Continuous-time Finance Model Research Articles

Machine Learning for Continuous-Time Finance

Abstract We develop an algorithm for solving a large class of nonlinear high-dimensional continuous-time models in finance. We approximate value and policy functions using deep learning and show that a combination of automatic differentiation and Ito’s lemma allows for the computation of exact expectations, resulting in a negligible computational cost that is independent of the number of state variables. We illustrate the applicability of our method to problems in asset pricing, corporate finance, and portfolio choice and show that the ability to solve high-dimensional problems allows us to derive new economic insights.

Macro, Finance, and Macro Finance: Solving Nonlinear Models in Continuous Time with Machine Learning

This paper proposes a global algorithm to solve a large class of nonlinear continuous-time models in finance and economics. Using tools from machine learning, I recast problem of solving the corresponding nonlinear partial differential equations as a sequence of supervised learning problems. To illustrate the scope of the method, I solve nontrivial benchmark models and compare the numerical solution with the analytical ones. Furthermore, I propose a setting to test and evaluate solution methods. In the context of a neoclassical growth model, given any value function, the productivity function is reverse engineered so that the Hamilton-Jacobi-Bellman equation corresponding to the optimization problem is identically zero. This provides a testing ground for solution methods and an objective way of comparing them. Results indicate that the method is accurate and can handle nonlinear models with as many as 10 dimensions. Finally, I provide an open source library that implements the proposed algorithm.

Generalized spectral testing for multivariate continuous-time models

We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed-form or can be accurately approximated for many multivariate continuous-time models in finance and economics. The proposed test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Our tests are also applicable to discrete-time distribution models. Simulation studies show that the tests provide reliable inference in finite samples.

Investor Protection and Ownership Decentralization

On the premise that the expropriation of corporate assets by controlling shareholders would generate a residual loss, we studied how the laws of investor protection influenced on the decentralization of corporate ownership and what role the litigation cost play in this story by continuous time finance model. There are several interested conclusions we reached. At first, the decentralization of ownership is the result of the effective legal protection of minority shareholders’ rights and interest, and this result is irrelevant to whether there is a litigation cost or who pays for this cost. Second, effective investor protection leads to lower private benefits of control. Third, our results demonstrated that an efficient investor protection system would evolve through a self-reinforcing pathway. This means that if the law can effectively protect the minority shareholders, then this protection will become increasingly strong. In contrast, if the investor protection system is ineffective, then this protection system will be practically dead in the end. Fourth, the conclusions about the judicial procedure of how to share the litigation cost means that if penalty remain unchanged, an effective investor protection would lead to the increase of litigation costs over time, while an ineffective investor protection would lead to a decrease over time. This result is independent of who will pay the cost. Finally, our results showed that if the minority shareholders within the legal timeliness of the investor protection law pay the litigation cost, an increase in the penalty rate would lead to an increase in litigation costs. The opposite result we obtain if the controlling shareholders pay the litigation cost; this will lead to a negative correlation between the penalty rate and litigation costs.

Estimating and Testing Continuous-Time Models in Finance: The Role of Transition Densities

This article surveys recent developments to estimate and test continuous-time models in finance using discrete observations on the underlying asset price or derivative securities' prices. Both parametric and nonparametric methods are described. All these methods share a common focus on the transition density as the central object for inference and testing of the model.

Market clearing, utility functions, and securities prices

We prove the existence of equilibrium in a continuous-time finance model; our results include the case of dynamically incomplete markets as well as dynamically complete markets. In addition, we derive explicitly the stochastic process describing securities prices. The price process depends on the risk-aversion characteristics of the utility function, as well as on the presence of additional sources of wealth (including endowments and other securities). With a single stock, zero endowment in the terminal period, and Constant Relative Risk Aversion (CRRA) utility, the price process is geometric Brownian motion; in essentially any other situation, the price process is not a geometric Brownian motion.

Diagnosing Multivariate Continuous-Time Models with Application to Affine Term Structure Models

Multivariate continuous-time models have been playing important roles in finance and economics. We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed form or can be accurately approximated for many multivariate continuous-time models in finance and economics. Unlike the existing methods in the literature, the proposed omnibus test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in finance and economics. The omnibus test has all-round power against various model misspecifications and the diagnostic tests can reveal useful information about the nature and type of model misspecification. In an application, we find that there is room for further improving upon a popular class of multivariate affine term structure models for monthly U.S. Treasury bond yields. It is documented that while there exists little dynamic structure in conditional means, there exist neglected dynamic structures in conditional variances and conditional correlations of yields with different maturities.

Market clearing and derivative pricing

Summary. We develop a method of assigning unique prices to derivative securities, including options, in the continuous-time finance model developed in Raimondo (2001). In contrast with the martingale method of valuing options, which cannot distinguish among infinitely many possible option pricing processes for a given underlying securities price process when markets are dynamically incomplete, our option prices are uniquely determined in equilibrium in closed form as a function of the underlying economic data.

Modelling Stock Market Volatility: Bridging the Gap to Continuous Time

Understanding And Specifying The Discrete Time Model: D.B. Nelson, Modelling Stock Market Volatility Changes. D.B. Nelson, Stationarity and Persistence in the GARCH(I,I) Model. D.B. Nelson, Conditional Heteroskedasticity in Asset Returns: A New Approach. P.A. Braun, D.B. Nelson and A.M. Sunier, Good News, Bad News, Volatility, and Betas. Continuous Time Limits And Optimal Filtering For ARCH Models: D.B. Nelson, ARCH Models as Diffusion Approximations. D.B. Nelson, Filtering and Forecasting with Misspecified ARCH Models I: Getting the Right Variance with the Wrong Model. D.B. Nelson and D.P. Foster, Filtering and Forecasting with Misspecified ARCH Models II: Making the Right Forecast with the Wrong Model. D.B. Nelson and D.P. Foster, Asymptotic Filtering Theory for Univariate ARCH Models. D.B. Nelson, Asymptotic Filtering Theory for Multivariate ARCH Models. D.B. Nelson and D.B. Nelson, Continuous Record Asymptotics for Rolling Sample Variance Estimators. Specification and Estimation of Continuous Time Processes: R.F. Engle and G.G.J. Lee, Estimating Diffusion Models of Stochastic Volatility. A.R. Gallant and G. Tauchen, Specification Analysis of Continuous Time Models in Finance. L.P. Hansen and J.A. Scheinkman, Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes. Y.Ait-Sahalia, Nonparametric Pricing of Interest Rate Derivative Securities. Index.

Consumption and portfolio turnpike theorems in a continuous-time finance model

Abstract This paper investigates consumption and portfolio turnpike theorems in a continuous-time model. When the inverse functions of the derivative of utility functions for consumption and investment belong to a special subclass of regularly varying functions, it is shown that optimum portfolio, final wealth and consumption processes for these utility functions can be approximated arbitrarily closely in a suitable sense by those for the corresponding power utility functions. As an immediate consequence, the consumption and investment turnpike theorem is established. Conversely, it is shown that the sufficient condition is also necessary for the turnpike property. Our results generalize those of Cox and Huang (1992) .

Existence and uniqueness of optimal consumption and portfolio rules in a continuous-time finance model with habit formation and without short sales

This paper studies the solution to optimal consumption and portfolio rules in a continuous-time finance model in which short sales are prohibited. In this model the preference of the agent exhibits habit formation and consumption durability which captures the effect of past consumption choices on current utility. It is demonstrated that any continuous, non-decreasing and concave utility function dominated by a linear function admits a solution. Furthermore, when the utility function is strictly concave, the optimal solution is unique.

Specification Analysis of Continuous Time Models in Finance

The paper describes the use of the Gallant-Tauchen efficient method of moments (EMM) technique for diagnostic checking of stochastic differential equations (SDEs) estimated from financial market data. The EMM technique is a simulation-based method that uses the score function of an auxiliary model as the criterion to define a generalized method of moments (GMM) objective function. The technique can handle multivariate SDEs where the state vector is not completely observed. The optimized GMM objective function is distributed as chi-square and may be used to test model adequacy. Elements of the score function correspond to specific parameters and large values reflect features of data that a rejected SDE specification does not describe well. The diagnostics are illustrated by estimating a three-factor model to weekly, 1962-1995, term structure data comprised of short (3 month), medium (12 month), and long (10 year) Treasury rates. The Yield-Factor Model is sharply rejected, although an extension that permits the local variance function to be a convex function of the interest rates comes much closer to describing the data.

Security Markets: Stochastic Models

Most modern financial research can be characterized as theoretical or empirical, or a mix of the two. Theoretical finance can be divided in the way that theoretical economics is. Neoclassical research stems from the neoclassical economics of perfectly competitive markets that have no imperfections or frictions, such, as taxes, transaction costs, externalities, or information asymmetries. By definition, imperfect markets research includes all other topics, such as the study of taxes, institutions, and topics stemming from information economics. Darrell Duffie's new book is an introduction to neoclassical finance that emphasizes topics in the intersection of theoretical finance and the mathematical economics of general equilibrium. The book is designed to take the student from first principles to the frontiers of finance-oriented general equilibrium theory. The book contains a rigorous introduction to the necessary topics in probability and stochastic control and is a good source for that material. It also contains a good summary of general equilibrium theory, including the required background mathematics. Each theorem from the general equilibrium literature is presented in a form that is general enough to be applicable to continuous-time models in finance but is as simple as possible given that level of generality. Other variations are mentioned in the literature reviews at the end of each section.