Introduction to Finite Difference Methods for Option Pricing

Finite-difference methods is the generic term for a large number of techniques that can be used for solving differential equations, approximating derivatives with finite differences.

It consists of a discretization of both spatial (underlying price in our case) and time intervals to obtain a finite grid.

We find solutions for a differential equation by approximating every partial derivative with numerical methods and we apply the solution to the points of the grid.

This numerical method is very useful in finance to solve partial differential equations (PDE) which model the evolution of option prices, such as the Black-Scholes PDE.

The three main methods are:

– the explicit method which goes backward

– the implicit method which goes forward

– the Crank-Nicolson method which is an average of the two

The explicit method is equivalent to a trinomial tree approach, it is easy to formulate but it can be unstable, requiring an important number of time steps to avoid stability issues.

The Implicit and Crank-Nicolson are stable ones, but more complex to formulate requiring the use of matrix calculations, the later one being the most efficient with a faster rate of convergence.

Presentation

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List of Quant Next Courses

We summarize below quantitative finance training courses proposed by Quant Next. Courses are 100% digital, they are composed of many videos, quizzes, applications and tutorials in Python.

Complete training program:

Options, Pricing, and Risk Management Part I: introduction to derivatives, arbitrage free pricing, Black-Scholes model, option Greeks and risk management.

Options, Pricing, and Risk Management Part II: numerical methods for option pricing (Monte Carlo simulations, finite difference methods), replication and risk management of exotic options.

Options, Pricing, and Risk Management Part III:  modelling of the volatility surface,  parametric models with a focus on the SVI model, and stochastic volatility models with a focus on the Heston and the SABR models.

A la carte:

Monte Carlo Simulations for Option Pricing: introduction to Monte Carlo simulations, applications to price options, methods to accelerate computation speed (quasi-Monte Carlo, variance reduction, code optimisation).

Finite Difference Methods for Option Pricing: numerical solving of the Black-Scholes equation, focus on the three main methods: explicit, implicit and Crank-Nicolson.

Replication and Risk Management of Exotic Options: dynamic and static replication methods of exotic options with several concrete examples.

Volatility Surface Parameterization: the SVI Model: introduction on the modelling of the volatility surface implied by option prices, focus on the parametric methods, and particularly on the Stochastic Volatility Inspired (SVI) model and some of its extensions.

The SABR Model: deep dive on on the SABR (Stochastic Alpha Beta Rho) model, one popular stochastic volatility model developed to model the dynamic of the forward price and to price options.

The Heston Model for Option Pricing: deep dive on the Heston model, one of the most popular stochastic volatility model for the pricing of options.