Martingale Methods in Financial Modelling

Abstract

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Correlated Poisson Processes and Their Applications in Financial Modeling

“…The modern financial theory often recommends jump-diffusion models to describe dynamics of the individual risk factors, such as interest rates, foreign exchange rates, stock indices, and volatility surfaces (Kou, 2002;Lipton and Rennie, 2008;Merton, 1976;Musiela and Rutkowsky, 2008), One of the most popular model of jumps, the Poisson model, requires introduction of a codependence structure in the multivariate setting.…”

“…The multivariate Gaussian diffusion models are traditionally popular in financial applications (Musiela and Rutkowsky, 2008). In this class of models, the dynamics of the risk factors are described by the Gaussian stochastic processes.…”

Nonparametric Bayesian Estimation of a HHlder Continuous Diffusion Coefficient

“…Furthermore, in many contexts, e.g. when pricing financial derivatives, knowledge of the drift coefficient is in fact of no interest, whereas the dispersion coefficient is of paramount importance; see Musiela and Rutkowski (2005). This motivates us to completely ignore the drift coefficient in our estimation procedure by intentionally misspecifying the model and acting as if the drift were equal to zero.…”

“…Problem description. Stochastic differential equations (SDEs) have been widely used as models in numerous applications ranging from physics (see for example Allen (2007)) to engineering (see Wong and Hajek (1985)) and to finance (see Musiela and Rutkowski (2005)). We assume observations from an SDE of the form…”

Integral equation characterization of the Feynman–Kac formula for a regime-switching diffusion

“…Conditional expectations of the form in (1.2) arise in many applications, particularly in the pricing of financial contracts (see e.g. [10] and references therein). This expectation can be approximated by numerically solving the coupled PDE in (1.1) by finite-difference methods ( [3], [5], [8]), by trinomial tree methods ( [9]), or by Monte Carlo simulation ( [7]).…”