Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deep Quadratic Hedging

Alessandro Gnoatto, Silvia Lavagnini, Athena Picarelli

TL;DR

A deep learning-based BSDE solver is applied to compute the entire optimal hedging strategies paths, overcoming the curse of dimensionality and extending the scope of applicability of quadratic hedging in high dimension.

Abstract

We propose a novel computational procedure for quadratic hedging in high-dimensional incomplete markets, covering mean-variance hedging and local risk minimization. Starting from the observation that both quadratic approaches can be treated from the point of view of backward stochastic differential equations (BSDEs), we (recursively) apply a deep learning-based BSDE solver to compute the entire optimal hedging strategies paths. This allows us to overcome the curse of dimensionality, extending the scope of applicability of quadratic hedging in high dimension. We test our approach with a classic Heston model and with a multiasset and multifactor generalization thereof, showing that this leads to high levels of accuracy.

Deep Quadratic Hedging

TL;DR

A deep learning-based BSDE solver is applied to compute the entire optimal hedging strategies paths, overcoming the curse of dimensionality and extending the scope of applicability of quadratic hedging in high dimension.

Abstract

We propose a novel computational procedure for quadratic hedging in high-dimensional incomplete markets, covering mean-variance hedging and local risk minimization. Starting from the observation that both quadratic approaches can be treated from the point of view of backward stochastic differential equations (BSDEs), we (recursively) apply a deep learning-based BSDE solver to compute the entire optimal hedging strategies paths. This allows us to overcome the curse of dimensionality, extending the scope of applicability of quadratic hedging in high dimension. We test our approach with a classic Heston model and with a multiasset and multifactor generalization thereof, showing that this leads to high levels of accuracy.
Paper Structure (25 sections, 12 theorems, 120 equations, 11 figures, 3 tables)

This paper contains 25 sections, 12 theorems, 120 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setting and main assumptions
  3. The market model
  4. Self-financing property and financing costs
  5. Mean-variance hedging
  6. Local risk minimisation
  7. The multidimensional Heston model
  8. Existence and uniqueness results for the BSRE
  9. Our methodology
  10. The deep BSDE solver by E, Han and Jentzen
  11. Details on the parametrization of the solver
  12. Deep mean-variance hedging
  13. Deep local risk minimization
  14. Comparison with the Deep Hedging approach of bgtw2019
  15. Convergence analysis
  16. ...and 10 more sections

Key Result

Proposition 3.1

If the BSDEs eq:StochasticRiccati and hedgingproblemX admit unique solutions $(L,\Lambda)\in L^{\infty}_{\mathbb{F}}(\Omega;{C}([0,T];\mathbb{R})) \times L^2_{\mathbb{F}}\left([0, T];\mathbb{R}^{m+d}\right)$ and $(\tilde{X}^{\mathrm{mv}},\eta^{\mathrm{mv}})\in L^{2}_{\mathbb{F}}(\Omega;{C}([0,T];\ma is the unique optimal control for the stochastic control problem hedgingproblemH, where $\tilde{V}^

Figures (11)

  • Figure 1: Logarithm of the loss functional as a function of the iteration number for the different experiment configurations presented in Table \ref{['table:resultsMV']}. The black curves represent the log-loss for the first BSDE and follow the black grid on the left-hand side. The red curves represent the log-loss for the second BSDE and follow the red grid on the right-hand side.
  • Figure 2: Mean-variance hedging results with portfolio dimension $m = 100$, when we reduce the learning rates values and improve the experiments in Table \ref{['table:resultsMV']} and Figure \ref{['MV_logloss']}.
  • Figure 3: Deep solver solution (solid line) and benchmark solution (dashed line) for the mean-variance hedging in a $10$ points discretization grid for $5$ random samples in the interval $[0, 1]$. Upper panel: the shares of risky asset (left) and the units of cash account (right); lower panel: the call option price.
  • Figure 4: Deep solver solution (solid line) and benchmark solution (dashed line) for the mean-variance hedging in a $50$ points discretization grid for $5$ random samples in the interval $[0, 1]$. Upper panel: the shares of risky asset (left) and the units of cash account (right); lower panel: the call option price.
  • Figure 5: Deep solver solution (solid line) and benchmark solution (dashed line) for the mean-variance hedging in a $100$ points discretization grid for $5$ random samples in the interval $[0, 1]$. Upper panel: the shares of risky asset (left) and the units of cash account (right); lower panel: the call option price.
  • ...and 6 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Theorem 4.1: Theorem 1.6 in Schweizer08
  • Theorem 4.2: Proposition 5.2 in Schweizer08
  • proof
  • Proposition 4.3
  • ...and 22 more