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Questions tagged [vasicek]

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Vasicek Short Rate

Consider a spot rate curve: 1% 1Yr, 2% 2Yr, 3% 3Yr. Suppose today issue a 3 year zero coupon bond, the price shall be 100 / (1+ 3%) ^3. My first question is, suppose the spot rate curve keeps the same ...
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Vasicek 2 factors model calibration

I am working on to calibrate two factors Vasicek interest rate model. Can I start by asking a really stupid question? For the stochastic model $dr_t=\kappa(\mu−r_t)dt+\sigma dWP(t)$ with the real ...
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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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1answer
91 views

Derive a mathematical equation for Eurodollar future rate

If we suppose that r(t) follows a Vasicek model, which is: $$dr(t) = (\mu - \kappa r(t))dt + \sqrt\sigma dW(t)$$ How to derive an expression for Eurodollar future rate?
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1answer
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Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model

I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation): $$ dr_t = a(b-r_t)dt + \sigma dW_t $$ Every solution is proceeding to multiply both sides of the equation by the ...
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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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1answer
103 views

Vasicek short rate: Risk-neutral measure into real-world measure

I consider the Vasicek model under the risk-neutral measure $\mathbb{Q}$: $$ dr_t=\kappa(\theta−r_t) dt+\sigma dW^{\mathbb{Q}}_t.$$ I have already determined $$\mathbb{E}^{\mathbb{Q}}\left[e^{−\int\...
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1answer
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Two papers - two different solutions of the Ornstein-Uhlenbeck process

Bernal 2016 says that the solution of $$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$ equals $$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
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Bond prices at future times under Vasick one-factor model

In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time $t$ conditional on the information at this time is $$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-...
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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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Black-Schole PDE for european put option under Cox-Ingersoll-Ross model

For a european put option, starting from the classical Black-Scholes PDE (assuming constant rate), how do we come up with the Black-Scholes PDE under the Cox-Ingersoll-Ross model (CIR) such as the ...
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Difference between Standard VaR and VaR with partial set of Risk Factors

Ciao, I'm working on VaR and Expected Shortfall and this question came out. For a given portfolio VaR can be computed w.r.t. all the risk factors or just for a subset. Infact you can decide to 'freeze'...
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1answer
357 views

Hull-White Extension of Vasicek Model

I am reading the book Interest Rate Models by Brigo and Mercurio and try to understand the Hull White Model Extended Vasicek Model. They start off by defining the instantaneous short-rate process ...
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1answer
109 views

Term structure used in Geometric Brownian Motions under Risk Neutral Measure?

When using a GBM under a risk-neutral measure to simulate stock prices, we have to use the risk-free interest rate, but how exactly do you determine what interest rate to use? I have used the Vasicek ...
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CIR IR model - Building in a “dynamic mean rate level”

I am currently running a Cox Ingersoll Ross model to predict LIBOR from today. I have used data from 1978 to calibrate the model. (This is a separate question, as I know there will be a vast variaton ...
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1answer
39 views

compute r(t) in Vasiceck model, what is $e^{at}r$

I know how to solve the exercise using the hint. But I do not understand where the hint is coming from. Is it just continous compounding? Can anybody explain $f(t,r) = e^{at}r$? What does it stand ...
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1answer
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time step choice impact in Vasicek model simulations

I am trying to make some computations using Vasicek short rate model. Especially I a trying to compare exact expectation(obtained with the formula) and the expectation from Monte Carlo simulation. ...
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1answer
884 views

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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Help evaluating covariance integral when deriving vasiceks model

Im working through a solution to evaluating pricing for Vasiceks model However i dont understand the u∧t terms and how that behaves under the integrals...any help?? Cheers
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1answer
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Difference between the Basel IRB and the Vasicek formula

The well known Basel IRB formula is as follows: $${\displaystyle K=LGD*\left[N\left({\sqrt {\frac {1}{1-R}}}*G(PD)+{\sqrt {\frac {R}{1-R}}}*G(0.999)\right)-PD\right]}$$ where the term below is the ...
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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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Price of a Bond-Call option in the defaultable framework

I would like to compute the price for a Call option written on a defaultable bond as underlying. Suppose you have the following dynamic under the risk free measure $\mathcal{Q}$ for the interest rate: ...
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1answer
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How to find the transition distribution functions of these two processes?

This question was asked by another user, but was deleted. As it may be useful for others, I re-post it here. What are the transition distribution (or density) functions of two processes defined by \...
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1answer
120 views

How to find the transition distribution functions of these two processes?

What are the transition distribution (or density) functions of processes defined by $dX_t=\mu dt +\sigma dW_t$ and $dX_t= \theta(\mu-X_t) dt +\sigma dW_t,$ where $\theta>0$, $\mu$ is a real ...
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381 views

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 with Vasicek model on basket of credit spreads

I would appreciate help with a valuation of a fixed income derivative, with an embedded exit option. Summary: Goal is to provide valuation of a fixed schedule of quarterly cash flows with an option ...
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1answer
48 views

How to understand the following limits when kapa limits to Zero

The equation is quite simple, however it is not very obvious to me to have the following relationship: $$\begin{equation} \frac{1-exp(-\kappa(T-t))}{\kappa}\rightarrow(T-t) \quad \rm{when\space} \...
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474 views

How to show that this process is “normally distributed”?

Say we have following SDE (Vasicek): $$dr(t) =(b-ar_t) dt + \sigma dW_t$$ I am able to reach an integral form of this SDE : $$r(t) = r(0) e^{-at} + \frac{b}{a}[1 - e^{-at}] + \sigma e^{-at}\int_0^t e^...
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1answer
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Why does the correlation between r and V in Longstaff and Schwartz 1992 model is positive?

I am reading the Longstaff and Schwartz's 1992 and 1993. From $r = \alpha x + \beta y$ and $V = \alpha^2 x + \beta^2 y$. It was mentioned in the paper that the $r$ is positive correlated with $V$. ...
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1answer
126 views

What is the limiting distribution of loss portfolio?

I am working through this paper on Vasicek's portfolio loss distribution. On page 3 he mentions that by the law of large numbers, $$\lim_{n\to\infty}\sum_{k=0}^{\lfloor nx \rfloor} \binom{n}{k}s^k(1-...
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1answer
610 views

LIBOR rates from Vasicek/Hull-White model?

I am somehow puzzled by the following problem: LIBOR rates are forward rates for an interbank loan for 1M or 3M (let's limit the range of possibilities to these two cases). Assuming that I have ...
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1answer
339 views

LIBOR 3M and 1M from Vasicek model

I would like to discuss my approach toward modelling of interest rates with respect to its downsides and advantages. My problem is to forecast daily LIBOR 3M and LIBOR 1M over a particular time ...
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2answers
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How to show that the exponential Vasicek model is not an affine term-structure model?

From the pricing formula, we know that the value at time $t\in [0,T]$ of a zero coupon bond maturing at time $T$ is $$ B(t,T)=E\left(\exp{\left(-\int_{t}^{T}r_sds\right)}\bigg|\mathcal{F}_t\right). $$...
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1answer
355 views

Vasicek model problem

I am analyzing a problem where the below is given Vasicek model with risk-neutral dynamics $$dr_t = \kappa (\theta - r_t)dt + \sqrt{r_t} dW_t \quad \quad (1) $$ bond prices $$P(t,T)=e^{A(t,T)-B(t,T)...
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On the construction of a Brownian motion from a Gaussian process

Let $X$ a Gaussian process defined by $$ X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T] $...
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How to price a stock under Q and stochastic interest rates?

I am interested in pricing a stock under $\mathbb{Q}$ when I assume that $$dS(t) = \mu(S(t))dt + \sigma(S(t))dW(t)$$ where $W(t)$ is a Wiener process under $\mathbb{P}$ and $$dr(t) = a(b-r(t))dt +...
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1answer
294 views

Timesteps in Vasicek model

When simulating stocks one can easily use GBM with only one random variable per simulation to create a new stock price in say 5 years, you don't need to create the whole asset paths if you don't need ...
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1answer
88 views

simulation and timestep

Suppose I have a stochastic process i.e. a Vasicek process with parameteres estimated with monthly (RW measure) data and want simulate the process using a daily timestep. Is this a good practice?
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50 views

MLE for two independent factor CIR

Following the maximun likelihood estimation as done in Klavidko I would like to generalize this to more independent factors . In first istance I would use the transition function at time t as a sum ...
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1answer
2k views

Pricing a zero with Vasicek model

I'm trying to understand bond pricing with the Vasicek interest rate model. I'm using McDonald's book for this purpose (not homework). Recall that Vasicek dynamics are \begin{equation*} \mathrm{d}...
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2answers
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What is the reasoning to derive this financial model called the Vasicek Model?

The model specifies that the instantaneous interest rate follows the stochastic differential equation $$\mathrm{d}r_t = a(b-r_t)\: \mathrm{d}t + \sigma \: \mathrm{d}W_t$$ where $W_{t}$ is a Wiener ...
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1answer
324 views

From $AR(p)$ to SDE

Let the Vasicek model to be $$\Delta r_{t}=k(\theta - r_{t-1})\Delta t+\sigma\Delta z_{t}$$ Due to the fact that $$\Delta r_{t}=r_{t}-r_{t-1}$$ if you let $\Delta t=1$, it is easy to see by ...
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3answers
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Is Vasicek risk neutral?

I am a bit new to this, and am trying to understand the concepts of the risk neutrality in interest-rate models. What I can't seem to understand is why the Vasicek model is risk-neutral? Following ...
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1answer
555 views

How to design back-testing (validation) for such modified Vasicek model?

Consider a classical Black Scholes model , $$\frac{dS}{S} = \mu dt + \sigma dW$$ , where $dW$ is a Brownian motion, that $W(t_1) - W(t_0) \sim N(0, t_1 - t_0)$. The back-testing strategy is straight-...
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2answers
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Why is the mean time-dependent in the Hull-White interest rate model?

In the Vasicek interest-rate model, the interest rate reverts to a constant mean. This makes sense to me. In my conception, the mean ought to be time-invariant, since interest rates don't follow an ...