PDEs for finance

J. Printems

Assessing the premium of a derivative on an underlying asset is initially seen by practitioners as a calculation of the expectation of the functional of the underlying asset (usually a stock). At first glance, therefore, we need to distinguish between the choice of underlying model (constant volatility, local volatility, stochastic volatility or other) and the choice of functional. The former is a modelling effort, left to the trader’s discretion; the latter is dictated by the contract, i.e. by the definition of the derivative in question (call or put option, with or without early exercise).

In most cases, this probabilistic interpretation has a deterministic counterpart through the resolution of a certain PDE (Feynman-Kac representation) dictated by the choice of model. This dual interpretation can be fruitful both theoretically and numerically. The aim of this course is to study the deep links between these two interpretations from a numerical point of view. The emphasis will be on the numerical resolution of PDEs, with comparisons with the probabilistic approach wherever possible.

This begins with a brief study of some of the simplest stochastic processes (Brownian motion), at the origin of a new type of differential calculus (Itô calculus). We then study the connections between heat PDEs and Brownian motion. Secondly, we’ll look at the general case of option pricing by PDE in the simplest models with the most liquid options (European and American). Finally, we will attempt to generalize these calculations to more exotic options.

I. Introduction to standard Brownian motion.

1. Definition

2. Numerical approximation (random walk, Fourier, etc.) and simulation.

3. Stochastic integral with deterministic integrand: the Wiener integral. Itô isometry.

4. Generalization to a stochastic integrand: stochastic differential equations. Martingales and the beginning of stochastic calculus.

II. Heat PDE and random walk.

1. Probabilistic representation of the solution of a heat equation. Numerical approximation by Monte Carlo.

2. Finite-difference method and random walks. Study of method convergence. Comparison with the probabilistic method.

III. European options and Black-Scholes PDE - General finite-difference method.

1. Pricing European options in a local volatility model. Derivations of the BS PDE.

2. Probabilistic representation of the solution.

3. Constant volatility case: binomial tree approximation.

4. General case using the finite-difference method.

5. Application to Greeks calculus.

IV. Early exercise options (American options).

1. Variational equations. A first “toy-model”.

2. Variational inequalities in BS frame.

3. Numerical resolution using the finite element method.

4. Comparison with binomial tree/DF for constant volatility.

5. Comparison with American Monte Carlo.

V. Trajectory-dependent options.

1. Asian options, lookback.