Hull-White Research Articles

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.

Pricing of a Binary Option Under a Mixed Exponential Jump Diffusion Model

This paper focuses on the pricing problem of binary options under stochastic interest rates, stochastic volatility, and a mixed exponential jump diffusion model. Considering the negative interest rates in the market in recent years, this paper assumes that the stochastic interest rate follows the Hull–White (HW) model. In addition, we assume that the stochastic volatility follows the Heston volatility model, and the price of the underlying asset follows the jump diffusion model in which the jumps follow the mixed exponential jump model. Considering these factors comprehensively, the mixed exponential jump diffusion of the Heston–HW (abbreviated as MEJ-Heston–HW) model is established. Using the idea of measure transformation, the pricing formula of binary call options is derived by the martingale method, eigenfunction, and Fourier transform. Finally, the effects of the volatility term and the parameters of the mixed-exponential jump diffusion model on the option price in the O-U process are analyzed. In the numerical simulation, compared with the double exponential jump Heston–HW (abbreviated as DEJ-Heston–HW) model and the Heston–HW model, the mixed exponential jump model is an extension of the double exponential jump model, which can approximate any distribution in the sense of weak convergence, including arbitrary discrete distributions, normal distributions, and various thick-tailed distributions. Therefore, the MEJ-Heston–HW model adopted in this paper can better describe the price of the underlying asset.

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.

European Call Option under Stochastic Interest Rate in a Fractional Brownian Motion with Transaction Cost

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price.

Efficient simulation and valuation of equity-indexed annuities under a two-factor G2++ model

Equity-indexed annuities (EIAs) with investment guarantees are pension products sensitive to changes in the interest rate environment. A flexible and common choice for modelling this risk factor is a Hull–White model in its G2++ variant. We investigate the valuation of EIAs in this model setting and extend the literature by introducing a more efficient framework for Monte-Carlo simulation. In addition, we build on previous work by adapting an approach based on scenario matrices to a two-factor G2++ model. This method does not rely on simulations or on Fourier transformations. In numerical studies, we demonstrate its fast convergence and the limitations of techniques relying on the independence of annual returns and the central limit theorem.

Efficient Scheme for the Economic Heston–Hull–White Problem Using Novel RBF-FD Coefficients Derived from Multiquadric Function Integrals

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD estimations on three-point non-uniform stencils to construct the final approximation on a tensor grid for the 3D Heston–Hull–White (HHW) PDE, which is relevant in economics and mathematical finance. Numerical evidence and comparative analyses validate the results and the proposed scheme.

PROFITABILITY CALCULATION AND ANALYSIS FOR INTEREST RATE SWAP USING THE HULL WHITE MODEL

The London Inter-Bank Offered Rate (LIBOR) volatility had resulted in higher interest rate risks faced by many big companies and financial institutions whose assets depend on the interest rate. Eventually, there was an appearance of the new financial product development that can be used for hedging, such as interest rate swap, one of the most popular methods, utilized by most financial institutions and big companies which use LIBOR as their benchmark. In fact, it is not uncommon for numerous companies to gain negative return from this swap transaction. Therefore, in this paper, we used the Hull-White model to predict the LIBOR rate, since this model has a decent level of accuracy, calculated using Root Mean Squared Error (RMSE) method. Furthermore, the estimation of LIBOR rate was used to calculate the net value of interest rate swap transaction, in three scenarios, using, respectively, the minimum value, the average value, and the maximum value of the LIBOR’s estimation results to provide an analysis of potential P/L (Profit & Loss) exposure due to the realization of interest rate swaps.

Modeling Interest Rate Dynamics for the Bank of Ghana Rates using the Hull-White Model

Modeling Interest Rate Dynamics for the Bank of Ghana Rates using the Hull-White Model

CVA with wrong-way risk and correlation between defaults: An application to an interest rate swap

This paper presents a counterparty credit risk adjustment model to value over-the-counter financial derivatives. To do so, a bilateral credit valuation adjustment with wrong-way risk (WWR) and dependency between the defaults of both contract parties is developed in line with the Hull-White model (2012), which calculates default probabilities using a hazard rate modelled as an exponential function dependent on the value of the derivative. The model proposed incorporates a modified hazard rate for each entity, which includes the company’s own exposure to credit risk attributable to the other entity’s default. By so doing, a correlation between the respective defaults of the entities party to the financial derivative is added. The model developed is also applied to obtain the fair value of an interest rate swap and the results obtained, using Monte Carlo simulation, demonstrate that the value of this swap adjusted to the credit risk falls when the dependency between the entities’ defaults is considered.

Valuation of callable range accrual linked to CMS Spread under generalized swap market model

In this paper, we price a widely-used financial instrument, the callable range accrual linked to constant maturity swap (CMS) spread, with the least square Monte Carlo method (LSMC) under the generalized swap market model (GSMM). This method, based on the swap rate, does not only provide an intuitive pricing solution, but also captures the characteristics of the swap market, which helps market participants better face the challenge of the London interbank offered rate (LIBOR) phase-out. By choosing the prices valued by the two-factor Hull–White model in Bloomberg as a benchmark, our method provides an accurate result, following the economic intuitions. Last but not least, we examine the impacts from the shifting of the yield curve and volatility term structure by sensitivity analysis. We find that the price positively correlates with the volatility term structure among 1% shifts, while the relationship can be uncertain in larger shifts. The changes in the estimated parameters, such as the yield curve and volatility term structure, might reflect the occurrences of macroeconomic events, such as the COVID-19 pandemic.

Extension of Short Rate Model Under a Lévy Process

Extension of Short Rate Model Under a Lévy Process

Caplets/Floorlets with Backward-Looking Risk-Free Rates under the One- and Two-Factor Hull-White Models

Caplets/Floorlets with Backward-Looking Risk-Free Rates under the One- and Two-Factor Hull-White Models

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.

Optimal investment strategy for DC pension plan with inflation risk under the hybrid stochastic volatility model

In this paper, we investigate an optimal investment strategy for defined-contribution (DC) pension plan under hybrid stochastic volatility (Heston–Hull–White) model, taking account of the inflation risk and the stochastic salary. The fund wealth is invested in financial market consisting of a risk-free asset, an inflation-indexed bond and a stock with hybrid Heston–Hull–White model. The goal of the pension fund manager is to maximize the expected utility of the terminal real wealth. We derive the Hamilton–Jacobi–Bellman (HJB) equation through the dynamic programming principle, under the constant relative risk aversion (CRRA) utility function, the optimal investment strategy is obtained. Finally, a numerical example is presented to characterize the impacts of financial parameters on the optimal investment strategy.

Maximum likelihood estimation of the Hull–White model

Maximum likelihood estimation of the Hull–White model

A class of non-zero-sum stochastic differential games between two mean–variance insurers under stochastic volatility

This paper studies the open-loop equilibrium strategies for a class of non-zero-sum reinsurance–investment stochastic differential games between two insurers with a state-dependent mean expectation in the incomplete market. Both insurers are able to purchase proportional reinsurance contracts and invest their wealth in a risk-free asset and a risky asset whose price is modeled by a general stochastic volatility model. The surplus processes of two insurers are driven by two standard Brownian motions. The objective for each insurer is to find the equilibrium investment and reinsurance strategies to balance the expected return and variance of relative terminal wealth. Incorporating the forward backward stochastic differential equations (FBSDEs), we derive the sufficient conditions and obtain the general solutions of equilibrium controls for two insurers. Furthermore, we apply our theoretical results to two special stochastic volatility models (Hull–White model and Heston model). Numerical examples are also provided to illustrate our results.

Efficient Pricing of Spread Options with Stochastic Rates and Stochastic Volatility

Spread options are notoriously difficult to price without the use of Monte Carlo simulation. Some strides have been made in recent years through the application of Fourier transform methods; however, to date, these methods have only been applied to specific underlying processes including two-factor geometric Brownian motion (gBm) and three-factor stochastic volatility models. In this paper, we derive the characteristic function for the two-asset Heston–Hull–White model with a full correlation matrix and apply the two-dimensional fast Fourier transform (FFT) method to price equity spread options. Our findings suggest that the FFT is up to 50 times faster than Monte Carlo and yields similar accuracy. Furthermore, stochastic interest rates can have a material impact on long-dated out-of-the-money spread options.

Pricing for a vulnerable bull spread options using a mixed modified fractional Hull–White–Vasicek model

Pricing for a vulnerable bull spread options using a mixed modified fractional Hull–White–Vasicek model

On the Deterministic-Shift Extended CIR Model in a Negative Interest Rate Framework

In this paper, we propose a new exogenous model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model with a perfect fit to the observed term-structure. We use the difference between two independent CIR processes and apply the deterministic-shift extension technique. To allow for a fast calibration to the market swaption surface, we apply the Gram–Charlier expansion to calculate the swaption prices in our model. We run several numerical tests to demonstrate the strengths of this model by using Monte-Carlo techniques. In particular, the model produces close Bermudan swaption prices compared to Bloomberg’s Hull–White one-factor model. Moreover, it finds constant maturity swap (CMS) rates very close to Bloomberg’s CMS rates.

Dynamic initial margin estimation based on quantiles of Johnson distributions

The estimation of dynamic initial margin (DIM) is a challenging problem. We describe an accurate new approach using Johnson-type distributions, which are fitted to conditional moments, estimated using a least-squares Monte Carlo simulation (the Johnson least-squares Monte Carlo (JLSMC) algorithm). We compare the JLSMC DIM estimates with those computed using an accurate nested Monte Carlo simulation as a benchmark, and with another method that assumes portfolio changes are Gaussian. The comparisons reveal that the JLSMC algorithm is accurate and efficient, producing results that are comparable with nested Monte Carlo with an order of magnitude less computational effort. We provide illustrative examples using the Hull–White and Heston models for different derivatives and portfolios. A further advantage of our new approach is that it relies only on the readily available data that is needed for any exposure or value adjustment calculation.