Questions tagged [short-rate]

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Deriving the variance of G2++ Model

I'm studying G2++ Model in Brigo(2007)'s book. The model constructed as follows, $$ r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\ $$ with the dynamics of $dx(t)$ and $dy(t)$ described by: \begin{align}...
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short rate, yield curve and zero-coupon bond price formula under CIR mode: How to calibrate the market price of risk

I recently read a document posted by a user in QF, who said that "In the past, I have calibrated simple short rate models to the term structure by using maximum likelihood to get the parameters ...
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1 answer
106 views

Difference HJM Framework versus Short rate model

Recently I study some interest rate models. When I moved on to forward rate models, I see this documents https://en.wikipedia.org/wiki/Heath-Jarrow-Morton-_framework It said "HJM-type models ...
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60 views

What instruments can be used to calibrate short-rate models?

What type of debt instruments can be used to estimate short-rate model parameters? How can I find the history of short-rate? My guesses are overnight LIBOR/SOFR or extrapolation through the yield ...
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1 vote
0 answers
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Why are/were Euribor rates consistently lower than ECB deposit facility rates since 2020?

Why are unsecured term deposits in EUR yielding lower than a simple overnight deposit at the ECB ? Isn't it supposed to be the opposite ? The 3 month Euribor just broke -0,50% a few days ago, why was ...
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3 votes
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Is the G2++ model apt to use when one needs estimates of longer term refinance rates for mortgages and can the model be created with Monte Carlo?

I am currently in the process of developing an interest rate model that would be used to price mortgage-backed securities and develop an OAS estimate. Referring to Brigo and Mercurio (2006) I'm ...
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Inferring a term structure when using a short-rate model

I'm relatively new to working with interest rate models and I am having some conceptual difficultly considering how a short-rate model can be utilized when calculating OAS of a mortgage-based bond, ...
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2 votes
1 answer
181 views

Obtaining the dynamics of the Vasicek model using Itô

Consider the following expression for the short-term interest rate $$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$ which is ...
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Calibrating the mean reversion parameter of the short-rate-model Black-Karasinski

When modelling the term structure of interest rates, one widespread possibility is using the Black-Karasinski(BK) model, which is given by the following stochastic process $$dln\,r=[θ(t)−a\,ln\,r]dt+σ(...
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LIBOR Rate in Short-Rate Models

Hey I have problem with understanding the relation between short rate $r$ and LIBOR rates (which we need to calculate payoff from FRA, Caps, Swaption etc.). We know that Zero-Coupon Bond price is $$P(...
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what's the difference between instantaneous short rate and instantaneous forward rate?

In the short rate models, sometimes it models the instantaneous short rate and sometimes it models the instantaneous forward rate. Does instantaneous short rate = F(0, t + tau) and instantaneous ...
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1 answer
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What's the difference between short rate and the bootstrapped interest rate?

This thing confused me for a long time, since we can a have a curve (e.g. LIBOR 3M) bootstrapped from the market quotes of instruments (e.g. FRA, SWAP), and we can get the spot rates and also the ...
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Vasicek Model Expected Short Rate

Please note I am new to quantitative finance and more so to stochastic calculus. I have what should be a relatively simple problem using the Vasicek model for estimating future parameters of the short ...
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1 vote
1 answer
113 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. ...
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2 answers
211 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 ...
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Why does adding a negative risk premium to the short rate avoid the occurrence of inverse yield curves?

I am reading about the Vasicek One Factor short rate model and how to implement a change in measure from a risk-neutral to real-world measure, when I came across this comment: Adding a negative risk ...
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1 vote
1 answer
152 views

Calibration of Heston model with stochastic short rate

I have following Heston model with stochastic short rate: \begin{eqnarray*}dS\left(t\right)&=&r\left(t\right)S\left(t\right)dt+\nu\left(t\right)S\left(t\right)dW^{S}\left(t\right)\\dr\left(t\...
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11 votes
2 answers
928 views

Differences between main classes of interest pricing derivatives models

There seems to be 3 main classes of interest rate pricing models: 1) Short rate models, 2) Heath Jarrow models and 3) Libor Market Model. My book doesnt seem to explain why we need all these different ...
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3 votes
2 answers
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What is the Q-dynamics of affine bond prices when r is described by the given model?

Assuming an Affine term structure model, where bond prices arebe defined as: $$P(t,T)=\exp({A(t,T)-B(t,T)r_t)}$$ and describing the Q-dynamics of the short rate according to the model: $$dr_t=ar_tdt+\...
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  • 221
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1 answer
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Simulating the path for Interest Rate

There are many ways to short term rates like Ho-lee process, HW process. However I failed to understand how this information can ...
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0 answers
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Short rate models practical textbook

Currently working on a validation and testing of a yield curve model (one factor short rate model). Have been reading Andersen and Piterbarg, and Mercurio and Brigo. Good for true understanding, but ...
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2 answers
175 views

Affine Structure Resolution for the Vasicek model

I would like to now how to solve the PDE of the affine structure under Vasicek.I am delineating the steps: First let's posit the OU process under a Risk Neutral Measure such as : \begin{align*} \...
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  • 446
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117 views

Longstaff and Schwartz example in their paper

I was looking at the well known Longstaff and Schwartz paper "Valuing American Options by Simulation: A Simple Least-Squares Approach". There are a couple of examples where they applied the ...
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Euro short-term rate (€STR) question

Based on the latest data published by ECB,€STR = -0.56%. Is this the rate a bank would pay to borrow overnight or it's an annualised overnight rate so the actual overnight rate can be approximated ...
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  • 341
2 votes
1 answer
1k views

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-...
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1 vote
0 answers
348 views

Vasicek Short rate simulation - analytical formula vs discretization

I've been using two approaches to simulate Vasicek short rate paths and I'm wondering if one of them is more correct than the other. The first approach is based on the analytical formula (see code ...
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Option Valuation With Hard To Borrow Rates

How would you include -in a simple way- high borrow rates, say 10%. Intuitively, for PUTs I'd set r as r - borrow_rate, to include the negative carry of the borrow. So If I'm selling puts, value would ...
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1 vote
1 answer
68 views

Hazard process and affine term structure

How can I extrapolate the hazard processes and calibrate an affine term-structure model from the historical series of curves (1y, 2y, ..., 10y tenors) of the CDs spreads of different entities?
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1 vote
1 answer
340 views

What is gsr model for short term interest rate

I am looking for a good definition for the GSR model for short rate. As mentioned in the page of https://rkapl123.github.io/QLAnnotatedSource/db/dd8/...
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2 votes
1 answer
132 views

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 ...
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Bond Options Calibration to market volatility using SABR Model

I'm trying to calibrate bond option implied volatility from SABR model to market volatilities, I tried calibration in python but the smile isn't correctly matching with market volatility? Any help is ...
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1 vote
0 answers
26 views

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 ...
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1 vote
1 answer
231 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 ...
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2 votes
1 answer
824 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 ...
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0 votes
1 answer
83 views

How are non-equity derivatives handled in monte carlo Value at Risk simulations

If you have a portfolio of stocks and options it's straight forward enough to generate correlated stock paths and evaluate the positions at the end of the time horizon, but what do you do if your ...
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0 votes
1 answer
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Calibrating g2++ in negative interest rate environment

I am working on a g2++ model in a dualcurve setup for both Euribor and EONIA. I have the model built, but have some issues in calibrating it - I get a perfect fit with a Nelder-Mead algorithm, but it ...
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-1 votes
1 answer
87 views

Cox Ingersoll Ross (1985) Model [closed]

How can I convert the following process to a standard Brownian Motion? $$\mathrm{d}r_t=(a-br_t)\mathrm{d}t+\sigma\sqrt{r_t}\mathrm{d}W_t$$
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0 votes
2 answers
421 views

Negative values in CIR model

I'm having difficulty understanding the well known property of the CIR model that it can't go below zero. Wikipedia says that this is because the random shock on the rate will grow very small as r ...
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4 votes
1 answer
176 views

How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process?

I wonder how I can determine the components $A(t,T)$ and $B(t,T)$ for the zero-coupon bond price process $p(t,T)=e^{A(t,T)-r(t)B(t,T)}$? The components are defined in the following link: https://en....
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1 answer
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Cox-Ingersoll-Ross Zero Bond Put Option

according to Brigo & Mercurio (2006): But how is the Zero bond Put of the CIR model? I couldn't find any information about that. Thanks in advance. Regards Chris
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4 votes
1 answer
993 views

Why isn't the Vasicek model arbitrage-free?

Could anyone explain why the Vasicek model isn't an arbitrage-free model? Additionally, which interest rate model is arbitrage-free and why?
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  • 41
2 votes
1 answer
319 views

Stochastic Processes (Applying Ito's Lemma on Ho-Lee Model )

I seek a basic form (SDE) to understand the Ho-Lee model. I already understand the models from Vasicek, Merton and Cox-Ingereoll-Ross, etc.. For example, \begin{align*} dX_t &= -1/2 \alpha X_t ...
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0 answers
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Markovian short rate in HJM framework

In Bjork it is proven in proposition 20.5 that a forward rate dynamics: \begin{equation} f(t,T) = f(0,T) + \int_0^t\alpha(s,T)ds + \int_0^t\sigma(s,T)dW(s) \end{equation} imply a dynamics for the ...
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1 vote
0 answers
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Derive the discount bond prices of the Vasicek model by the PDE approach

The question is shown above. Anyone can help me?
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1 vote
1 answer
605 views

Understanding Front-End Spreads (terminology, lingo, convention)

Would appreciate a clear explanation as to what the OIS/Tsy spread and the TU OIS spread is. I've seen it being talked about in Wall St research reports but can't seem to find good explanations on ...
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3 votes
1 answer
202 views

Bond Option Hedging

(My question) Please show me how to solve from (2) to (4) with computation processes. These are too difficult to solve. Thank you for your help in advance. (Cross-link) I have posted the same ...
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  • 179
0 votes
1 answer
99 views

If short rates $r(t)$ do not determine the bond prices $P(t, T)$, then what is the basis for short rate models?

The question title says it all: We know that in general, specifying the short rate $r(t)$ does not specify the bond prices $P(t, T)$. So how can a model for short rates—for example the Vasicek model—...
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2 votes
2 answers
205 views

Cumulative Integration with regard to Vasicek Model's Bond Price and its Forward Price

(My Question) Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M. $$E\left[ \exp \left( - \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \...
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  • 179
2 votes
1 answer
178 views

The Riccatti equation for The Cox-Ingerson-Ross Model

(My Question) I went through the calculations halfway, but I cannot find out how to calculate the following Riccatti equation. Please tell me how to calculate this The Riccatti equation with its ...
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  • 179
2 votes
0 answers
152 views

The Ho-Lee Model (1986)

(My question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M. (Thank you for your ...
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