13 votes
Accepted

Hull-White formula on wikipedia, correct?

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)} ...
  • 20.5k
9 votes
Accepted

How to get set the theta function in the Hull-White model to replicate the current yield curve

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 ...
9 votes
Accepted

Proof behind solution for theta in Hull-White with time-dependent volatility and mean reversion?

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, \,...
  • 20.5k
8 votes
Accepted

Extended Hull White Interest Rate Model for Zero Coupon Bond

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 ...
  • 20.5k
8 votes

Hull-White model applied in practice

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 ...
  • 14k
7 votes

Calibrating Hull-White model

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 ...
6 votes

Hull-White model applied in practice

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 ...
  • 14.5k
5 votes

Why does the short rate in the Hull White model follow a normal distribution?

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(...
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5 votes

Black Derman Toy model: from tree to differential equation

From the gentleman and scholar Emanuel Derman. Emanuel states "the last two pages answer the question asked". https://www.dropbox.com/s/cg299qsbquuqdru/TwitterNotesOnBDT.2017.pdf?dl=0&m= Please ...
5 votes

Zero-coupon bond price volatility with one factor Hull White interest rate model

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 ...
  • 20.5k
5 votes
Accepted

Quantlib: How do I price a ZC bond using the Hull White model?

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)...
  • 2,856
5 votes
Accepted

What is the definition of "co-terminal swaptions"? why they are important in the calibration process?

This question has partially already been answered here. Let's do a simple example to illustrate the idea though. Take a 5y Bermudan callable S/A USD bond. How would you reconstruct the multi-call ...
  • 940
4 votes

Consequence of negative mean reversion of hull white one factor model

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} + \...
4 votes

Why Hull White 2 Factor model can't capture vol skew?

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) ...
4 votes
Accepted

Instruments for calibrating Hull White Model

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 ...
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4 votes
Accepted

why calibrate volatility and fix the mean reversion

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 ...
4 votes
Accepted

Hull-White Monte Carlo simulation - mean reversion function

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) + ...
  • 711
3 votes
Accepted

LIBOR rates from Vasicek/Hull-White model?

In practice, you can calibrate to either 1 month libor or 3 month libor, but not both. That's because there's a basis swap between 1 month libor and 3 month libor that can't be explained by your ...
  • 14.3k
3 votes
Accepted

FX Hull-White model

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 ...
  • 20.5k
3 votes
Accepted

zero coupon bond pricing formula using Hull White

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 ...
  • 20.5k
3 votes

Applicability of PCA to get historical volatilities to calibrate interest rates trees

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 ...
  • 739
3 votes
Accepted

Hull White Stochastic Volatility Model in Matlab

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 ...
  • 14.1k
3 votes
Accepted

Why does the short rate in the Hull White model follow a normal distribution?

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 $\{...
  • 20.5k
3 votes
Accepted

Lattice pricing of derivatives under multi curve framework (OIS and LIBOR)

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 ...
3 votes

Hull White help needed

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. ...
3 votes

Hull white model Monte Carlo simulation Zero Coupon Bond

You do not model $v(t)$ by Monte-Carlo! As your excerpt explains $\phi(t)$ is a deterministic function of the initial yield curve and accordingly $v(t)$ is deterministic as well. Two further remarks: (...
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3 votes

Hull-White model: match between HJM framework and short model formulation

Note that \begin{align*} f(t, T) = f(0, T) + \int_0^t\alpha(u,T)du+\int_0^t\sigma e^{-a(T-u)}dW_u, \end{align*} where, based on this question, \begin{align*} f(0, T) = \int_0^T \theta(u) e^{-a(T-u)} ...
  • 20.5k
3 votes
Accepted

QuantLib - Calibrating Hull White one-factor on negative interest rates

When building a SwaptionHelper, you have to tell QuantLib what kind of volatility you are inputting. There are three options: Black Vol, Shifted Black Vol and Normal Vol. Since you don't have black ...
  • 5,375

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