Questions tagged [short-rate]

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3
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1answer
84 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 Dothan model: $$dr_t=...
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67 views

Which interest rate model is the most popular

Hey on wikpedia (https://en.wikipedia.org/wiki/Short-rate_model) there are quite a few short rate models listed, but which models are the most commonly used now? Because such simple models as Vasicek ...
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1answer
48 views

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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0answers
47 views

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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2answers
117 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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0answers
55 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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0answers
27 views

Interest rate futures notional

I was wondering what is the notional used to calculate tick values when the underlying of the futures is the average of an overnight rate (eg 1m SONIA futures, 1m EONIA futures, etc.)? When the ...
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1answer
65 views

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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1answer
233 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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0answers
74 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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0answers
35 views

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 ...
1
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1answer
49 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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1answer
140 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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0answers
66 views

Calibration of the Ho Lee model in Excel only based on historical treasury rates (3 months)

I would like to calibrate the Ho Lee model $dr_t=θ_tdt+σdW_t$ with $θ(t)= F_t(0,t)+σ^2$ in Excel based on a time series of 3-month treasury rates and then compare the real time series with that ...
2
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1answer
110 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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0answers
96 views

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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0answers
19 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 ...
1
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1answer
139 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 ...
1
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1answer
316 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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1answer
55 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 ...
0
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1answer
72 views

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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1answer
75 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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2answers
176 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 ...
4
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1answer
141 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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1answer
154 views

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
4
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1answer
645 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?
2
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1answer
202 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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0answers
84 views

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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0answers
85 views

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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1answer
279 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 ...
3
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1answer
173 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 ...
0
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1answer
81 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—...
2
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2answers
115 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 \...
2
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1answer
120 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 ...
2
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0answers
96 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 ...
2
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1answer
62 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.
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0answers
136 views

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 ...
2
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1answer
357 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 ...
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0answers
70 views

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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1answer
285 views

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 ...
2
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2answers
262 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 $...
1
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1answer
234 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 ...
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1answer
192 views

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?
5
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1answer
284 views

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(...
1
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1answer
125 views

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} \...
6
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0answers
138 views

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\...
3
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1answer
349 views

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 ...
1
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1answer
274 views

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 ...
2
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1answer
125 views

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 ...
1
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0answers
172 views

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 ...