Efficient Learning of Nested Deep Hedging using Multiple Options

ArXiv ID: 2305.12264 “View on arXiv”

Authors: Unknown

Abstract

Deep hedging is a framework for hedging derivatives in the presence of market frictions. In this study, we focus on the problem of hedging a given target option by using multiple options. To extend the deep hedging framework to this setting, the options used as hedging instruments also have to be priced during training. While one might use classical pricing model such as the Black-Scholes formula, ignoring frictions can offer arbitrage opportunities which are undesirable for deep hedging learning. The goal of this study is to develop a nested deep hedging method. That is, we develop a fully-deep approach of deep hedging in which the hedging instruments are also priced by deep neural networks that are aware of frictions. However, since the prices of hedging instruments have to be calculated under many different conditions, the entire learning process can be computationally intractable. To overcome this problem, we propose an efficient learning method for nested deep hedging. Our method consists of three techniques to circumvent computational intractability, each of which reduces redundant computations during training. We show through experiments that the Black-Scholes pricing of hedge instruments can admit significant arbitrage opportunities, which are not observed when the pricing is performed by deep hedging. We also demonstrate that our proposed method successfully reduces the hedging risks compared to a baseline method that does not use options as hedging instruments.

Keywords: Deep Hedging, Derivatives, Market Frictions, Arbitrage, Neural Networks, Derivatives

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper presents a complex nested optimization problem involving deep neural networks for hedging and pricing, requiring advanced stochastic calculus and machine learning theory, while also providing empirical evidence through experiments on simulated data comparing arbitrage opportunities and risk reduction.

  flowchart TD
    A["Research Goal: Develop Efficient Nested Deep Hedging"]
    B["Methodology: Fully-Deep Approach<br/>Pricing hedging instruments via NNs"]
    C["Input Data: Market Frictions &<br/>Target & Hedging Options"]
    D{"Computational Challenge<br/>Intractability of Pricing Under Frictions"}
    E["Proposed Solution: Efficient Learning Method<br/>- Sub-network Reuse<br/>- Shared Initial Layers<br/>- Reduced Sampling Steps"]
    F["Computational Process:<br/>Arbitrage-Free Training"]
    G["Key Findings"]
    
    A --> B
    B --> C
    C --> D
    D --> E
    E --> F
    F --> G
    
    subgraph G_Outcomes ["Outcomes"]
        G1["Arbitrage Present in Black-Scholes Pricing"]
        G2["Risk Reduced vs Baseline"]
    end
    
    G --> G1
    G --> G2