Machine Learning in Financial Mathematics

Lecturer

Dr Kihun Nam, Monash University

Synopsis

Recent advances in machine learning have enabled the use of novel numerical techniques in solving challenging problems in financial mathematics. This course will introduce the basics of stochastic calculus and machine learning, establish connections between probabilistic and PDE formulations of stochastic models, and demonstrate how all these elements can be combined to solve financial mathematics problems such as derivative pricing and portfolio selection.

Course Overview

Week 1: Brownian Motion and Stochastic Calculus

  • To model the price dynamics of financial assets, we will introduce the concept of Brownian motion and its associated stochastic calculus. We will explore key properties such as distributional characteristics, the Markov property, quadratic variation, and Ito’s isometry.

Week 2: Derivative Pricing and Monte Carlo Method

  • Using the no-arbitrage principle, we will derive the probabilistic representation of the fair price of European-style financial derivatives. Both theoretical and numerical pricing methods will be discussed, with an emphasis on the Least Squares Monte Carlo (LSMC) method. The connection to parabolic partial differential equations via the Feynman-Kac theorem will also be introduced, providing insight into how classical numerical methods can be applied.

Week 3: Neural Networks and Stochastic Gradient Descent

  • Neural networks offer an alternative approach to pricing derivatives, particularly in high-dimensional settings. After introducing the fundamental architecture and concepts of neural networks, we will discuss stochastic gradient descent (SGD) as an efficient optimization method for training models.

Week 4: Derivative Pricing with Deep Learning

  • We will apply deep neural networks to the problem of derivative pricing. Using Python and PyTorch, we will construct and train models to approximate the prices of European-style options.

Prerequisites

  • Linear Algebra (level 1/2)
  • Multivariable Calculus (level 1/2)
  • Probability Theory (level 2)
  • Basic knowledge in Python and Numpy package

Assessment

  • 3x assignments (due weeks 2, 3 & 4) – 20% each
  • Take home exam – 40%

Resources/pre-reading

  • Basic Measure-theoretic Probability Theory including probability space as a measure space, random variable, distribution, expectation, and conditioning with sigma algebra.
    e.g. Shreve Stochastic Calculus for Finance 2: Continuous-Time Models Chapter 1 – 2
  • Basic python knowledge
    e.g. https://www.youtube.com/watch?v=K5KVEU3aaeQ

Not sure if you should sign up for this course?

Check back for pre-enrolment QUIZ details so you can self-evaluate and get a measure of the key foundational knowledge required.

Dr Kihun Nam, Monash University

Dr Kihun Nam is a lecturer studying financial mathematics at Monash University, specialising in backward stochastic differential equations (BSDEs) in stochastic optimization and stochastic differential games. His expertise lies in the stability analysis of BSDE solutions with respect to underlying noise, employing advanced techniques rooted in high-dimensional analysis. Recent works of Dr. Nam explores deep connections between BSDEs and parabolic or elliptic PDEs. Through both teaching and research, he advances the integration of machine learning and stochastic analysis.