Quantitative Finance

Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive Marginal Quantization of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. As part of this generalization a simple matrix formulation is presented, allowing efficient implementation. We further extend the applicability of recursive marginal quantization by showing how absorption and reflection at the zero boundary may be incorporated, when this is necessary. To illustrate the improved accuracy of the higher order schemes, various computations are performed using geometric Brownian motion and its generalization, the constant elasticity of variance model. For both processes, we show numerical evidence of improved weak order convergence and we compare the marginal distributions implied by the three schemes to the known analytical distributions. By pricing European, Bermudan and Barrier options, further evidence of improved accuracy of the higher order schemes is demonstrated.

A number of optimal decision problems with uncertainty can be formulated into a stochastic optimal control framework. The Least-Squares Monte Carlo (LSMC) algorithm is a popular numerical method to approach solutions of such stochastic control problems as analytical solutions are not tractable in general. This paper generalizes the LSMC algorithm proposed in Shen and Weng (2017) to solve a wide class of stochastic optimal control models. Our algorithm has three pillars: a construction of auxiliary stochastic control model, an artificial simulation of the post-action value of state process, and a shape-preserving sieve estimation method which equip the algorithm with a number of merits including bypassing forward simulation and control randomization, evading extrapolating the value function, and alleviating computational burden of the tuning parameter selection. The efficacy of the algorithm is corroborated by an application to pricing equity-linked insurance products.

Interest in agent-based models of financial markets and the wider economy has increased consistently over the last few decades, in no small part due to their ability to reproduce a number of empirically-observed stylised facts that are not easily recovered by more traditional modelling approaches. Nevertheless, the agent-based modelling paradigm faces mounting criticism, focused particularly on the rigour of current validation and calibration practices, most of which remain qualitative and stylised fact-driven. While the literature on quantitative and data-driven approaches has seen significant expansion in recent years, most studies have focused on the introduction of new calibration methods that are neither benchmarked against existing alternatives nor rigorously tested in terms of the quality of the estimates they produce. We therefore compare a number of prominent ABM calibration methods, both established and novel, through a series of computational experiments in an attempt to determine the respective strengths and weaknesses of each approach and the overall quality of the resultant parameter estimates. We find that Bayesian estimation, though less popular in the literature, consistently outperforms frequentist, objective function-based approaches and results in reasonable parameter estimates in many contexts. Despite this, we also find that agent-based model calibration techniques require further development in order to definitively calibrate large-scale models. We therefore make suggestions for future research.

This study contributes to understanding Valuation Adjustments (xVA) by focussing on the dynamic hedging of Credit Valuation Adjustment (CVA), corresponding Profit & Loss (P&L) and the P&L explain. This is done in a Monte Carlo simulation setting, based on a theoretical hedging framework discussed in existing literature. We look at hedging CVA market risk for a portfolio with European options on a stock, first in a Black-Scholes setting, then in a Merton jump-diffusion setting. Furthermore, we analyze the trading business at a bank after including xVAs in pricing. We provide insights into the hedging of derivatives and their xVAs by analyzing and visualizing the cash-flows of a portfolio from a desk structure perspective. The case study shows that not charging CVA at trade inception results in an expected loss. Furthermore, hedging CVA market risk is crucial to end up with a stable trading strategy. In the Black-Scholes setting this can be done using the underlying stock, whereas in the Merton jump-diffusion setting we need to add extra options to the hedge portfolio to properly hedge the jump risk. In addition to the simulation, we derive analytical results that explain our observations from the numerical experiments. Understanding the hedging of CVA helps to deal with xVAs in a practical setting.

In the aftermath of the financial crisis, supervisory authorities have considerably improved their approaches in performing financial stress testing. However, they have received significant criticism by the market participants due to the methodological assumptions and simplifications employed, which are considered as not accurately reflecting real conditions. First and foremost, current stress testing methodologies attempt to simulate the risks underlying a financial institution's balance sheet by using several satellite models, making their integration a really challenging task with significant estimation errors. Secondly, they still suffer from not employing advanced statistical techniques, like machine learning, which capture better the nonlinear nature of adverse shocks. Finally, the static balance sheet assumption, that is often employed, implies that the management of a bank passively monitors the realization of the adverse scenario, but does nothing to mitigate its impact. To address the above mentioned criticism, we introduce in this study a novel approach utilizing deep learning approach for dynamic balance sheet stress testing. Experimental results give strong evidence that deep learning applied in big financial/supervisory datasets create a state of the art paradigm, which is capable of simulating real world scenarios in a more efficient way.

Financial portfolio management is the process of constant redistribution of a fund into different financial products. This paper presents a financial-model-free Reinforcement Learning framework to provide a deep machine learning solution to the portfolio management problem. The framework consists of the Ensemble of Identical Independent Evaluators (EIIE) topology, a Portfolio-Vector Memory (PVM), an Online Stochastic Batch Learning (OSBL) scheme, and a fully exploiting and explicit reward function. This framework is realized in three instants in this work with a Convolutional Neural Network (CNN), a basic Recurrent Neural Network (RNN), and a Long Short-Term Memory (LSTM). They are, along with a number of recently reviewed or published portfolio-selection strategies, examined in three back-test experiments with a trading period of 30 minutes in a cryptocurrency market. Cryptocurrencies are electronic and decentralized alternatives to government-issued money, with Bitcoin as the best-known example of a cryptocurrency. All three instances of the framework monopolize the top three positions in all experiments, outdistancing other compared trading algorithms. Although with a high commission rate of 0.25% in the backtests, the framework is able to achieve at least 4-fold returns in 50 days.

We propose a heterogeneous simultaneous graphical dynamic linear model (H-SGDLM), which extends the standard SGDLM framework to incorporate a heterogeneous autoregressive realised volatility (HAR-RV) model. This novel approach creates a GPU-scalable multivariate volatility estimator, which decomposes multiple time series into economically-meaningful variables to explain the endogenous and exogenous factors driving the underlying variability. This unique decomposition goes beyond the classic one step ahead prediction; indeed, we investigate inferences up to one month into the future using stocks, FX futures and ETF futures, demonstrating its superior performance according to accuracy of large moves, longer-term prediction and consistency over time.

In this paper, finite element method is applied to Leland's model for numerical simulation of option pricing with transaction costs. Spatial finite element models based on P1 and/or P2 elements are formulated in combination with a Crank-Nicolson-type temporal scheme. The temporal scheme is implemented using the Rannacher approach. Examples with several sets of parameter values are presented and compared with finite difference results in the literature. Spatial-temporal mesh-size ratios are observed for controlling the stability of our method. Our results compare favorably with the finite difference results in the literature for the model.

Time-series calibrations often suggest that the GARCH diffusion model could also be a suitable candidate for option (risk-neutral) calibration. But unlike the popular Heston model, it lacks a fast, semi-analytic solution for the pricing of vanilla options, perhaps the main reason why it is not used in this way. In this paper we show how an efficient finite difference-based PDE solver can effectively replace analytical solutions, enabling accurate option calibrations in less than a minute. The proposed pricing engine is shown to be robust under a wide range of model parameters and combines smoothly with black-box optimizers. We use this approach to produce a first PDE calibration of the GARCH diffusion model to SPX options and present some benchmark results for future reference.

One popular approach to option pricing in Lévy models is through solving the related partial integro differential equation (PIDE). For the numerical solution of such equations powerful Galerkin methods have been put forward e.g. by Hilber et al. (2013). As in practice large classes of models are maintained simultaneously, flexibility in the driving Lévy model is crucial for the implementation of these powerful tools. In this article we provide such a flexible finite element Galerkin method. To this end we exploit the Fourier representation of the infinitesimal generator, i.e. the related symbol, which is explicitly available for the most relevant Lévy models. Empirical studies for the Merton, NIG and CGMY model confirm the numerical feasibility of the method.