Mathematical methods in finance

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Proposed projects:1. We are interested in the framework of financial models with proportional transaction costs. We would like to build up an appropriate duality theory for markets in continuous time, and to prove an equivalence between an appropriate concept of no arbitrage and the existence of dual variables in continuous time.2. We are interested in the calibration problem of delay financial models. We would like to obtain an approximate solution provided by a regularized calibration problem. We consider a finite difference method for solving parabolic integro-differential equations with possibly singular kernels; we focus on localization to a finite domain and provide an estimate for the localization error.3. We shall provide expectation formulas for the calculation of Greeks for jump-diffusion models. We apply methods from Malliavin calculus, and we assume that the linkage opertaor is invertible. We apply these ideas to the jump-diffusion models.4. We will analyze the evolution of prices in deregulated electricty markets. We estimate the parameters of the model using historical data from the Canadian and US markets, and demonstrate how it can be used for option pricing via Monte-Carlo simulation.5. Using the saddlepoint approximation technique, we compute the prices of various European-style options for models driven by Levy processes. We expect to obtain a better approximation to the Gaussian base.