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Generative Pricing of Basket Options via Signature-Conditioned Mixture Density Networks

Hasib Uddin Molla, Antony Ware, Ilnaz Asadzadeh, Nelson Mesquita Fernandes

TL;DR

A generative framework for pricing European-style basket options by learning the conditional terminal distribution of the log arithmetic-weighted basket return, and a Mixture Density Network that acts as a reusable surrogate distribution.

Abstract

We present a generative framework for pricing European-style basket options by learning the conditional terminal distribution of the log arithmetic-weighted basket return. A Mixture Density Network (MDN) maps time-varying market inputs encoded via truncated path signatures to the full terminal density in a single forward pass. Traditional approaches either impose restrictive assumptions or require costly re-simulation whenever inputs change, limiting real-time use. Trained on Monte Carlo (MC) under GBM with time-varying volatility or local volatility, the MDN acts as a reusable surrogate distribution: once trained, it prices new scenarios by integrating the learned density. Across maturities, correlations, and basket weights, the learned densities closely match MC (low KL) and produce small pricing errors, while enabling \emph{train-once, price-anywhere} reuse at inference-time latency.

Generative Pricing of Basket Options via Signature-Conditioned Mixture Density Networks

TL;DR

A generative framework for pricing European-style basket options by learning the conditional terminal distribution of the log arithmetic-weighted basket return, and a Mixture Density Network that acts as a reusable surrogate distribution.

Abstract

We present a generative framework for pricing European-style basket options by learning the conditional terminal distribution of the log arithmetic-weighted basket return. A Mixture Density Network (MDN) maps time-varying market inputs encoded via truncated path signatures to the full terminal density in a single forward pass. Traditional approaches either impose restrictive assumptions or require costly re-simulation whenever inputs change, limiting real-time use. Trained on Monte Carlo (MC) under GBM with time-varying volatility or local volatility, the MDN acts as a reusable surrogate distribution: once trained, it prices new scenarios by integrating the learned density. Across maturities, correlations, and basket weights, the learned densities closely match MC (low KL) and produce small pricing errors, while enabling \emph{train-once, price-anywhere} reuse at inference-time latency.

Paper Structure

This paper contains 31 sections, 47 equations, 19 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. From Risk-Neutral Expectation to Density Learning
  3. Mixture Density Networks for Distribution Approximation
  4. Signature-Conditioned MDNs for Basket Option Pricing
  5. Background on Path Signatures
  6. Mixture Density Networks and Path Signatures
  7. Mixture Density Model (Univariate)
  8. Mixture Density Network (Maximum Likelihood)
  9. Path Signatures
  10. The Mechanism of Training an MDN
  11. Asset Price Model
  12. Geometric Brownian Motion (GBM)
  13. Correlation Structure of the Basket
  14. Target Distribution
  15. Training Data Generation
  16. ...and 16 more sections

Figures (19)

  • Figure 1: Mixture Density Network
  • Figure 2: MDN Training Workflow
  • Figure 3: (a) Time-varying interest rate, dividend yields, and volatilities. (b) MDN vs. MC distributions of average basket return with KL divergence at different maturities.
  • Figure 4: Relative percentage error in European call and put option prices based on MDN vs. MC pricing.
  • Figure 5: (a) Time-varying interest rate, dividend yields, and volatilities. (b) MDN vs. MC distributions of average basket return with KL divergence at different maturities.
  • ...and 14 more figures