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:

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.