Questions tagged [hullwhite]

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Hull White Model and Local Volatility

What is the practical use case for Hull White 1F model. In its crude form , it allows us to calibrate mean levels to yield curve and the speed of MR and Vol to a single value which describes the term ...
Kavinkumar R's user avatar
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Bond option price under hull-white model with different settlement and expiration dates

I am aware of bond option (lets say, call option) price formula under Hull-White model, for example, here - https://www.applied-financial-mathematics.de/sites/default/files/Teaching/...
Madhuresh's user avatar
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Euribor 3M simulation

I am required to simulate the trajectory of the Euribor3M rate as it is crucial for determining the future cash flows of my derivative instrument. I've received guidance to employ the Hull-White model....
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Forward ZC bond with Hull-White model

I am interested in finding the price of a forward zero coupon bond B(t0,t1,t2) under the Hull-White model. To arrive at this result, it makes sense to proceed in this way: calculate ...
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Multiple factor Hull-While and yield curve deformation

I am currently studying rate models and I understand that the One-Factor model has some incompleteness: The yield-curve can only be shifted. But I don’t understand what parameter controls this shift ( ...
Adel Chakir's user avatar
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Pricing a swaption in a Hull-White model with two curves

Let's say the forward swap rate $s_t$ is equal to $$s_t = \frac{\sum_{j=1}^N \delta_j^{\textrm{float}} P_{t,T_j^{\textrm{float}}}^{\textrm{disc}} L_t^{[T_{j-1}^{\textrm{float}},T_j^{\textrm{float}}]}}{...
11house's user avatar
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Instantaneous forward rate function to use in HJM framework

HJM framework uses the instantaneous forward rate $f(t,T)$ in the resulting dynamics and pricing formulas (like in Hull-White or Ho-Lee model). But clearly market does not have an $f(t,T)$ formula, so ...
Oliver Mohr Bonometti's user avatar
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Derive the convexity adjustment for inflation YoY swap with unconventional payoff

I'm trying to solve for the convexity adjustment for an inflation YoY swap with unconventional payoff, where $I_i$ is CPI at time i: $Notional * ([I_i/I_{i-1}]^{Day Count Fraction} - 1)$ In the normal ...
bphone's user avatar
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1 answer
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Are instantaneous short rates compatible across models?

If I calibrate the Vasicek's yield curve to the Nelson-Siegel's (NS) yield curve, can I assume that $r_V(0) = r_{NS}(0) = \beta_0 + \beta_1$ or not? NS short rate: $r_{NS}(S) = β_0 + β_1 e^{-S/\tau} + ...
Евгений Данилов's user avatar
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Simulating Hull-White Model in Python

I first simulated the short rate in the Vasicek model using the following code, which is equivalent to simulating the following normal distribution $r_{t} \sim N\left(r_{0}e^{-at} + b\left(1-e^{-at}\...
Guyon Van Rooij's user avatar
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Zero-coupon bond price in the risk-neutral word

In Hull's technical note (http://www-2.rotman.utoronto.ca/~hull/technicalnotes/TechnicalNote31.pdf), on page 3, in the third row from the bottom, a plus sign suddenly appears before σ dz in an ...
Nitram's user avatar
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Convexity adjustment for inflation

I'd like to prove the following equation: $\mathbb{E}\left[\frac{e^{\int_0^{T_1} y_s d s}}{e^{\int_0^{T_2} r_s d s}}\right]=\frac{\mathbb{E}\left[e^{\int_0^{T_2} r_s d s}\right]}{\mathbb{E}\left[e^{\...
ice_fox21's user avatar
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QuantLib: null term structure set to this instance of index

I'm playing around with QuantLib and trying to price an interest rate cap using HW 1F model. ...
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Replicating strategy for the zero-coupon bond in the Hull-White model

I would like to derive a self-financing, replicating strategy for a call option on a zero-coupon bond in the Hull-White model. I tried to use Proposition 21.10 in 1, in which zero-coupon bonds are ...
Jorge's user avatar
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Is it possible to estimate 'future zero curves' using Hull-White 1 factor model?

With regards to the following: https://www.mathworks.com/help/fininst/price-bermudan-swaptions-with-different-interest-rate-models.html It seems to me that the HW1 model is used to generate the ...
user160738's user avatar
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Correlating multiple two-factor hull white processes in the context of CVA

Let's say we have two rates processes (EUR and USD) both following a 2-factor hull white. Since it is in the context of CVA calculation, we calibrate both processes using Cap/Floors and swaptions (or ...
oumda's user avatar
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How to calibrate short-rate model (Hull-White) using historical domestic IBOR curve without other derivative price? [duplicate]

I'm trying to calibrate Hull White model in VietNam market to value IRS, CSS products which are not publicly traded. dr(t)=(θ(t)−αr(t))dt+σ(t)dW(t) I only ...
Quý Nguyễn's user avatar
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Interest Rate Calibration and Backtesting under Fed's raising rates 2022-2023

With the Fed raising interest rates so fast and so drastically, classical interest rate models such as the Black Karasinski (BK), Hull-White (HW), etc., may have trouble calibrating to current rate ...
RMA's user avatar
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2 answers
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Workaround for Hull-White short rate model in market without swaptions

Every time I search calibration methods in short-rate models such as Hull-White, I always find information on how to do it with swaptions market data. Yet, I can't find anything about how to do it in ...
Oliver Mohr Bonometti's user avatar
3 votes
1 answer
226 views

Derivations of the pricing PDE for the Heston-Hull-White or Heston-CIR models

Consider the hybrid model given by $$dS=(r-q) S dt + \sqrt{v} S dZ_1$$ $$dv = \kappa_v (\theta_v - v) dt + \sigma_v \sqrt{v} dZ_2$$ $$dr = \kappa_r (\theta_r - r) dt + \sigma_r r^p dZ_3$$ with ...
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Approximating second derivatives at boundary of finite difference scheme

The Question I am implementing a finite difference scheme for the Heston-Hull-White PDE: \begin{align} \frac{\partial u}{\partial t} &= \frac{1}{2}s^2v\frac{\partial^2 u}{\partial s^2 } + \frac{1}{...
user59093's user avatar
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How do I calculate Hull White's Theta from the discount curve?

The Question I'm currently implementing the a finite difference method for the Hull-White model, shown below: $$\mathrm{d}r(t)=\lambda[\theta(t) − r(t)]\mathrm{d}t + \sigma\mathrm{d}W(t)\tag{1}$$ This ...
user59093's user avatar
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Dealing with the ru term in an ADI Finite Difference Scheme

I'm trying to code up the algorithm from this paper. The paper presents an ADI algorithm for pricing options in the Heston-Hull-White model. The starting point is the Heston-Hull-White PDE, given ...
user54908's user avatar
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1 answer
143 views

Why should future short rates tend towards the current term structure of interest rates?

I'm currently looking at the Hull-White model reproduced below: $$\mathrm{d}r = \lambda(\theta(t)-r)\mathrm{d}t + \sigma\mathrm{d}W(t)\text{.}\tag{1}$$ I have a simplistic understanding of the model. ...
user54908's user avatar
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4 votes
1 answer
591 views

Deriving the Heston-Hull-White PDE

I'm trying to derive the Heston-Hull-White PDE. The correct backwards PDE is equation (1.3) of this paper on page (2). I will begin deriving the forward PDE, but switching between the two is trivial. ...
user54908's user avatar
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2 answers
462 views

Calculating the short rate from the discount curve

I'm currently looking at some code that implements the Hull-White model. As one of the inputs, the code accepts a table of discount factors at various dates. Time in Years Discount Factor 0 1 0.003 ...
user54908's user avatar
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1 answer
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Hull White 1 Factor Formulas with Time Dependent Variables

In John Hull's "Options Futures and Other Derivatives" I see that bond prices in Hull White 1 Factor model are specified as the following: $P(t,T) = A(t,T)e^{-B(t,T)r(t)}$ where $B(t,T) = \...
JoeBass's user avatar
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1 answer
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Calibrating Hull-White 1 Factor

I have been trying to learn HW1F on my own, out of nothing more than genuine curiosity during my twilight years, and I'm confused on the issue of calibrating. I don't know why, but all the research ...
JoeBass's user avatar
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1 answer
2k views

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

could anyone help me understand the definition of "co-terminal" swaptions? What are they? Can you provide an example to illustrate? And why such instruments are important in model ...
pqsn's user avatar
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1 vote
2 answers
1k views

Calibrate Hull-white one factor model with swaption in analytical formula

I've been trying to calibrate Hull-white one factor model with swaption but I have a trouble making closed form solution of swaption Below is the part of paper I've been referencing to https://people....
SHyou's user avatar
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1 answer
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Implication of Humped Spot Curve on future spot curve(s)

I'm currently implementing a G++ model (Two Factor Hull & White model with constant parameters) on zero curve bootstrapped from USD IRS. Currently, USD IRS is humped at 30 years; swap rate goes up ...
HumpedCurve's user avatar
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how to get 3 month Forward rates from Hull white model simulation?

I implemented the Hull White one factor model in Monte Carlo simulation, and got the short rate on each node (time step =1month). my question is how to get the forward rate from the short rate? I am ...
marietta's user avatar
1 vote
1 answer
148 views

Hybrid Models - Hull white with Heston / SchobelZhu / BS

I was looking at literature and found that for hybrid models, most of the literature only gives hybrid models where the volatility of the interest rate process(e.g Hull White) is constant. Is there a ...
Benedict's user avatar
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1 Factor Hull And White Swaption Calibration

I'm trying to calibrate a Hull and White model with constant volatility, mean reversion and theta such that the model can reproduce the initial Term Structure. I'm using this python code adapted from &...
Hilbert's user avatar
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1 answer
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QuantLib - Calibrating Hull White one-factor on negative interest rates

I have been working with the QuantLib Python package for some days now. Currently, I am working on calibrating a Hull White one-factor model for short rates. I am calibrating the model on the yield-...
Michielap's user avatar
1 vote
0 answers
348 views

Hull white model calibration - constant mean reverse factor and sigma

I setup a HW 1F model using Monte Carlo simulation with constant mean reversion and volatility factors. When I calibrate to a series of swaptions ( 1x4yr;2x3yr;3x2yr;4x1yr),the last three swaption ...
marietta's user avatar
2 votes
1 answer
780 views

Hull-White Monte Carlo simulation - mean reversion function

Quite new to implementing Hull white model in Monte Carlo simulation, hope to get help for 1. how to get the function $\theta$ in the following formula (the function used to match initial term ...
marietta's user avatar
6 votes
2 answers
8k views

Hull-White model applied in practice

I'm reading about the Hull-White model, I understand the math behind it and logic but what I am struggling to understand is how it's actually used in practice ? How can we combine it with technics ...
Gogo78's user avatar
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1 answer
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Quantlib: How do I price a ZC bond using the Hull White model?

I am trying to use QuantLib to model short rate and looks like QL has some material here http://gouthamanbalaraman.com/blog/hull-white-simulation-quantlib-python.html I have been able to simulate ...
TRex's user avatar
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1 answer
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Instantaneous correlation in the 2 factor Hull White model

I'm trying to understand which parameter controls the instantaneous correlation in the 2 F HW model. As in, correlation b/w 2 rates observed at the same time. My thinking is as follows: $$Rate(1)=P(t,...
Arshdeep's user avatar
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Hedge robustness of the one factor Hull White model

I recently came across a quote in a book: "All single factor models share the limitation that shifts in curve levels cause shifts in the package of vanilla options that are a good hedge for the ...
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A forward contract to buy a foreign currency can be handled by a linear model

In Hull's book, he says that: "An example of a derivative that can be handled by the linear model is a forward contract to buy a foreign currency." Then he continues with, "For the ...
Guess601's user avatar
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1 answer
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Implication of forward-rate dynamics when the short-rate follows a normal process

In the section 3.2.3 of the second edition of "Interest Rate Models - Theory and Practice" by Brigo and Mercurio, the forward-rate dynamics implied by the CIR model is derived as follow: The ...
tnk's user avatar
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Hull & White 1F - What is the appropriate calibration portfolio for Libor indexed structured note?

I'm wondering what is the best swaptions or caps portfolio I could use to calibrate the two parameters of H&W 1F model for a structured note with optionality on Libor underlying. Let's suppose ...
Samuel's user avatar
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1 vote
1 answer
2k views

Trinomial Trees for Hull-White model

I am studying trinomial trees and trying to implement them in Python to compare them to the monte carlo simulation. I searched 3-4 hours in the web; but can't find any implementation on binomial or ...
a.hilary's user avatar
1 vote
1 answer
381 views

Current discount rate of Hull White One-Factor Monte Carlo Simulation

I have a question about the Hull-White One-Factor Monte Carlo Simulation. As we know under the Hull-White One-Factor Model, the short rate follows a random process. So basically, every simulation path ...
Kay's user avatar
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2 votes
1 answer
2k views

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

I need to show that the Hull-White model $$dr=(\theta(t)-ar)dt+\sigma dW^Q$$ corresponds to the Heath-Jarrow-Morton formulation $$df(t,T)=\alpha(t,T)dt+\sigma e^{-a(T-t)}dW^Q.$$ I obtained the drift ...
Giulio Carlo Venturi's user avatar
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2 answers
937 views

Two Factor Hull White Model Calibrate

I have a question about the optimizer method to calibrate the parameters of two factor hull white model. I have the analytical pricing formula for cap and market cap price. There are five parameters ...
wen's user avatar
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Pricing of the compound coupon bond with PDE

I am now studying finance math using Steven E.Shereve's book. Using Interest Rate models, We can the price for zero-coupon with maturity price $1$ under Hull-White interest rate model[page 274] and ...
user avatar
2 votes
2 answers
899 views

Discount factor in Hull-White model

Consider a Hull-White model $dr(t)=\left(\theta(t)-a(t) r(t)\right) dt + \sigma dW(t)$ with parameters $a=0.1$ $\sigma=0.3$ $\theta(t)$ was calibrated to match $P(0,t)=\exp(-\mu t)$ with: $\mu=0.2$...
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