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Questions tagged [short-rate]

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

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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 ...
Trajan's user avatar
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10 votes
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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+\...
Milan's user avatar
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9 votes
2 answers
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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 ...
Chinny84's user avatar
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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 ...
David Addison's user avatar
7 votes
0 answers
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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\...
FunnyBuzer's user avatar
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7 votes
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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 $...
A.Boh's user avatar
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6 votes
3 answers
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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 ...
Robert Brown's user avatar
6 votes
0 answers
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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 ...
Jose Pedro Melo's user avatar
6 votes
0 answers
829 views

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 ...
Lisa Ann's user avatar
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5 votes
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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(...
JohnLord's user avatar
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CIR model: is the short rate really non-central $\chi^2$ distributed?

Probably simple question. Consider the CIR (1985) model for interest rates $$ dr = k(\theta - r)dt + \sigma \sqrt{r}dz $$ Then it is known in closed form the conditional pdf $f(r(s),s|r(t),t)$ ($s \...
Gabriele Pompa's user avatar
5 votes
1 answer
1k views

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(\...
reteip's user avatar
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2 answers
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Extensions of CIR

I could need some advice on extensions of the CIR model. The standard CIR reads $dr(t)=\kappa(\theta-r(t))dt + \sigma \sqrt{r(t)} dW(t)$. A possible extension, if we would like the short-rate to ...
A.Boh's user avatar
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4 votes
1 answer
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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 ...
lbf_1994's user avatar
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1 answer
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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?
Karry's user avatar
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4 votes
1 answer
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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....
Dreason94's user avatar
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4 votes
1 answer
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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 ...
InnocentR's user avatar
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4 votes
2 answers
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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 ...
MikeRand's user avatar
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4 votes
1 answer
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HJM simulation problem

I'm trying to simulate a 3-factor HJM model. I got the algorithms from Glasserman book. In my case, I have $3$ maturity:$ 0.25y, 0.5y, 0.75y$. So my time grid is: $t_0=0,t_1=0.25,t_2=0.5,t_3=0.75$. ...
user7778's user avatar
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3 votes
3 answers
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Basic LIBOR curve question

I'm new to the quant finance and have a very basic question about LIBOR curve. LIBOR is published every day for 4 different tenors (1M, 3M, 6M, 1Y), and each rate means how much annual interest ...
brian kim's user avatar
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3 votes
1 answer
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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.
dmitry's user avatar
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3 votes
2 answers
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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 ...
L.Chau's user avatar
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3 votes
1 answer
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CallableFloatingRateBond in QuantLib: just a matter of multiple inheritance?

I would like to know what are the issues related to a possible CallableFloatingRateBond class in QuantLib and to have some hints on implementation. My (very ...
Lisa Ann's user avatar
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3 votes
2 answers
282 views

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+\...
Mr Frog's user avatar
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3 votes
1 answer
246 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 ...
koji's user avatar
  • 279
3 votes
1 answer
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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 ...
user8465900's user avatar
3 votes
1 answer
568 views

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 ...
Adam's user avatar
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3 votes
2 answers
749 views

Cox-Ingersoll-Ross

I am looking at a displaced CIR model and try to calibrate it to market data. I think my results looks reasonable but would like to sense-check with other studies. Does anyone know what "reasonable" ...
A.Boh's user avatar
  • 353
3 votes
1 answer
449 views

Solving the Jamshidian Zhu (1997) PCA short rate model

This is my first time posting a question. I have very limited experience in the field of stochastic calculus and interest rate modelling. I have been tasked with implementing the short rate model ...
NewToThis's user avatar
3 votes
0 answers
57 views

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 ...
Simon's user avatar
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2 votes
1 answer
830 views

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 ...
Feynman_kac's user avatar
2 votes
2 answers
517 views

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 ...
siwy9's user avatar
  • 63
2 votes
2 answers
562 views

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 $...
tosik's user avatar
  • 476
2 votes
1 answer
2k 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-...
Michielap's user avatar
2 votes
1 answer
2k 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 ...
Giulio Carlo Venturi's user avatar
2 votes
1 answer
81 views

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.
Michael's user avatar
  • 133
2 votes
1 answer
450 views

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 ...
VVKK77's user avatar
  • 149
2 votes
1 answer
497 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 ...
Mr Frog's user avatar
  • 233
2 votes
1 answer
213 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 ...
tnk's user avatar
  • 23
2 votes
2 answers
372 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 \...
koji's user avatar
  • 279
2 votes
1 answer
237 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 ...
koji's user avatar
  • 279
2 votes
1 answer
611 views

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, ...
Jiem's user avatar
  • 436
2 votes
2 answers
416 views

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 ...
Oliver Mohr Bonometti's user avatar
2 votes
1 answer
200 views

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}...
user13232877's user avatar
2 votes
1 answer
879 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 ...
Xman's user avatar
  • 267
2 votes
1 answer
173 views

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 ...
Trajan's user avatar
  • 2,512
2 votes
1 answer
449 views

Black–Karasinski - Market Price of Risk

In the past I have calibrated simple short rate models to the term structure by using maximum likelihood to get the parameters of the Vasicek/CIR sde, and then use the ZCB formula and the current ...
RonRich's user avatar
  • 303
2 votes
0 answers
60 views

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 ...
stokboi's user avatar
  • 71
2 votes
0 answers
90 views

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 ...
user3081005's user avatar
2 votes
0 answers
159 views

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 ...
Don Abbondio's user avatar