This self-contained volume brings together a collection of chapters by some of the most distinguished researchers and practitioners in the fields of mathematical finance and financial engineering. Presenting state-of-the-art developments in theory and practice, the Festschrift is dedicated to Dilip B. Madan on the occasion of his 60th birthday. Specific topics covered include:* Theory and application of the Variance-Gamma process* Lévy process driven fixed-income and credit-risk models, including CDO pricing* Numerical PDE and Monte Carlo methods* Asset pricing and derivatives valuation and hedging* Itô formulas for fractional Brownian motion* Martingale characterization of asset price bubbles* Utility valuation for credit derivatives and portfolio managementAdvances in Mathematical Finance is a valuable resource for graduate students, researchers, and practitioners in mathematical finance and financialengineering. Contributors: H. Albrecher, D. C. Brody, P. Carr, E. Eberlein, R. J. Elliott, M. C. Fu, H. Geman, M. Heidari, A. Hirsa, L. P. Hughston, R. A. Jarrow, X. Jin, W. Kluge, S. A. Ladoucette, A. Macrina, D. B. Madan, F. Milne, M. Musiela, P. Protter, W. Schoutens, E. Seneta, K. Shimbo, R. Sircar, J. van der Hoek, M.Yor, T. Zariphopoulou