Pricing path-dependent derivative securities: new approach

• Ano de defesa: 2019
• Autor(a) principal: Rodriguez Otazú, Juan Bladimiro
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Laboratório Nacional de Computação Científica; Coordenação de Pós-Graduação e Aperfeiçoamento (COPGA); Brasil; LNCC; Programa de Pós-Graduação em Modelagem Computacional
• Palavras-chave em Português: Derivativos (Finanças); Processos, Lévy; No-arbitrage pricing; Path-dependent derivatives; CNPQ::CIENCIAS SOCIAIS APLICADAS::ECONOMIA::ECONOMIA MONETARIA E FISCAL
• Link de acesso: https://tede.lncc.br/handle/tede/297

The pricing and hedging of derivatives in stock and fixed income markets is a challenging task both from a theoretical and empirical point of view. Although the risk-neutral pricing method is a well-established theory, its practical application is not trivial. Products available in the market can be very complex with a diversity of exotic payoffs. A large amount of academic literature has been dedicated to propose models to circumvent the practical issues of the pricing and hedging derivatives in stocks and fixed income markets. In special cases it is possible to find a closed formula to its price. However, in more general setups this is not the case and one must rely in numerical methods to find the derivative price. In this thesis, we address the pricing problem by proposing a new method in series format to pricing path-dependent derivatives. The idea is to produce a time and value discretization of the stochastic process on which the derivative is subscribed and that appears in the risk-neutral expectation for the price. The theoretical novelty of the method is that we benefit from the Feynman-Kac formula not to obtain prices - as it is the case of standard PDE methods, but instead, conditional probabilities - which appears as weighted factors in our series. The corresponding Feynman- Kac PDEs (or PIDEs) linked to our method are one dimensional with prescribed terminal conditions of very simple shape to be solved in just a unitary time lag. Payoffs play no role neither in the terminal condition nor anywhere related to the PDEs (or PIDEs), and the method is insensitive to the shape of the path-dependent payoff. To the best of our knowledge this is the first time that such approach is used. Our approach allows parallel computing. It is quite general, since it deals with continuous time as well as discrete monitoring path-dependent derivatives, of arbitrary format, on diffusions and Levy processes. It requires only two weak and natural assumptions. First, the market to be arbitrage-free which allows us to use the risk-neutral pricing technique. Second, the diffusion or the Levy process must be consistent with the Feynman-Kac formula. The expectation under the risk neutral probability - the one that renders the underlying prices into Martingales - is considered as an abstract stochastic formula and then applied to the financial scenario. Our discretization method provide therefore a formula, to be solved numerically, for calculating the exact price of derivatives in stock markets and fixed income markets. We illustrate the method by pricing an Asian type interest rate option, called Interbank Deposit Index (IDI) option, discretely updated. This is a standardized derivative product traded at the Securities and Futures Exchange in the Brazilian fixed income market.