Data-driven Option Pricing

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Data-driven Option Pricing

Min Dai
Hanqing Jin
Xi Yang
Published on 1/20/2024
Options
Derivatives
Machine learning
Deep learning
Risk management

This paper introduces a novel data-driven option pricing methodology that relies exclusively on historical underlying asset prices, bypassing the need for option market data. The key idea is to recover a pricing kernel that transforms option pricing from the risk-neutral world to the objective world, where historical data is available. The pricing kernel is obtained by solving a logarithmic utility maximization problem, whose optimal wealth process is shown to be the reciprocal of the pricing kernel. This optimization is performed using deep learning, where portfolio strategies are parameterized by neural networks and trained on historical price paths. Once the pricing kernel is learned, option prices are computed by estimating conditional expectations under the objective measure, which is reformulated as a functional optimization problem minimizing a martingale loss function. The method ensures arbitrage-free prices and requires only a single training phase for all derivatives on the same underlying. Numerical experiments confirm the effectiveness of the approach, making it particularly useful for emerging markets with limited option data.

Highlights

  • 1Proposes a data-driven option pricing method using only historical underlying asset prices, eliminating the need for option market data.
  • 2Recovers the pricing kernel via a logarithmic utility maximization problem, linking objective and risk-neutral measures.
  • 3Leverages deep learning to solve optimization problems over historical price paths, enabling pricing without model calibration.
  • 4Ensures arbitrage-free prices through martingale theory, unlike deep hedging methods.
  • 5Single training phase for pricing kernel applies to all derivatives on the same underlying, reducing computation.

Methods

  • M
    Logarithmic utility maximization to recover the pricing kernel as the reciprocal of optimal wealth.
  • M
    Deep learning with neural networks to parameterize portfolio strategies and minimize loss functions.
  • M
    Martingale loss function for conditional expectation estimation, inspired by Jia and Zhou (2022).
  • M
    Discrete-time approximation of continuous-time stochastic integrals using historical price paths.

Results

  • R
    The optimal portfolio for logarithmic utility is proportional to the Sharpe ratio over variance, and the optimal wealth equals the inverse pricing kernel.
  • R
    The maximizer of the expected log utility is independent of the initial stock price distribution, allowing use of overlapping paths.
  • R
    The pricing kernel can be recovered without knowledge of drift and volatility functions.
  • R
    Option prices are obtained by minimizing a martingale loss function, which is equivalent to conditional expectation estimation.
  • R
    Numerical experiments demonstrate the efficiency of the proposed data-driven algorithms.
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