Deep Learning-Based Option Pricing Models: Comparison and Innovation with Traditional Models

Problem Description

Option pricing is a core problem in financial engineering. Traditional models (such as the Black-Scholes model) rely on strict assumptions (e.g., perfect markets, constant volatility), which have limitations in practical applications. In recent years, deep learning models (e.g., neural networks) have been introduced into the field of option pricing, aiming to capture market non-linear features in a data-driven manner to improve pricing accuracy. This topic requires an understanding of the basic principles of deep learning option pricing, its differences from traditional models, and key challenges in practical applications.

Step 1: Limitations Analysis of Traditional Option Pricing Models

Objective: Clarify the assumptions and shortcomings of traditional models to provide direction for improving deep learning models.

• Black-Scholes Model:
  • Core Formula:

\[ C = S_0 N(d_1) - K e^{-rT} N(d__2) \]

where $ d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}} $, $ d_2 = d_1 - \sigma \sqrt{T} $.  

• Limitations:
  1. Assumes constant volatility, but actual market volatility changes with time and price (exhibiting the "volatility smile" phenomenon).
  2. Ignores real-world factors such as transaction costs and market frictions.
  3. Poor applicability to non-European options (e.g., American options) or complex derivatives.
• Improved Models (e.g., Heston model, local volatility model):
  • Enhance flexibility through stochastic volatility or time-varying parameters, but still rely on accurate parameter estimation and involve complex computations.

Step 2: Basic Framework of Deep Learning Models

Objective: Construct a data-driven pricing model to learn the mapping relationship between option prices and market variables in an end-to-end manner.

• Input Feature Design:
  • Traditional parameters: underlying asset price (S), strike price (K), time to maturity (T), risk-free interest rate (r), etc.
  • Extended features: historical volatility, market sentiment indicators, underlying asset yield curve, etc.
• Neural Network Architecture Selection:
  • Fully Connected Network (FCN): Processes structured inputs, suitable for learning non-linear relationships.
  • Recurrent Neural Network (RNN/LSTM): If inputs include time-series data (e.g., historical price sequences), can capture temporal dependencies.
  • Attention Mechanism: Used to handle importance weights of multi-dimensional features (e.g., the impact of volatility at different maturities).
• Loss Function Design:
  • Mean Squared Error (MSE): Minimizes the difference between model prices and market prices.
  • Regularization Terms: Prevents overfitting (e.g., L2 regularization or Dropout).

Step 3: Model Training and Optimization Process

Objective: Train the model with real data and address special issues in financial data.

• Data Preparation:
  • Source: Option market trading data (e.g., daily closing prices of European options).
  • Preprocessing: Normalize input features (e.g., scaling S, K to the same magnitude), handle missing values.
• Training Techniques:
  • Combating Overfitting: Use Early Stopping, cross-validation.
  • Handling Class Imbalance: Apply weighted loss functions for in-the-money/out-of-the-money option samples.
• Hyperparameter Tuning:
  • Network depth, learning rate, batch size, etc., determined via grid search or Bayesian optimization.

Step 4: Comparative Analysis with Traditional Models

Objective: Evaluate the advantages and disadvantages of deep learning models from dimensions such as accuracy, generalizability, and interpretability.

• Accuracy Comparison:
  • Calculate Root Mean Square Error (RMSE) on the test set; deep learning models typically outperform the Black-Scholes model, especially during periods of high market volatility.
  • Example result: Deep learning model RMSE is 0.15, Black-Scholes model RMSE is 0.25.
• Generalization Ability:
  • Deep learning models can adapt to different market regimes (e.g., bull/bear markets) but require sufficient historical data support.
  • Traditional models are more stable when data is scarce (due to theoretical foundations).
• Interpretability Challenges:
  • Deep learning is a "black box"; tools like SHAP, LIME are needed to analyze feature importance (e.g., finding volatility features have the highest weight).

Step 5: Challenges and Innovation Directions in Practical Applications

Objective: Understand key issues during model deployment and cutting-edge solutions.

• Insufficient Data:
  • Solutions: Transfer learning (pre-training with other asset data), Generative Adversarial Networks (GAN) for synthetic data generation.
• Dynamic Market Changes:
  • Online Learning: Periodically update model parameters with new data to adapt to changing market structures.
• Risk Control:
  • Models need to be combined with traditional methods, e.g., using Black-Scholes results as boundary constraints for deep learning outputs.

Summary

Deep learning option pricing relaxes strict assumptions and introduces non-linear mapping, improving pricing capability in complex market environments. However, its success depends on high-quality data, carefully designed feature engineering, and integration with traditional financial theory. Future directions include incorporating physics-inspired models (e.g., neural stochastic differential equations) to enhance interpretability, and exploring architectures like Transformers for processing high-frequency data.