Questions tagged [short-rate]
A short-rate model is a mathematical model that describes the evolution of interest rates
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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 ...
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How to convert the parameters of multi-factors cheyette model (quasi-Gaussian model) from tenors to factors?
The book "Interest Rate Modeling" by Andersen and Piterbarg is an extermely fascinating book on interest rate derivatives.
Recently, I have encoutered some issues while reading this book.
...
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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} + ...
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Vasicek model calibration to bond prices or rates (no swaptions)
I need to calibrate Vasicek's model $dr_{t} = a(\theta - r_{t})dt + \sigma dW_{t}$ in a market with no swaptions. I was thinking in estimating $\sigma$ with historic data, but I'm in the doubt with ...
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Delta hedge call option on short rate
Usually delta hedging an european call option in the black-scholes model is constructed of three assets; a call option, the underlying stock and the risk-free asset often assumed to have constant ...
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Half-life of short rate
The SDE for the short rate r(t) in the Vasicek model is given by:
$$ d(r) = k(r^* - r)dt + \sigma dW $$
The deterministic part of the above SDE is the following ODE
$$ d(r) = k(r^* - r)dt, $$
where $k$...
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Kalman Filtering to estimate parameters of G2++ Model
I'm trying to use Kalman Filtering to estimate the parameters of the G2++ short rate model. For this, I've been using Implementing Short Rate Models: A Practical Guide by F.C. Park.
For reference, he ...
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Black-Karasinski trinomial tree implementation
I have implemented the Black-Karasinski model aiming to fit the interest rate curve for particular dates. The way I implemented it was:
Defined the volatility.
Defined $\Delta x = \sigma \sqrt{3\...
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How can I use Monte Carlo to price a Zero-coupon bond in the Cox-Ingersoll-Ross model?
Let me prefix this by saying that, yes, Cox-Ingersoll-Ross (C.I.R.) is deprecated when used to model interest rates. Yet integrals of the form
$$P(0,T) = E\left(\exp\left(-\int_0^Tr_s ds\right)\right) ...
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Black-Karasinski & Market Price of Risk [closed]
I have implemented the Black-Karasinski model using trinomial trees and calibrated following Brigo (2007) page 29. However, the results do not fit the interest rate curve practiced in the market. As I ...
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Bond-pricing under the Vasicek short rate model
I'm currently studying the Vasicek model of the short interest rate
$$dr_t=a(\mu-r_t)dt+\sigma dW_t$$
I know how to solve this stochastic differential equation (SDE) and how to find expectation and ...
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How to calibrate short-rate model (Hull-White) using historical domestic IBOR curve without other derivative price? [duplicate]
I'm trying to calibrate Hull White model in VietNam market to value IRS, CSS products which are not publicly traded.
dr(t)=(θ(t)−αr(t))dt+σ(t)dW(t)
I only ...
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Workaround for Hull-White short rate model in market without swaptions
Every time I search calibration methods in short-rate models such as Hull-White, I always find information on how to do it with swaptions market data. Yet, I can't find anything about how to do it in ...
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Simulating of short rate model
I'm trying to simulate the risk factor of PFE from the interest rate model.
For example, under Vasicek model :
$$dr_t = k(\theta-r_t)dt + {\sigma}dW_t$$
with the analytic solution, we can simulate N ...
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Term Structure Modelling - Why model the state prices and not an asset or rate
When modelling stocks we specify the model in terms of the dynamics of the stock itself (e.g. in Black-Scholes, Heston and SABR - often denoted $S$).
However, as I am reading about Term Structure ...
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Problem with pricing a call option using the Monte Carlo Vasicek model
I am trying to price a call option on a zero coupon under the Vasicek Model using Monte Carlo method:
$$C_0 = B(0,\theta) \ \mathbb{E}^{\mathbb{Q}_T}[(B(\theta,T)-K)^{+}]$$
The problem is that the ...
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How do I estimate volatility for MPR historical data
How can I estimate volatility with historical data for Monetary Policy Rate (MPR) to use in a short rate model?
I could use regular techniques like simple standard deviation or max likelihood, but the ...
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Deriving the variance of G2++ Model
I'm studying G2++ Model in Brigo(2007)'s book.
The model constructed as follows,
$$
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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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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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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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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Ho and lee derivation for short rates model
A silly question that is bugging me. I am working my way through Baxter and Rennie (again) and I am getting my wires crossed on the short rate models in particular the straight forward Ho and Lee ...
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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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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},$$
which is ...
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what's the difference between instantaneous short rate and instantaneous forward rate?
In the short rate models, sometimes it models the instantaneous short rate and sometimes it models the instantaneous forward rate. Does instantaneous short rate = F(0, t + tau) and instantaneous ...
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What's the difference between short rate and the bootstrapped interest rate?
This thing confused me for a long time, since we can a have a curve (e.g. LIBOR 3M) bootstrapped from the market quotes of instruments (e.g. FRA, SWAP), and we can get the spot rates and also the ...
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Why should future short rates tend towards the current term structure of interest rates?
I'm currently looking at the Hull-White model reproduced below:
$$\mathrm{d}r = \lambda(\theta(t)-r)\mathrm{d}t + \sigma\mathrm{d}W(t)\text{.}\tag{1}$$
I have a simplistic understanding of the model. ...
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Calculating the short rate from the discount curve
I'm currently looking at some code that implements the Hull-White model. As one of the inputs, the code accepts a table of discount factors at various dates.
Time in Years
Discount Factor
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How to price Swaptions with short rate models?
I have specified a (Lognormal) short-rate model (non-affine) under the Risk-Neutral measure $Q$ as a shifted exponential vasicek:
$ r(t) = e^{y(t)} + \phi(t)\\
\text{with} \quad dy(t) = \kappa(\...
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Why does adding a negative risk premium to the short rate avoid the occurrence of inverse yield curves?
I am reading about the Vasicek One Factor short rate model and how to implement a change in measure from a risk-neutral to real-world measure, when I came across this comment:
Adding a negative risk ...
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Vasicek model: joint simulation with discount factor
In Vasicek model, we have the following relation to get Discount factors given the value of short rate:
$$P(t\,,T)={{e}^{A(t,T)\,-\,B(t,T){{r}_{t}}\,}}$$
So, Discount factors are known as soon as we ...
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Volatility considerations with interest rate derivatives
I am a bit confused about the practical use of vol surfaces used for derivative pricing. We know that the two main products that best represent market volatility are caps and swaptions, from which ...
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Calibration of Heston model with stochastic short rate
I have following Heston model with stochastic short rate:
\begin{eqnarray*}dS\left(t\right)&=&r\left(t\right)S\left(t\right)dt+\nu\left(t\right)S\left(t\right)dW^{S}\left(t\right)\\dr\left(t\...
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Differences between main classes of interest pricing derivatives models
There seems to be 3 main classes of interest rate pricing models: 1) Short rate models, 2) Heath Jarrow models and 3) Libor Market Model. My book doesnt seem to explain why we need all these different ...
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What is the Q-dynamics of affine bond prices when r is described by the given model?
Assuming an Affine term structure model, where bond prices arebe defined as: $$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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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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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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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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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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Vasicek model calibration
I am trying to calibrate Vasicek model, i.e. to determine the parameters $\kappa, \mu, \bar{\mu}$ and $\sigma$ where the process dynamics are given through
$$ dr_t=\kappa\left( \mu - r_t\right) dt+\...
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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-...
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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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What is the purpose of short rate models?
Just venturing into quantitative finance and studying short rate models (Vasicek, CIR, Hull-White etc.). Wanted to ask a very simple intuitive question. How would a practitioner use these models? I ...
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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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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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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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Deriving interest rate term structure in a short rate model
I have often seen a statement that we can model only a short rate process $r(t)$ and then use it to derive a term structure $R(t,T)$ for every $t$. Could someone please elaborate? Say, I’ve simulated $...
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