Quantitative Finance

In this note we derive the backward (automatic) differentiation (adjoint [automatic] differentiation) for an algorithm containing a conditional expectation operator. As an example we consider the backward algorithm as it is used in Bermudan product valuation, but the method is applicable in full generality. The method relies on three simple properties: 1) a forward or backward (automatic) differentiation of an algorithm containing a conditional expectation operator results in a linear combination of the conditional expectation operators; 2) the differential of an expectation is the expectation of the differential d dx E(Y)=E( d dx Y) ; 3) if we are only interested in the expectation of the final result (as we are in all valuation problems), we may use E(A⋅E(B|F))=E(E(A|F)⋅B) , i.e., instead of applying the (conditional) expectation operator to a function of the underlying random variable (continuation values), it may be applied to the adjoint differential. \end{enumerate} The methodology not only allows for a very clean and simple implementation, but also offers the ability to use different conditional expectation estimators in the valuation and the differentiation.

This paper is concerned with the process of risk allocation for a generic multivariate model when the risk measure is chosen as the Value-at-Risk (VaR). Making use of Malliavin calculus, we recast the traditional Euler contributions from an expectation conditional to an event of zero probability to a ratio of conditional expectations, where both the numerator and the denominator's conditioning events have positive probability. For several different models we show empirically that the estimator using this novel representation has no perceivable bias and variance smaller than a standard estimator used in practice.

We showcase how dropout variational inference can be applied to a large-scale deep learning model that predicts price movements from limit order books (LOBs), the canonical data source representing trading and pricing movements. We demonstrate that uncertainty information derived from posterior predictive distributions can be utilised for position sizing, avoiding unnecessary trades and improving profits. Further, we test our models by using millions of observations across several instruments and markets from the London Stock Exchange. Our results suggest that those Bayesian techniques not only deliver uncertainty information that can be used for trading but also improve predictive performance as stochastic regularisers. To the best of our knowledge, we are the first to apply Bayesian networks to LOBs.

In this paper, we present a backward deep BSDE method applied to Forward Backward Stochastic Differential Equations (FBSDE) with given terminal condition at maturity that time-steps the BSDE backwards. We present an application of this method to a nonlinear pricing problem - the differential rates problem. To time-step the BSDE backward, one needs to solve a nonlinear problem. For the differential rates problem, we derive an exact solution of this time-step problem and a Taylor-based approximation. Previously backward deep BSDE methods only treated zero or linear generators. While a Taylor approach for nonlinear generators was previously mentioned, it had not been implemented or applied, while we apply our method to nonlinear generators and derive details and present results. Likewise, previously backward deep BSDE methods were presented for fixed initial risk factor values X 0 only, while we present a version with random X 0 and a version that learns portfolio values at intermediate times as well. The method is able to solve nonlinear FBSDE problems in high dimensions.

In the wake of the still ongoing global financial crisis, bank interdependencies have come into focus in trying to assess linkages among banks and systemic risk. To date, such analysis has largely been based on numerical data. By contrast, this study attempts to gain further insight into bank interconnections by tapping into financial discourse. We present a text-to-network process, which has its basis in co-occurrences of bank names and can be analyzed quantitatively and visualized. To quantify bank importance, we propose an information centrality measure to rank and assess trends of bank centrality in discussion. For qualitative assessment of bank networks, we put forward a visual, interactive interface for better illustrating network structures. We illustrate the text-based approach on European Large and Complex Banking Groups (LCBGs) during the ongoing financial crisis by quantifying bank interrelations and centrality from discussion in 3M news articles, spanning 2007Q1 to 2014Q3.

Using data on 17 listed public banks from Russia over the period 2008 to 2016, we analyze whether international oil prices affect the bank stability in an oil-dependent country. We posit that a decrease in international oil prices has a negative long-run macroeconomic impact for an oil-exporting country, which further deteriorates the bank financial stability. More specifically, a decrease in international oil prices leads for an oil-exporting country as Russia to a currency depreciation and to a deterioration of the fiscal stance. In addition, given the positive correlation of oil and stock prices documented by numerous previous studies, a decrease in international oil prices represents a negative signal for the stock markets investors, negatively affecting banks' share prices and thus, their capacity to generate sustainable earnings. In this context, the bank financial stability can be menaced. With a focus on public listed banks and using a Pool Mean Group (PMG) estimator, we show that an increase in international oil prices and in the price to book value ratio has a long-run positive effect on Russian public banks stability, and conversely. While positive oil-price shocks contribute to bank stability in the long run, an opposite effect is recorded for negative shocks. However, no significant impact is documented in the short run. Our findings are robust to different bank stability specifications, different samples and control variables.

We refine the analysis of hedging strategies for options under the SABR model carried out in [2]. In particular, we provide a theoretical justification of the empirical observation made in [2] that the modified delta ("Bartlett's delta") introduced there provides a more accurate and robust hedging strategy than the conventional SABR delta hedge.

The hybrid Monte Carlo algorithm (HMCA) is applied for Bayesian parameter estimation of the realized stochastic volatility (RSV) model. Using the 2nd order minimum norm integrator (2MNI) for the molecular dynamics (MD) simulation in the HMCA, we find that the 2MNI is more efficient than the conventional leapfrog integrator. We also find that the autocorrelation time of the volatility variables sampled by the HMCA is very short. Thus it is concluded that the HMCA with the 2MNI is an efficient algorithm for parameter estimations of the RSV model.

The aim of this study is to devise numerical methods for dealing with very high-dimensional Bermudan-style derivatives. For such problems, we quickly see that we can at best hope for price bounds, and we can only use a simulation approach. We use the approach of Barraquand & Martineau which proposes that the reward process should be treated as if it were Markovian, and then uses this to generate a stopping rule and hence a lower bound on the price. Using the dual approach introduced by Rogers, and Haugh & Kogan, this approximate Markov process leads us to hedging strategies, and upper bounds on the price. The methodology is generic, and is illustrated on eight examples of varying levels of difficulty. Run times are largely insensitive to dimension.

The purpose of this research paper it is to present a new approach in the framework of a biased roulette wheel. It is used the approach of a quantitative trading strategy, commonly used in quantitative finance, in order to assess the profitability of the strategy in the short term. The tools of backtesting and walk-forward optimization were used to achieve such task. The data has been generated from a real European roulette wheel from an on-line casino based in Riga, Latvia. It has been recorded 10,980 spins and sent to the computer through a voice-to-text software for further numerical analysis in R. It has been observed that the probabilities of occurrence of the numbers at the roulette wheel follows an Ornstein-Uhlenbeck process. Moreover, it is shown that a flat betting system against Kelly Criterion was more profitable in the short term.