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# Tag Info

10

For the Hull-White model, where \begin{align*} dr_t = (\theta(t)-a r_t)dt+ \sigma dW_t, \end{align*} under the risk-neutral measure, we have that, for $t\ge s \ge 0$, \begin{align*} r_t = e^{-a(t-s)} r_s + \int_s^t \theta(u)e^{-a(t-u)} du + \int_s^t \sigma e^{-a(t-u)} dW_u. \end{align*} Then, if $\theta$ is a constant, \begin{align*} r_t \mid r_s &\sim ...

9

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 ...

9

Concerning your first question, this depends on what curve, currency, etc. you are interested in. The general method for constructing yield curves is called bootstrapping which allows you to derive spot, zero-coupon rates from the known price of coupon-bearing instruments $-$ such as bonds or swaps. In general: You start picking short-term (typically less ...

8

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 ...

8

We assume that the process $\{r(t), \, t \ge 0\}$ satisfies an SDE of the form \begin{align*} dr(t) = \big( \theta(t) - a(t) r(t) \big)dt + \sigma(t) dW_t, \quad t > 0, \end{align*} where $\{W_t, \, t \ge 0 \}$ is a standard Brwonian motion. Note that \begin{align*} d\left(e^{\int_0^t a(u) du}r(t) \right) &=a(t) e^{\int_0^t a(u) du}r(t) dt + e^{\int_0^...

7

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 ...

7

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

5

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 ...

5

I find your approach to calibration (training an ANN to learn the inverse function f-1 from a training set of 'market_prices = f(model_parameters)' interesting, novel (at least this is the first time I am hearing about it) and definitely worth investigating further. If you make it work, you have almost instantaneous calibration and a methodology applicable ...

5

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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 ...

4

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 ...

4

A negative mean reversion makes the dynamics of the asset explode. If the model is: $$dr=[\theta-\alpha r]dt+\sigma dW$$ The expected value in this model is: $$\mathbb{E}(r)= r(0) e^{-\alpha t} + \frac{\theta}{\alpha} (1-e^{-\alpha t} )$$ If $\alpha<0$ $\mathbb{E}(r)$ goes to $\infty$ or $-\infty$, depending on if $r(0)$ is above or below the "long ...

3

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 ...

3

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 - ...

3

Based on this question, for the Hull-White model of the form \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 a standard Brownian motion, the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value is given by \begin{align*} P(t, ...

3

There are many resources describing how to build a trinomial tree for the Hull & White model (for instance http://www-2.rotman.utoronto.ca/~hull/downloadablepublications/TreeBuilding.pdf), and finite differences schemes are popular as well. These apply to the single curve case. To deal with the multi curve case while keeping everything 1 factor, the ...

3

Fixing the mean reversion, and parameterizing the volatility as a step function or as a piecewise linear function, the volatility can be bootstrapped exactly to a set of vanilla options sorted by expiries. This is a very stable and fast procedure, akin to the bootstrapping of a discount curve onto rate instruments. For instance when pricing a bermuda ...

2

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 components,...

2

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.

2

That's right. Calibration is done in other to obtain one 'a' (mean reversion rate) and one sigma for the entire curve. This case is referred to as the Hull White model with constant mean reversion, i.e. the parameters are not time dependent.

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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

2

I took a look at the paper and would contend that it is a typo. I would assume he just copy-pasted the equation - for it is exactly the same for the two factor model cf. eq (157) and eq (41) If you follow his reasoning and his notation it would make no sense to use the observed sample variance. He always denotes the variace by $\sigma^2$ and the standard-...

2

There are two test methods described in Aronson's book; White's Reality Check and a Permutation test. At the heart of both is the idea of a "position vector," e.g. a numerical vector of a series of 1, -1 or 0, which correspond to long, short or neutral positions. For example, a vector of [ 1 1 1 1 1 0 0 0 -1 -1 -1 -1 ] would represent being long for 5 ...

2

Under the Hull-White interest rate model, the short rate $r_t$ satisfies a risk-neutral SDE of the form \begin{align*} dr_t = (\theta(t)-a r_t)dt+ \sigma dW_t. \end{align*} The price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value is then given by \begin{align*} P(t, T) &= A(t, T) e^{-B(t, T) r_t}, \end{align*} where \begin{...

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

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 $t>0$,...

2

In order to compute $$P_0 = \mathbb {E}[C (\hat{V})]$$ where $$\hat{V} = \frac {1}{T} \int_0^T \sigma^2_s ds$$ and $$d\sigma_t = \sigma_t (\alpha dt + \gamma dW_t)$$ using Monte Carlo, you should: Generate stochastic volatility paths over $[0,T]$ by discretising the above SDE (which here defines a GBM, not a Hull & White diffusion) Calculate the ...

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