Numerical Methods in Financial Mathematics

1 Topics

1. SDE Simulation
   1. Euler-Maruyama Method
   2. Milstein Method
   3. SDE Simulation Error
   4. SDE Simulation Examples:
      1. Multiple Black-Scholes SDE Simulation
      2. Heston Model SDE Simulation
      3. Jump Diffusion SDE Simulation
      4. Brownian Bridge SDE Simulation
2. Monte-Carlo Variance Reduction
   1. [[antithetic-variates]]
   2. Control Variates
   3. Importance Sampling
3. Multi-Level Monte-Carlo (MLMC)
4. PDE Option Pricing
   1. Finite-Difference (FD) Methods
      1. Explicit Scheme
      2. Implicit Scheme
         1. Solving Linear Systems
      3. Crank-Nicolson Method
      4. Boundary Conditions
      5. Convergence and Stability of Finite Difference Schemes
      6. FD Method Examples:
   2. Multidimensional PDE
   3. Asian Options - PDE Approach
5. [[surrogate-methods]]
   1. Gaussian Process Regression
   2. Deep Galerkin Method (DGM)
6. [[american-options]]
   1. [[american-options|Finite Difference Methods for American Options]]
7. Option Greeks
   1. Bump-and-Revalue
   2. Pathwise Estimation
   3. Likelihood Ratio Networks
   4. Neural Networks for SDE Simulation

2 References

1. The Mathematics of Financial Derivatives
2. [[higham-kloeden]]
3. [[glasserman|Monte Carlo Methods in Financial Engineering]]

3 Backlinks

• Financial Mathematics