The previous article outlined the basic C ++ books required for a good foundation in quantitative programming. Now is the time to discuss useful books for learning numerical methods, in particular, Finite Difference Methods (FDM) and Monte Carlo Methods (MCM).
Finite difference methods
Finite difference methods are a class of numerical methods used to provide an approximate and discrete solution to various partial differential equations, in particular the Black-Scholes PDE. Finite difference methods work by discretizing the derived terms in the PDE, so that they can be implemented algorithmically. An explicit FDM has the quantities in the next time step calculated in terms of the values from the previous step. An implicit FDM has the quantities in the next time step calculated in terms of the values of the next time step and the previous time step. Schema stability is an important concept.
The following are some of the best known (and recommended!) Textbooks on finite difference methods:
- Finite Difference Methods in Financial Engineering: A Partial Differential Equations Approach – Duffy
- Financial Instrument Pricing with C ++ – Duffy
- Numerical Solution of Partial Differential Equations: Finite Difference Methods – Smith
- Financial Instrument Pricing: The Finite Difference Method – Tavella and Randall
- Price Option: Mathematical Models and Computation -Wilmott et al.
Monte Carlo methods
The Monte Carlo methods are based on the concept of risk neutral valuation to set the price of derivatives. In essence, many underlying random asset price paths are calculated and the associated derivative payment is calculated for each path. The average of the payments is taken and then the price is discounted to the current price. This will give an approximation of the option price. Greater precision can be obtained by increasing the number of randomized trials.
Here are some of the best MCM books on financial modeling:
- C ++ Design Patterns and Derivative Prices – Joshi
- Monte Carlo Methods in Financial Engineering – Glasserman
- Monte Carlo Structures: Building High-Performance Customizable 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 numerical / C ++ standpoint are the books “Financial Instruments Pricing with C ++” by Duffy and “C ++ Design Patterns and Derivatives Pricing” by Joshi. . In fact, Joshi can be read in conjunction with his “Concepts and Practice of Mathematical Finance”. They will bring you up to speed on intermediate use of C ++ and give you a taste of FDM and MCM. Depending on how you support it (FDM or MCM), you may want to continue with Wilmott’s “Option Pricing” or Glasserman’s “Monte Carlo Methods in Financial Engineering” and Duffy’s “Monte Carlo Frameworks” .