Questions tagged [short-rate]

A short-rate model is a mathematical model that describes the evolution of interest rates

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Produce volatility smile/skew with G2++ model

Suppose I have a G2++ short rate model: $$r(t)=x(t)+y(t)+\phi(t), \quad r(0)=r_0$$ with $$dx(t)=-ax(t)dt+\sigma dW_1(t), \quad x(0)=0$$ $$dy(t)=-bx(t)dt+\eta dW_2(t), \quad y(0)=0$$ $$d\langle W_1,W_2\...
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simulating from the CIR++

I am looking at the CIR++ model which is described in interest rate models by Brigo et al, and was wondering on how to actually simulate from this model. The model reads $$r_t=x_t+\phi(t),$$ where $...
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Pricing interest rate options in emerging markets

I've been thinking how to price the early payment of mortgages in banks from emerging markets, where swaptions/caps/floors aren't available, and how to hedge this kind of options. At first I thought ...
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Callable bond price sensitivity to Hull-White volatility changes

I'm using classic Hull-White model for short term interest rate dynamic: $$dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$$ (Notation is quite intuitive, anyway I am using the same as Wikipedia ...
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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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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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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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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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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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Basic Interest Rate Modelling Ques

I have got a question regarding the Vasicek Model and the corresponding Bond Pricing Equation (BPE). Starting with a short-rate process (under measure $P$ or real world drift $u(r,t)$) of the form: $...
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Complete Algorithm of Calibration with Vasicek Model using Term-Structure Dynamics over Time

As there are so many different sccenarios about Vaicek Calibration but there has not been a clear example with data shown, I am totally Confused about how should I do it. so I am bringing the question ...
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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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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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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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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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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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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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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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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 ...
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Correlation between Two Factor Gaussian Shortrate Model and Black Scholes Model

I want to implement a two factor Gaussian Shortrate Model \begin{align} r(t) & = x(t) + y(t) + \phi(t), \\ dx(t) & = -ax(t)dt + \sigma dB_1 (t), \\ dy(t) & = -by(t)dt + \eta dB_2(t), \end{...
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basic difference between interest rate models

I am reading up on interest rate models, but currently confused about difference in the two types of models: no arb models like ho-lee, vasicek etc. others like nelson siegel, pca models etc. While ...
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Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + \sigma^2t(t-\...
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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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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 ...
Oliver Mohr Bonometti's user avatar
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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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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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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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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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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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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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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 ...
Ashwin Mvs's user avatar
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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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Vasicek Model - Should I simulate short-rate under the real-world or risk-neutral measure if I am interested in simulating future bond prices

In the classic Vasicek model, the market's short rate process $(r_t)_{t \geq 0 }$ is given through the SDEs: $$ dr_t=\alpha \left( \bar{\mu} - r_t\right) dt+\sigma d W^{\mathbb{P}}(t), $$ $$ dr_t=\...
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