Hull-White Interest Rate Model Research Articles

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.

Valuation of general GMWB annuities in a low interest rate environment

Variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees.

Exploring Dynamic Asset Pricing within Bachelier’s Market Model

This paper delves into the dynamics of asset pricing within Bachelier’s market model (BMM), elucidating the representation of risky asset price dynamics and the definition of riskless assets. It highlights the fundamental differences between BMM and the Black–Scholes–Merton market model (BSMMM), including the extension of BMM to handle assets yielding a simple dividend. Our investigation further explores Bachelier’s term structure of interest rates (BTSIR), introducing a novel version of Bachelier’s Heath–Jarrow–Morton model and adapting the Hull–White interest rate model to fit BMM. This study concludes by examining the applicability of BMM in real-world scenarios, such as those involving environmental, social, and governance (ESG)-adjusted stock prices and commodity spreads.

Valuation of callable accreting interest rate swaps: Least squares Monte-Carlo method under Hull-White interest rate model

Using the Hull-White interest rate model, this paper proposes a valuation method of callable accreting interest rate swap (CAIRS) and how it can be used for managing the risk of zero callable bonds (ZCBs). Firstly, CAIRS can be decomposed into accreting payer interest rate swaps and Bermudan options. Considering the financial valuation of both components, the former can be valued directly while the latter has no close-form due to its early exercise characteristics. Using the Least Squares Monte-Carlo method (LSM) proposed by Longstaff and Schwartz (2001), we find that the two options embedded in ZCB and CAIRS have the same exercise strategy since the terms of the swaps will include the bonds in practice. However, the cash flow of risk management in swaps and bonds can differ when considering the time value. Hence, CAIRS is not the best financial instrument for managing risks of ZCB under the current design.

Dual-curve Hull–White interest rate model with stochastic volatility

The sub-prime mortgage crisis of 2008 had a great effect on the financial market; it led to new market features and regulations. In particular, the dual curve feature has appeared in the over-the-counter market. As a result, the yield curve for computing forward values and the yield curve for discounting are no longer the same due to counterparty default risk. In addition, deleveraging regulation encourages financial institutions to invest in widely known simple products, typically at a single short rate. The classical Hull–White model, which is typically useful for simple interest rate products, is inadequate for the dual-curve situation. This study extends the Hull–White model with stochastic volatility to the dual-curve economy. Analytical solutions are derived for vanilla securities such as risk bearing zero bonds, bond options, and caplets. Our formulas facilitate the calibration of parameters by simultaneously fitting to the dual curves and the implied volatility surfaces.

Optimal dynamic asset-liability management with stochastic interest rates and inflation risks

This paper considers an optimal asset-liability management problem with stochastic interest rates and inflation risks under the expected utility maximization framework, where the stochastic interest rate follows the Hull-White interest rate model and the inflation risk is modelled by an additional stochastic process. The investor can invest in n+1 assets: cash, a default-free zero-coupon bond, an inflation-indexed bond and n−2 stocks. The liability process is given by a geometric Brownian motion rather than a Brownian motion to ensure a definite liability value. Applying the stochastic control theory and partial differential equation approach, we obtain the explicit solutions of optimal investment strategies for the power utility and exponential utility functions. We also provide numerical examples to show the effects of model parameters on the optimal investment strategies.

Valuation of correlation options under a stochastic interest rate model with regime switching

We consider the valuation of a correlation option, a two-factor analog of a European call option, under a Hull-White interest rate model with regime switching. More specifically, the model parameters are modulated by an observable, continuous-time, finite-state Markov chain. We obtain an integral pricing formula for the correlation option by adopting the techniques of measure changes and inverse Fourier transform. Numerical analysis, via the fast Fourier transform, is provided to illustrate the practical implementation of our model.

Pricing for options in a mixed fractional Hull–White interest rate model

In this paper, we present a pricing model for European options in a mixed fractional Hull–White interest rate model. By using the variable transform techniques and mathematical physics methods, we derive closed-form pricing formulas for this pricing problem, which are the main contribution of this paper and expand the relevant literature’s conclusions. Moreover, we provide numerical examples to illustrate the effects of main parameters of the mixed fractional interest rate model on the option price. Numerical results show that the long memory property of interest rates plays an important role in determining the option price and cannot be neglected in option pricing.

An FFT approach for option pricing under a regime-switching stochastic interest rate model

ABSTRACTIn this article, we investigate the pricing of European-style options under a Markovian regime-switching Hull–White interest rate model. The parameters of this model, including the mean-reversion level, the volatility of the stochastic interest rate, and the volatility of an asset’s value, are modulated by an observable, continuous-time, finite-state Markov chain. A closed-form expression for the characteristic function of the logarithmic terminal asset price is derived. Then, using the fast Fourier transform, a price of a European-style option is computed. In a two-state Markov chain case, numerical examples and empirical studies are presented to illustrate the practical implementation of the model.

Market-Consistent Valuation and Funding of Cash Balance Pensions

Cash balance pension benefits are accumulated at guaranteed crediting rates, usually based on yields on government securities. Viewed as a financial liability, the benefit is a form of interest rate derivative, which can be valued using financial models and theory. In this article, we derive the market value for a range of commonly used crediting rates, assuming the accrued benefit liability comprises the past contributions, allowing for full interest credits up to a known future retirement date. We use the Hull-White interest rate model to determine crediting rates and to determine the market value. We explore the risks associated with different crediting rate choices by evaluating the liability using market data from 1998 to 2013. We propose two other approaches to the accrued benefit. The first approach assumes the accrued benefit comprises past contributions with interest up to the valuation date, but no future interest credits. Future credits on past contributions are assumed funded through future contributions. The second method projects all contributions and interest to retirement and assumes equal units of accrual of this projected benefit in each year of service. We compare the three approaches through numerical examples.

Calculations in the Hull-White Model

In this note many trivial (and some advanced) calculations in the Hull-White interest rate model are performed. Especially the theta function is derived. This function ensures that prices of zero coupon bonds in the model fit market values of zero coupon bonds. In addition, a section explaining how to estimate the model using maximum likelihood is included.

A fractal version of the Hull–White interest rate model

This paper develops a new version of the Hull–White's model of interest rates, in which the volatility of the short term rate is driven by a Markov switching multifractal model. The interest rate dynamics is still mean reverting but the constant volatility of the Brownian motion is replaced by a multifractal process so as to capture persistent volatility shocks. In this setting, we infer properties of the short term rate distribution, a semi-closed form expression for bond prices and their dynamics under a forward measure. Finally, our work is illustrated by a numerical application in which we assess the exposure of a bonds portfolio to the interest risk.

A Fractal Version of the Hull-White Interest Rate Model

This paper develops a new version of the Hull-White's model of interest rates, in which the volatility of the short term rate is driven by a Markov switching multifractal model. The interest rate dynamics is still mean reverting but the constant volatility of the Brownian motion is replaced by a multifractal process so as to capture persistent volatility shocks. In this setting, we infer properties of the short term rate distribution, a semi closed form expression for bond prices and their dynamics under a forward measure. Finally, our work is illustrated by a numerical application in which we assess the exposure of a bonds portfolio to the interest risk.

Efficient calibration of the Hull White model

SUMMARY In this paper we study the calibration of the Hull White interest rate model for the pricing of Bermudan swaptions and Bermudan bond options. A common modelling approach for the calibration of the Hull White model is to choose the model parameters such that market prices of corresponding European derivatives are replicated by the model. This requires that a multidimensional non-linear optimization problem has to be solved. The focus of this paper lies in the efficient formulation and numerical solution of this optimization problem. We investigate the numerical properties of the iterative solution of the optimization problem by means of a Gauss Newton and an Adjoint Broyden quasi-Newton method. Required derivatives of the objective function are evaluated by techniques of Automatic Differentiation. The algorithms are benchmarked to Minpack's general purpose solver HYBRD. Numerical results for the calibration of long-term Bermudan swaptions are reported. Copyright © 2011 John Wiley & Sons, Ltd.