of increasing the number of tradings and we make some robustness tests.Hedging, Lévy models, Fast Fourier Transform

A Multivariate Jump-Driven Financial Asset Model

We discuss a Lévy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behavior of a series of stocks or indexes and to study a multi-firm, value-based default model. Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a sto- chastic time-change technique and provide the details for a Gamma change. The main feature of the model is the fact that - opposite to other, non jointly Gaussian settings - its risk neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit.Lévy processes, multivariate asset modelling, copulas, risk neutral dependence.

A Lévy Option Pricing model of FFT-Based High-order Multinomial Tree

This paper studies the method of constructing high order recombined multinomial tree based on fast Fourier transform (FFT), and applies multinomial tree option pricing under the Lévy process. First, the Lévy option pricing model and Fourier transform are introduced. Then, the network model based on FFT (Markov chain) is presented. After that, a method of constructing a recombined multinomial tree based on FFT is given. It is proved that the discrete random variables corresponding to the multinomial tree converge to the Lévy distributed continuous random variable. Next, we obtain the European option pricing formula of FFT multinomial tree pricing, and apply the reverse iteration method to the American option pricing. Finally, under the Jump-diffuse process, the difference between the computational accuracy and computational efficiency of the Semi-analytical solution of European Option and Merton European Call Option which are priced under FFT is compared. The results show that the method of constructing a high-order recombined multinomial tree based on FFT has very high calculation precision and calculation speed, which can solve the problem of traditional risk-neutral multinomial tree construction and it is a promising pricing method for derivative products

Valuation of dynamic fund protection under levy processes.

Lam, Ka Wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.Includes bibliographical references (leaves 51-55).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Levy Processes --- p.6Chapter 2.1 --- Definition --- p.6Chapter 2.2 --- Levy-Khinchine formula --- p.7Chapter 2.3 --- Applications of Levy Processes in Finance --- p.10Chapter 2.4 --- Option pricing under Levy Processes --- p.12Chapter 2.4.1 --- Black-Scholes Formula with Characteristic Function --- p.12Chapter 2.4.2 --- Fast Fourier Transform --- p.14Chapter 2.4.3 --- Other Payoff Functions --- p.16Chapter 3 --- Dynamic Fund Protection --- p.19Chapter 3.1 --- Discrete Dynamic Fund Protection --- p.20Chapter 3.2 --- Link DFP to Discrete Lookback Options --- p.22Chapter 4 --- Spitzer´ةs Identity --- p.25Chapter 4.1 --- Applications of Spitzer's Identity --- p.25Chapter 4.2 --- Discrete Lookback Options --- p.29Chapter 5 --- Pricing Discrete DFP --- p.32Chapter 5.1 --- Girsanov´ةs Theorem --- p.32Chapter 5.2 --- Equivalent Martingale Measure in DFP --- p.34Chapter 5.3 --- Pricing DFP at any Time Points --- p.36Chapter 5.4 --- The Main Algorithm --- p.38Chapter 6 --- Numerical Results --- p.40Chapter 6.1 --- Simulation of Discrete DFP --- p.40Chapter 6.2 --- Numerical Implementation --- p.42Chapter 7 --- Conclusion --- p.50Bibliography --- p.5

Structural models in credit risk

Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2014O principal objetivo desta tese é modelizar o risco de crédito de uma determinada instituição financeira utilizando modelos estruturais. Neste sentido, propomos a análise de dois modelos - Merton(1974) e CreditGrades - que são apresentados de acordo com a sua evolução temporal. Em cada modelo é calculada a fórmula fechada para a probabilidade de default neutra face ao risco, assim como o credit spread para uma empresa de referência. No entanto, antes da implementação prática dos modelos estruturais, é apresentado um referencial teórico que visa fornecer, de forma gradual, informações consideradas indispensáveis para a compreensão dos modelos em causa. O modelo de Merton (1974) apresenta uma grande inovação que reside no modo de tratar o capital próprio de uma companhia como uma opção de compra sobre os seus ativos, permitindo assim a aplicação de métodos de avaliação de opções, tais como os modelos de Black e Scholes (1973) e de Merton (1973). As vantagens reconhecidas do modelo são não apenas a quantidade reduzida de parâmetros a estimar, como também a simplicidade de o colocar em prática. No capítulo I são também apresentadas algumas vantagens e desvantagens do modelo. O facto de o processo do valor dos ativos da empresa não ser observável no mercado constitui a maior dificuldade na implementação dos modelos estruturais. Estudos académicos propõem metodologias de estimação avançadas para determinar os parâmetros deste processo. Com efeito, um dos inconvenientes deste modelo é assumir o valor dos ativos (V t) e a respetiva volatilidade (svt), como parâmetros de input ao modelo, uma vez que não são diretamente observáveis no mercado. Neste trabalho, são apresentadas duas aproximações ao modelo no que se refere à estimação dos parâmetros: uma aproximação iterativa e outra como solução de um sistema de equações não-lineares. Em 2002, foi construído um modelo baseado na completa transparência de mercado - CreditGrades - para comparar os spreads modelados com os spreads observados no mercado e calcular a probabilidade de sobrevivência de uma determinada empresa. Construído sobre a estrutura do modelo Black e Cox (1976), o qual relativiza algumas das premissas presentes no modelo standard de Merton (1974), permite que um evento de default possa ocorrer antes da maturidade T (se o valor dos ativos da empresa tocar na barreira de default). Outra das vantagens relevantes é o facto de a dívida financeira ser expressa por ação e estimada com base em dados financeiros provenientes de demonstrações consolidadas. Por outro lado, os poucos inputs do modelo são todos observáveis no mercado. Na segunda parte deste trabalho, é apresentada uma aplicação destes modelos a um caso real: trata-se de um banco português que recentemente entrou em default. Pretende-se assim mostrar a probabilidade de default e o credit spread do banco em estudo, num cenário financeiro adverso, permitindo observar e analisar a adequação destes modelos ao mundo real. Por outro lado, o uso prévio destes modelos não teria evitado a situação de bancarrota do banco, mas daria uma boa percepção do risco de crédito ao longo do tempo. Concluindo, o objetivo geral desta tese é informar o leitor sobre o modo possível de construir modelos de risco de crédito, dando-se um ênfase especial aos métodos práticos que um banco e/ou uma seguradora, nas respetivas áreas de corporate banking e atuariado, podem fazer uso, num processo de desenvolvimento de um novo modelo de credit rating.The main objective of this thesis is to model the credit risk of a certain financial institution under the structural model approach. In this setting, we propose the analysis of two models - Merton (1974) and CreditGrades - which are presented according to its temporal evolution. Each model provides the closed-form formulae for the risk-neutral default probability and credit spread of a reference _rm. However, before the practical implementation of the structural models, it is presented a theoretical framework that aims to provide, gradually, information considered essential to the understanding of the models under analysis. The Merton (1974) model offers a huge innovation that lies in the way of treating a company's equity as a call option on its assets, thus allowing for applications of Black and Scholes (1973) and Merton (1973) option pricing methods. The advantages recognized for the model are not only the few parameters to estimate but also the simplicity of putting it into practice. In Chapter I, we also present some advantages and disadvantages of the model. The unobservability of the firm's assets value is a major difficulty in the implementation of structural models. Academic studies propose advanced estimation methodologies to determine the parameters of this process. In fact, one of the shortcomings of Merton's model (1974) is to assume the value of company assets (V t) and the respective volatility (svt) as parameters of input to the model, since they are not directly observable in the market. In this work, we presented two approaches to the model regarding the estimation of parameters: an iterative approach and other as a solution of a system of nonlinear equations. In 2002, it was developed a completely transparent market based model – CreditGrades - to match modeled spreads with the observed spreads and which calculate the survival probabilities of a reference firm. Built on the framework of the Black and Cox (1976) model, which relaxes some of the assumptions present in the standard Merton model, it enables a default event to occur before maturity T (if the value of company assets hits the default barrier). Others relevant advantages is the fact that the financial debt expressed on a per-share basis and estimated based on financial data from consolidated statements. On the other hand, the inputs of the model are all observable in the market. In the second part of this work, it is presented an application of these models to a real case, i.e. a portuguese bank that recently went into default. The aim is to show the default probability and the credit spread of the bank in study, in an adverse financial scenario, allowing to observe and analyze the adequacy of these models to the real world. Moreover, previous use of these models would not have avoided the situation of bankruptcy of the bank, but give a good insight of credit risk over time. The general purpose of this thesis is to inform the reader on how it is possible to construct credit rating models. Special emphasis is made on the practical methods that a bank or insurance in the respective areas corporate banking sector and actuarial could make use of in the development process of a new credit rating model