Hull-White

Absorbents, wood pulp (sawdust), wood flour (very fine sawdust), apricot pits, ground coarse or fine, com flour (very fine com meal), bagasse (sugar cane pitch), oat hulls, white starch (usually com ormilo), chalk, zinc oxide, ground coal, charcoal, sulfur, and lampblack. 

Even British steam warsliips of the Victorian era were not camouflaged or painted in any way to render them less visible. Their colour schemes, as revealed by paintings of contemporary Naval Reviews at Spithead, comprised black hull, white upperworks and buff or black funnels, colours that even had a rather commercial or mercantile look about them. 

The traditional one-, two- and multi-factor equilibrium models, known as ajfine term structure models (see James and Webber, 2000 or Duffie, 1996, p. 136). These include Gaussian affine models such as Vasicek, Hull-White and Steeley, where the model describes a process with constant volatility and models that have a square-root volatility such as Cox-Ingersoll-Ross (CIR) ... 

The most commonly used models are the Hull-White type models which are relatively straightforward to implement, although HJM models are also more... 

It might be considered to be more reahstic to consider that there are no constant parameters for the drift rate and the standard deviation that would ensure that the price of a zero-coupon bond at any time is exactly the same as that suggested by observed market yields. For this reason, a modified version of the Vasicek model has been described by Hull and White (1990), known as the Hull-White or extended Vasicek model, which we will consider later. 

The Hull-White (1990) model is an extension of the Vasicek model designed to produce a precise fit with the current term structure of rates. It is also known as the extended Vasicek model, with the interest rate following a process described by Equation (3.48) ... 

The Hull-White model can be fitted to an initial term structure, and also a volatility term structure. A comprehensive analysis is given in Pelsser (1996) as well as James and Webber (2000). 

The forward rate function/at time t is not static and is a function of the short-rate r at time f, the time t and the time to maturity T. The Hull-White model can be calibrated in terms of the forward rate /. That is, at time t the information (parameters) required to implement this are the short-rate r(t), the standard deviation a of the short-rate, the forward rate / and the standard deviations Bj t, T)a t) of the forward rates at time t. If the forward rates are known in a form that allows their first differential to be calculated with respect to t, using the other information, it is possible to calculate the function Bj, the derivative of this function and thereby the value for a f), using the relationship in Equation (3.56) ... 

To recap on the issues involved in fitting the extended Vasicek model or Hull-White model this describes the short-rate process as following the form... 

The Vasicek, Cox-Ingersoll-Ross, Hull-White and other models incorporate mean reversion. As the time to maturity increases and as it approaches infinity, the forward rates converge to a point at the long-run mean reversion level of the current short-rate. This is the limiting level of the forward rate and is a function of the volatility of the current short-rate. As the time to maturity approaches zero, the short-term forward rate converges to the same level as the instantaneous short-rate. In the Merton and Vasicek models, the mean of the short-rate over the maturity period T is assumed to be constant. The same constant for the mean, or the drift of the interest rate, is described in the Ho-Lee model, but not the extended Vasicek or Hull-White model. 

Cap prices can also be valued analytically using the Hull-White model. The cap prices calculated using the implied volatilities of interest rate caps and the Black-Scholes model serve as the calibrating instruments. After the Hull-White model has been calibrated, the parameters a and o that minimize a goodness-of-fit measure can be used to solve for the convexity bias. 

The estimated or assumed values of the parameters a and o can be used in combination with the Hull-White convexity adjustment term to estimate the convexity bias embedded in futures rates. 

The NX CR Engine also relies on multiple models, depending on the type of product being modeled. For example, it utilizes 1-dimensional models for valuing callable credit default swaps and uses more complicated models (such as quasi-multiperiod, diffusion multiperiod (Hull-White), Copula function, etc.) for baskets and CDOs. 

Equation (4.3) describes a stochastic short-rate process modified to include the direction of change. To be more realistic, it should also include a term describing the tendency of interest rates to drift back to their long-run average level. This process is known as mean reversion and is perhaps best captured in the Hull-White model. Adding a general specification of mean reversion to (4.3) results in (4.4). 

The academic literature and market participants have proposed a large number of alternatives to the Vasicek term-structure model and models, such as the Hull-White model, that are based on it. Like those they seek to replace, each of the alternatives has advantages and disadvant es. 

To model these phenomena, different modification of the Black Scholes model have been proposed. The most two common models are the stochastic volatility model (see Hull White process (Hull White 1987)), CIR process (see (Cox Ross 1976)) and Ornstein-Uhmenbeck process (see (Heston 1993, Stein Stein 1991)) and the jump diffusion model. 

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