Stochastic Interest Rate Model Research Articles

ANALYTICALLY PRICING EUROPEAN OPTIONS UNDER A TWO-FACTOR STOCHASTIC INTEREST RATE MODEL WITH A STOCHASTIC LONG-RUN EQUILIBRIUM LEVEL

Abstract We construct a new stochastic interest rate model with two stochastic factors, by introducing a stochastic long-run equilibrium level into the Vasicek interest rate model which follows another Ornstein–Uhlenbeck process. With the interest rate under the Black–Scholes model being assumed to follow the newly proposed model, a closed-form representation of European option prices is successfully presented, when the analytical characteristic function of the underlying log-price under a forward measure is derived. To assess the model performance, a preliminary empirical study is conducted using S&P 500 index and its options, with the Vasicek model and an alternative two-factor Vasicek model taken as benchmarks.

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.

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.

On the Bias of the Unbiased Expectation Theory

The unbiased expectation theory stipulates that long-term interest rates are determined by the market’s expectations of future short-term interest rates. According to this hypothesis, if investors have unbiased expectations about future interest rate movements, the forward interest rates should be good predictors of future spot interest rates. This hypothesis of the term structure of interest rates has long been a subject of debate due to empirical and theoretical challenges. Despite extensive research, a satisfactory explanation for the observed systematic difference between future spot interest rates and forward interest rates has not yet been identified. In this study, we approach this issue from an arbitrage theory perspective, leveraging on the connection between the expectation hypothesis and changes in probability measures. We propose that the observed bias can be explained by two adjustments: a risk premia adjustment, previously considered in the literature, and a stochastic adjustment that has been overlooked until now resulting from two measure changes. We further demonstrate that for specific instances of the Vasicek and Cox, as well as the Ingersoll and Ross, stochastic interest rate models, quantifying these adjustments reveals that the stochastic adjustment plays a significant role in explaining the bias, and ignoring it may lead to an overestimation of the required risk premia/aversion adjustment. Our findings extend beyond the realm of financial economic theory to have tangible implications for interest rate modelling. The capacity to quantify and distinguish between risk and stochastic adjustments empowers modellers to make more informed decisions, leading to a more accurate understanding of interest rate dynamics over time.

APPRAISAL OF STOCHASTIC PRICING FORMULA FOR HYBRID EQUITY WARRANTS

An evaluator of warrant models is essential for enhancing the current procedures used in the country's stock market and economy. This would provide additional insight for investors to make informed investment decisions and help to better understand the dynamics of warrant pricing in Malaysia. It has been established that the conventional process for valuing equity warrants using call option pricing models, like the Black-Scholes model, has numerous flaws, including the presumption of constant volatility and interest rates. As a matter of fact, existing alternative models were more concerned with providing techniques for pricing than with conducting empirical tests of warrant pricing models. Consequently, it is crucial to have a mathematical model that takes into account stochastic volatility and stochastic interest rates when pricing and analyzing equity warrants. Recent research by Roslan, Ibrahim, Jameel, and Ibrahim (2020) and Roslan, Ibrahim, Karim (2020) examined the pricing of hybrid equity warrants in the presence of stochastic factors. There was no comparative analysis conducted in these studies to ascertain the relative efficacy of the two methods used. This study's main goal is to empirically investigate how well equity warrant pricing models perform in the Malaysian market. This evaluation will take into account the impact of stochastic interest rates, using the Cox-Ingersoll-Ross (CIR) stochastic interest rate model, and stochastic volatility, using the Heston stochastic volatility model. To perform the analysis, the study will make use of a number of mathematical techniques, such as partial differential equations, Fourier transform approaches, and the change of measure method. The research methodology is divided into three main phases. It is anticipated that the research's findings will provide new perspectives that go beyond the domain of warrant pricing, creating chances for more widespread applications in fields like foreign exchange, insurance, and mortgage contracts.

Optimal Reinsurance–Investment Strategy Based on Stochastic Volatility and the Stochastic Interest Rate Model

This paper studies insurance companies’ optimal reinsurance–investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer’s surplus process and the variance premium principle to calculate premiums. In this paper, we assume that insurance companies can invest in risk-free assets, risky assets, and zero-coupon bonds, where the Cox–Ingersoll–Ross model describes the dynamic change in stochastic interest rates and the Heston model describes the price process of risky assets. The analytic solution of the optimal reinsurance–investment strategy is deduced by employing related methods from the stochastic optimal control theory, the stochastic analysis theory, and the dynamic programming principle. Finally, the influence of model parameters on the optimal reinsurance–investment strategy is illustrated using numerical examples.

Analytically pricing European options under a hybrid stochastic volatility and interest rate model with a general correlation structure

AbstractIn this paper, an additional factor is introduced into the Heston–Hull–White (HHW) hybrid model, which originally combines the Heston stochastic volatility model and the Hull–White stochastic interest rate model, to capture the correlation between the underlying price and the interest rate, while at the same time preserve the analytical tractability. With the analytical solution to the characteristic function of the underlying price being successfully derived, a closed‐form pricing formula for European options under the two‐factor HHW hybrid model is presented. Numerical experiments are also carried out to show the effect of the newly introduced factor. To further demonstrate the importance of introducing the correlation, we have also conducted an empirical study, comparing the performance of our model and that of the HHW model, based on European options written on S&P 500 index.

Comparison of Zillmer and Premium Sufficiency Reserve Method using the Vasicek Stochastic Interest Rate Model

An accurate calculation of premium reserves will ensure that the insurance company can pay claims. Premium reserves are funds collected by insurance companies which are the difference between the sum insured and the value of payments during the insurance period prepared for claim payments. There are several methods for calculating premium reserves, but the methods that are the focus of this study are the Zillmer method and the Premium Sufficiency method, extensions of the prospective method. This study aims to compare the two methods using the Vasicek model to determine the stochastic interest rate. Then the Ordinary Least Square method is used to estimate the Vasicek interest rate parameter. In comparing the two premium reserve methods, this study builds a simulation for reference male and female insureds using Bank Indonesia reference rate data for 2017-2021 and the Indonesian Mortality Table IV from 2019. The results of this study indicate that premiums reserve calculations using the Zillmer method yield less value than the Premium Adequacy method for both male and female insureds.

The riskiness of stock versus money market investment with stochastic rates

To efficiently assess the performance of investing in stocks rather than in a bank account for the long run, stochastic interest rate modelling is advocated. We introduce a correlated stochastic interest rate model that addresses this problem. We derive analytic formulas for general spectral risk measures in our setting, and apply our results to Value at Risk, Expected Shortfall and GlueVaR. We characterize the short- and long-term behaviour of these risk measures. We fit our model to financial markets, perform an empirical study and evaluate risk numbers for realistic scenarios in the future. Our results reveal sizeable sensitivities on parameter estimation, but we may conclude that holding stocks for less than a few decades bears significant risk.

Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS Model

In this paper, we reconstruct the time-dependent volatility function of the underlying asset and the mean-reverting parameter γ of the interest rate for European options under the fractional Chan–Karolyi–Longstaff–Sanders (CKLS) stochastic interest rate model. Tikhonov regularization is used to solve the ill-posedness of the inverse problem. The existence and stability of the solution of the regularization problem are given. We employ the alternating direction method of multipliers (ADMM) to iteratively optimize the volatility function and the parameter γ. Finally, numerical simulations and the empirical analysis are presented to illustrate the efficiency of the proposed method.

CALIBRATING LOCAL VOLATILITY MODELS WITH STOCHASTIC DRIFT AND DIFFUSION

We propose Monte Carlo calibration algorithms for three models: local volatility with stochastic interest rates, stochastic local volatility with deterministic interest rates and finally stochastic local volatility with stochastic interest rates. For each model, we include detailed derivations of the corresponding SDE systems and list the required input data and steps for calibration. We give conditions under which a local volatility can exist given European option prices, stochastic interest rate model parameters, and correlations. The models are posed in a foreign exchange setting. The drift term for the exchange rate is given as a difference of two stochastic short rates, domestic and foreign; each modeled by a Gaussian one-factor model with deterministic shift (G1[Formula: see text]) process. For stochastic volatility, we model the variance for the exchange rate by a Cox–Ingersoll–Ross (CIR) process. We include tests to show the convergence and the accuracy of the proposed algorithms.

Vulnerable European option pricing in a Markov regime-switching Heston model with stochastic interest rate

This paper considers pricing of European-style vulnerable options under the Heston stochastic volatility and stochastic interest rate model in which the mean-reversion levels of both variance and interest rate processes are modulated by a continuous-time Markov process with a finite state space. An analytical pricing formula is derived by using the Esscher transform, joint characteristic function and multivariate Fourier transform technique, where the closed-form solution of the characteristic function is obtained by the law of iterated expectation. Then we provide the efficient approximation to calculate the analytical pricing formula of option using the FFT approach and examine the accuracy of the approximation by Monte Carlo simulation. Finally, the sensitivity analysis of different parameters in the proposed model on the vulnerable call option price and its Delta value are provided, and the difference between the proposed model and the Heston and stochastic interest rate model with non-Markov regime-switching are presented by some numerical experiments, which shows the influence of introducing regime-switching into Heston model with stochastic interest rate.

Vasicek interest rate model under Lévy process and pricing bond option

In this paper, we concentrate on the European option pricing problem in incomplete markets by taking Vasicek stochastic interest rate model into account and expanding it in accordance with a Lévy process named Variance Gamma (VG) process. In doing so, first, we prove the existence and uniqueness of the solution to the stochastic differential equation of this model. Then, by the means of the hedging and replicating techniques, pricing formulas for the bond and, in succession, for European options are established. Finally, from a numerical point of view, we will analyze the value of a bond based on our consideration model and also, we will study a European call option bond price. All these results are reported through the Monte-Carlo simulation method.

Optimal Asset Allocation for CRRA and CARA Insurers under the Vasicek Interest Rate Model

This paper considers the reinsurance‐investment problem with interest rate risks under constant relative risk aversion and constant absolute risk aversion preferences, respectively. Stochastic control theory and dynamic programming principle are applied to investigate the optimal proportional reinsurance‐investment strategy for an insurer under the Vasicek stochastic interest rate model. Solving the corresponding Hamilton‐Jacobi‐Bellman equation via the Legendre transform approach, the optimal premium allocation strategies maximizing the expected utilities of terminal wealth are derived. In addition, several sensitivity analyses and numerical illustrations are given to analyze the impacts of different risk preferences and interest rate fluctuation on the optimal strategies. We find that the asset allocation and reinsurance ratio of the insurer are correlated with risk preference coefficient and interest rate fluctuation, and the insurance company may adjust the reinsurance‐investment strategy to deal with interest rate risk.

A quick operator splitting method for option pricing

This paper proposes a new efficient operator splitting method for option pricing problem under the Heston model, which is very popular in financial engineering. The key idea of this method is relying on eliminating the cross derivative term in partial differential equation in two dimension by some variable transformation techniques, and then decomposes the original equation in two dimensions into two partial differential equations in one dimension, which can be numerically solved efficiently. Moreover, this method not only keeps the differentiability of model parameters, but also preserves the positivity, monotonicity and convexity of the option prices. Numerical results for a European put option show that this method achieves accuracy of second-order in space and first-order in time, which are coinciding with the theoretical analysis results. Since the algorithm of this paper can be parallelized easily, the option pricing problems in high-dimension can also be dealt with, such as the Basket option written on several assets and etc. Our method can also be applied to pricing American options, Asian options and option pricing problems in stochastic interest-rate models.

Meshless approach for pricing Islamic Ijarah under stochastic interest rate models

Nowadays, the fixed interest rate financing method is commonly used in the capitalist financial system and in a wide range of financial liability instruments, the most important of which is bond. In the Islamic financial system, using these instruments is considered as usury and has been prohibited. In fact, Islamic law, Shariah, forbids Muslims from receiving or paying off the Riba. Therefore, using customary financial instruments such as bond is not acceptable or applicable in countries which have a majority of Muslim citizens. In this paper, we introduce one financial instrument, Sukuk, as a securities-based asset under stochastic income. These securities can be traded in secondary markets based on the Shariah law. To this end, this paper will focus on the most common structure of the Islamic bond, the Ijarah and its negotiation mechanism. Then, by presenting the short-term stochastic model, we solve fixed interest rate and model the securities-based asset by the stochastic model. Finally, we approximate the resulting model by radial basis function method, as well as utilizing the Matlab software.

A Comprehensive Literature Review on Pricing Equity Warrants Using Stochastic Approaches

Prior studies revealed that most researchers tend to employ the Black Scholes model to price equity warrants. However, the Black Scholes model was found deficient by contributing to large estimation errors and mispricing of equity warrants. Therefore, issues involving equity warrants are discussed in this paper, by focusing on specific topics and respective stochastic models to provide a basis for improvements in future research. In recent years, stochastic approaches have been used to a great extent among researchers due to the expansive applications in both theoretical and practical sense. Subsequently, this paper provides the results of a comprehensive literature review on various stochastic modelling methods and its applications for pricing financial derivatives in terms of applications, modifications of methods, comparisons with other methods, and general related researches. Focus is given on two types of stochastic models namely stochastic volatility and stochastic interest rate models, along with the discussions associating these two types of models. This paper acts as a valuable source of information for academic researchers and practitioners not only for pricing financial instruments, but also in various other fields involving stochastic techniques.

Valuation of European options with stochastic interest rates and transaction costs

The celebrated Black–Scholes model is well known for its elegant pricing formula for European options. However, like many other models, the Black–Scholes model is not perfect, which is largely due to the fact that assumptions in the model are idealized and not all empirically valid. One of the assumptions is that the market does not have transaction costs, which is not satisfied in a real market. Leland [H. Leland, Option pricing and replication with transactions costs, J. Financ. 40 (1985), pp. 1283–1301] pioneers a modified replicating strategy for European options by incorporating transaction costs. In this paper, we further consider the problem of pricing European options under a stochastic interest rate and stochastic volatility model with transaction costs, and derive a nonlinear partial differential equation (PDE) from this model. Then, we apply the finite-difference scheme to solve this PDE and conduct numerical experiments.

Put Options with Linear Investment for Hull-White Interest Rates

We derive a Put Option price associated with selling strategy of the underlying security in a random interest rate environment. This extends Put Option pricing under linear investment strategy from the Black-Scholes setting to Hull-White stochastic interest rate model. As an application, Call Option price for the linear investment strategy in the Hull-White model is established. Our results address recent emergence of developing dynamic investment strategies for the purpose of reducing the investor risk exposure associated with European-type options.

Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $ H \gt \frac{1}{2}$. The drift term of the equation is locally Lipschitz and unbounded in the neighbourhood of the origin. We show the existence, uniqueness and positivity of the solutions. The estimates of moments, including the negative power moments, are given. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, by using Lamperti transformation, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear Aït-Sahalia type model.