Numerical Methods

We cover the main numerical methods used in options pricing theory.
We start off with the theory of binomial trees and trinomial trees as simple ways to approximate the underlying stochastic process for the stock. We then move on to finite difference methods for solving the partial differential equation satisfied by the option price for constant volatility, time dependent volatility and local volatility. We also describe advanced numerical techniques to calculate prices when we have stochastic volatility. The second section explains Monte Carlo techniques and various ways to improve accuracy and reduce error.

Trees

1. Binomial Trees – construction and pricing vanilla options
2.Trinomial Trees – construction and comparison with Binomial method

Finite Difference Methods

1. Explicit scheme, Implicit scheme, and Crank Nicolson Scheme.
2. Alternating Direction Method (ADI) for solving prroblems with stochastic volatility

Monte Carlo Methods

1. Euler and Milstein discretisation scheme.
2. Pricing error and convergence.
3. Variance Reduction Techniques – Control Variates and Antithetic Variates

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