Interest Rate Model Research Articles

Robust mean-variance precommitment strategies of DC pension plans with ambiguity under stochastic interest rate and stochastic volatility

. This article studies a robust optimal investment problem with an ambiguity-averse manager (AAM) for a defined contribution (DC) plan with multiple risks under the mean-variance criterion. In the pension accumulation stage, the interest rate, the volatility, and the salary level are considered to be stochastic. The financial market consists of a risk-free asset, a risky asset, and a rolling bond. We assume that the term structure of interest rates is driven by an affine interest rate model, while the stock price and the salary level are modeled by the stochastic volatility model with stochastic interest rate. The goal of an AAM is to find a robust optimal strategy to maximize the expectation of terminal wealth and minimize the variance of terminal wealth in the worst-case scenario. By applying the Lagrange dual theory and the robust optimal control approach, we obtain closed-form expressions of the robust precommitment strategy and the efficient frontier, and subsequently some special cases are derived. Finally, a numerical example is given to illustrate the results obtained.

Evaluating the impact of stochastic interest rates and COVID-19 on financial performance under IFRS 17

Abstract The emergence of COVID-19 has resulted in a notable rise in mortality rates, consequently affecting various sectors, including the insurance industry. This paper analyzes the reflections of a sudden increase in mortality rates on the financial performance of a survival benefit scenario under the International Financial Reporting Standard 17. For this purpose, we thoroughly examined a single insurance scenario under four different states by modifying the interest and jump elements. We use Poisson-log bilinear Lee–Carter and Vasicek models for mortality and stochastic interest rate, respectively. Integrating the mortality model with a jump model that incorporates COVID-19 deaths we constructed a temporary mortality jump model. As a result, the temporary mortality jump model reflects the effects of the pandemic more realistically. We observe that even in this case mortality has a minor impact, whereas interest rates significantly still affect the financial position and performance of insurance companies.

Pensions and protestants: or why everything in retirement can’t be optimized

Abstract A common narrative among insurance actuaries and business economists is that national or regional pension systems can be finetuned, optimized, and improved simply by tinkering with demographic and financial parameters; all within the context of the “right” mathematical model. Indeed, recent papers in the actuarial literature have offered technical fixes around savings rates, retirement ages, decumulation strategies as well as more refined mortality and interest rate models. But alas, not everything in the world of pensions and retirement can be optimized, in particular as it relates to the history, background culture, or religion of the underlying population. This paper documents a statistically significant relationship between a region’s pension plan “health status” and the fraction of the region’s population identifying as Protestant Christians (PC). We begin the analysis at the national level using a well-known pension quality index and then obtain similar results for the actuarial funded status of U.S. state pension plans. Overall, this work is within the sphere of recent literature that indicates historical religious beliefs, values, and culture matter for financial economic outcomes; a factor which obviously can’t be optimized within a mathematical Hamilton–Jacobi–Bellman (HJB) equation. In other words, some things in retirement are truly beyond control.

Predictive Models for Interest Rate Forecasting Using Machine Learning: A Comparative Analysis and Practical Application

Forecasting interest rates is a fundamental task in financial planning, investment strategies, and policy-making. Traditional statistical models, while widely used, often fail to adequately capture complex non-linear relationships and temporal dependencies inherent in financial data. This study addresses these limitations by exploring the potential of machine learning models to improve the accuracy and reliability of interest rate forecasts. The primary objective of this research is to evaluate and compare the performance of multiple machine learning models, including linear regression, support vector machines, and deep learning techniques, in predicting interest rate trends. Historical data spanning two decades was collected and preprocessed, ensuring data quality and consistency. The models were trained and tested on this dataset using well-defined evaluation metrics such as mean absolute error and root mean squared error to ensure robust performance assessments. The results revealed that machine learning approaches, particularly deep learning models, outperformed traditional methods in capturing complex patterns and delivering more accurate forecasts. The findings further discuss the practical implications of implementing machine learning techniques in real-world financial contexts, highlighting both opportunities and challenges. In conclusion, this study provides actionable insights and a robust framework for integrating machine learning into interest rate forecasting, contributing to the advancement of predictive modeling in finance.

Calibration of the Ueno’s Shadow Rate Model of Interest Rates

Shadow rate models of interest rates are based on the assumption that the interest rates are determined by an unobservable shadow rate. This idea dates back to Fischer Black, who understood the interest rate as an option that cannot become negative. Its possible zero values are consequences of negative values of the shadow rate. In recent years, however, the negative interest rates have become a reality. To capture this behavior, shadow rate models need to be adjusted. In this paper, we study Ueno’s model, which uses the Vasicek process for the shadow rate and adjusts its negative values when constructing the short rate. We derive the probability properties of the short rate in this model and apply the maximum likelihood estimation method to obtain the parameters from the real data. The other interest rates are—after a specification of the market price of risk—solutions to a parabolic partial differential equation. We solve the equation numerically and use the long-term rates to fit the market price of risk.

A Comparison of the Vasicek and Cox, Ingersoll, and Ross Interest Rate Models in Valuation of Insurance Assets, Liabilities and Surplus

Insurance Company’s cash flows are subjected to the risk of interest rate (C-3 risk). To curb the effect of this risk, Insurance companies normally adopts an interest period model that predicts the movement of the rates of interest. The most common models adopted by the Insurance Companies are the vasicek (1977) model and The Cox, Ingersoll and Ross (1985) Model. These two models are stochastic single period short-rate models; however, they exhibit different assumptions and because of this, the future values of insurance Assets and liabilities are likely to differ when these models are applied to estimate their values. Valuing of Insurance Assets and liabilities, especially in the Kenyan market is very challenging because of the tremendous fluctuations of interest rates as a result of gradual increments of the rate of inflation. In order for insurance companies to correctly value their insurance policies, they need to have a substantive Knowledge of their cash flows. The current valuation methods of insurance assets, liabilities and Surplus based on a stochastic interest rate models do not consider the possibility of occurrence of model risk, and therefore there is a possibility of either under estimating the future values of insurance assets and liabilities or over estimating. In this research paper, Geometric simulation was used to explore the effect of model risk By creating a comparison between The vacisek and the Cox, Ingersoll and Ross interest rate model. First, we evaluated the value of an insurance company’s assets and liabilities by assuming that the interest rate process is followed by the Cox, Ingersoll and Ross model and The vasicek (1977). Model risk arose by the different Values obtained for both the vacisek and the Cox, Ingersoll and Ross model. The results of the simulation showed that the cox, Ingersoll and Ross interest rate model provided a better fit of interest as compared to The Vasicek model.

The determinants of funding liquidity risk in decentralized lending

Decentralized lending in the DeFi ecosystem mirrors traditional financial intermediation but poses significant risks, particularly funding liquidity risk, due to the volatility and composbility of digital assets, high leverage, and the absence of regulatory protections. This study applies traditional financial intermediation theories to DeFi lending and empirically test which internal factors such as interest rates and user market power, as well as external factors like the USD Index, influence funding liquidity risk in DeFi lending. Analyzing high-frequency blockchain data using the ARDL model and a novel dynamic ARDL simulation from major pools such as Wrapped Bitcoin (WBTC) and Wrapped Ethereum (WETH), the research finds that current algorithmic interest rate models fail to function as effective self-stabilization mechanisms. Additionally, lower deposit concentration in these pools may exacerbate, rather than mitigate, funding liquidity risk.

Interest Rate Risk Modelling Using Semi-Heavy Tail Distributions of Normal Variance-Mean Mixtures: A Study on Central Bank of Kenya Interest Rates

Derivative prices such as options and bond prices as well as swaps depend on the distributional assumptions of the underlying economic variables, normally, interest rates. The risk associated with changes in interest rates may worsen the value of the contract that depend on it since the values of these assets (derivative contracts) are affected directly by the fluctuations in interest rates. The distribution of interest rates, therefore, needs to be well understood to reduce the risks of losses associated with it. The Binomial Option pricing model assumes that interest rates are constant throughout the life of the option. Another common assumption of the underlying economic variables is that their returns are normally distributed with constant volatility. These assumptions have been used in pricing derivatives and currencies and has led to over-pricing and in some cases under-pricing. The models of the Normal Variance-Mean Mixtures used in this research shows better performance than the normal distribution. Weekly 91-day and commercial bank interest rates are used from 1991 to 2021. This study will provide a better model for interest rates to avoid mispricing. We find that the 91-day Treasury Bills interest rates follow a Generalized Hyperbolic distribution while the Commercial Bank interest rates follows a Normal Inverse Gaussian distribution. G.A.R.C.H model is then used in forecasting under each of the above findings. A 99pc Value at Risk is then computed for the two models and calculates the minimum expected returns in the subsequent months. This research forms a foundation for the development of advanced pricing models that incorporates the fluctuations of interest rate in the pricing industry. We conclude that the Normal Inverse Gaussian and Generalized hyperbolic distributions provide better fits than the assumed normal distribution.

The geometry of multi-curve interest rate models

We study the problems of consistency and the existence of finite-dimensional realizations for multi-curve interest rate models of Heath–Jarrow–Morton type, generalizing the geometric approach developed by T. Björk and co-authors for the classical single-curve setting. We characterize when a multi-curve interest rate model is consistent with a given parameterized family of forward curves and spreads and when a model can be realized by a finite-dimensional state process. We illustrate the general theory in a number of model classes and examples, providing explicit constructions of finite-dimensional realizations. Based on these theoretical results, we perform the calibration of a three-curve Hull–White model to market data and analyse the stability of the estimated parameters.

Practice of Python in Programming and Optimization of Quantitative Analysis Model of Fixed Income

Python has become an indispensable tool in fixed income trading. This paper introduces the principles and general process of data analysis visualisation by analysing the basic operation and performance of personal investment and finance, combined with data analysis in the era of big data. The library of data analysis tools using Python has become an indispensable tool in fixed income trading. In fixed income trading, it supports fixed income traders to handle tasks such as bond pricing, interest rate modelling, and credit risk analysis Using Python's powerful data processing capabilities, traders can simplify data collection, analysis, and reporting to make better decisions. Based on the Python language, financial data is acquired from different platforms, processed and analysed step-by-step, ensuring that the model remains accurate and stable in different market conditions. Backtesting and ongoing performance evaluations have revealed that Python can be a valuable asset in the dynamic world of fixed income trading by refining trading strategies and effectively managing risk.

Revisiting elastic string models of forward interest rates

Twenty five years ago, several authors proposed to describe the forward interest rate curve (FRC) as an elastic string along which idiosyncratic shocks propagate, accounting for the peculiar structure of the return correlation across different maturities. In this paper, we revisit the specific ‘stiff’ elastic string field theory of Baaquie and Bouchaud [Stiff field theory of interest rates and psychological future time. Wilmott Mag., 2004, 2–6] in a way that makes its micro-foundation more transparent. Our model can be interpreted as capturing the effect of market forces that set the rates of nearby tenors in a self-referential fashion. The model is parsimonious and accurately reproduces the whole correlation structure of the FRC over the time period 1994–2023, with an error around 1% and with only one adjustable parameter, the value of which being very stable across the last three decades. The dependence of correlation on time resolution (also called the Epps effect) is also faithfully reproduced within the model and leads to a cross-tenor information propagation time on the order of 30 minutes. Finally, we confirm that the perceived time in interest rate markets is a strongly sub-linear function of real time, as surmised by Baaquie and Bouchaud [Stiff field theory of interest rates and psychological future time. Wilmott Mag., 2004, 2–6]. In fact, our results are fully compatible with hyperbolic discounting, in line with the recent behavioral Finance literature Farmer and Geanakoplos [Hyperbolic Discounting is Rational: Valuing the Far Future with Uncertain Discount Rates, Cowles Foundation Discussion Papers, 2009 (Cowles Foundation for Research in Economics, Yale University)].

Enhancing valuation of variable annuities in Lévy models with stochastic interest rate

This paper extends the valuation and optimal surrender framework for variable annuities with guaranteed minimum benefits in a Lévy equity market environment by incorporating a stochastic interest rate described by the Hull-White model. This approach frames a more dynamic and realistic financial setting compared to previous literature. We exploit a robust valuation mechanism employing a hybrid numerical method that merges tree methods for interest rate modeling with finite difference techniques for the underlying asset price. This method is particularly effective for addressing the complexities of variable annuities, where periodic fees and mortality risks are significant factors. Our findings reveal the influence of stochastic interest rates on the strategic decision-making process concerning the surrender of these financial instruments. Through comprehensive numerical experiments, and by comparing our results with those obtained through the Longstaff-Schwartz Monte Carlo method, we illustrate how our refined model can guide insurers in designing contracts that equitably balance the interests of both parties. This is particularly relevant in discouraging premature surrenders while adapting to the realistic fluctuations of financial markets. Lastly, a comparative statics analysis with varying interest rate parameters underscores the impact of interest rates on the cost of the optimal surrender strategy, emphasizing the importance of accurately modeling stochastic interest rates.

Pricing exchange options under hybrid stochastic volatility and interest rate models

This paper investigates the pricing of exchange options under hybrid models integrating stochastic volatility and stochastic interest rates. It aims to achieve two primary objectives. First, we derive a closed-form pricing formula for exchange options under a two-factor Heston–Hull–White hybrid model, which accounts for long-term volatility and exhibits relatively broad correlations among the dynamics of asset prices, volatilities, and interest rates. Second, we explore the Heston model’s integration with a generalized single-factor stochastic interest rate model, illustrating that the price is not dependent on the specific form of the interest rate process. A closed-form pricing formula for exchange options under this framework is also derived. Our numerical experiments support the proposed formulas and elucidate the effects of various parameters on option prices.

Random Processes and Their Applications in Financial Mathematics

Random processes are fundamental tools in modern financial mathematics, widely applied in fields such as financial derivatives pricing, risk management, and interest rate models. This paper systematically introduces the basic theory of random processes, including definitions and classifications, Brownian motion and geometric Brownian motion, and stochastic differential equations. Furthermore, the paper explores specific applications of random processes in financial mathematics, focusing on financial derivatives pricing, risk management and portfolio optimization, and applications in interest rate models and credit risk. Through theoretical and empirical research, this paper reveals the significant role and development prospects of random processes in financial mathematics, providing new ideas and methods for risk management and pricing models in financial markets.

Implementasi Metode New Jersey dalam Perhitungan Cadangan Premi dengan Suku Bunga Stokastik dan Konstan

Premium reserve allocation represents an obligation undertaken by insurance companies to set aside funds for future claims payment to policyholders. Some insurance companies have faced operational challenges, leading to their closure, primarily due to inaccurate premium reserve computations. This research aims to calculate premium reserve in lifelong insurance using the New Jersey method, an improvement upon the Illionis method. The New Jersey method initiates the premium reserve at the beginning or end of the first year at zero dollars. The majority of premium reserve calculations still rely on constant interest rates. However, in reality, this approach inadequately reflects future fluctuations in interest rates, which are crucial for long-term life insurance products. Therefore, this study implements a more realistic approach using stochastic elements, using the Vasicek stochastic interest rate model to determine premium reserve values. From this research, it was found that there was quite a significant difference between the New Jersey method premium reserve value and the two interest rates. The calculation graph shows that the premium reserve value using the Vasicek model of stochastic interest rates tends to be lower than when using constant interest rates. This can be caused by the results of non-constan variations in interest rates in the Vasicek model which ultimately results in fluctuations in interest rates which wffect the calculation of premium reserve.

Pricing of discretely sampled arithmetic Asian options, under the Hull–White interest rate model

This paper studies the pricing of discrete arithmetic Asian options (AAOs) with fixed strikes under the Hull–White interest rate model. For the pricing of AAOs, we first investigate the stochastic dynamics of the price of the underlying asset under the T-forward measure, and then study the distribution of the discrete arithmetic average of the underlying asset price. Specifically, we provide the first three moments of the discrete arithmetic average under the T-forward measure. Then, we derive approximate pricing formulas for AAOs using the three-moment matching method. Furthermore, we calculate the first three conditional moments of the discrete arithmetic average, given the final value of the underlying asset, under the T-forward measure. These conditional moments can be used to improve the accuracy of the approximation of the AAO prices. The numerical results show that our three-moment matching approximations are very accurate. Additionally, the accuracy can be further improved by combining the conditioning approach with the three-moment matching method. Our procedure is also applied to the computation of deltas of AAOs.

On testing the changes in trends of stock market index and rates

Calibration of interest rate models benefits from grouping data to homogenous classes. Such an approach is typical in many financial time series. Preliminaries have been developed for Cox–Ingersoll–Ross models but this issue remains an open problem for many more realistic interest rate models. Here we develop such a strategy for general class interest rate and classes are based on p-value thresholds for testing for normality and gamma distributions. We use as the benchmark financial series of Chilean stock market index IPSA (Indice de precios selectivo de acciones) and its log-returns. We also study the relationship between interest rate and the market returns represented by the IPSA indicator, with positive correlation in some lags which reveals some interesting facts in the contrary to the conventional theory.

A Study on Modeling Financial Mathematics by Computational Program and Its Applications

Financial mathematics plays a pivotal role in various aspects of modern economics and finance. This paper provides an introduction to the fundamental concepts, theories, and applications of financial mathematics. It begins by outlining the basic principles of financial mathematics, including the time value of money, interest rates, and compounding. Computational programs enhance these mathematical models, offering robust solutions and efficient computation for complex financial problems. This study explores the integration of computational programs with financial mathematics, their methodologies, and applications in the finance sector. The results underscore the significance of computational methods in improving the accuracy, speed, and scalability of financial models, ultimately contributing to better decision-making and risk management. We explore fundamental concepts, models, and techniques employed in financial mathematics, aiming to provide a comprehensive understanding of their applications and significance in real-world financial scenarios. This paper provides a comprehensive overview of the application of differential equations in financial mathematics, highlighting key models such as the Black-Scholes model, interest rate models, and optimal investment strategies.

Application of Stochastic Processes in Financial Market Models

This paper explores the significant role of stochastic processes in financial modeling, tracing the evolution from basic Brownian motion to sophisticated stochastic differential equations used in modern financial markets. Beginning with the historical development of Brownian motion, identified by Robert Brown and later mathematically modeled by Louis Bachelier for stock price fluctuations, the paper outlines its foundational influence on the Efficient Market Hypothesis and the random walk theory. The extension of these concepts in the Black-Scholes model for option pricing highlights the practical applications of these theories in predicting financial outcomes. The discussion progresses to geometric Brownian motion (GBM) and its crucial role in stock price modeling, emphasizing its use in Monte Carlo simulations for option pricing. The limitations of the Black-Scholes model and GBM in dealing with real market conditions such as stochastic volatility and jump-diffusion processes are addressed, showcasing the evolution of more complex models like the Heston model and GARCH. Interest rate models like the Vasicek and Cox-Ingersoll-Ross models are evaluated for their real-world applicability, particularly in scenarios of low and negative interest rates. This comprehensive review not only underscores the theoretical advancements in financial modeling but also its practical implications in contemporary financial markets.

Pricing of geometric average Asian option under the sub-diffusion Merton interest rate model

. Asian option is an essentially new type of option. In existing option pricing models, the Brownian motion is generally the stochastic driving source of changes in the underlying asset price. This article explores the pricing problem of geometric average Asian option under the sub-diffusive Merton interest rate model. The results obtained the partial differential equations satisfied by the geometric average Asian option via constructing portfolios and utilizing the delta hedging technique. Finally, the findings provide display expressions for the geometric average Asian option under the sub-diffusive Merton interest rate model, combined with boundary conditions and variable substitutions.