# Questions tagged [hullwhite]

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### Zero Coupon Bond prices in One Factor Hull White model

I implemented the one factor Hull White model for educational purposes and I calibrated the model from a given (made up!) yield curve: The Zero Coupon Bond Prices from this yield curve are: Taking ...
555 views

### Calibration of Theta, A(t) and B(t) of Hull White 1Factor model

I’m simulating interest rates via the HullWhite One factor model. To simulate the short rate I’m using the code from the Quantlib Python Cookbook, chapter 15 and beyond (By Goutham Balaraman and Luigi ...
62 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 ...
66 views

### Formula for quantiles of swaprates in the 1-factor Hull-White model

Is there a closed formula to approximate the quantiles of swaprates in the 1-factor Hull White model? Background The Hull-White is a Gaussian model for the short rate. Its mean and covariance ...
47 views

### 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 ...
82 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: $\mu=0.2$...
481 views

### Monte-Carlo simulation Hull-White process: physical and risk-neutral measure

From Monte-Carlo simulation Hull-White process I get paths in risk-neutal measure. How can I get paths in physical measure?
115 views

### Hull-White calibration volatility as a function of time

I need some help for the parametrization of the volatility parameter in the Hull-White model. I have the necessary Caplet vols and I calibrated the HW model to match the Caplet and hence the Cap ...
145 views

### Cap price as bond options

I am currently struggling with model calibration of the Hull-White (or Vasicek) model to Caps and Floors. My main problem is that I am confused about the notation. In Brigo & Mercurio (2006, p. ...
Consider Hull White model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha t}\... 1answer 61 views ### Hull White Cap/Floor calibration I have a problem and I hope someone could help me. I calibrated the Hull-White model to Caps and Floors from t=0 so the bond prices are equivalent to todays term structure. See: Hull-White zero-... 1answer 172 views ### Why co-terminal swaptions are that important? Usually Hull & White is calibrated to co-terminal swaptions. When asking why specifically co-terminal, I get the response that it is just a choice and it depends on the use we intend to do with ... 2answers 627 views ### Hull White help needed I've been trying to calibrate Hull-White 1 Factor & 2 Factor model using Caps but I've some major doubts about my methodology, and would really appreciate some help. I am using these formulas ... 0answers 43 views ### Swap rate in the annuity measure and Martingale Representation Theorem As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is ... 1answer 2k views ### How to get set the theta function in the Hull-White model to replicate the current yield curve I want to calibrate the HW one factor model to current market data. How do I set the function$\theta(t)in $$\mathrm{d}r(t) = \kappa(\theta(t)-r(t))\mathrm{d}t+\sigma\mathrm{d}W(t)$$ to ... 2answers 1k views ### Fixing mean reversion parameter in the 1F HW model I am trying to calibrate the 1 factor Hull White model to ATM swaptions. The strategy which I use is to minimise the sum of squared difference between model and market prices for the swaptions on the ... 1answer 874 views ### FX Hull-White model A model for FX, presented in Interest Rate Models, Brigo and Mercurio (2006), has the following dynamics: \begin{align} dr_t^d&= \lambda_d(\theta_d(t)-r_t^d)dt+\eta_d dW_t^d\\ dr_t^f&= [\... 1answer 296 views ### Implementation of the Hull and White short rate model This is the first time I'm using quantlib, and I wanted to compare the velocity of quantlib with my own Python code. I found a tutorial about Hull and White to generate the short rate paths with ... 0answers 295 views ### Zero Coupon Bond Price under Hull White Model (One Factor) While pricing Zero coupon bond using One Factor Hull White model: $$dr(t) = \left(\theta(t) - a r \right)dt + \sigma dW(t)$$ How to determine the value of\theta(t)$using real world example: $$θ(t)=... 0answers 68 views ### Hull White Equation Derivation Hello I need your help. I found the formula for deriving A(t,T) and B(t,T) in Hull White paper is like this BB_{tT} - B_{t}B_{T} - B_{T} = 0 and ABA_{tT} - BA_{t}A_{T} - AA_{t}B_{T} + \frac{1}... 0answers 82 views ### How to solve these SDE Problems Quuestion1. I make a solution r(t) used by Ito's lemma r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u) Is this right? and I try to make ... 1answer 184 views ### Proof of the Hull & White Model calibration I have a question about the demonstration of the formula which states that: If we have an Hull & White Model for the short rate diffusion such that Then the model is fully calibrated if and only ... 2answers 196 views ### Hull-White zero-coupon bond price does not depend on the volatility? So, today I started pricing zero-coupon bonds using the Hull-White model. An interesting feature is that when t = 0 the bond price does not actually depend on the volatility since the last term of A(0,... 0answers 106 views ### Fitting to Market Data in Extended Vasicek / Hull White I need your help for my task. I need to calibrate to the market data for Hull White model for Zero Coupon Bond Price. I refer to John Hull and Alan White paper. I want to ask you a few questions and ... 0answers 388 views ### Implementation of Generalized Hull-White interest rate model I am looking for implementation in R (or Matlab) of the Generalized Hull-White interest rate trinomial tree according to the following paper: Interest Rate Trees: Extensions and Applications (2017) . ... 0answers 162 views ### Calibration of 1-factor Hull-White model using Jamshidians trick - see my code So, I'm trying to calibrate the Hull-White 1-factor model given Black swaption volatilities that I have from the Bloomberg terminal. I'm following the Jamshidian method as described in this thesis (3.... 1answer 347 views ### Hull white model Monte Carlo simulation Zero Coupon Bond I am trying to use Hull White Model to price a zero coupon bond by Monte Carlo Simulation. The basic idea is under this equation: Under Hull White Model, I want to generate every short rate (r) and ... 1answer 766 views ### Hull-White Extension of Vasicek Model I am reading the book Interest Rate Models by Brigo and Mercurio and try to understand the Hull White Model Extended Vasicek Model. They start off by defining the instantaneous short-rate process ... 0answers 122 views ### European Call Option Modelling under 2 factor Hull White interest rates I have modelled the yield curve through the two factor Hull White Model. Now I want to implement in Matlab the price development of a ATM-Call-Option (European). Has someone an idea how to combine ... 1answer 442 views ### why calibrate volatility and fix the mean reversion I have had a few experiences or chats with teammates about the Hull-White model. The famous model has 2 parameters : The volatility The mean reversion Very often I hear that the mean reversion has ... 1answer 656 views ### Why Hull White 2 Factor model can't capture vol skew? Is there a way to stay with the short rate model (like HW2F or G2++) but extend it to capture vol term structure (vol smile or skew). What happens if I calibrate HW2F to OTM swaptions? (I don't want ... 1answer 853 views ### Calibrating a two-factor Hull-White model using Neural Networks So I have the following short-rate model$$dX_t = a_1X_tdt + \sigma_1dW_tdY_t = a_2Y_tdt + \sigma_2dB_tr_t = X_t + Y_t + f(t)$$with X_0 = Y_0 = 0 where W and B are Brownian motions ... 1answer 250 views ### Hybrid Heston-Hull White Model I am wondering if anyone could recommend a few good papers on hybrid heston-hull white models, in particular with respect to the approximation of model European options for calibration. Literature on ... 2answers 1k views ### Implementation of one-factor Hull-White short interest rate model I am looking for implementation in R, VBA, C++, Python (or in any other programming language) of one-factor Hull-White short rate interest model according to the following article: Hull J. and White ... 0answers 129 views ### How can I estimate the time-varying θ term in the Hull-White one factor model? I am trying to simulate the prices of bond indexes (e.g. Barclays Aggregate, IBOXX sovereign, IBOXX corporates) using Monte Carlo assuming that they follow the SDE given by the Hull-White model (one-... 1answer 1k views ### How does the 2-factor Hull White model propagate the forward rates curve? I've been trying to get a grasp on some of the basics of interest rate modeling, and am looking to simulate rates using the 2 factor Hull White model, which I am aware offers a more realistic model of ... 0answers 68 views ### Convert Short rate from HW simulation into Swap rates I am trying to price an exotic option that requires me to simulate 10 yr swap rates. I have calibrated a 1 factor HW model to swaption prices. However, my understanding is that the HW model describes ... 2answers 633 views ### Whites Reality Check for Pair Trading I want to use the Monte Carlo Method described in Aronsons book Evidence based Technical Analysis to test if a given pairs trading strategy is useless. First step there is to randomize the returns of ... 1answer 444 views ### Estimation of market price of risk of short interest rate under the Hull-White model I think I am a bit confused. I intend to estimate the market price of risk the short interest rate, say, under the Hull-White model. I have the following two questions. Is it correct to state state ... 2answers 602 views ### Time dependent parameters in Hull-White model Hull-White:$$d r = [\theta(t) - ar]d t + \sigma d W_t.$$There is a statement in John Hull's book: The advantage of making a or \sigma, or both, functions of ... 0answers 69 views ### A basic question about short-rate models Sorry, if it's a very rudimentary question. I mainly practice tax but have to deal with financial transactions from time to time where I have to benchmark option prices. I have usually used Hull's ... 0answers 60 views ### Relation between BDT volatility and Hull-White one factor Volatility Is there any mathematical relationship between the volatility of spot rates calibrated from Lognormal model and the volatility of spot rates calibrated from HW one factor model? 2answers 13k views ### How to calibrate Hull-White from zero curve? I am interested in calibrating a Hull-White model to the market. I do not, however, have data on anything except the market zero curves, as all derivatives are being traded OTC. My plan is to ... 1answer 702 views ### Hull White Tree Calibration 2 This is actually to extend the question I asked previously and to follow up Bernd's answers. This is the original link: Instruments for calibrating Hull White Model 1. As Bernd mentioned, it's ... 1answer 980 views ### Instruments for calibrating Hull White Model I have a few questions regarding hull white calibration, specificly for the trinomial tree model. 1.I am wondering what are the ideal instruments could be used for hull white model calibration? Cap, ... 2answers 4k views ### Historical calibration of Hull-White model I have a question concerning 1-factor Hull-White model. For my master project I need to calibrate it to compute Counterparty credit risk metrics. I know that the model might be calibrated either for ... 1answer 245 views ### Lattice pricing of derivatives under multi curve framework (OIS and LIBOR) My goal is to price various derivatives resetting to 1M and 3M LIBOR via using a lattice. I have calculated an OIS curve for discounting and adjusted 1M and 3M LIBOR forward curves to be consistent ... 1answer 637 views ### Proof behind solution for theta in Hull-White with time-dependent volatility and mean reversion? I'm studying the following paper on Hull-White model calibration: Hull-White paper In this paper they study the general form of the HW model with time-dependent mean reversion and volatility:$$dr(t) ... 1answer 4k views ### Extended Hull White Interest Rate Model for Zero Coupon Bond Let's take the following three SDEs: $$dr=u(r,t)dt + w(r,t)dX$$ $$u(r,t)=a(t)-br$$ $$w(r,t)=c$$ where$b$and$c$are constants and$a(t)$an arbitrary function of time$t$. If Zero Coupon Bond$Z(...
Consider the one-factor Hull-White model $$\mathrm{d}r(t) = (\theta(t)-\kappa r(t))\mathrm{d}t + \sigma\mathrm{d}W(t)$$ When one calibrates the model to market data one chooses  \theta(t) = \...