Mean-reverting Stochastic Process Research Articles

Network dynamical stability analysis reveals key “mallostatic” natural variables that erode homeostasis and drive age-related decline of health

Using longitudinal study data, we dynamically model how aging affects homeostasis in both mice and humans. We operationalize homeostasis as a multivariate mean-reverting stochastic process. We hypothesize that biomarkers have stable equilibrium values, but that deviations from equilibrium of each biomarker affects other biomarkers through an interaction network—this precludes univariate analysis. We therefore looked for age-related changes to homeostasis using dynamic network stability analysis, which transforms observed biomarker data into independent “natural” variables and determines their associated recovery rates. Most natural variables remained near equilibrium and were essentially constant in time. A small number of natural variables were unable to equilibrate due to a gradual drift with age in their homeostatic equilibrium, i.e. allostasis. This drift caused them to accumulate over the lifespan course and makes them natural aging variables. Their rate of accumulation was correlated with risk of adverse outcomes: death or dementia onset. We call this tendency for aging organisms to drift towards an equilibrium position of ever-worsening health “mallostasis”. We demonstrate that the effects of mallostasis on observed biomarkers are spread out through the interaction network. This could provide a redundancy mechanism to preserve functioning until multi-system dysfunction emerges at advanced ages.

Foreign exchange trading and management with the stochastic dual dynamic programming method

We present a novel tool for generating speculative and hedging foreign exchange (FX) trading policies. Our solution provides a schedule that determines trades in each rebalancing period based on future currency prices, net foreign account positions, and incoming (outgoing) flows from business operations. To obtain such policies, we construct a multistage stochastic programming (MSP) model and solve it using the stochastic dual dynamic programming (SDDP) numerical method, which specializes in solving high-dimensional MSP models. We construct our methodology within an open-source SDDP package, avoiding implementing the method from scratch. To measure the performance of our policies, we model FX prices as a mean-reverting stochastic process with random events that incorporate stochastic trends. We calibrate this price model on seven currency pairs, demonstrating that our trading policies not only outperform the benchmarks for each currency, but may also be close to ex-post optimal solutions. We also show how the tool can be used to generate more or less conservative strategies by adjusting the risk tolerance, and how it can be used in a variety of contexts and time scales, ranging from intraday speculative trading to monthly hedging for business operations. Finally, we examine the impact of increasing trade policy uncertainty (TPU) levels on our findings. Our findings show that the volatility of currencies from emerging economies rises in comparison to currencies from developed markets. We discover that an increase in the TPU level has no effect on the average profit obtained by our method. However, the risk exposure of the policies increases (decreases) for the group of currencies from emerging (developed) markets.

Bayesian Filtering for Dynamic Anomaly Detection and Tracking

This article presents a Bayesian approach for sequential detection of anomalies in the motion of a target and joint tracking. The anomaly is modeled as a binary (on/off) switching unknown control input that goes into action (begins to exist, or “switches on”) thereby modifying the object dynamics; and by ceasing its activity (becoming nonexistent, or “switching off”) returns the dynamics to nominal. The developed Bayesian framework brings together random finite set (RFS) theory to represent the switching nature of the anomaly, and optimal joint input and state estimation to sequentially update a hybrid state that incorporates a random vector for the kinematic state and a Bernoulli RFS for the unknown control input. In addition, a closed-form solution, the Gaussian-mixture hybrid Bernoulli filter (GM-HBF), has been developed to provide a customized solution for dynamic anomaly detection in the maritime domain characterized by linear Gaussian target dynamics. Based on the Ornstein–Uhlenbeck dynamic model for vessels, where the evolution of the object velocity is governed by a piecewise mean-reverting stochastic process, the anomaly can be represented by a change in the long-run mean velocity (i.e., the unknown control input) from the nominal condition that forces the vessel to deviate from its standard route. We demonstrate the effectiveness of the proposed GM-HBF in both simulated and real-world maritime surveillance applications and test its performance in the face of false measurements, detection uncertainty, and sensor data gaps.

A closed-form pricing formula for European options in an illiquid asset market

This article addresses the problem of pricing European options when the underlying asset is not perfectly liquid. A liquidity discounting factor as a function of market-wide liquidity governed by a mean-reverting stochastic process and the sensitivity of the underlying price to market-wide liquidity is firstly introduced, so that the impact of liquidity on the underlying asset can be captured by the option pricing model. The characteristic function is analytically worked out using the Feynman–Kac theorem and a closed-form pricing formula for European options is successfully derived thereafter. Through numerical experiments, the accuracy of the newly derived formula is verified, and the significance of incorporating liquidity risk into option pricing is demonstrated.

Stochastic Modeling of Stratospheric Temperature

This study suggests a stochastic model for time series of daily zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a combination of a deterministic seasonality function and a Lévy-driven multidimensional Ornstein–Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous-time autoregressive model, meaning that the stratospheric temperature is modeled to be directly dependent on the temperature over four preceding days, while the model’s longer-range memory stems from its recursive nature. This study is based on temperature data from the European Centre for Medium-Range Weather Forecasts ERA-Interim reanalysis model product. The residuals of the autoregressive model are well represented by normal inverse Gaussian-distributed random variables scaled with a time-dependent volatility function. A monthly variability in speed of mean reversion of stratospheric temperature is found, hence suggesting a generalization of the fourth-order continuous-time autoregressive model. A stochastic stratospheric temperature model, as proposed in this paper, can be used in geophysical analyses to improve the understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modeling and prediction of large-scale middle atmospheric events, such as sudden stratospheric warming. Through stratosphere–troposphere coupling, the stratosphere is hence a source of extended tropospheric predictability at weekly to monthly timescales, which is of great importance in several societal and industry sectors.

A closed-form pricing formula for European options with market liquidity risk

In this paper, the impact of liquidity on the underlying asset is taken into account when pricing European options through a discounting factor which depends on two factors, i.e., market liquidity risk modeled as a mean-reverting stochastic process and the sensitivity of the underlying to market liquidity. A closed-form pricing formula for liquidity-adjusted European options is derived in the form of an infinite series using Karhunen–Loève expansion for the Ornstein–Uhlenbeck process. The convergence of the series solution is theoretically proved to guarantee closedness so that market practitioners can adopt the new formula when they need to account in market liquidity risk. The speed of convergence is demonstrated through numerical experiments. Finally, the accuracy of the newly derived formula is shown by comparing option prices calculated with our formula and those obtained from Monte-Carlo simulation, and various properties of our formula are also investigated.

Equilibrium investment strategy for a DC pension plan with learning about stock return predictability

This paper investigates a time-consistent investment strategy under the mean-variance criterion for an investor who accumulates retirement savings through a defined contribution (DC) pension plan with stock and bond investment opportunities. The expected return rate on the stock is modulated by an unobservable predictor which follows a mean-reverting stochastic process. The evolution of the instantaneous interest rate is described by the Vasicek model. In addition, the contribution rate of the DC pension plan is stochastic and correlated with financial risks coming from the stochastic interest rate and stock price. In a game theoretic framework, we derive a closed-form equilibrium investment strategy and corresponding equilibrium value function for the mean-variance criterion by adopting the filtering technique and the stochastic control method. Furthermore, we provide an equilibrium investment strategy and equilibrium value function when the expected return rate of the stock is completely observable. Finally, some numerical examples are presented to demonstrate the sensitivity analysis of the equilibrium investment strategy and equilibrium efficient frontier. Numerical analysis confirms that there is non-negligible information loss on the equilibrium investment strategy and equilibrium value function due to partial observation in the stock price dynamics.

The continuous limit of weak GARCH

GARCH models are called ‘strong’ or ‘weak’ depending on the presence of parametric distributional assumptions for the innovations. The symmetric weak GARCH(1, 1) is the only model in the GARCH class that has been proved to be closed under the temporal aggregation property . This property is fundamental in two respects: (a) for a time-series model to be invariant to the data frequency; and (b) for a unique option-pricing model to exist as a continuous-time limit. While the symmetric weak GARCH(1, 1) is temporally aggregating precisely because it makes no parametric distributional assumptions, the lack of these also makes it harder to derive theoretical results. Rising to this challenge, we prove that its continuous-time limit is a geometric mean-reverting stochastic volatility process with diffusion coefficient governed by a time-varying kurtosis of log returns. When log returns are normal the limit coincides with Nelson’s strong GARCH(1, 1) limit. But unlike strong GARCH models, the weak GARCH(1, 1) has a unique limit because it makes no assumptions about the convergence of model parameters. The convergence of each parameter is uniquely determined by the temporal aggregation property. Empirical results show that the additional time-varying kurtosis parameter enhances both term-structure and smile effects in implied volatilities, thereby affording greater flexibility for the weak GARCH limit to fit real-world data from option prices.

Closed-form analytical solutions for options on agricultural futures with seasonality and stochastic convenience yield

Abstract Stochastic commodity price models play a significant role in the pricing and hedging of commodity derivatives and real assets. This paper proposes a commodity pricing model that extends the Ewald and Ouyang two-factor model by adding a time-varying function into the volatility term of convenience yield process to describe seasonality. The stochastic factors in the model are the spot price of the commodity is assumed to follow geometrical Brownian motion process with a stochastic drift, the instantaneous convenience yield follows a mean-reverting Ornstein-Uhlenbeck stochastic process with time-varying seasonal volatility. The model we propose is analytically tractable and allows us to derive closed-form pricing formulas for futures and options by using the Mellin transform method. In addition, we provide the experiments results to illustrate the important properties of commodity futures options with numerical table and graphs.

Estimating the dynamics of systems with noisy boundaries

In a smooth dynamical system the characteristics of a given reference trajectory can be determined, to lowest order, by examining the linearised system about the reference trajectory. In other words, we can approximate the deviations of trajectories after a given time, with starting points in a neighbourhood of the reference trajectory, by multiplying the initial deviations by the corresponding fundamental matrix solution.This form of analysis cannot be used directly in nonsmooth systems as the vector field is either not everywhere differentiable or the flow function is not continuous. To account for this, one can derive the zero-time discontinuity mapping (ZDM) associated with the discontinuity boundary. The Jacobian of this mapping is known as the saltation matrix and its properties can tell us how the crossing of the discontinuity boundary affects the deviations of trajectories from a reference trajectory. In particular, this matrix can be composed with the fundamental matrix solutions of the individual flows on either side of the discontinuity boundary in order to determine the overall fundamental matrix solution of a trajectory that crosses the boundary.In this paper we derive a saltation matrix for a piecewise-smooth dynamical system in which the position of the discontinuity boundary oscillates according to a mean-reverting stochastic process. The derived saltation matrix contains the entire effect of both the discontinuity and the uncertainty introduced into the system by the noisy boundary, and is composable with the deterministic fundamental matrix solutions of the individual flows to give the overall fundamental matrix solution of a crossing trajectory.We also present some simple examples of piecewise-smooth systems with stochastically varying boundaries, analysed using the derived noisy saltation matrix. In particular we focus on the analysis of a discontinuous variant of the Chua circuit. In this case we apply noise to the system’s discontinuity boundaries which are generated by the piecewise-linear nature of the voltage–current response of the Chua diode. We find that our method allows us to analyse the effects of boundary noise on periodic attractors close to bifurcation points. In particular we show that we can use the method to accurately predict the noise amplitudes required to destroy or merge periodic attractors.

A Stylized Model of the Oil Squeeze

The paper proposes a practical alternative to a traditional theory of storage, which is difficult to apply to oil markets due to structural differences, such as low-price elasticities of demand and supply, and the dominant role of oil as an investment asset. We model futures time spreads as a derivative of the observable state variable, oil inventories. Inventories, normalized by the storage capacity, are assumed to follow a mean-reverting stochastic process. The boundary condition at the futures expiry is specified by a pair of Dirac delta functions which correspond to the events of default by the futures trader when either oil inventory or the storage capacity are no longer available. The model is calibrated to data for Cushing, Oklahoma, the delivery location for WTI futures, and is used to analyze the market dynamics that led to negative oil prices. We argue that while the limitation of storage capacity was the initial catalyst leading to the event, the actual episode of negative prices was the result of the financial squeeze in the futures market.

A primer on the pricing of electric energy options in Brazil via mean-reverting stochastic processes

Pricing option contracts on electricity remains methodologically challenging, with a lack of clearly defined and robust methods. In particular, little is known about pricing options in Brazilian energy markets, despite their economic significance. Using weekly price data (R$/MWh) on four electrical subsystems from the Chamber for Commercialization of Electrical Energy, we estimate models to price Brazilian electricity energy options. This paper has three objectives: (i) to identify the occurrence of change-points (regime-switching) in time series of Brazilian energy spot prices; (ii) to determine the best Stochastic Differential Equation (SDE) with which to model Brazilian energy spot prices and (iii) to price five types of options used to manage electricity price risk in Brazil. We show that the change-point occurred between 2002 and 2018. During this period, the long-run marginal cost of production was the most affected. Furthermore, we find that the Ornstein–Uhlenbeck/Vasicek stochastic process and resulting SDE best explains electricity prices in Brazil, even with the occurrence of structural changes. Finally, our results indicate that Asian-style options are the least costly option contracts to manage electricity price risk in Brazil.

Detecting Anomalous Deviations From Standard Maritime Routes Using the Ornstein–Uhlenbeck Process

A novel anomaly detection procedure based on the Ornstein–Uhlenbeck (OU) mean-reverting stochastic process is presented. The considered anomaly is a vessel that deviates from a planned route, changing its nominal velocity $\boldsymbol{v}_0$ . In order to hide this behavior, the vessel switches off its automatic identification system (AIS) device for a time $T$ and then tries to revert to the previous nominal velocity $\boldsymbol{v}_0$ . The decision that has to be made is declaring that a deviation either happened or not, relying only upon two consecutive AIS contacts. Furthermore, the extension to the scenario in which multiple contacts (e.g., radar) are available during the time period $T$ is also considered. A proper statistical hypothesis testing procedure that builds on the changes in the OU process long-term velocity parameter $\boldsymbol{v}_0$ of the vessel is the core of the proposed approach and enables the solution of the anomaly detection problem. Closed analytical forms are provided for the detection and false alarm probabilities of the hypothesis test.

PRICING SOVEREIGN CONTINGENT CONVERTIBLE DEBT

We develop a pricing model for Sovereign Contingent Convertible bonds (S-CoCo) with payment standstills triggered by a sovereign’s Credit Default Swap (CDS) spread. We model CDS spread regime switching, which is prevalent during crises, as a hidden Markov process, coupled with a mean-reverting stochastic process of spread levels under fixed regimes, in order to obtain S-CoCo prices through simulation. The paper uses the pricing model in a Longstaff–Schwartz American option pricing framework to compute future state contingent S-CoCo prices for risk management. Dual trigger pricing is also discussed using the idiosyncratic CDS spread for the sovereign debt together with a broad market index. Numerical results are reported using S-CoCo designs for Greece, Italy and Germany with both the pricing and contingent pricing models.

The GARCH Linear SDE: Explicit Formulas and the Pricing of a Quanto CDS

We derive an efficient closed-form approximation for the moment generating function of the integral of a mean-reverting stochastic process, following a linear SDE, which we call GARCH. We then consider a financial application, namely the pricing of a quanto CDS under stochastic intensity of default and an FX devaluation model. Numerical results are finally showcased.

R&D Project Valuation Considering Changes of Economic Environment: A Case of a Pharmaceutical R&D Project

R&D project valuation is important for effective R&D portfolio management through decision making, related to the firm’s R&D productivity, sustainable management. In particular, scholars have emphasized the necessities of capturing option value in R&D and developed methods of real option valuation. However, despite suggesting various real option models, there are few studies on simultaneously employing mean-reverting stochastic process and Markov regime switching to describe the evolution of cash flow and to reflect time-varying parameters resulting from changes of economic environment. Therefore, we suggest a mean-reverting binomial lattice model under Markov regime switching and apply it to evaluate clinical development with project cases of the pharmaceutical industry. This study finds that decision making can be different according to the regime condition, thus the suggested model can capture risks caused by the uncertainty of the economic environment, represented by regime switching. Further, this study simulates the model according to rate parameter from 0.00 to 1.00 and risk-free interest rates for regimes 1 and 2 from ( r 1 = 4%, r 2 = 2%) to ( r 1 = 7%, r 2 = 5%), and confirms the rigidity of the model. Therefore, in practice, the mean-reverting binomial lattice model under Markov regime switching proposed in this study for R&D project valuation contributes to assisting company R&D project managers make effective decision making considering current economic environment and future changes.

Deep energy renovation strategies: A real option approach for add-ons in a social housing case study

In this paper a techno-economic evaluation method for deep renovation of buildings is introduced. We study the validation of an integrated design methodology based upon the creation of additional volume on existing buildings (Add-Ons) intended as roof-top or aside extensions, enclosures, parasites or façade transformations. Energy price uncertainty is modelled through a mean-reverting stochastic process, which well fits the data for natural gas prices, and the proposed evaluation method is based on real-option theory, which is an advancement of standard cost-benefit analysis, to be employed whenever uncertainty has a significant impact on the investment option. The new method to evaluate the economical feasibility of the Add-ons strategy for Deep Energy Renovation is tested through design experiments on a real case study of social housing in Reggio Emilia (Italy).

Modeling energy spreads with a generalized novel mean-reverting stochastic process

The spread between two related energy prices is a very important quantity throughout energy finance. Of particular interest are spreads between different energy types, different delivery points (location spreads) and different delivery times (calendar spreads). Each underlying price process may be modeled directly. At times, however, it is a useful simplification to consider the spread as a distinct process, which may itself be directly modeled. For this purpose, we investigate the continuous limit of a mean-reverting random walk and its extensions. Analytical results about the solution of this process, including its stationary distribution, are obtained. This new mean-reverting process is compared with the Vasicek process, and its advantages are discussed. We show that this new model for spread dynamics is capable of capturing kurtosis. It can also capture the possible skewness in the transition density of the price spread process. Since the analytical transition density is unknown for this nonlinear stochastic process, the local linearization method is deployed to estimate the model parameters. We apply this method to empirical data for modeling the spread between West Texas Intermediate (WTI) crude oil and West Texas Sour (WTS) crude oil.

Inference for a mean-reverting stochastic process with multiple change points

The use of an Ornstein-Uhlenbeck (OU) process is ubiquitous in business, economics and finance to capture various price processes and evolution of economic indicators exhibiting mean-reverting properties. The time at which structural transition representing drastic changes in the economic dynamics occur are of particular interest to policy makers, investors and financial product providers. This paper addresses the change-point problem under a generalised OU model and investigates the associated statistical inference. We propose two estimation methods to locate multiple change points and show the asymptotic properties of the estimators. An informational approach is employed in detecting the change points, and the consistency of our methods is also theoretically demonstrated. Estimation is considered under the setting where both the number and location of change points are unknown. Three computing algorithms are further developed for implementation. The practical applicability of our methods is illustrated using simulated and observed financial market data.

Modeling vessel kinematics using a stochastic mean-reverting process for long-term prediction

We present a novel method for predicting long-term target states based on mean-reverting stochastic processes. We use the Ornstein-Uhlenbeck (OU) process, leading to a revised target state equation and to a time scaling law for the related uncertainty that in the long term is shown to be orders of magnitude lower than under the nearly constant velocity (NCV) assumption. In support of the proposed model, an analysis of a significant portion of real-world maritime traffic is provided.