# Tag Info

11

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 $$\theta\mathrm{(t)=}\frac{\partial f(0,t)}{\partial T}+\alpha f(0,t)+\frac{\sigma^2}{2\alpha}(1-e^{-2\alpha t}),$$ ...

7

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 ...

6

The Hull-White model is an no-arbitrage short rate model. It is used to price interest rate derivatives such as caps and floors. It generalises the seminal equilibrium model from Vasicek (1977). The Model The model postulates that $$\mathrm{d}r_t=\kappa_t(\theta_t-r_t)\mathrm{d}t+\sigma_t \mathrm{d}W_t.$$ Two of the key model features are that the short ...

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

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, ...

5

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 ...

4

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

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 ...

4

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 ...

4

Here is the price in HW[4] for a ZCB at time $t$: \begin{align} P(t,T) &= A(t,T) e^{-B(t,T) r(t)}\\ A(t,T) &= {\frac {P(0,T)} {P(0,t)}} \exp \Bigl( B(t,T)F(0,t) - {\frac {\sigma^2} {4a}} B(t,T)^2(1-e^{-2at})\Bigr)\\ B(t,T) &= {\frac {1-e^{-a(T-t)}} {a}} \end{align} You seem to be simulating to rate $r(t)$ at time $t$ and putting that into your ...

4

The unembellished Hull-White model is not used very much in practice, because it is under-parameterized to handle a term structure of risk-free rates, and hence cannot be calibrated in any reasonable way. As you have probably remarked, in its usual form it starts the short rate $r$ at some single value, and evolves $r$ according to just a couple volatility ...

4

Given a initial discount bond $P^M(0, T)$ curve, the expression for $\theta(t)$ in the Hull White Short Rate model is a know result given by: $$\theta(t) = \frac{1}{\kappa} \cdot f'(0, t) + f(0, t) + \frac{1}{2} \cdot \left( \frac{\sigma}{\kappa} \right)^2 \cdot \left( 1 - e^{-2 \kappa t} \right).$$ I have used a notation where the spot rate dynamics is ...

3

For simplicity, we assume that $\alpha$ is a positive constant. You need to show that, for any $t>0$, \begin{align*} M_t = \int_0^t e^{\alpha u} dW_u \end{align*} is normally distributed, where $\{W_t, \, t \ge 0\}$ is a standard Brownian motion with respect to the filtration $\{\mathscr{F}_t,\, t \ge 0\}$. Here, we employ the time-changed Brownian ...

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

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{...

3

Let $K$ be the forward exchange rate determined at time $t$ for maturity $T$. Then the payoff at time $T$ is given by $S_T-K$, which has zero value at time $t$. Let $Q$ and $Q^f$ be the respective domestic and foreign risk-neutral measures, and $E^Q$ and $E^{Q^f}$ be the corresponding expectation operators. Moreover, let $B^d_T = e^{\int_0^t r^d_sds}$ and \$B^...

3

Local and/or stochastic vol extensions of HW (incl. multi-factor) were produced around the mid 1990s, more or less independently in a number of research papers, the most notable being Cheyette (1992) and Ritchken-Sankarasubramanian (1995). Quants generally call the one-factor extension "Cheyette Model" and the multi-factor version "Multi-Factor Cheyette" or ...

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

I assume you are asking for the popular Hull/White one-factor model. You could eiter calibrate them to Cap/Floor Volas or to swaption volas. Don't try to fit a model to both at the same time. You should decide this by the products you want to price. If you want to price caps/floors with the model, calibrate it to cap/floor volas and vice versa. Calibrate ...

3

On the Monte-Carlo Simulation of the Hull-White Model: You can find the specification of the Euler Scheme simulation in https://ssrn.com/abstract=2737091 . The paper gives the exact Euler step, i.e. the simulation step does not have a simulation time discretisation error. An implementation of this in Java is available as part of http://finmath.net/finmath-...

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