Quantitative Finance

In this paper, we propose a methodology based on piece-wise homogeneous Markov chain for credit ratings and a multivariate model of the credit spreads to evaluate the financial risk in European Union (EU). Two main aspects are considered: how the financial risk is distributed among the European countries and how large is the value of the total risk. The first aspect is evaluated by means of the expected value of a dynamic entropy measure. The second one is solved by computing the evolution of the total credit spread over time. Moreover, the covariance between countries' total spread allows understand any contagions in EU. The methodology is applied to real data of 24 countries for the three major agencies: Moody's, Standard and Poor's, and Fitch. Obtained results suggest that both the financial risk inequality and the value of the total risk increase over time at a different rate depending on the rating agency and that the dependence structure is characterized by a strong correlation between most of European countries.

In this paper, we propose a neural network-based method for approximating expected exposures and potential future exposures of Bermudan options. In a first phase, the method relies on the Deep Optimal Stopping algorithm, which learns the optimal stopping rule from Monte-Carlo samples of the underlying risk factors. Cashflow-paths are then created by applying the learned stopping strategy on a new set of realizations of the risk factors. Furthermore, in a second phase the risk factors are regressed against the cashflow-paths to obtain approximations of pathwise option values. The regression step is carried out by ordinary least squares as well as neural networks, and it is shown that the latter produces more accurate approximations. The expected exposure is formulated, both in terms of the cashflow-paths and in terms of the pathwise option values and it is shown that a simple Monte-Carlo average yields accurate approximations in both cases. The potential future exposure is estimated by the empirical α -percentile. Finally, it is shown that the expected exposures, as well as the potential future exposures can be computed under either, the risk neutral measure, or the real world measure, without having to re-train the neural networks.

To the best of our knowledge, the application of deep learning in the field of quantitative risk management is still a relatively recent phenomenon. In this article, we utilize techniques inspired by reinforcement learning in order to optimize the operation plans of underground natural gas storage facilities. We provide a theoretical framework and assess the performance of the proposed method numerically in comparison to a state-of-the-art least-squares Monte-Carlo approach. Due to the inherent intricacy originating from the high-dimensional forward market as well as the numerous constraints and frictions, the optimization exercise can hardly be tackled by means of traditional techniques.

We present an agent based model of a single asset financial market that is capable of replicating several non-trivial statistical properties observed in real financial markets, generically referred to as stylized facts. While previous models reported in the literature are also capable of replicating some of these statistical properties, in general, they tend to oversimplify either the trading mechanisms or the behavior of the agents. In our model, we strived to capture the most important characteristics of both aspects to create agents that employ strategies inspired on those used in real markets, and, at the same time, a more realistic trade mechanism based on a double auction order book. We study the role of the distinct types of trader on the return statistics: specifically, correlation properties (or lack thereof), volatilty clustering, heavy tails, and the degree to which the distribution can be described by a log-normal. Further, by introducing the practice of profit taking, our model is also capable of replicating the stylized fact related to an asymmetry in the distribution of losses and gains.

The interconnectedness of financial institutions affects instability and credit crises. To quantify systemic risk we introduce here the PD model, a dynamic model that combines credit risk techniques with a contagion mechanism on the network of exposures among banks. A potential loss distribution is obtained through a multi-period Monte Carlo simulation that considers the probability of default (PD) of the banks and their tendency of defaulting in the same time interval. A contagion process increases the PD of banks exposed toward distressed counterparties. The systemic risk is measured by statistics of the loss distribution, while the contribution of each node is quantified by the new measures PDRank and PDImpact. We illustrate how the model works on the network of the European Global Systemically Important Banks. For a certain range of the banks' capital and of their assets volatility, our results reveal the emergence of a strong contagion regime where lower default correlation between banks corresponds to higher losses. This is the opposite of the diversification benefits postulated by standard credit risk models used by banks and regulators who could therefore underestimate the capital needed to overcome a period of crisis, thereby contributing to the financial system instability.

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference method and the simulation method can be used for pricing under this model, but their speed is not satisfactory. So there is a need for fast but accurate approximation methods. In the case of Black-Merton-Scholes model, there are fast approximation methods, but they cannot be utilized for the variance gamma model. We develop a new fast method inspired by the quadratic approximation method, while reducing the error by making use of a machine learning technique on pre-calculated quantities. We compare the performance of our proposed method with those of the existing methods and show that this method is efficient and accurate for practical use.

We propose a fully data-driven approach to calibrate local stochastic volatility (LSV) models, circumventing in particular the ad hoc interpolation of the volatility surface. To achieve this, we parametrize the leverage function by a family of feed-forward neural networks and learn their parameters directly from the available market option prices. This should be seen in the context of neural SDEs and (causal) generative adversarial networks: we generate volatility surfaces by specific neural SDEs, whose quality is assessed by quantifying, possibly in an adversarial manner, distances to market prices. The minimization of the calibration functional relies strongly on a variance reduction technique based on hedging and deep hedging, which is interesting in its own right: it allows the calculation of model prices and model implied volatilities in an accurate way using only small sets of sample paths. For numerical illustration we implement a SABR-type LSV model and conduct a thorough statistical performance analysis on many samples of implied volatility smiles, showing the accuracy and stability of the method.

We study a hybrid tree-finite difference method which permits to obtain efficient and accurate European and American option prices in the Heston Hull-White and Heston Hull-White2d models. Moreover, as a by-product, we provide a new simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed methods

In this paper, the valuation of European and path-dependent options in foreign exchange (FX) markets is considered when the currency exchange rate evolves according to the Heston model combined with the Cox-Ingersoll-Ross dynamics for the stochastic domestic and foreign short interest rates. The mixed Monte Carlo/PDE method requires that we simulate only the paths of the squared volatility and the two interest rates, while an "inner" Black-Scholes-type expectation is evaluated by means of a PDE. This can lead to a substantial variance reduction and complexity improvements under certain circumstances depending on the contract and the model parameters. In this work, we establish the uniform boundedness of moments of the exchange rate process and its approximation, and prove strong convergence in L p ( p≥1 ) of the latter. Then, we carry out a variance reduction analysis and obtain accurate approximations for quantities of interest. All theoretical contributions can be extended to multi-factor short rates in a straightforward manner. Finally, we illustrate the efficiency of the method for the four-factor Heston-CIR model through a detailed quantitative assessment.

We introduce a model for the dynamics of stock prices based on a non quadratic path integral. The model is a generalization of Ilinski's path integral model, more precisely we choose a different action, which can be tuned to different time scales. The result is a model with a very small number of parameters that provides very good fits of some stock prices and indices fluctuations.