Spectral Methods for Numerical Solution of PDEs

J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge, 2007
D. A. Kopriva, Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers, Springer, 2009
C. Pozrikidis, Introduction to Finite and Spectral Element Methods using MATLAB, Chapman & Hall/CRC, 2005 (code source 1 or code source 2)
L. N. Trefethen, Spectral Methods in MATLAB, SIAM, 2001 (code)

Finite Difference Methods for Numerical Solution of PDEs

L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, Wiley, 1982
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 1994
Y. Shapira, Solving PDEs in C++, SIAM, 2006
G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edition, Oxford, 1986
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer, 1998
J. W. Thomas, Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations, Springer, 1999

Finite Element Methods for Numerical Solution of PDEs

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, 2nd edition, SIAM, 2002
T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover, 2000
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd edition, Springer, 2006
E. Thompson, Introduction to the Finite Element Method: Theory, Programming, and Applications, Wiley, 2005

J. Topper, Financial Engineering with Finite Elements, Wiley, 2005
O. C. Zienkiewicz and K. Morgan, Finite Elements and Approximation, Dover, 2006
O. C. Zienkiewicz, R. L. Taylor, and J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 6th edition, Butterworth-Heinemann, 2005

Introductory Mathematical Finance

M. Avellaneda and P. Lawrence, Quantitative Modeling of Derivative Securities, Chapman and Hall/CRC, 1999
M. Baxter and A. Rennie, Financial Calculus: An Introduction to Option Pricing, Cambridge, 1996
T. Björk, Arbitrage Theory in Continuous Time, Oxford, 2004
M. Capinski and T. Zastawniak, Mathematics for Finance : An Introduction to Financial Engineering, Springer Undergraduate Mathematics Series, 2004
V. Goodman and J. Stampfli, The Mathematics of Finance: Modeling and Hedging, Brooks Cole, 2000
H. Follmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 2nd Edition, Springer, 2004
J. C. Hull, Options, Futures, and Other Derivatives (with code), 7th edition, Prentice Hall, 2007
P. Hunt and J. Kennedy, Financial Derivatives in Theory and Practice, Wiley, 2004
J. E. Ingersoll, Theory of Financial Decision Making, Rowman & Littlefield Publishers, 1987
R. Jarrow and S. Turnbull, Derivative Securities, 2nd edition, South-Western College,1999
M. S. Joshi, The Concepts and Practice of Mathematical Finance, Cambridge, 2003
D. Lamberton and B. Lapeyre, Introduction to stochastic calculus applied to finance, Springer, 1996
D. G. Luenberger, Investment science, Oxford, 1997
M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, Springer, 1997
S. Neftci, An introduction to the mathematics of financial derivatives, 2nd edition, Academic Press, 2000
S. Neftci, Principles of Financial Engineering, Academic Press, 2004
S. R. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell, 1997
S. Ross, An Elementary Introduction to Mathematical Finance, 2nd edition, Cambridge University Press, 2002
S. E. Shreve, Stochastic calculus and Finance I: Binomial Model, Springer, 2004
S. E. Shreve, Stochastic calculus and Finance II: Continuous-time finance, Springer, 2004
J. M. Steele, Stochastic calculus and financial applications, Springer, 2000
P. Wilmott, Paul Wilmott on Quantitative Finance, 2nd edition, 3 volume set, Wiley, 2006
P. Wilmott, S. Howison, and J. Dewynne, The Mathematics of Financial Derivatives, Wiley

Advanced Mathematical Finance

C. Albanese and G. Campolieti, Advanced Derivatives Pricing and Risk Management: Theory, Tools, and Hands-On Programming Applications, Academic Press, 2005
S. Allen, Financial Risk Management, Wiley, 2003
T. Bielecki, Credit Risk Modeling, Valuation, and Hedging, Springer, 2002
J. R. Birge and V. Linetsky, Handbooks in Operations Research and Management Science: Financial Engineering, Volume 15, Elsevier, 2007
A. Bomfim, Understanding Credit Derivatives and Related Instruments, Elsevier, 2005
D. Brigo and F. Mercurio, Interest rate models - theory and practice, with smile, inflation, and credit, 2nd edition, Springer, 2006
R. Carmona and M. Tehranchi, Interest Rate Models: an Infinite Dimensional Stochastic Analysis Perspective, Springer, 2006
P. Carr, Derivatives Pricing: The Classic Collection, Risk Publications, 2004
M. Crouhy, R. Mark, and D. Galai, Risk Management, McGraw-Hill, 2000
R. Cont and P. Tankov, Financial Modeling with Jump Processes, Wiley, 2003
D. Duffie, Dynamic Asset Pricing Theory, 3rd edition, Princeton University Press, 2001
D. Duffie and K. Singleton, Credit Risk, Princeton University Press, 2003
R. J. Elliott, Mathematics of Financial Markets, Springer, 2005
P. Embrechts, C. Klüppelberg, and T. Mikosch, Modeling Extremal Events for Insurance and Finance, Springer, 1997
M. Fengler, Semiparametric Modeling of Implied Volatility Surfaces, Springer, 2005
E. R. Fernholz, Stochastic Portfolio Theory, Springer, 2002
D. Filipovic, Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, Springer, 2001
J-P. Fouque and G. Papanicolaou and K. R. Sircar, Derivatives in financial markets with stochastic volatility, Cambridge, 2000
J. Gatheral, The Volatility Surface: A Practitioner's Guide, Wiley, 2006
H. Geman, Commodities and Commodity Derivative Modeling, Wiley, 2005
E. Haug, The Complete Guide to Option Pricing Formulas, 2nd edition, McGraw-Hill, 2006
P. Henry-Labordère, Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing, Chapman & Hall, 2008
A. Javaheri, Inside Volatility Arbitrage: The Secrets of Skewness, Wiley, 2005
I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, 1998
A. Kyprianou, W. Schoutens, and P. Wilmott, Exotic Option Pricing and Advanced Lévy Models, Wiley 2005
D. Lando, Credit Risk Modeling, Princeton, 2004
A. Lipton, Mathematical methods for foreign exchange: a financial engineer's approach, World Scientific, 2001
H. Markowitz, Mean-Variance Analysis in Portfolio Choice and Capital Markets, Wiley, 2000
A. Meucci, Risk and Asset Allocation, Springer, 2008
R. C. Merton, Continuous-Time Finance, Blackwell Publishers, 1992
D. O'Kane, Modeling Single-Name and Multi-Name Credit Derivatives, Wiley, 2008
R. Rebonato, Volatility and Correlation: The perfect Hedger and the Fox, Wiley, 2004
R. Rebonato, K. McKay, and R. White, The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives, Wiley, 2009
P. J. Schönbucher, Credit Derivatives Pricing Models: Model, Pricing and Implementation, Wiley, 2003
W. Schoutens, Lévy Processes in Finance, Wiley, 2003
N. N. Taleb, Dynamic Hedging, Wiley, 1996
L. Wu, Interest Rate Modeling: Theory and Practice, Chapman & Hall, 2009

Stochastic Processes and Stochastic Differential Equations

D. Applebaum, Lévy processes and stochastic calculus, Cambridge, 2004
J. Bertoin, Lévy Processes, Cambridge University Press, 1998
A. N. Borodin and P. Salminen, Handbook of Brownian motion, 2nd edition, Birkhaüser, 2002
A. Friedman, Stochastic differential equations and applications, Academic, 1975
B. E. Øksendal, Stochastic Differential Equations, 6th edition, Springer, 2003
J. Jacod and A. N. Shirayev, Limit Theorems for Stochastic Processes, 2nd edition, Springer, 2002
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1997
S. Karlin and H. M. Taylor, A First Course on Stochastic Processes, Academic, 1975
S. Karlin and H. M. Taylor, A Second Course on Stochastic Processes, Academic, 1981
P. Malliavin, Stochastic Calculus of Variations in Mathematical Finance, Springer, 2006
D. Nualart, The Malliavin Calculus and Related Topics, Springer, 2006
P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springer, 2003
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, Springer, 1999
L. C. G. Rogers and D. Williams, Diffusions, Markov processes, and martingales I, II, Cambridge University Press, 2000
K-I. Saito, Lévy Processes and Infinitely Divisible Distributions , Cambridge University Press, 1999
D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, 1997

Probability, Statistics, and Time Series

P. Billingsley, Probability and Measure, Wiley, 3rd edition, 1995
W. Feller, An Introduction to Probability Theory and Applications I, II, Wiley
J. Jacod and P. Protter, Probability Essentials, Springer, 2003
A. Papoulis, Probability, Random Variables, and Stochastic Processes
S. Ross, A First Course in Probability, Prentice Hall, 2001