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Quant GANs: Deep Generation of Financial Time Series

Magnus Wiese, Robert Knobloch, Ralf Korn, Peter Kretschmer

arXiv:1907.06673v2q-fin.MFcs.LGq-fin.CPstat.ML

TL;DR

The paper addresses realistic asset price simulation for enriching limited real-world datasets and proposes Quant GANs, a data-driven adversarial approach using neural networks and temporal convolutional networks. Its numerical study reports agreement with the S&P 500 in distributional and dependence properties, while the architecture supports risk-neutral evaluation and heavy-tailed returns through specialized construction and preprocessing.

  • Problem

    The paper seeks to approximate a realistic asset price simulator that can extend limited real-world datasets for financial trading strategies.

  • Method

    Quant GANs use generative adversarial networks with temporal convolutional generators, including stochastic volatility and drift TCNs plus an innovation neural network.

  • Results

    The numerical study reports that SVNNs and TCNs outperform a typical GARCH model on distributional and dependence properties for S&P 500 data.

  • Takeaways & Limitations

    Quant GANs can generate financial paths with long-range dependencies, risk-neutral evaluation, and empirically heavier-tailed returns.

  • Takeaways & Limitations

    Using a heavier-tailed latent process may enable heavier tails but can cause optimization instabilities, and stabilizing preprocessing remains future work.

Abstract

from arXiv · show

Modeling financial time series by stochastic processes is a challenging task and a central area of research in financial mathematics. As an alternative, we introduce Quant GANs, a data-driven model which is inspired by the recent success of generative adversarial networks (GANs). Quant GANs consist of a generator and discriminator function, which utilize temporal convolutional networks (TCNs) and thereby achieve to capture long-range dependencies such as the presence of volatility clusters. The generator function is explicitly constructed such that the induced stochastic process allows a transition to its risk-neutral distribution. Our numerical results highlight that distributional properties for small and large lags are in an excellent agreement and dependence properties such as volatility clusters, leverage effects, and serial autocorrelations can be generated by the generator function of Quant GANs, demonstrably in high fidelity.

1 Introduction

The paper introduces Quant GANs as data-driven financial path simulators using adversarial training and TCN-based architectures. Its SVNN generator supports risk-neutral evaluation, while experiments on the S&P 500 show strong distributional and dependence-property performance.

  • Motivation and contribution: Quant GANs approximate realistic asset-price paths with a data-driven approach positioned between historical simulation and model-driven Monte Carlo methods.The simulator can extend limited real-world datasets for refining or robustifying trading strategies.
  • Adversarial architecture: GAN training uses a generator to create stock-price paths and a discriminator to judge whether paths resemble the data distribution.
  • Neural architectures: TCNs serve as generator architectures because they address recurrent extrapolation issues while modeling long-range dependencies, enabling parallelization, and guaranteeing stationarity.
  • Evaluation: The paper presents a rigorous mathematical definition of TCNs and applies Quant GANs to S&P 500 data from May 2009 to December 2018.The numerical study reports that SVNNs and TCNs outperform a typical GARCH model on distributional and dependence properties.
  • Neural architectures: SVNNs combine volatility and drift TCNs with an innovation NN, and their generated paths can be evaluated under a risk-neutral distribution.As a special case, SVNNs can constrain log returns to be conditionally normal.
  • Theoretical properties: SVNN theory establishes existence of all process moments, while the Lambert W transformation is used to generate heavy-tailed stock-price returns.

2 Generative modeling of financial time series

The paper situates data-driven neural modeling within financial time-series research and motivates it by stylized return properties and the difficulty of extending conventional models. It reviews prior neural approaches while identifying gaps in architectural detail, distributional specification, and reproducibility.

  • Financial stylized facts: Asset returns exhibit heavier tails, greater peakedness, volatility clustering, leverage effects, and negligible serial correlation despite dependence.
  • Conventional modeling: Financial modeling spans discrete-time GARCH models and continuous-time models such as Black–Scholes and Heston, but developing more sophisticated models can require decades.
  • Related work: Data-driven neural modeling is a relatively new area with potential to represent complicated, possibly unknown, financial dynamics.
  • Related work: Prior neural studies include GAN-based conditional modeling, Wasserstein and relativistic GANs, stylized-fact approximation, and adversarial generation of arbitrage-constrained option markets.
  • Open limitations: Pardo and Takahashi lack detailed neural-architecture descriptions and do not specify whether their algorithms approximate conditional or unconditional distributions.Code availability is also absent for the mentioned papers, making benchmark development difficult.

3 Neural network topologies

This section introduces MLPs and TCNs as the neural-network building blocks for SVNNs, emphasizing causal convolutions and dilation for modeling long-range dependencies. It also states a practical limitation: TCN sequence length is bounded by the receptive field size, potentially creating computational and data requirements.

  • 3.1 Multilayer perceptrons: MLPs compose affine transformations with component-wise Lipschitz continuous, monotonic activation functions.Their parameters are weight matrices and bias vectors across the network layers.
  • 3.2 Temporal convolutional networks: Increasing dilations expand a TCN’s receptive field, enabling it to capture longer sequence dependencies than constant dilation.With four layers, kernel size K = 2, and dilation D = 1, the network uses 5 sequence elements; with D = 2, it uses 16.
  • 3.2 Temporal convolutional networks: A dilated causal convolution uses kernel size K and dilation D to select present or past inputs at specified temporal distances.For K = D = 2, the information-carrying input elements are separated by two time steps.
  • 3.2 Temporal convolutional networks: TCNs compose dilated causal convolutional layers with activation functions, optionally using block modules and 1 × 1 convolutions.Causal convolutions make outputs depend only on past sequence elements.
  • 3.2 Temporal convolutional networks: TCNs restrict the processed time-series length to their receptive field size, so modeling long dependencies may require large, computationally demanding networks and sufficient data.The paper leaves extensions for handling this limitation to future work.

4 Generative adversarial networks

This section extends GANs from random-variable generation to stochastic-process generation using TCN-based generators and discriminators. The generator maps i.i.d. noise into sequences, while adversarial training makes generated sequences difficult for the discriminator to distinguish from real ones.

  • 4.1 GANs: GANs learn generative models by training a generator to produce samples resembling a target distribution and a discriminator to classify real versus generated samples.The generator and discriminator optimize opposing objectives in a zero-sum min-max game.
  • 4.1 GANs: The generator maps a latent noise variable through a neural network to obtain the generated random variable.The generated variable is denoted by ˜Xθ and depends on the generator parameters θ.
  • 4.3 Training: GAN parameters are trained by alternating discriminator and generator gradient updates, with multiple discriminator updates commonly used per iteration.Algorithm 1 specifies generator and discriminator learning rates, sample size, and discriminator optimization steps.
  • 4.2 GANs for stochastic processes: For stochastic processes, TCNs generate sequences from an i.i.d. noise process, and a TCN discriminator distinguishes generated sequences from real sequences.The generator aims to simulate sequences that the discriminator cannot distinguish from the real process.
  • 4.2 GANs for stochastic processes: The paper assumes each noise variable follows a multivariate standard normal distribution, Z_t ∼ N(0, I), before inference through the TCN generator.This noise process serves as the GAN latent prior.

5 The model

Quant GANs define financial path generators using stochastic volatility neural networks, combining TCN-based volatility and drift components with an innovation network. The model supports moment characterization, heavy-tail transformation, risk-neutral evaluation, and volatility-dependent price dynamics.

  • 5.1 Log return neural processes: SVNNs combine a volatility and drift TCN with a separate innovation network to define the log return neural process.The volatility and drift infer past latent noise, while the innovation network infers the current latent noise.
  • 5.2 Lp-space characterization of Rθ: Theorem 5.4 establishes that neural networks map Lp inputs to Lp outputs, and Corollary 5.5 extends this to all moments of SVNN log returns.Gaussian i.i.d. latent noise therefore yields finite mean, variance, and higher moments for the model components and returns.
  • 5.2 Lp-space characterization of Rθ: Gaussian i.i.d. latent noise supports finite back-propagated gradients in Lp, whereas heavy-tailed latent noise may destabilize optimization.The authors leave preprocessing-based stabilization for future work.
  • 5.3 Generating heavier-tails and modeling assumptions: The Lambert W probability transform is used to generate heavier-tailed returns while retaining normalized Gaussian data for the neural process.For δ > 0, the transformation is bijective and differentiable and produces heavier tails than the original random variable.
  • 5.4 Risk-neutral representation of Rθ: SVNNs are constructed so generated paths can be evaluated under a risk-neutral distribution and can model nonconstant volatility through the volatility generator.The risk-neutral parameter-inference problem remains unresolved because risk-neutral asset paths are unobservable; option-price calibration is proposed as future work.

6 Preprocessing

The preprocessing pipeline converts financial time series into normalized log returns and applies an inverse Lambert W transformation before discriminator training. Its transformations are invertible except for the rolling window, enabling generated samples to be post-processed into the desired form.

  • 6 Preprocessing: The pipeline preprocesses financial time series before passing them to the discriminator and includes the inverse Lambert W transformation.All transformations except the rolling window are invertible, allowing generated log-return samples to be reconstructed afterward.
  • Step 1: Log returns: Step 1 computes the log return series from the asset-price series.The section explicitly labels this operation as Step 1: Log returns r1:T.
  • 6 Preprocessing: The log returns are normalized to zero mean and unit variance for numerical reasons.
  • 6 Preprocessing: The transformed standardized S&P 500 log-return distribution approximately follows a standard normal distribution, avoiding direct generation of the original heavy tail.Figure 10 compares the original Lambert W × Gaussian distribution with the inverse-transformed Gaussian distribution.
  • Step 5: Rolling window: A rolling window matching the discriminator receptive-field length is applied with stride one to the preprocessed log-return sequence.The windowing step under-samples observations at the beginning and end of the series, creating bias that weighted batch sampling can correct.

7 Numerical results

Quant GAN models reproduce the S&P 500’s distributional and dependence properties more accurately than GARCH(1,1), with TCN generally performing best and constrained SVNN remaining close.

  • Quant GANs are evaluated against GARCH(1,1) using distributional and dependence metrics for S&P 500 daily returns.The models include a pure TCN, a constrained log return neural process, and GARCH(1,1).
  • The Lambert W transformation significantly improves the models’ ability to capture asset-return stylized facts, especially distributional tails.The underlying S&P 500 series is supported as stationary by an augmented Dickey-Fuller statistic of -10.87 and p-value 1.36 × 10^-19.
  • The TCN closely matches historical return densities across daily, weekly, monthly, and 100-day lags.The fit remains good even for 100-day lagged returns.
  • The TCN reproduces serial-return autocorrelation decay, persistent squared and absolute-return autocorrelations, and the leverage effect.The leverage effect appears as negative correlation between squared and non-squared returns at small lags.
  • The TCN performs best on all but two evaluated metrics and often outperforms GARCH(1,1) by a factor of 2-10.The comparison covers both distributional and dependence properties.
  • The constrained SVNN performs nearly as well as the TCN and significantly outperforms GARCH(1,1), while modeling the same broad properties.The TCN models squared- and absolute-return autocorrelation somewhat better than the constrained SVNN.
  • GARCH captures autocorrelation reasonably well but poorly matches return distributions and does not capture the leverage effect.Its normal-return assumption places too little probability mass near the peak and in the tails.

8 Conclusion and future work

The paper shows that neural architectures can approximate discrete-time financial time series through adversarial modeling. Future progress depends on improving tail extrapolation and evaluation across distributional and dependence properties.

  • Neural architectures can approximate discrete-time financial time series within an adversarial modeling framework.
  • Future work must improve exact modeling and extrapolation of generated tails, potentially using prior knowledge such as estimated tail indices.
  • A unified metric is needed to combine distributional metrics with dependence scores when benchmarking generator architectures.

A.1 Pure TCN

The pure TCN generates structurally diverse S&P 500 paths and closely reproduces historical return distributions and dependence patterns across multiple lags.

  • Five and fifty generated log paths exhibit reasonable patterns and structural diversity.
  • The pure TCN accurately models serial, squared-return, and absolute-return autocorrelations together with the leverage effect.The leverage effect is represented by negative correlation between squared and non-squared returns at small lags.

A.2 Constrained SVNN

The constrained SVNN generates diverse paths and reproduces the S&P 500’s distributional and dependence properties, with performance close to the pure TCN.

  • Five and fifty generated log paths show reasonable patterns and structural diversity.
  • Generated log-return densities are compared with historical S&P 500 densities across the evaluated time scales.
  • The constrained SVNN captures the same broad distributional and dependence properties as the pure TCN.The TCN models squared- and absolute-return autocorrelation somewhat better.
  • Fifty additional generated log paths provide another view of the constrained SVNN’s path diversity.
  • The constrained SVNN’s dependence behavior is assessed through autocorrelation functions and the leverage-effect plot.

B Architecture

The generator and discriminator use TCNs with skip connections and temporal blocks built from dilated causal convolutions and PReLUs. The reported architectures use mostly 80-dimensional hidden layers, mostly kernel size two, and a receptive field size of 127.

  • TCN design: Both the generator and discriminator use TCNs with skip connections and temporal blocks as their modular components.The architecture diagrams and configuration tables report these designs for the pure TCN and C-SVNN models.
  • Temporal blocks: Each temporal block contains two dilated causal convolutions and two PReLUs.The additional nonlinear operations are intended to make the TCN more expressive.
  • Temporal blocks: Temporal blocks are parameterized by the arguments (NI, NH, NO, K, D).The paper introduces this parameterization in Definition B.1.
  • Model configuration: 80 was the hidden dimension for all models except the C-SVNN generator.The kernel size was two for every temporal block except the first, and each TCN modeled a receptive field of 127.
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