Stochastic Modeling with Fractional and non-Fractional Noises: Applications to Finance and Insurance

In my doctoral work, I have developed stochastic models that use different type of noises, to price financial derivatives and insurance products. Stock prices have a random component in their behavior that characterizes statistical aspects of that particular asset. The type of noise chosen to model the dynamics of prices is key to reproduce empirical facts of assets. In this dissertation I have developed the continuous version for a new kind of noise with memory. One of its particularities is that it takes into account all the past states, in order to determine its current state and smoothness. This is a highly desirable feature in asset modeling. I have also proposed a new approximating formula for call options in the form of a series expansion that can be truncated to any desired order depending on the degree of accuracy needed. A second approximating formula is also provided for the case where the noise used to model stock prices and volatility has discontinuous trajectories. In the scope of a different project I have also proposed a model to price insurance products that takes into account the join risks arised from stock price volatility and interest rates