# Tag Info

7

The Hull-White model can represents the risk free rate as a stochastic process, that is, in terms of expected return and volatility. The zero curve only gives you expected returns and you have to find a source to calibrate volatility, as FQuant told you. Common volatility sources used for this calibration are historical series of the zero curve or ...

5

The one-factor Hull-White model is given by $$dr(t) = (\theta(t) - \alpha\; r(t))\,dt + \sigma\, dW(t)\,\!.$$ The zero curves are only sufficient for the calibration of the parameter $\theta(t)$, which is given in terms of them by $$\theta\mathrm{(t)=}\frac{\partial f(0,t)}{\partial T}+\alpha f(0,t)+\frac{\sigma^2}{2a}(1-e^{-2\alpha t}),$$ where ...

4

Milstein Scheme This scheme is described in Glasserman (2003) and in Kloeden and Platen (1992) for general processes.Hence, for simplicity, we can assume that the Stochastic Process is driven by the SDE \begin{align} &dX_t=\Xi(t,X_t)dt+\Sigma(t,X_t)dW_t\\ \end{align} Milstein discretization is, \begin{align} dX_{t+\Delta ...

4

I will refer to "Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit" by Damiano Brigo and Fabio Mercurio. In chapter 3 (One-factor short-rate models) they have a very nice table which lists some of the properties of instantaneous short rate models. In both of your models you know the distribution of $r_t$. The huge difference ...

3

In fact you can calibrate $\theta(t)$ piecewise constant and $\alpha$ and $\sigma$ to bond prices only. You don't need the swaption prices in mM. If you let $\sigma(t)$ depend on $t$ (this is called the generalized Hull-White model) then you need information about the options market. For the model as you write it you don't necessarily need MC to calculate ...

3

General knowledge: The reference for short rates models is: Interest Rate Models, by D. Brigo & F. Mercurio, Springer Worth the cost. You can find a summary of the propeties of the "dr" models p15 & p19: Interest Rate Models: Paradigm shifts in recent years, D. Brigo, Columbia University Seminar You will see the quote p19: "Pricing models need to ...

2

The claim that interest rates don't follow long term trends is not consistent with observed data. The idea of mean reversion is that interest rates do not rise or fall without bound, but are limited by economic and political factors. But there is no indication that this oscillation of short rates should happen around a constant mean. Allowing the mean ...

2

The first principle component of interest rates will not help you capture the term structure better at all. It will basically remove all term structure affects you are going to see. When we decompose the returns on interest rates you are going to get 3 PC's which explain 99.9% of the variance. PC1 - Level of the interest rates (~90% of variance) PC2 - ...

2

Once the single-factor Hull-White model is calibrated, you can compute zero-coupon bond prices in closed form (i.e., without running simulations). See http://en.wikipedia.org/wiki/Hull%E2%80%93White_model#Analysis_of_the_one-factor_model .

2

This is a special case of the question of why $$\int_0^T f(t) dW_t$$ is normally distributed for a continuous function $f(t).$ This Ito integral can be approximated by a sum $$\sum_{i=0}^{N-1} f(i T/N) (W_{(i+1)T/N} - W_{i T/N}) .$$ The Brownian increments $(W_{(i+1)T/N} - W_{i T/N})$ are independent normally distributed random variables. The key point ...

1

Here is a solution without using the PDE technique, which is preferred as we do not need to assume the affine form of a zero-coupon price from the start. we assume that, under the risk-neutral measure, \begin{align*} dr_t = (\theta(t)-a r_t) dt + \sigma dW_t, \end{align*} where $a$ and $\sigma$ are constants, $a(t)$ is a deterministic function, and $W_t$ is ...

1

For simplicity, We assume that $\alpha$ is a positive constant. You need to show that, for any $t>0$, \begin{align*} \int_0^t e^{\alpha u} dW_u \end{align*} is normally distributed. Consider the process $\{X_t, t \geq 0\}$, where \begin{align*} X_t = \frac{1}{\sqrt{\frac{1}{t}\int_0^t e^{2\alpha u} du}}\int_0^t e^{\alpha u} dW_u, \end{align*} for ...

1

Have you tried: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1514192 It should cover it, anyway Hull White fits the HJM framework so you should be able to calibrate it to swaptions or something if not the yield curve

1

The Heath-Jarrow-Morton representations of short interest rate models (such as Hull-White) will give you an expression for the evolution of the entire forward curve, but it doesn't make the problem any easier. The closed form ZC formulae you mention above are probably your best bet.

1

A way to go would be to linearly build indepedant interest rates to eliminate correlation effects. How do you do that ? You linearly build orthogonal interest rates from your starting ones. This is totaly equivalent to diagonalising correlation matrix, which is the principle of PCA. Using information criteria you can then choose to remove lowest ...

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