Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of d risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process X that ensure ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to O(ε−1), and with constants implied in O(⋅) which grow polynomially in d, thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps. In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide ε expression error in DNN size O(|log(ε)|a) with exponent a proportional to d, but with constants implied in O(⋅) growing exponentially with respect to d. Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.ISSN:0949-2984ISSN:1432-112

Innovative derivative pricing and time series simulation techniques via machine and deep learning

There is a growing number of applications of machine learning and deep learning in quantitative and computational finance. In this thesis, we focus on two of them. In the first application, we employ machine learning and deep learning in derivative pricing. The models considering jumps or stochastic volatility are more complicated than the Black-Merton-Scholes model and the derivatives under these models are harder to be priced. The traditional pricing methods are computationally intensive, so machine learning and deep learning are employed for fast pricing. I n Chapter 2, we propose a method for pricing American options under the variance gamma model. We develop a new fast and accurate approximation method inspired by the quadratic approximation to get rid of the time steps required in finite difference and simulation methods, while reducing the error by making use of a machine learning technique on pre-calculated quantities. We compare the performance of our method with those of the existing methods and show that this method is efficient and accurate for practical use. In Chapters 3 and 4, we propose unsupervised deep learning methods for option pricing under Lévy process and stochastic volatility respectively, with a special focus on barrier options in Chapter 4. The unsupervised deep learning approach employs a neural network as the candidate option surface and trains the neural network to satisfy certain equations. By matching the equation and the boundary conditions, the neural network would yield an accurate solution. Special structures called singular terms are added to the neural networks to deal with the non-smooth and discontinuous payoff at the strike and barrier levels so that the neural networks can replicate the asymptotic behaviors of options at short maturities. Unlike supervised learning, this approach does not require any labels. Once trained, the neural network solution yields fast and accurate option values. The second application focuses on financial time series simulation utilizing deep learning techniques. Simulation extends the limited real data for training and evaluation of trading strategies. It is challenging because of the complex statistical properties of the real financial data. In Chapter 5, we introduce two generative adversarial networks, which utilize the convolutional networks with attention and the transformers, for financial time series simulation. The networks learn the statistical properties in a data-driven manner and the attention mechanism helps to replicate the long-range dependencies. The proposed models are tested on the S&P 500 index and its option data, examined by scores based on the stylized facts and are compared with the pure convolutional network, i.e. QuantGAN. The attention-based networks not only reproduce the stylized facts, including heavy tails, autocorrelation and cross-correlation, but also smooth the autocorrelation of returns

How to handle the COS method for option pricing

The Fourier Cosine Expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the Greeks. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results

Estimation of Default Probabilities with Support Vector Machines

Predicting default probabilities is important for firms and banks to operate successfully and to estimate their specific risks. There are many reasons to use nonlinear techniques for predicting bankruptcy from financial ratios. Here we propose the so called Support Vector Machine (SVM) to estimate default probabilities of German firms. Our analysis is based on the Creditreform database. The results reveal that the most important eight predictors related to bankruptcy for these German firms belong to the ratios of activity, profitability, liquidity, leverage and the percentage of incremental inventories. Based on the performance measures, the SVM tool can predict a firms default risk and identify the insolvent firm more accurately than the benchmark logit model. The sensitivity investigation and a corresponding visualization tool reveal that the classifying ability of SVM appears to be superior over a wide range of the SVM parameters. Based on the nonparametric Nadaraya-Watson estimator, the expected returns predicted by the SVM for regression have a significant positive linear relationship with the risk scores obtained for classification. This evidence is stronger than empirical results for the CAPM based on a linear regression and confirms that higher risks need to be compensated by higher potential returns.Support Vector Machine, Bankruptcy, Default Probabilities Prediction, Expected Profitability, CAPM.