Quantitative Finance Reading List - Numerical Methods

In the previous article the core C++ books required for a good grounding in quantitative programming were outlined. Now it is time to discuss the books useful for learning numerical methods, in particular Finite Difference Methods (FDM) and 190-701Monte Carlo Methods (MCM).
Finite Difference Methods
Finite Difference Methods are a class of numerical methods used to provide an approximate, discrete solution to various partial differential equations, in particular the Black-Scholes PDE. Finite Difference Methods work by discretising the derivative terms in the PDE, such that they can be implemented algorithmically. An explicit FDM has the quantities at the next time step calculated in terms of the values at the previous step. An implicit FDM has the quantities at the next time step calculated in terms of both the values of the next time step and the previous time step. Stability of the scheme is an important concept.
The following are some of the more well known (and recommended!) text books on Finite Difference Methods:
Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach - Duffy
Financial Instrument Pricing Using C++ - Duffy
Numerical Solution of Partial Differential Equations: Finite Difference Methods - Smith
Pricing Financial Instruments: The Finite Difference Method - Tavella and Randall
Option Pricing: Mathematical Models and Computation -Wilmott et al.

Monte Carlo Methods
Monte Carlo 190-710Methods rely on the concept of risk neutral valuation in order to price derivatives. In essence, many underlying random asset price paths are calculated and the associated derivative payoff is calculated for each path. The mean of the payoffs are taken and then the price is discounted to today's price. This will give an approximation of the the option price. Further accuracy can be obtained by increasing the number of random trials.
Here are some of the top financial modelling MCM books:
C++ Design Patterns and Derivatives Pricing - Joshi
Monte Carlo Methods in Financial Engineering - Glasserman
Monte Carlo Frameworks: Building Customisable High-performance C++ Applications - Duffy et al.
Monte Carlo Methods in Finance - Jaeckel
Monte Carlo Methodologies and Applications for Pricing and Risk Management - Dupire

Suggested Reading
The best books to start with from a C++/numerical point of view are Duffy's "Financial Instrument Pricing Using C++" and Joshi's "C++ Design Patterns and Derivatives Pricing" books. In fact, Joshi's can be read in conjunction with his "Concepts and Practice of Mathematical Finance". They will get you up to speed on intermediate usage of C++ as well as give you an insight into both FDM and MCM. Depending on which way you lean (FDM or MCM), you may wish to continue with Wilmott's190-711 "Option Pricing" or with Glasserman's "Monte Carlo Methods in Financial Engineering" and Duffy's "Monte Carlo Frameworks.