Page 1. Acceleration of Market Value-at-Risk Estimation Matthew Dixon Department of Computer Scie... more Page 1. Acceleration of Market Value-at-Risk Estimation Matthew Dixon Department of Computer Science 3060 Kemper Hall UC Davis, CA 95616 mfdixon@ucdavis. edu Jike Chong Department of Electrical Engineering and ...
Deep learning for option pricing has emerged as a novel methodology for fast computations with ap... more Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.
Deep learning applies hierarchical layers of hidden variables to construct nonlinear high dimensi... more Deep learning applies hierarchical layers of hidden variables to construct nonlinear high dimensional predictors. Our goal is to develop and train deep learning architectures for spatio-temporal modeling. Training a deep architecture is achieved by stochastic gradient descent (SGD) and drop-out (DO) for parameter regularization with a goal of minimizing out-of-sample predictive mean squared error. To illustrate our methodology, we predict the sharp discontinuities in traffic flow data, and secondly, we develop a classification rule to predict short-term futures market prices as a function of the order book depth. Finally, we conclude with directions for future research.
We propose a simple non-equilibrium model of a financial market as an open system with a possible... more We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the "Quantum Equilibrium-Disequilibrium" (QED) model. The QED model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, "classical" GBM scal...
Modeling counterparty risk is computationally challenging because it requires the simultaneous ev... more Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all the trades with each counterparty under both market and credit risk. We present a multi-Gaussian process regression approach, which is well suited for OTC derivative portfolio valuation involved in CVA computation. Our approach avoids nested simulation or simulation and regression of cash flows by learning a Gaussian metamodel for the mark-to-market cube of a derivative portfolio. We model the joint posterior of the derivatives as a Gaussian process over function space, with the spatial covariance structure imposed on the risk factors. Monte-Carlo simulation is then used to simulate the dynamics of the risk factors. The uncertainty in portfolio valuation arising from the Gaussian process approximation is quantified numerically. Numerical experiments demonstrate the accuracy and convergence properties of our approach for CVA computations.
A key challenge for Bitcoin cryptocurrency holders, such as startups using ICOs to raise funding,... more A key challenge for Bitcoin cryptocurrency holders, such as startups using ICOs to raise funding, is managing their FX risk. Specifically, a misinformed decision to convert Bitcoin to fiat currency could, by itself, cost USD millions. In contrast to financial exchanges, Blockchain based crypto-currencies expose the entire transaction history to the public. By processing all transactions, we model the network with a high fidelity graph so that it is possible to characterize how the flow of information in the network evolves over time. We demonstrate how this data representation permits a new form of microstructure modeling - with the emphasis on the topological network structures to study the role of users, entities and their interactions in formation and dynamics of crypto-currency investment risk. In particular, we identify certain sub-graphs ('chainlets') that exhibit predictive influence on Bitcoin price and volatility, and characterize the types of chainlets that signify...
The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the a... more The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by AUG84 who discretized Hamilton's action principle with a free-Lagrange data structure. The purpose of this article is to assess the long-time conservation properties of the VFL method for regularized shallow water which are useful for climate simulation. Long-time regularized shallow water simulations show that the VFL method exhibits no secular drift in the (i) energy error through the application of symplectic integrators; and (ii) the potential vorticity error through the construction of discrete curl, divergence and gradient operators which satisfy semi-discrete divergence and potential vorticity conservation laws. These diagnostic semi-discrete equations augment the description of the VFL method by characterizing the evolution of its respective irr...
The era of modern financial data modeling seeks machine learning techniques which are suitable fo... more The era of modern financial data modeling seeks machine learning techniques which are suitable for noisy and non-stationary big data. We demonstrate how a general class of exponential smoothed recurrent neural networks (α-RNNs) are well suited to modeling dynamical systems arising in big data applications such as high frequency and algorithmic trading. Application of exponentially smoothed RNNs to minute level Bitcoin prices and CME futures tick data, highlight the efficacy of exponential smoothing for multi-step time series forecasting. Our α-RNNs are also compared with more complex, “black-box”, architectures such as GRUs and LSTMs and shown to provide comparable performance, but with far fewer model parameters and network complexity.
arXiv: Exactly Solvable and Integrable Systems, 2006
The body and spatial representations of rigid body motion correspond, respectively, to the convec... more The body and spatial representations of rigid body motion correspond, respectively, to the convective and spatial representations of continuum dynamics. With a view to developing a unified computational approach for both types of problems, the discrete Clebsch approach of Cotter and Holm for continuum mechanics is applied to derive (i) body and spatial representations of discrete time models of various rigid body motions and (ii) the discrete momentum maps associated with symmetry reduction for these motions. For these problems, this paper shows that the discrete Clebsch approach yields a known class of explicit variational integrators, called discrete Moser-Veselov (DMV) integrators. The spatial representation of DMV integrators are Poisson with respect to a Lie-Poisson bracket for the semi-direct product Lie algebra. Numerical results are presented which confirm the conservative properties and accuracy of the numerical solutions.
This paper investigates the Data for Development (D4D) challenge [3], an open challenge set by th... more This paper investigates the Data for Development (D4D) challenge [3], an open challenge set by the French mobile phone company, Orange, who have provided anonymized records of their customers in the Ivory Coast. This data spans a 5 month (150 day) horizon spread across 4 different sets containing antenna-to-antenna traffic, trace data for 50,000 customers at varying spatial resolution, and social graphs for 5,000 customers. By leveraging cloud-based and open-source analytics infrastructure to (1) merge the D4D datasets with Geographic Information System (GIS) data and (2) apply data mining algorithms, this paper presents a number of techniques for detecting mobility patterns of Orange customers in the Ivory Coast. By applying a k-medoid clustering algorithm to the antenna locations and their average distance to nearby antennas, we show how the high spatial resolution mobile phone dataset reveals a number of daily mobility patterns and properties, including trends in week-day versus ...
It is well known that the Black-Scholes model fails to accurately predict option prices. This fai... more It is well known that the Black-Scholes model fails to accurately predict option prices. This failure can be parametrized by the so-called volatility surface. To do this one observes the option price, strike, stock price, time to maturity, and interest rate and then inverts the Black-Scholes equation to compute the implied volatility. The Black-Scholes model assumes that the volatility is a fixed property of the underlying asset. If this assumption were correct then the volatility surface would be a constant across options and static in time. However, implied volatilities are observed to vary both with strike and time to maturity. Furthermore, the surface fluctuates in time. The cross-sections of the volatility surface that vary with strike are known as smiles or smirks, because of their characteristic shape. The cross-setions that vary with maturity are known as the term structure. The smile is believed to be a consequence of market participants adjusting for large stock movements,...
Deep fundamental factor models are developed to automatically capture non-linearity and interacti... more Deep fundamental factor models are developed to automatically capture non-linearity and interaction effects in factor modeling. Uncertainty quantification provides interpretability with interval estimation, ranking of factor importances and estimation of interaction effects. With no hidden layers we recover a linear factor model and for one or more hidden layers, uncertainty bands for the sensitivity to each input naturally arise from the network weights. Using 3290 assets in the Russell 1000 index over a period of December 1989 to January 2018, we assess a 49 factor model and generate information ratios that are approximately 1.5x greater than the OLS factor model. Furthermore, we compare our deep fundamental factor model with a quadratic LASSO model and demonstrate the superior performance and robustness to outliers. The Python source code and the data used for this study are provided.
The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the a... more The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by \cite{AUG84} who discretized Hamilton's action principle with a free-Lagrange data structure. The purpose of this article is to assess the long-time conservation properties of the VFL method for regularized shallow water which are useful for climate simulation. Long-time regularized shallow water simulations show that the VFL method exhibits no secular drift in the (i) energy error through the application of symplectic integrators; and (ii) the potential vorticity error through the construction of discrete curl, divergence and gradient operators which satisfy semi-discrete divergence and potential vorticity conservation laws. These diagnostic semi-discrete equations augment the description of the VFL method by characterizing the evolution of its respect...
Modeling counterparty risk is computationally challenging because it requires the simultaneous ev... more Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all the trades with each counterparty under both market and credit risk. We present a multi-Gaussian process regression for estimating portfolio risk, which is well suited for OTC derivative portfolios, in particular CVA computation. Our spatiotemporal modeling approach avoids nested MC simulation by learning a ’kernel pricing layer’. The pricing layer is flexible we model the joint posterior of the derivatives as a Gaussian over function space, with the spatial covariance structure imposed only on the risk factors. Monte-Carlo (MC) simulation is then used to simulate the dynamics of the risk factors. Our approach quantifies uncertainty in portfolio risk arising from uncertainty in point estimates. Numerical experiments demonstrate the accuracy and convergence properties of our approach for CVA estimation.
Page 1. Acceleration of Market Value-at-Risk Estimation Matthew Dixon Department of Computer Scie... more Page 1. Acceleration of Market Value-at-Risk Estimation Matthew Dixon Department of Computer Science 3060 Kemper Hall UC Davis, CA 95616 mfdixon@ucdavis. edu Jike Chong Department of Electrical Engineering and ...
Deep learning for option pricing has emerged as a novel methodology for fast computations with ap... more Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.
Deep learning applies hierarchical layers of hidden variables to construct nonlinear high dimensi... more Deep learning applies hierarchical layers of hidden variables to construct nonlinear high dimensional predictors. Our goal is to develop and train deep learning architectures for spatio-temporal modeling. Training a deep architecture is achieved by stochastic gradient descent (SGD) and drop-out (DO) for parameter regularization with a goal of minimizing out-of-sample predictive mean squared error. To illustrate our methodology, we predict the sharp discontinuities in traffic flow data, and secondly, we develop a classification rule to predict short-term futures market prices as a function of the order book depth. Finally, we conclude with directions for future research.
We propose a simple non-equilibrium model of a financial market as an open system with a possible... more We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the "Quantum Equilibrium-Disequilibrium" (QED) model. The QED model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, "classical" GBM scal...
Modeling counterparty risk is computationally challenging because it requires the simultaneous ev... more Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all the trades with each counterparty under both market and credit risk. We present a multi-Gaussian process regression approach, which is well suited for OTC derivative portfolio valuation involved in CVA computation. Our approach avoids nested simulation or simulation and regression of cash flows by learning a Gaussian metamodel for the mark-to-market cube of a derivative portfolio. We model the joint posterior of the derivatives as a Gaussian process over function space, with the spatial covariance structure imposed on the risk factors. Monte-Carlo simulation is then used to simulate the dynamics of the risk factors. The uncertainty in portfolio valuation arising from the Gaussian process approximation is quantified numerically. Numerical experiments demonstrate the accuracy and convergence properties of our approach for CVA computations.
A key challenge for Bitcoin cryptocurrency holders, such as startups using ICOs to raise funding,... more A key challenge for Bitcoin cryptocurrency holders, such as startups using ICOs to raise funding, is managing their FX risk. Specifically, a misinformed decision to convert Bitcoin to fiat currency could, by itself, cost USD millions. In contrast to financial exchanges, Blockchain based crypto-currencies expose the entire transaction history to the public. By processing all transactions, we model the network with a high fidelity graph so that it is possible to characterize how the flow of information in the network evolves over time. We demonstrate how this data representation permits a new form of microstructure modeling - with the emphasis on the topological network structures to study the role of users, entities and their interactions in formation and dynamics of crypto-currency investment risk. In particular, we identify certain sub-graphs ('chainlets') that exhibit predictive influence on Bitcoin price and volatility, and characterize the types of chainlets that signify...
The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the a... more The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by AUG84 who discretized Hamilton's action principle with a free-Lagrange data structure. The purpose of this article is to assess the long-time conservation properties of the VFL method for regularized shallow water which are useful for climate simulation. Long-time regularized shallow water simulations show that the VFL method exhibits no secular drift in the (i) energy error through the application of symplectic integrators; and (ii) the potential vorticity error through the construction of discrete curl, divergence and gradient operators which satisfy semi-discrete divergence and potential vorticity conservation laws. These diagnostic semi-discrete equations augment the description of the VFL method by characterizing the evolution of its respective irr...
The era of modern financial data modeling seeks machine learning techniques which are suitable fo... more The era of modern financial data modeling seeks machine learning techniques which are suitable for noisy and non-stationary big data. We demonstrate how a general class of exponential smoothed recurrent neural networks (α-RNNs) are well suited to modeling dynamical systems arising in big data applications such as high frequency and algorithmic trading. Application of exponentially smoothed RNNs to minute level Bitcoin prices and CME futures tick data, highlight the efficacy of exponential smoothing for multi-step time series forecasting. Our α-RNNs are also compared with more complex, “black-box”, architectures such as GRUs and LSTMs and shown to provide comparable performance, but with far fewer model parameters and network complexity.
arXiv: Exactly Solvable and Integrable Systems, 2006
The body and spatial representations of rigid body motion correspond, respectively, to the convec... more The body and spatial representations of rigid body motion correspond, respectively, to the convective and spatial representations of continuum dynamics. With a view to developing a unified computational approach for both types of problems, the discrete Clebsch approach of Cotter and Holm for continuum mechanics is applied to derive (i) body and spatial representations of discrete time models of various rigid body motions and (ii) the discrete momentum maps associated with symmetry reduction for these motions. For these problems, this paper shows that the discrete Clebsch approach yields a known class of explicit variational integrators, called discrete Moser-Veselov (DMV) integrators. The spatial representation of DMV integrators are Poisson with respect to a Lie-Poisson bracket for the semi-direct product Lie algebra. Numerical results are presented which confirm the conservative properties and accuracy of the numerical solutions.
This paper investigates the Data for Development (D4D) challenge [3], an open challenge set by th... more This paper investigates the Data for Development (D4D) challenge [3], an open challenge set by the French mobile phone company, Orange, who have provided anonymized records of their customers in the Ivory Coast. This data spans a 5 month (150 day) horizon spread across 4 different sets containing antenna-to-antenna traffic, trace data for 50,000 customers at varying spatial resolution, and social graphs for 5,000 customers. By leveraging cloud-based and open-source analytics infrastructure to (1) merge the D4D datasets with Geographic Information System (GIS) data and (2) apply data mining algorithms, this paper presents a number of techniques for detecting mobility patterns of Orange customers in the Ivory Coast. By applying a k-medoid clustering algorithm to the antenna locations and their average distance to nearby antennas, we show how the high spatial resolution mobile phone dataset reveals a number of daily mobility patterns and properties, including trends in week-day versus ...
It is well known that the Black-Scholes model fails to accurately predict option prices. This fai... more It is well known that the Black-Scholes model fails to accurately predict option prices. This failure can be parametrized by the so-called volatility surface. To do this one observes the option price, strike, stock price, time to maturity, and interest rate and then inverts the Black-Scholes equation to compute the implied volatility. The Black-Scholes model assumes that the volatility is a fixed property of the underlying asset. If this assumption were correct then the volatility surface would be a constant across options and static in time. However, implied volatilities are observed to vary both with strike and time to maturity. Furthermore, the surface fluctuates in time. The cross-sections of the volatility surface that vary with strike are known as smiles or smirks, because of their characteristic shape. The cross-setions that vary with maturity are known as the term structure. The smile is believed to be a consequence of market participants adjusting for large stock movements,...
Deep fundamental factor models are developed to automatically capture non-linearity and interacti... more Deep fundamental factor models are developed to automatically capture non-linearity and interaction effects in factor modeling. Uncertainty quantification provides interpretability with interval estimation, ranking of factor importances and estimation of interaction effects. With no hidden layers we recover a linear factor model and for one or more hidden layers, uncertainty bands for the sensitivity to each input naturally arise from the network weights. Using 3290 assets in the Russell 1000 index over a period of December 1989 to January 2018, we assess a 49 factor model and generate information ratios that are approximately 1.5x greater than the OLS factor model. Furthermore, we compare our deep fundamental factor model with a quadratic LASSO model and demonstrate the superior performance and robustness to outliers. The Python source code and the data used for this study are provided.
The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the a... more The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by \cite{AUG84} who discretized Hamilton's action principle with a free-Lagrange data structure. The purpose of this article is to assess the long-time conservation properties of the VFL method for regularized shallow water which are useful for climate simulation. Long-time regularized shallow water simulations show that the VFL method exhibits no secular drift in the (i) energy error through the application of symplectic integrators; and (ii) the potential vorticity error through the construction of discrete curl, divergence and gradient operators which satisfy semi-discrete divergence and potential vorticity conservation laws. These diagnostic semi-discrete equations augment the description of the VFL method by characterizing the evolution of its respect...
Modeling counterparty risk is computationally challenging because it requires the simultaneous ev... more Modeling counterparty risk is computationally challenging because it requires the simultaneous evaluation of all the trades with each counterparty under both market and credit risk. We present a multi-Gaussian process regression for estimating portfolio risk, which is well suited for OTC derivative portfolios, in particular CVA computation. Our spatiotemporal modeling approach avoids nested MC simulation by learning a ’kernel pricing layer’. The pricing layer is flexible we model the joint posterior of the derivatives as a Gaussian over function space, with the spatial covariance structure imposed only on the risk factors. Monte-Carlo (MC) simulation is then used to simulate the dynamics of the risk factors. Our approach quantifies uncertainty in portfolio risk arising from uncertainty in point estimates. Numerical experiments demonstrate the accuracy and convergence properties of our approach for CVA estimation.
With the proliferation of algorithmic trading, derivative usage and highly leveraged hedge funds,... more With the proliferation of algorithmic trading, derivative usage and highly leveraged hedge funds, there is increasing need to accelerate financial market Value-at-Risk (VaR) estimation to measure the severity of potential portfolio losses in real time. However, VaR estimation of portfolios uses the Monte Carlo method, which is a computationally intensive method. GPUs provide the scale of performance improvement to enable " on demand " deployment of financial market VaR estimates rather than as an overnight batch job. This chapter allows quantitative financial application developers in the capital markets industry, who have some knowledge of GPU computing and finance, to gain insights into implementation challenges and solutions in risk analysis and the Monte Carlo method. Quantitative technology researchers and managers in the finance industry with limited knowledge of GPU computing can also get an overview of the key areas of concern in developing a high performance risk analysis engine based on the Monte Carlo method. GPU computing researchers and developers with no background in quantitative finance will find this chapter useful as (i) a source of guidance on leveraging the CUDA SDK for implementing Monte Carlo methods and as (ii) an entry point for applying their own work to performance critical quantitative finance applications. 25.1 INTRODUCTION, PROBLEM STATEMENT, AND CONTEXT Financial institutions seek quantitative risk infrastructure which is able to provide " on demand " reporting of global financial market risk while managing thousands of risk factors and considering hundreds of thousands of future market scenarios [16]. A fast implementation can improve the responsiveness of risk management systems, enable risk analysts to perform more comprehensive VaR estimates on an adhoc basis (e.g., pre-deal limit checking), and can give financial institutions a competitive edge in algorithmic trading. It will also allow institutions to effectively step up to the more pervasive systematic stress testing standards recently imposed by market risk regulators. The ability to closely monitor and control an institution's market risk exposure is critical to the performance of trading units, which rely on complex risk analysis in order to structure and execute trades. Yet, despite tightening legal requirements on market risk estimation, the industry is still far from adopting a standardized and comprehensive framework for market risk analysis. As a result, GPU Computing Gems
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