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The Algotrading Book

Chapter 155: Neural Implicit Finance (INRs & SIREN)

Chapter 155: Neural Implicit Finance (INRs & SIREN)

Overview

Traditional financial modeling represents data on discrete grids (e.g., historical prices at specific timestamps, volatility surfaces at specific strike/maturity grid points). Implicit Neural Representations (INRs) change this paradigm by representing financial data as a continuous function parameterized by a neural network.

In this chapter, we explore SIREN (Sinusoidal Representation Networks) to model the Implied Volatility (IV) Surface. Unlike standard ReLU-based networks, SIREN uses periodic activation functions, allowing it to capture high-frequency details and, more importantly, its derivatives (Greeks) with extreme accuracy.

Why Implicit Representations for Finance?

  1. Resolution Independence: Once trained, you can query the model at any coordinate (moneyness, time-to-expiry), not just the grid points used for training.
  2. Analytical Greeks via Autograd: Since the model is a continuous, differentiable function, Greeks like Delta ($\Delta$), Gamma ($\Gamma$), and Vega ($\nu$) can be calculated directly via backpropagation through the network coordinates.
  3. Arbitrage-Free Constraints: Real-world volatility surfaces must satisfy no-arbitrage conditions (butterfly, calendar). INRs allow us to incorporate these constraints directly into the loss function or architecture.
  4. Memory Efficiency: A complex volatility surface can be compressed into the weights of a small MLP.

SIREN: The Periodic Powerhouse

SIREN uses the $\text{sin}(\omega_0 \cdot \phi)$ activation function. This is critical because:

  • The derivative of a sine is a cosine (another periodic function).
  • This allows the network to maintain its “expressiveness” through multiple layers of differentiation.
  • It is uniquely suited for modeling surfaces where smooth second-order derivatives (like Gamma) are required.

Contents

  • python/model.py: Implementation of SIREN architecture in PyTorch.
  • python/train.py: Fitting sparse market quotes to a continuous IV surface.
  • python/backtest.py: Calculating Greeks and verifying arbitrage constraints via autograd.
  • rust/src/: Optimized Rust implementation for real-time inference and calibration.

References

  1. Sitzmann, V., Martel, J., Bergman, A., Lindell, D., & Wetzstein, G. (2020). Implicit Neural Representations with Periodic Activation Functions. arXiv:2006.09661.
  2. HyperIV: Real-time implied volatility smoothing. University of Edinburgh research.