Calculations in the Hull-White Model

We present an extension of stochastic volatility equity models by a stochastic Hull–White interest rate component while assuming non-zero correlations between the underlying processes. We place these systems of stochastic differential equations in the class of affine jump-diffusion–linear quadratic jump-diffusion processes so that the pricing of European products can be efficiently performed within the Fourier cosine expansion pricing framework. We compare the new stochastic volatility Schöbel–Zhu–Hull–White hybrid model with a Heston–Hull–White model, and also apply the models to price hybrid structured derivatives that combine the equity and interest rate asset classes.

This research attempts to propose closed‐form solutions for prices of credit‐risky bonds, assuming a nonzero correlation between interest rates and credit spreads. The times of default of a credit‐risky bond are modelled as the jump times of a Cox process, following the method of Lando, with an intensity that follows a Hull and White model, correlated with a similar model of the risk‐free interest rate. Under the fractional recovery of market value assumption of Duffie and Singleton, the partial differential equation (PDE) for the price of the zero‐coupon credit‐risky bond is derived. Then this PDE is analytically solved, using the method of separation of variables, and easy‐to‐implement closed‐form solutions are found. Finally, numerical examples are presented to show how these closed‐form solutions can identify the magnitude and the direction of the credit‐risky bond mispricing under different parameter assumptions.

In 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.

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

We consider the valuation of collateralized derivative contracts such as bond option or Caplet contracts. We allow for posting different collaterals such as securities or cash for the derivatives and its hedges. The pricing is based on modeling the joint evolution of collateral rate and the spread between collaterals. The Hull–White models are applied to collateral rate and spread to generate the closed pricing formula for zero coupon bond option. We also derive the pricing formula for Caplet under the Libor Market model and SABR model framework.

Although the HJM term structure model is widely accepted as the mostgeneral, and perhaps the most consistent, framework under which to studyinterest rate derivatives, the earlier models of Vasicek,Cox–Ingersoll–Ross, Hull–White, andBlack–Karasinski remain popular among both academics andpractitioners. It is often stated that these models are special cases ofthe HJM framework, but the precise links have not been fully establishedin the literature. By beginning with certain forward rate volatilityprocesses, it is possible to obtain classes of interest models under theHJM framework that closely resemble the traditional models listed above.Further, greater insight into the dynamics of the interest rate processemerges as a result of natural links being established between the modelparameters and market observed variables.

The objective of this thesis is to apply the Hull-White model and CIR++ model to price the range accrual note by using Monte Carlo simulation practically. This thesis also studies the behavior of the model by providing the sensitivity of the note’s price, when changing the model’s parameters. As a conclusion, the method used in this thesis is well applied to price the range accrual note by using Monte Carlo simulation. Models’ parameters can be practically computed from the interpolated historical data which are daily 6M USD LIBOR and weekly U.S. swap curve. These models’ prices are little affected by changing their mean reversion and speed of mean reversion in case of low volatility, but they are highly affected by changing them in case of high volatility. In addition, the changing of range of strike rates highly affects the prices of range accrual note. The narrow range provides the note’s prices are lower than the wide range for all volatility cases. Moreover, range of strike rates which is near the movement of reference rate makes the note’s prices more sensitive than other ranges. Therefore, the initial short rate highly affects the range accrual note’s prices too. Lastly, the normal distribution function of Hull-White model makes the range accrual note’s prices are higher than the chi-squares distribution function of CIR++ model.

This empirical paper investigates the effect of different distributional assumptions governing defaultable bond price uncertainty on the price of a credit default swap. We value a credit default swap using the two-factor Hull-White (1994) model for the term structure of default-free Spot interest rates and the credit spread process of a Baa-rated bond index and use the fractional recovery model of Duffie-Singleton (1999) and its multiple default extension as given in Schonbucher (1996,1998). The model is implemented using a tree algorithm outlined in Schonbucher (1999) where one factor is used for the spot interest rate and the other factor is for the credit spread or the intensity rate of a Cox process. This tree representation permits correlation between the spot rate and the credit spread dynamics of the Baa bond index, enabling us to provide a model that better fits the term structure of default-free and defaultable bond prices. We compare the values of a credit default swap obtained when the underlying risk factors are modeled with Gaussian and stable non-Gaussian distributions.

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.

Fast likelihood calculation for multivariate Gaussian phylogenetic models with shifts

Ordinal regression is used to model the ordinal response variable as functions of several explanatory variables. The most commonly used model for ordinal regression is the proportional odds model (POM). The classical technique for estimating the unknown parameters of this model is the maximum likelihood (ML) estimator. However, this method is not suitable for solving problems with extreme observations. A robust regression method is needed to handle the problem of extreme points in the data. This study proposes Huber M-estimator as a robust method to estimate the parameters of the POM with a logistic link function and polytomous explanatory variables. This study assesses ML estimator performance and the robust method proposed through an extensive Monte Carlo simulation study conducted using statistical software, R. Measurement for comparisons are bias, RMSE, and Lipsitzs' goodness of fit test. Various sample sizes, percentages of contamination, and residual standard deviations are considered in the simulation study. Preliminary results show that Huber estimates provide the best results for parameter estimation and overall model fitting. Huber's estimator has reached a 50% breakdown point for data containing extreme points that are quite far from most points. In addition, the presence of extreme points that have only a distance of two times far from most points has no major impact on ML estimates. This means that the estimates for ML and Huber may yield the same results if the model's residual values are between -2 and 2. This situation may also occur for data with a percentage of contamination below 5%.

The high persistence of interest rates has important implications for the preferred method used to estimate term structure models. We study the finite-sample properties of two standard dynamic simulation methods—efficient method of moments (EMM) and indirect inference—when they are applied to an first order autoregressive (AR[1]) process with Gaussian innovations. When simulated data are as persistent as interest rates, the finite-sample properties of EMM differ both from their asymptotic properties and from the finite-sample properties of indirect inference and maximum likelihood. EMM produces larger confidence bounds than indirect inference and maximum likelihood, yet is much less likely to contain the true parameter value. This is primarily because the population variance of the data plays a much larger role in the EMM conditions than in the moment conditions for either indirect inference or maximum likelihood. These results suggest that, under Gaussian assumptions, indirect inference (if practical) is preferable to EMM when working with persistent data such as interest rates. EMM's emphasis on the population variance strongly enforces stationarity on the underlying process, so this same reasoning suggests that EMM may be preferable in settings where stability and stationarity are important and difficult to impose. Copyright The Author 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

In this article, the authors propose an alternative approach for pricing bond options and swaptions under the one-factor Hull–White model. Their proposal differs from the existing models used to evaluate these type of instruments when the evolution of the term structure of interest rates is modeled by short-rate models. Assuming that the forward price of a coupon bond is a martingale under the forward risk-neutral measure, the proposed model involves deriving the volatility of the coupon bond as a function of its stochastic duration. Once the volatility function is defined, the price of options on coupon bonds can be derived by simply applying the standard no-arbitrage pricing theory in the context of the one-factor Hull–White model. Given the equivalence between the price of a coupon bond and the price of the corresponding swaptions, this approach can be adopted to calibrate the parameters of the one-factor Hull–White model using swaptions quoted in the market. The new solution proposed for the calibration can be considered as an alternative with respect to the existing approaches proposed in the literature and used currently by practitioners. Numerical results are provided in order to analyze the features of the proposed model for calibration purposes. <b>TOPICS:</b>Fixed income and structured finance, derivatives applications, quantitative methods

The problem of estimating parameters theta that determine the mean mu ( theta ) of a Gaussian-distributed observation X is considered. It is noted that the maximum-likelihood (ML) estimate-in this case, the least-squares estimate-has desirable statistical properties but can be difficult to compute when mu ( theta ) is a nonlinear function of theta . An estimate formed by combining ML estimates based on subsections of the data vector X is proposed as a computationally inexpensive alternative. It is shown that this alternative estimate, termed the divide-and-conquer estimate, has ML performance in the small-error region when the data vector X is appropriately subdivided. As an example application, an inexpensive range-difference-based position estimator is derived and shown by Monte-Carlo simulation to have small-error-region mean-square error equal to the Cramer-Rao bound. >

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.