=Paper=
{{Paper
|id=Vol-3041/381-386-paper-70
|storemode=property
|title=On Deep Learning for Option Pricing in Local Volatility Models
|pdfUrl=https://ceur-ws.org/Vol-3041/381-386-paper-70.pdf
|volume=Vol-3041
|authors=Sergey Shorokhov
}}
==On Deep Learning for Option Pricing in Local Volatility Models==
https://ceur-ws.org/Vol-3041/381-386-paper-70.pdf
Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
ON DEEP LEARNING FOR OPTION PRICING
IN LOCAL VOLATILITY MODELS
S.G. Shorokhov
Peoplesβ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow,
117198, Russia
E-mail: shorokhov-sg@rudn.ru
We study neural network approximation of the solution to boundary value problem for Black-Scholes-
Merton 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.
Keywords: partial differential equation, local volatility model, option price, neural network
Sergey Shorokhov
Copyright Β© 2021 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
1. Introduction
In a local volatility model (LVM) [1], in contrast to Black-Scholes constant volatility model
[2, 3], the volatility depends on underlying asset price π and time π‘. Well-known LVM with exact
closed form solutions include CEV model [4], shifted lognormal model [5] and normal model (a clone
of OrnsteinβUhlenbeck model [6]).
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 [7] and the references
therein), however, the need to improve the quality of option price approximation stimulates further
research. Deep Galerkin Method (DGM), introduced recently in [8] for the solution of PDEs, makes
use of the neural network of special architecture and stochastic gradient descent (SGD) [9] on a
sequence of random time and space points. The method has been successfully applied to various PDEs
[8], including BSM PDE with constant volatility.
Our goal is to study application of DGM approach [8] to option pricing when volatility
function is not constant and evaluate the quality of ANN approximation for an LVM with known exact
analytical closed form solution.
2. Deep option pricing with local volatility models
For pricing of a European call option with strike-price πΎ and expiry day π in LVM with a
volatility function Ο(π, π‘ ) and a risk-free interest rate π > 0, the solution of BSM PDE [2, 3]
βu βu 1 2 β2 u
+ππ + Ο (π, π‘) π 2 2 β π u = 0#(1)
βπ‘ βπ 2 βπ
with terminal condition
u(π, π, πΎ, π ) = max (π β πΎ, 0) #(2)
is to be determined. Strike-price πΎ and expiry day π are assumed to be fixed.
To determine ANN approximation to PDE (1) with condition (2) using DGM approach the
neural network of special architecture [8] is to be built and trained. The architecture of the ANN is
similar to architectures of LSTM [10] and Highway [11] networks. It consists of the layers in Fig.1: an
input layer, π hidden (LSTM) layers and an output layer.
Figure 1. Architecture of DGM neural network
The input to DGM ANN is a set of randomly sampled price-time points x = (π, π‘). In the input
layer, the price-time points x are transformed into the output X 0
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Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
X 0 = Ο(w0 x + b0 )
with a nonlinear activation function Ο 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 X iβ1 are processed with the following transformations:
Zi = Ο(uzi x + wiz Xiβ1 + bzi ), Ri = Ο(uri x + wir Xiβ1 + bri ),
g g g
Gi = Ο(ui x + wi Xiβ1 + bi ), Hi = Ο(uhi x + wih (X iβ1 β Ri ) + bhi ),
where β denotes element-wise multiplication, and the outputs of the layer are
X i = (1 β Gi ) β Hi + Zi β X iβ1 .
In the output layer, the outputs of the last LSTM layer X d are transformed into the neural
network outputs y with a linear transform
y = f(x; ΞΈ) = w β² X d + bβ² ,
where w β² and bβ² 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.
Figure 2. Architecture of LSTM layer in DGM network
The number of parameters (weights and biases) in DGM ANN can be calculated as follows.
Let π be the number of neurons (nodes) in each hidden layer of DGM neural network.
In the input layer, the shape of the weight parameter w0 is 2 Γ π and the shape of the bias
parameter b0 is 1 Γ π.
g
In hidden LSTM layers, the shape of the weight parameters uzi , ui , uri , uhi is 2 Γ π, the shape
g g
of the weight parameters wiz, wi , wir, wih is π Γ π, the shape of the bias parameters bzi , bi , bri , bhi is
1 Γ π.
In the output layer, the shape of the weight parameter w β² is π Γ 1 and bβ² is a scalar
parameter. The neural network parameter set π contains all weight and bias parameters mentioned
above. Thus, the total number of parameters in DGM neural network is equal to
|π | = 4 d (N + 1)2 + 4 N + 1.
DGM neural network is trained with adaptive algorithm Adam [12], which is an extension to
classical SGD algorithm. General outline of DGM algorithm for the solution of BSM PDE (1)-(2) is
shown below in Algorithm 1.
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Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
Algorithm 1: Approximation of option price in local volatility model with DGM neural network
Data: a volatility function π(π, π‘), strike-price πΎ, time to maturity π, risk free interest rate π, distributions
π1 and π2 , absolute tolerance π > 0;
Result: optimal parameter set π β for the approximation of option price in LVM;
β choose initial parameter set π0 and learning rate πΌ0 ;
repeat
β generate random time-price points (ππ, π‘π) from πΊ Γ [0, π] with distribution π1 and random price
points ππβ² from πΊ with distribution π2 , πΊ = [π π , π β ] β β;
β calculate the loss function πΏ(ππ , ππ ) at the randomly sampled points ΞΎπ = {(ππ, π‘π ), ππβ² } , where
πΏ(ππ, ππ)
ππ(ππ, π‘π ; ππ ) ππ(ππ, π‘π ; ππ )
β( + π ππ
ππ‘ ππ
2
2 π π(ππ, π‘π ; ππ )
2
1
+ Ο2 (ππ, π‘π ) ππ β π π(ππ, π‘π; ππ ))
2 ππ 2
2
+ (π(ππβ², π; ππ )β max (ππβ²β πΎ)) .
β update parameter set π for a gradient descent step at the random points ππ for the learning rate πΌπ
with adaptive algorithm Adam [12]:
ππ+1 β ππ β πΌπ π»π πΏ(ππ , ππ ).
until βππ+1 β ππ β < π;
π β β ππ+1;
As a result of Algorithm 1, an approximation of the price of a European call option in LVM
with volatility function Ο(π, π‘) is obtained in the form u(π‘, π) = π(π‘, π; π β ).
3. Computational experiment for hyperbolic sine LVM
Consider hyperbolic sine LVM [13] with underlying asset price driven by SDE
ππ = π π ππ‘ + β2 π π 2 + π2 ππ, π > 0, π > 0#(3)
with the following BSM PDE for a derivative price
βu βu 1 β2 u
+ ππ + (2 π π 2 + π2 ) 2 β π u = 0. #(4)
βπ‘ βπ 2 βπ
The goal is to find a DGM estimate of a European option price in hyperbolic sine LVM (3),
compare it with known analytical option price [13] and evaluate the quality of approximation.
DGM ANN for PDE (4) was implemented with TensorFlow framework [14].
The computational experiment is performed with the following parameters: the number π of
hidden layers is 3, the number π 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, Ξ» = 0.25, K = 50, T = 1, S0 = 0.5. Quality of obtained ANN approximation is
characterized by the following metrics:
ο· mean absolute error (MAE) is 0.2014;
ο· mean squared error (MSE) is 0.2483;
ο· coefficient of determination (π
2) is 99.93%.
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Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
Absolute Error Surface of DGM Option Price Estimate
The resulting error of approximation, i.e. difference between exact analytical option prices in
hyperbolic sine LVM (3) [13] and option prices, predicted by DGM ANN, is visualized in Fig. 3.
4.63
4.17
Absolut e Opt ion Price Error
3.70 4
3.24
2.78
3
2.32
1.85
1.39 2
0.93
0.46
1
0.00
100
80 0
60
Pr 1.0
i ce 40 0.8
S 0.6
20 0.4 t
0.2 T i me
0
0.0
Figure 3. Absolute Error Surface of DGM Option Price Approximation
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.
4. Conclusion and Future plans
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 (π, π‘, πΎ, π) instead of price-time points (π, π‘). This
transition from input (π, π‘) to input (π, π‘, πΎ, π) may require different architecture of ANN and another
strategy of its training.
As noted in [15,16], the loss function in the form of energy functional (potential) is preferable
for loss minimization, so construction of variational formulation for BSM PDE (1) can contribute to
deep option pricing. Energy functional (potential) for BSM PDE (1) may be obtained using methods of
the inverse problem of the calculus of variations [17].
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.
385
Proceedings of the 9th International Conference "Distributed Computing and Grid Technologies in Science and
Education" (GRID'2021), Dubna, Russia, July 5-9, 2021
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