Mean Reversion In Rates Research Articles

FROM CALENDAR TIME TO BUSINESS TIME: THE CASE OF COMMODITY MARKETS

We address the problem of modeling commodity forward curves, while preserving empirical features observed in commodity markets. By letting a commodity market to “ live” in the tempo of a business rather than calendar clock, we create a model with a rich but realistic set of features, such as stochastic volatility, stochastic rate of mean reversion and various shapes of forward curves such as backwardation and contango. The model, when applied to extensive historical datasets of crude oil and natural gas forward curves, shows a remarkably good fit to the observed futures prices, also in periods of high volatility and negative prices. The model is developed in such a way that it can be used for a wide variety of applications, ranging from exotic derivatives pricing to risk management of commodity portfolios.

SOFR term structure dynamics—Discontinuous short rates and stochastic volatility forward rates

AbstractThe Secured Overnight Funding Rate (SOFR) has become the risk‐free rate benchmark in US dollars, thus term structure models should reflect key features exhibited by SOFR and forward rates implied by SOFR futures. We construct a multifactor, stochastic volatility term structure model which incorporates these features. Calibrating to options on SOFR futures, we achieve a reasonable fit to the market across maturities and strikes in a single model. This also provides novel insights into SOFR term rate behavior (and implied volatilities) within their accrual periods, and a model mechanism by which interest rate mean reversion arises from monetary policy.

Comparison of volatility and mean reversion among developed, developing and emerging countries

PurposeThis research paper aims to analyze the stock exchanges of developed, emerging and developing countries to investigate the volatility in stock markets and to evaluate the rate of mean reversion.Design/methodology/approachThe stock exchanges included in the research are NASDAQ, Tokyo stock exchange, Shanghai stock exchange, Bombay stock exchange, Karachi stock exchange and Jakarta stock exchange. Secondary daily data from Bloomberg are used to conduct the research for the period from January 2011 to December 2018. Generalized autoregressive conditional heteroskedasticity (GARCH) (1,1) model was applied to examine volatility and the half-life formula was used to calculate mean reversion in days.FindingsThe research concluded that all the stock exchanges included in the research satisfy the assumptions of mean reversion. Developing countries have the lowest volatility while emerging countries have the highest volatility which means that the rate of mean reversion is fastest in developing countries and slowest in emerging countries.Research limitations/implicationsFuture studies can determine the reasons for fastest rate of mean reversion in developing countries and slowest rate of mean reversion in emerging countries.Practical implicationsDeveloping countries show the lowest mean reversion in days while the emerging countries show the highest mean reversion in days indicating that developing countries take less time to revert to their mean position.Originality/valueThe majority of previous studies on univariate volatility models are mostly on applications of the models. Only a few researchers have taken the robustness of the models into account when applying them in emerging countries and not in developed, developing and emerging countries in one place. This makes the current study unique and more rigorous.

Modelling Temperature Using CARMA Processes with Stochastic Speed of Mean Reversion for Temperature Insurance Pricing

In this paper, we present a continuous time autoregressive moving average (CARMA) model with stochastic speed of mean reversion. This model allows the mean reversion rates to behave stochastically and governed by an Ornstein-Uhlenbeck process. We provide closed-form solution to the CARMA with stochastic speed of mean reversion and formulate the price of temperature insurance using spot-forward relationship framework. We demonstrate the insurance pricing based on the cumulative average temperatures (CAT) index by simulating the temperature variations. We found that our proposed model may explain the temperature evolution well and the price of CAT-based index insurance looks reasonable.

A Murine Model of Waning Scrub Typhus Cross-Protection between Heterologous Strains of Orientia tsutsugamushi.

Orientia tsutsugamushi, the etiologic agent of the life-threatening febrile disease scrub typhus, is an obligately intracellular small coccobacillary bacterium belonging to the family Rickettsiaceae and is transmitted by the parasitic larval stage of trombiculid mites. Progress towards a vaccine for protection against scrub typhus has been impeded by characteristics of the pathogen and the infection. There are numerous strains of O. tsutsugamushi in the Asia-Pacific region with geographical overlap. In human cases immunity has been described as poor against heterologous strains of the pathogen, as well as short-lived against the homologous strain, with a mean antibody reversion rate of less than one year. Animal models of cross-protection as well as of deterioration of this cross-protection are needed to enhance understanding of transient immunity to scrub typhus. To build upon current understanding of this ineffective protection we sought to utilize our recently developed models, sublethal intradermal infection followed by challenge via ordinarily lethal hematogenous dissemination. Mice that were initially infected sublethally with O. tsutsugamushi Gilliam strain and were challenged with an ordinarily lethal dose of heterologous Karp strain were protected from death by a robust immune response at one month after the primary infection as evidenced by an abundance of mononuclear cellular infiltrates in target organs such as lung, liver, and kidney; maintenance of body weight; and low bacterial loads in the organs. Waning protection from lethal Karp strain challenge indicated by weight loss mirroring that observed in naïve mice was observed as early as 9 months after primary Gilliam strain infection, and higher bacterial loads, severe disease, and eventual death in some mice was observed after challenge with Karp strain at 14 months post-initial heterologous infection.

Robust Optimal Investment-Reinsurance Strategies for a Jump-Diffusion Risk Model with Mean-reverting Stock Return

Objectives: This paper proposes a strategy of robust optimal investment reinsurance for insurance companies. It was assumed that the surplus procedure of the insurance company satisfies the jump-diffusion procedure. Insurance companies could invest their surplus funds in the financial market consisted of both risk assets and one risk-free asset. The price procedure of risk assets satisfies the stochastic procedure with a mean reversion rate. Considering the uncertainty of the model, the ambiguity-averse insurance firm aims to enhance the exponential utility of insurance surplus at terminal time. This paper has investigated the problem of robust optimal investment reinsurance and obtained the differential equation supported by the value function.

Four-factor model of Quanto CDS with jumps-at-default and stochastic recovery

We modify the model of Itkin, Shcherbakov and Veygman (ISV), proposed for pricing Quanto CDS and risky bonds, in several ways. First, the recovery rate could significantly vary right before or at default, therefore, here we treat it as stochastic. Second, we assume the domestic interest rate to be deterministic, because, as shown by ISV, its volatility does not contribute much to the Quanto CDS spread. Finally, to solve the corresponding systems of 4D PDEs we use a flavor of the RBF method which is a combination of localized RBF and finite-difference methods. Results of our numerical experiments demonstrate that the influence of volatility of the recovery rate is significant if the correlation between the recovery rate and the log-intensity of the default is non-zero. Also, the impact of the recovery mean-reversion rate on the Quanto CDS spread could be comparable with the impact due to jump-at-default in the FX rate.

The economics of time as it is embedded in the prices of options§

Risk neutral variance term structures are characterized by their time elasticities. They are synthesized by space scaling and time changing self-decomposable laws at unit time. Monotone concave or convex time elasticities are modeled using exponential functions while gamma functions permit changes in curvature. Results for both cases as time changes are followed by those with simultaneous space-scaling and time-changing. Space scaling contributes towards front end options while time changing works on the back end. Splitting the space scaling and time changing for the positive and negative moves delivers models with rising absolute skewness and kurtosis. The space scaled and time changed densities are those of additive processes. The space scaled process scales a solution to a time varying OU equation driven by a time changed Lévy process taken at log time. The mean reversion rates for the OU process are the variance time elasticities. The two processes are termed the space scaled and time change components and their relative contributions, space to time are determined to be twice the ratio of their variance elasticities. In particular, the space scaling elasticity synthesizes the effects of perpetual motion as captured by mean reversion in the underlying OU equation.

Modelling the dynamics of long-term bonds with Kalman filter

We construct a time-consistent and arbitrage-free three-factor Vasicek model for long-term bonds. A new methodology based on a stochastic mean reversion rate which captures uncertainty in long-term bond yields is presented. To allow measurement errors to be accounted for in observed yields, the model is expressed in a state space form. Kalman filtering is then applied to filter uncertainty in the observed yields. An appropriate set of transition equations on state variables and measurement equations on observed yields are derived. Using historical market data from the US Treasury daily interest rates (March 2006 to June 2020), Germany Government bond yields (August 2000 to 15 January 2021) and Canada Government bond yields (16 January 2020 to 14 January 2021), parameters of one-, two- and three-factor models are estimated. The results indicate that the constructed Vasicek model can fit the US, Germany and Canada term structure of interest rates.

Modelling the dynamics of long-term bonds with Kalman filter

We construct a time-consistent and arbitrage-free three-factor Vasicek model for long-term bonds. A new methodology based on a stochastic mean reversion rate which captures uncertainty in long-term bond yields is presented. To allow measurement errors to be accounted for in observed yields, the model is expressed in a state space form. Kalman filtering is then applied to filter uncertainty in the observed yields. An appropriate set of transition equations on state variables and measurement equations on observed yields are derived. Using historical market data from the US Treasury daily interest rates (March 2006 to June 2020), Germany Government bond yields (August 2000 to 15 January 2021) and Canada Government bond yields (16 January 2020 to 14 January 2021), parameters of one-, two- and three-factor models are estimated. The results indicate that the constructed Vasicek model can fit the US, Germany and Canada term structure of interest rates.

Application of OU processes to modelling temporal dynamics of the human microbiome, and calculating optimal sampling schemes

BackgroundThe vast majority of microbiome research so far has focused on the structure of the microbiome at a single time-point. There have been several studies that measure the microbiome from a particular environment over time. A few models have been developed by extending time series models to accomodate specific features in microbiome data to address questions of stability and interactions of the microbime time series. Most research has observed the stability and mean reversion for some microbiomes. However, little has been done to study the mean reversion rates of these stable microbes and how sampling frequencies are related to such conclusions. In this paper, we begin to rectify this situation. We analyse two widely studied microbial time series data sets on four healthy individuals. We choose to study healthy individuals because we are interested in the baseline temporal dynamics of the microbiome.ResultsFor this analysis, we focus on the temporal dynamics of individual genera, absorbing all interactions in a stochastic term. We use a simple stochastic differential equation model to assess the following three questions. (1) Does the microbiome exhibit temporal continuity? (2) Does the microbiome have a stable state? (3) To better understand the temporal dynamics, how frequently should data be sampled in future studies? We find that a simple Ornstein–Uhlenbeck model which incorporates both temporal continuity and reversion to a stable state fits the data for almost every genus better than a Brownian motion model that contains only temporal continuity. The Ornstein–Uhlenbeck model also fits the data better than modelling separate time points as independent. Under the Ornstein–Uhlenbeck model, we calculate the variance of the estimated mean reversion rate (the speed with which each genus returns to its stable state). Based on this calculation, we are able to determine the optimal sample schemes for studying temporal dynamics.ConclusionsThere is evidence of temporal continuity for most genera; there is clear evidence of a stable state; and the optimal sampling frequency for studying temporal dynamics is in the range of one sample every 0.8–3.2 days.

The Private Housing Market Cyclical Price Dynamics

Two types of heterogeneous investors (momentum and disposition) form a unique difference model to interpret housing price dynamics. Three parameters are crucial: auto-correlation, the rate of mean reversion and the contemporaneous adjustment towards long-term equilibrium price. For Singapore, we examine the dynamic structures that oscillate and/or diverge from equilibrium. Disposition investors predominate although the interaction between momentum and disposition investors acts as a key determinant of private housing price dynamics for a given time in a specific market. Key implication is that Singapore’s private housing market is low risk, offering stable returns owing to virtually no divergence even in the speculative 1990s. The best way to invest is to consider the momentum strategy and avoid the herd behavior for profit sustainability. For policy-makers, the Singapore private housing market is over-damped in the long run. Predominating disposition investors contribute to the market mechanism, which automatically adjusts private housing market prices. It is imperative to relax government intervention in Singapore’s private housing market to enhance its efficiency.

Numerical Ross Recovery for Diffusion Processes Using a PDE Approach

ABSTRACT We develop and analyse a numerical method for solving the Ross recovery problem for a diffusion problem with unbounded support, with a transition independent pricing kernel. Asset prices are assumed to only be available on a bounded subinterval . Theoretical error bounds on the recovered pricing kernel are derived, relating the convergence rate as a function of to the rate of mean reversion of the diffusion process. Our suggested numerical method for finding the pricing kernel employs finite differences, and we apply Sturm–Liouville theory to make use of inverse iteration on the resulting discretized eigenvalue problem. We numerically verify the derived error bounds on a test bench of three model problems.

Forecasting of the Nigeria Stock Returns Volatility Using GARCH Models with Structural Breaks

This study examines the stock returns series using Symmetric and Asymmetric GARCH models with structural breaks in the presence of some varying distribution assumptions. Volatility models of Symmetric GARCH (1,1), Asymmetric Power GARCH (1,1) and GJR-GARCH(1,1) models were considered in estimating and measuring shock persistence, leverage effects and mean reversion rate with structural breaks considering dummy variable for these structural changes and varying distributions . The skewed student-t distribution is considered best distribution for the models; moreover findings showed the high persistence of shock in returns series for the estimated models. However, when structural breaks were incorporated in the estimated models by including dummy variable in the conditional variance equations of all the models, there was significant reduction of shock persistence parameter and mean reversion rate. The study found the GJR-GARCH (1,1) with skewed student-t distribution best fit the series. The volatility was forecasted for 12 months period using GJR-GARCH (1,1) model and the values are compared with the actual values and the results indicates a continuous increase in unconditional variance.

Bank of America Stock Price Research

The stock price is always an interesting topic. On the one hand, from intuition, the stock price of bank is relatively stable since bank usually does not take risk behavior. From 2007 to 2014, the price of stock of Bank of American is relatively stable by checking the RRV (relative realized volatility). There is only one day has relatively high RRVs during 2007 to 2014. The date May 6th 2010, which called crash day, is a special day that needs to analyze separately. On the other hand, to find the pattern of frequency of trading, there are different sample sizes tested and compared with Poisson distribution. The result is that we can use Poisson distribution to predict probability of no arrival trade when second gap is relatively small. In addition, when plotting the daily 100 seconds accumulated RV (realized volatility) and daily average RV, there was found strong linear relationship between these two variables. In the end, using the Heston model to verify if there exist linear relationship between daily average and mean reversion rate. Then comparing trend of alpha with weekly VIX from Yahoo Finance. When using 5 days as a period to calculate the daily average RV and mean reversion rate, the significance of linear relationship is stronger. It proved the statistic intuition that larger sample size tends to decrease the volatility. The overall trend of VIX from Yahoo Finance is similar to the shape of five-day period alpha.

An SPDE model for systemic risk with endogenous contagion

We propose a dynamic mean field model for `systemic risk' in large financial systems, which we derive from a system of interacting diffusions on the positive half-line with an absorbing boundary at the origin. These diffusions represent the distances-to-default of financial institutions and absorption at zero corresponds to default. As a way of modelling correlated exposures and herd behaviour, we consider a common source of noise and a form of mean-reversion in the drift. Moreover, we introduce an endogenous contagion mechanism whereby the default of one institution can cause a drop in the distances-to-default of the other institutions. In this way, we aim to capture key `system-wide' effects on risk. The resulting mean field limit is characterized uniquely by a nonlinear SPDE on the half-line with a Dirichlet boundary condition. The density of this SPDE gives the conditional law of a non-standard `conditional' McKean--Vlasov diffusion, for which we provide a novel upper Dirichlet heat kernel type estimate that is essential to the proofs. Depending on the realizations of the common noise and the rate of mean reversion, the SPDE can exhibit rapid accelerations in the loss of mass at the boundary. In other words, the contagion mechanism can give rise to periods of significant systemic default clustering.

Management strategies for a defined contribution pension fund under the hybrid stochastic volatility model

In this paper, we introduce the Heston–Hull–White (the hybrid) model in the pension fund management. The optimal investment and benefit payments policies for the DC pension fund with an income drawdown option are presented explicitly. In this study, the pension fund manager is allowed to invest the fund wealth in riskless and risky assets. The risk asset price dynamics evolve according to the Hybrid Stochastic Volatility (Heston–Hull–White) model. The goal of the pension fund manager is to maintain the standard of lives for the pension fund members after retirement. We derived the HJB equation and established the closed-form solution for the optimal control policies using stochastic dynamic programming principle. To illustrate the economic implications of the model, we provide a numerical example and simulations. The results reveal that the optimal investment and benefit payments strategies convergence successful with time t. The risk asset price volatility and interest rate mean reversion are inversely proportional to investment policy. The interest rate volatility and risk preference are both directly proportional to investment and benefit payments. The pension fund managers are advised to control the investment and the benefit payments policies to achieve the goals of the pension fund members.

SPEED OF MEAN REVERSION: AN EMPIRICAL ANALYSIS OF KSE, LSE AND ISE INDICES

The purpose of this study is to determine the presence of mean reversion in the stock markets indices of Pakistan, moreover, to measure, and compare the speed of mean reversion of the stock markets indices across Pakistan. In order to carry out the research study, the daily data of three stock indices of Pakistan such as: KSE-100, LSE-25 and ISE-10 are collected from 2003 to 2014. After the application of tests such as ARCH and GARCH, it was found that returns series of KSE-100, LSE-25 and ISE-10 indices exhibit mean reversion, indicating that the returns revert back to their historical value after reaching an extreme value. Further, the mean reversion rate shows that KSE-100 index has the slowest mean reversion, however, the ISE-10 index has the fastest mean reversion among the three indices. Therefore, the results of the study concluded that KSE-100 index, due to the slowest mean reversion rate has higher volatility over a longer period of time. On the contrary, since, ISE-10 index has exhibited the fastest mean reversion with the lowest volatility as compared to others. But due to fast mean reversion rate, it will help investors to gain profits over a shorter period of time. Thus, it can be recommended that the investor willing to bear the risk of time and looking for long-term investment should invest in KSE-100 index. However, investors looking for higher profits in a shorter period can invest in the ISE-10 index but with higher risk-returns trade-off.

Impacts of CECL

We study the impact of Current Expected Credit Loss (CECL) standards on various metrics, measuring the timing, level and volatility of CECL reserves in comparison to existing GAAP rules using historical industry-level bank data. We find that CECL standards could significantly increase the reserves set for loan losses. The impact would be larger for longer maturity exposures, such as 30-year residential mortgages. In addition to expected life of loan estimation, we analyze the relative importance of other CECL parameters such as Reasonable & Supportable period, the speed of mean reversion and longer-term charge off rates. Our study covers two underlying assumptions. Initially, CECL impact is simulated under perfect foresight assumption where the loss forecasts and macroeconomic projections assumed be very accurate. Then we relax this assumption and build biases in our loss forecasting capability. Our analysis show that CECL could bring in pro-cyclical variations in reserve setting which could exacerbate a downturn by limiting the availability of credit and loan growth.

US Bank Credit Spreads During the Financial Crisis

This paper argues that first passage time models are likely to better than affine hazard rate models in modelling stressed credit markets and confirms their superior performance in explaining the behavior of Credit Default Swap rates for the major US banking groups over the period of the financial crisis. Affine models find it hard to deal with periods of exceptionally high or low default risk given their assumption of a constant rate of mean reversion in the hazard rate. In contrast, first passage time models are specified in terms of the distance to default rather than the hazard rate. The persistence of shocks varies with the distance to default, allowing the default curve to invert sharply (compress) when the distance to default is low (high). I use an empirical version of the Collin-Dufresne et al. (2003) model, which contains a smoothing parameter that allows it to control the relative effect of these shocks on the short spreads and can be interpreted as an information lag.