Deep Learning Algorithms for Hedging with Frictions

Overview

Deep Learning Algorithms for Hedging with Frictions

This repository contains the Forward-Backward Stochastic Differential Equation (FBSDE) solver and the Deep Hedging, as described in reference [2]. Both of them are implemented in PyTorch.

Basic Setup

The special case with following assumptions is considered:

  • the dynamic of the market satisfies that return and voalatility are constant;
  • the cost parameter is constant;
  • the endowment volatility is in the form of where is constant;
  • the frictionless strategy satisfies that and

On top of that, we consider two calibrated models: a quadratic transaction cost models, and a power cost model with elastic parameter of 3/2. In both experiments, the FBSDE solver and the Deep Hedging are implemented, as well as the asymptotic formula from Theorem 3.6 in reference [2].

For the case of quadratic costs, the ground truth from equation (3.7) in reference [2] is also compared. See Script/sample_code_quadratic_cost.py for details.

For the case of 3/2 power costs, the ground truth is no longer available in closed form. Meanwhile, in regard to the asymptotic formula g(x) in equation (3.8) in reference [2], the numerical solution by SciPy is not stable, thus it is solved via MATHEMATICA (see Script/power_cost_ODE.nb). Consequently, the value of g(x) corresponding to x ranging from 0 to 50 by 0.0001, is stored in table Data/EVA.txt. Benefitted from the oddness and the growth conditions (equation (3.9) in reference [2]), the value of g(x) on is obatinable. Following that, the numerical result of the asymptotic solution is compared with two machine learning methods. See Script/sample_code_power_cost.py for details.

The general variables and the market parameters in the code are summarized below:

Variable Meaning
q power of the trading cost, q
S_OUTSTANDING total shares in the market, s
TIME trading horizon, T
TIME_STEP time discretization, N
DT
GAMMA risk aversion,
XI_1 endowment volatility parameter,
PHI_INITIAL initial holding,
ALPHA market volatility,
MU_BAR market return,
LAM trading cost parameter,
test_samples number of test sample path, batch_size

FBSDE solver

For the detailed implementation of the FBSDE solver, see Script/sample_code_FBSDE.py;
The core dynamic is defined in the method System.forward(), and the key variables in the code are summarized below:

Variable Meaning
time_step time discretization, N
n_samples number of sample path, batch_size
dW_t iid normally distributed random variables with mean zero and variance ,
W_t Brownian motion at time t,
XI_t Brownian motion at time t,
sigma_t vector of 0
sigmaxi_t vector of 1
X_t vector of 1
Y_t vector of 0
Lam_t 1
in_t input of the neural network
sigmaZ_t output of the neural network ,
Delta_t difference between the frictional and frictionless positions (the forward component) divided by the endowment parameter,
Z_t the backward component,

Deep Hedging

For the detailed implementation of the Deep Hedging, see Script/sample_code_Deep_Hedging.py;
The core dynamic of the Deep Hedging is defined in the function TRAIN_Utility(), and the key variables in the code are summarized below:

Variable Meaning
time_step time discretization, N
n_samples number of sample path, batch_size
PHI_0_on_s initial holding divided by the total shares in the market,
W collection of the Brownian motion, throughout the trading horizon,
XI_W_on_s collection of the endowment volatility divided by the total shares in the market, throughout the trading horizon,
PHI_on_s collection of the frictional positions divided by the total shares in the market, throughout the trading horizon,
PHI_dot_on_s collection of the frictional trading rate divided by the total shares in the market, throughout the trading horizon,
loss_Utility minus goal function,

Example

Here we proivde an example for the quadratic cost case (q=2) with the trading horizon of 21 days (TIME=21).

The trading horizon is discretized in 168 time steps (TIME_STEP=168). The parameters are taken from the calibration in [1]:

Parameter Value Code
agent risk aversion GAMMA=1.66*1e-13
total shares outstanding S_OUTSTANDING=2.46*1e11
stock volatility ALPHA=1.88
stock return MU_BAR=0.5*GAMMA*ALPHA**2
endowment volatility parameter XI_1=2.19*1e10
trading cost parameter LAM=1.08*1e-10

And these lead to the optimal trading rate (left panel) and the optimal position (right panel) illustrated below, leanrt by the FBSDE solver and the Deep Hedging, as well as the ground truth and the Leading-order solution based on the asymptotic formula:

TR=21_q=2
With the same simulation with test batch size of 3000 (test_samples=3000), the expectation and the standard deviation of the goal function and the mean square error of the terminal trading rate are calculated, as summarized below:

Method
FBSDE
Deep Q-learning
Leading Order Approximation
Ground Truth

See more examples and discussion in Section 4 of paper [2].

Acknowledgments

Reference

[1] Asset Pricing with General Transaction Costs: Theory and Numerics, L. Gonon, J. Muhle-Karbe, X. Shi. [Mathematical Finance], 2021.

[2] Deep Learning Algorithms for Hedging with Frictions, X. Shi, D. Xu, Z. Zhang. [arXiv], 2021.

Owner
Xiaofei Shi
Xiaofei Shi
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