Questions tagged [short-rate]
A short-rate model is a mathematical model that describes the evolution of interest rates
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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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2
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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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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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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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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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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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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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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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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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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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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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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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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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0
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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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1
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Negative Libor Simulation
Can LIBOR rates be simulated using short rate models?
If no, what is the reason behind it?
What is a simple model to simulate LIBOR rates? Especially in a negative rate environment.
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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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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 ...
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0
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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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answer
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Ho-Lee short rate model under the Heath-Jarrow-Morton framework
Under the Heath-Jarrow-Morton (HJM) framework the dynamics of the Ho-Lee short rate model are defined as following: $$dr(t)=\theta(t)dt+\sigma dW^{\mathbb{Q}}(t)$$ with $\mathbb{Q}$ the risk-neutral ...
user40884
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2
answers
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Deriving interest rate term structure in a short rate model
I have often seen a statement that we can model only a short rate process $r(t)$ and then use it to derive a term structure $R(t,T)$ for every $t$. Could someone please elaborate? Say, I’ve simulated $...
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Short rate models
On the short rate model in Wikipedia
https://en.m.wikipedia.org/wiki/Short-rate_model
Why is the first function, the P(t,T) given? This is not the short rate model this is generating prices for a ...
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What's the difference between the short rate model projection and the 3M forward curve?
A term structure has a forward curve
So what is it that the short rate model is projecting exactly?
Why is it needed?
How are they different?
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Bond dynamics in Ho Lee model
The short rate in the Ho-Lee model is given by :
$$dr_t=\left( \frac{df(0,t)}{dt} +\sigma^2t\right)dt + \sigma dW_t$$
I'm trying to find the bond dynamics given by :
$$dP(t,T)/P(t,T)=r_tdt-\sigma(...
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1
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why $f(t,u) \neq E_t^Q [r(u)]$ when $r$ is random?
If I suppose the short rate $r$ deterministic, and the risk neutral measure $Q$, I can write the following :
$$f(t,u) = -\frac{d}{du}\ln P(t,u) = -\frac{d}{du} E_t^Q \left[ e^{-\int_t^{u}r_sds} \...
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Produce volatility smile/skew with G2++ model
Suppose I have a G2++ short rate model:
$$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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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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CIR calibration
I'm using a CIR short rate model to forecast interest rate paths.
I've been thinking and also searching online about different ways of estimating its parameters (a, b and sigma). While there are a ...
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1
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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 ...
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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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"Standard" Model for Effective Fed Funds Rate
Is there a "standard" model used to model the Effective Fed Funds Rate? I know that BGM is often used for LIBOR but haven't found a similar application to the Effective Fed Funds Rate.
Do ...
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HJM or Short rates model?
When market practitioners do prefer HJM models to short rates models when it comes to pricing derivatives (other than swaptions and caps, let say light exotics to exotics) ?
To be more specific, ...
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Why Arent There Long Rate Models?
You have short rate models, https://en.wikipedia.org/wiki/Short-rate_model, but there doesnt seem to be any long rate models.
I find this weird as in options modelling you model the whole smile, not ...
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2
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Ho Lee model in Baxter&Rennie
I am currentyl reading Baxter&Rennie and I have a difficulty with understanding a derivation of formula for one function, $g(x,t,T)$ (this can be found on page 152 in the book). I know that there ...
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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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Details of calibration of Hull-White model
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
$$
\theta(t) = \...
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votes
1
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How to get set the theta function in the Hull-White model to replicate the current yield curve
I want to calibrate the HW one factor model to current market data. How do I set the function $\theta(t)$ in
$$
\mathrm{d}r(t) = \kappa(\theta(t)-r(t))\mathrm{d}t+\sigma\mathrm{d}W(t)
$$
to ...
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0
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How are short rate models used to construct the whole of the yield curve? [closed]
There are a number of short rate models that give $r(t)$.
How can those be used to construct the whole of the yield curve $y(t,T)$ (where $y(t, 0) = r(t)$)?
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Short-rate models: Risk-premium of $T$-bonds
Following "Arbitrage Theory in Continuous Time" by Thomas Bjork, a standard one-factor short-rate model is of the form
\begin{align*}
dr_t = \mu(t,r_t)dt + \sigma(t,r_t)dW_t.
\end{align*}
The only ...
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Vasicek model calibration
I am trying to calibrate Vasicek model, i.e. to determine the parameters $\kappa, \mu, \bar{\mu}$ and $\sigma$ where the process dynamics are given through
$$ dr_t=\kappa\left( \mu - r_t\right) dt+\...
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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:
$$ dr_t=\alpha \left( \bar{\mu} - r_t\right) dt+\sigma d W^{\mathbb{P}}(t), $$
$$ dr_t=\...
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votes
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Volatility considerations with interest rate derivatives
I am a bit confused about the practical use of vol surfaces used for derivative pricing. We know that the two main products that best represent market volatility are caps and swaptions, from which ...
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0
answers
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Basic Interest Rate Modelling Ques
I have got a question regarding the Vasicek Model and the corresponding Bond Pricing Equation (BPE).
Starting with a short-rate process (under measure $P$ or real world drift $u(r,t)$) of the form:
$...
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Why is logarithmic mean equal to the arithmetic expectation less one-half its variance?
I've taken it as gospel that the following equality is true:
$$\mathbb{E}[\mu_x] = m_x - \frac{1}{2}\sigma_x^2 $$
where:
$\mathbb{E}[\mu_x]$ is the expected value of the logarithmic mean of some ...
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QuantLib Gsr model
Almost spent the whole day. Could anyone give a link to the Gsr model specification that is implemented in QuantLib? Or give an explanation? Any help is highly appreciated.
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Risk neutral measure of short rate model
As we all know, all affine term-structure models are members of HJM model. Under HJM model, there is a unique risk-neutral measure in both forward-rate process and bond evolving process. Hence, the ...
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How to price Swaptions with short rate models?
I have specified a (Lognormal) short-rate model (non-affine) under the Risk-Neutral measure $Q$ as a shifted exponential vasicek:
$ r(t) = e^{y(t)} + \phi(t)\\
\text{with} \quad dy(t) = \kappa(\...
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vote
1
answer
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Variance of the Cox-Ingersoll-Ross short rate
Shreve II page 151, the Cox-Ingersoll-Ross model is defined as
$$dr_t=(\alpha-\beta r_t)dt+\sigma\sqrt{r_t}dW_t$$
By applying Ito's Lemma, we obtain
\begin{align}
r_t&=r_0e^{-\beta t}+\frac{\alpha}...
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1
answer
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Volatility in short-rate models and vol practical issues
I am slightly confused about the volatility term when pricing zero coupon bonds in the Ho-Lee model (and generally about where to get vol from in these kind of short rate models).
A particular ...
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votes
0
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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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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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