Mini-courses on Adjoints and Modern Sequence Models in Computational Finance

31 August 09:45
Aramugam Building, St Catherine's College, Oxford, Manor Road, OX1 3UJ

This mini-course precedes the International Conference on Computational Finance, but is bookable separately through the University Online Store.

Fees (including course attendance, 3-course served lunch, refreshments):

Aramugam Building, St Catherine's College, Oxford, Manor Road, OX1 3UJ

Morning: Prof. Mike Giles, An introduction to the use of adjoints in computational finance

 

AAD (Adjoint Algorithmic Differentiation, or sometimes Adjoint Automatic Differentiation)) is used extensively in computational finance for estimating sensitivities (Greeks), especially when estimating the sensitivity of a single option value to changes in a large number of input parameters (such as future interest rates or correlation coefficients).  The mathematics is also the same as back-propagation in machine learning, computing the sensitivity of the average mis-match to training data to changes in all of the neural network coefficients.

This set of three lectures (each about 50 mins long) will give an introduction to this subject for students and others, with no prior knowledge assumed other than a basic knowledge of Monte Carlo and finite difference methods in computational finance.  Those who are interested solely in Monte Carlo simulation are welcome to skip the third lecture.

9.45-10. Registration

Aramugam Building.

10-10.50. Lecture 1: The mathematical basics

  • A simple example of matrix multiplication
  • A black-box view -- forward and reverse mode
  • Automatic differentiation
  • Adjoints for linear algebra
  • Fixed-point iteration

 

Coffee/tea

 

11.10-12. Lecture 2: Monte Carlo calculations
  • Pathwise sensitivity analysis
  • SDE approximation for European options
  • Path-dependent options
  • Multiple options
  • Binning for expensive pre-computations
  • Discontinuous payoffs
  • Black-box assembly for multi-stage calculations
12.10-1. Lecture 3: Finite difference methods
  • Forward/backward Kolmogorov PDEs
  • Use of adjoints for European option pricing (not sensitivities)
  • Sensitivity calculations
  • Calibration to European prices
  • What can go wrong?

 

1-2. Lunch

Afternoon: Dr Hans Buehler, Prof. Blanka Horvath, Prof. Mikko Pakkanen, Dr Ben Walker, Modern sequence models and their applications in computational finance

 

This joint mini-course offers an accessible introduction to modern sequence models — from the mathematical foundations of transformers and tokenisation through to state space models and neural differential equations — with a particular focus on their emerging role in computational finance. The course is designed for PhD students and researchers with a background in probability, stochastic analysis, or quantitative finance who wish to understand both the theoretical underpinnings and the practical impact of these architectures.

The three lectures (each approximately 50 minutes) are designed to be self-contained but form a coherent arc: from the landscape of current machine learning models, through the mechanics of transformers and tokenisation, to applications in finance and the differential-equations perspective on sequence modelling.

 

2-2.50. Lecture 1: A View of Current Models in Machine Learning

The Machine Learning Landscape

  • From classical statistics to deep learning: a brief history
  • Key architectural families briefly revisited 
  • The rise of foundation models and large-scale pretraining
  • Sequence Models — Why RNNs struggle and what came next
  • The Attention Mechanism: Self-attention as a kernel: queries, keys, and values
  • Multi-head attention and its role in learning multiple representations
  • The Transformer Architecture: Encoder, decoder, and encoder-decoder designs
  • Tokenisation — in NLP: byte-pair encoding (BPE) and WordPiece
  • Tokenising financial time series: discretisation, binning, and patch embeddings
  • Path signatures as tokens: the rough-paths perspective on sequential data
    • Signature features as canonical embeddings of streams
    • Connection to the 'Generating financial markets with signatures' framework

 

3-3.50. Lecture 2: Applications in Finance -- Deep Hedging, Calibration, and Market Generation

Part A: Deep Hedging and Optimal Execution

  • Hedging as a sequential decision problem: the Deep Hedging setup
  • Market frictions, transaction costs, and path-dependent payoffs
  • Risk measures as natural non-linear robust convex objectives in reinforcement learning: CVaR, expected shortfall, and beyond
  • Recurrent hedging agents and how to address their challenges

Part B: Model Calibration with Deep Learning

  • The calibration problem: fitting model parameters to market prices
  • Neural network surrogates for option pricing
  • Calibration as inverse problem: from surface to parameters
  • Generative calibration: differentiating through the pricing model

Part C: Generative Models for Financial Markets

  • What makes a financial time series hard to model?
  • GANs, VAEs, and score-based models for market data generation
  • Signature-based market generation: the conditional expectation view
  • Evaluation metrics: stylised facts, rank correlations, and downstream task performance
  • Practical considerations: data scarcity, non-stationarity, and regime changes

 

Coffee/Tea

 

4.10-5. Lecture 3: State Space Models, Neural Differential Equations, and Sequence Modelling

Structured State Space Models and Neural Differential Equations

  • Neural ODEs: continuous-depth networks and adjoint sensitivity
  • Controlled differential equations (CDEs) as sequence models
    • The CDE as a principled continuous-time RNN
    • Log-ODE methods and signature-based solvers
  • Neural SDEs: stochastic dynamics and latent variable models
  • Irregular time series: handling asynchronous, missing, and multi-rate data