We study neural network approximation of the solution to boundary value problem for Black-ScholesMerton partial differential equation for a European call option price, when model volatility is a function of underlying asset price and time (local volatility model). Strike-price and expiry day of the option are assumed to be fixed. An approximation to option price in local volatility model is obtained via deep learning with deep Galerkin method (DGM), making use of the neural network of special architecture and stochastic gradient descent on a sequence of random time and underlying price points. Architecture of the neural network and the algorithm of its training for option pricing in local volatility models are described in detail. Computational experiment with DGM neural network is performed to evaluate the quality of neural network approximation for hyperbolic sine local volatility model with known exact closed form option price. The quality of the neural network approximation is estimated with mean absolute error, mean squared error and coefficient of determination. The computational experiment demonstrates that DGM neural network approximation converges to a European call option price of the local volatility model with acceptable accuracy.
In a local volatility model (LVM) [
When evaluating derivatives with LVM, the boundary (terminal) value problem for Black Scholes-Merton (BSM) partial differential equation (PDE) is to be solved. Exact closed form solutions of terminal value problem for BSM PDE are known only in a few special cases, therefore, in general case application of numerical methods such as binomial trees, Monte Carlo simulations, Fourier or finite difference methods is required. Alternatively, derivative prices in LVM can be approximated with an artificial neural network (ANN).
The idea to use ANNs for option pricing is several decades old (see [
Our goal is to study application of DGM approach [
For pricing of a European call option with strike-price and expiry day in LVM with a
volatility function σ( , ) and a risk-free interest rate > 0, the solution of BSM PDE [
To determine ANN approximation to PDE (1) with condition (2) using DGM approach the
neural network of special architecture [
X0 = σ(w0 x + b0) with a nonlinear activation function σ and input layer parameters w0 and b0.
Each hidden (LSTM) layer receives as an input the original set of price-time points x and the output of the previous layer. In hidden layers, the price-time points x and the outputs of the previous layer Xi−1 are processed with the following transformations:
Zi = σ(uiz x + wiz Xi−1 + biz), Gi = σ(uig x + wig Xi−1 + big),
Ri = σ(uir x + wir Xi−1 + bir), Hi = σ(uih x + wih (Xi−1 ⊙ Ri ) + bih ), where ⊙ denotes element-wise multiplication, and the outputs of the layer are
Xi = (1 − Gi) ⊙ Hi + Zi ⊙ Xi−1.
In the output layer, the outputs of the last LSTM layer Xd are transformed into the neural network outputs y with a linear transform
y = f(x; θ) = w′ Xd + b′, where w′ and b′ are the output layer parameters. The output of the DGM neural network y is the approximation of option price u at the initial price-time points x.
In hidden LSTM layers, the shape of the weight parameters uiz, uig, uir, uih is 2 × , the shape of the weight parameters wiz, wig, wir, wih is × , the shape of the bias parameters biz, big, bir, bih is 1 × .
In the output layer, the shape of the weight parameter w′ is × 1 and b′ is a scalar parameter. The neural network parameter set contains all weight and bias parameters mentioned above. Thus, the total number of parameters in DGM neural network is equal to
| | = 4 d (N+ 1)2 + 4 N + 1.
DGM neural network is trained with adaptive algorithm Adam [
+ 12 σ2( , ) 2 2 ( , 2 ; )– ( , ; ))
2 + ( ( ′, ; )– max( ′ – )) .
2
update parameter set for a gradient descent step at the random points for the learning rate
with adaptive algorithm Adam [
+1 ← − ( , ).
Data: Result: repeat – – – until ‖ +1 − ‖ < ; ∗ ← +1;
As a result of Algorithm 1, an approximation of the price of a European call option in LVM with volatility function σ( , ) is obtained in the form u( , ) = ( , ; ∗).
Consider hyperbolic sine LVM [
The computational experiment is performed with the following parameters: the number of hidden layers is 3, the number of nodes (neurons) per hidden layer is 50, the number of training stages is 100 with 10 SGD steps in each stage, and other parameters of the model are as follows r = 0.05, λ = 0.25, K = 50, T = 1, S0 = 0.5. Quality of obtained ANN approximation is characterized by the following metrics: mean absolute error (MAE) is 0.2014; mean squared error (MSE) is 0.2483; coefficient of determination ( 2) is 99.93%. Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
Absolut e Error Surface of DGM Opt ion Price Est imat e
The resulting error of approximation, i.e. difference between exact analytical option prices in
hyperbolic sine LVM (3) [
The computational experiment shows that the approximation, obtained with DGM ANN, predicts option prices in hyperbolic sine LVM (3) with acceptable accuracy, but the quality of approximation deteriorates for options ATM (At The Money) at expiry day and for options ITM (In The Money) with long time to maturity.
Generally, option exchange trades various options on the same underlying with a range of exercise (strike) prices and expiry days, so to price all these options ANN shall receive as an input the set of price-time-strike price-expiry day points ( , , , ) instead of price-time points ( , ). This transition from input ( , ) to input ( , , , ) may require different architecture of ANN and another strategy of its training.
As noted in [
The computational experiment with deep option pricing in hyperbolic sine LVM demonstrates that the algorithm converges to exact analytical European call option price of the LVM with acceptable accuracy, but oscillating behavior of the option price approximation makes it desirable to modify the neural network architecture for smoothing its output.