Questions tagged [vasicek]
The vasicek tag has no usage guidance.
2
votes
0answers
76 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
vote
1answer
103 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?
2
votes
1answer
117 views
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 ...
1
vote
0answers
91 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 ...
1
vote
1answer
158 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 $$\mathbb{E}^{\mathbb{Q}}\left[e^{−\int\...
1
vote
1answer
142 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ 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)...
2
votes
0answers
54 views
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^{-...
3
votes
0answers
83 views
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 ...
0
votes
0answers
84 views
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 ...
0
votes
0answers
20 views
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'...
1
vote
1answer
487 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 ...
1
vote
1answer
132 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 ...
0
votes
1answer
41 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 ...
0
votes
1answer
96 views
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.
...
4
votes
1answer
1k 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+\...
0
votes
0answers
219 views
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:
$$ dr_t=\alpha \left( \bar{\mu} - r_t\right) dt+\sigma d W^{\mathbb{P}}(t), $$
$$ dr_t=\...
0
votes
1answer
129 views
Help evaluating covariance integral when deriving vasiceks model
Im working through a solution to evaluating pricing for Vasiceks model
However i dont understand the u∧t terms and how that behaves under the integrals...any help??
Cheers
1
vote
1answer
275 views
Difference between the Basel IRB and the Vasicek formula
The well known Basel IRB formula is as follows:
$${\displaystyle K=LGD*\left[N\left({\sqrt {\frac {1}{1-R}}}*G(PD)+{\sqrt {\frac {R}{1-R}}}*G(0.999)\right)-PD\right]}$$
where the term below is the ...
3
votes
0answers
110 views
Basic Interest Rate Modelling Ques
I have got a question regarding the Vasicek Model and the corresponding Bond Pricing Equation (BPE).
Starting with a short-rate process (under measure $P$ or real world drift $u(r,t)$) of the form:
$...
3
votes
0answers
140 views
Price of a Bond-Call option in the defaultable framework
I would like to compute the price for a Call option written on a defaultable bond as underlying.
Suppose you have the following dynamic under the risk free measure $\mathcal{Q}$ for the interest rate:
...
1
vote
1answer
67 views
How to find the transition distribution functions of these two processes?
This question was asked by another user, but was deleted. As it may be useful for others, I re-post it here.
What are the transition distribution (or density) functions of two processes defined by
\...
1
vote
1answer
120 views
How to find the transition distribution functions of these two processes?
What are the transition distribution (or density) functions of processes defined by
$dX_t=\mu dt +\sigma dW_t$
and
$dX_t= \theta(\mu-X_t) dt +\sigma dW_t,$
where $\theta>0$, $\mu$ is a real ...
2
votes
0answers
488 views
Complete Algorithm of Calibration with Vasicek Model using Term-Structure Dynamics over Time
As there are so many different sccenarios about Vaicek Calibration but there has not been a clear example with data shown, I am totally Confused about how should I do it. so I am bringing the question ...
1
vote
0answers
148 views
Pricing with Vasicek model on basket of credit spreads
I would appreciate help with a valuation of a fixed income derivative, with an embedded exit option.
Summary:
Goal is to provide valuation of a fixed schedule of quarterly cash flows with an option ...
1
vote
1answer
48 views
How to understand the following limits when kapa limits to Zero
The equation is quite simple, however it is not very obvious to me to have the following relationship:
$$\begin{equation}
\frac{1-exp(-\kappa(T-t))}{\kappa}\rightarrow(T-t) \quad \rm{when\space} \...
8
votes
1answer
536 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 :
$$r(t) = r(0) e^{-at} + \frac{b}{a}[1 - e^{-at}] + \sigma e^{-at}\int_0^t e^...
1
vote
1answer
69 views
Why does the correlation between r and V in Longstaff and Schwartz 1992 model is positive?
I am reading the Longstaff and Schwartz's 1992 and 1993.
From $r = \alpha x + \beta y$ and $V = \alpha^2 x + \beta^2 y$. It was mentioned in the paper that the $r$ is positive correlated with $V$.
...
2
votes
1answer
131 views
What is the limiting distribution of loss portfolio?
I am working through this paper on Vasicek's portfolio loss distribution.
On page 3 he mentions that by the law of large numbers,
$$\lim_{n\to\infty}\sum_{k=0}^{\lfloor nx \rfloor} \binom{n}{k}s^k(1-...
1
vote
1answer
710 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 ...
1
vote
1answer
367 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 ...
2
votes
2answers
745 views
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
$$
B(t,T)=E\left(\exp{\left(-\int_{t}^{T}r_sds\right)}\bigg|\mathcal{F}_t\right).
$$...
1
vote
1answer
413 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 $$P(t,T)=e^{A(t,T)-B(t,T)...
1
vote
0answers
83 views
On the construction of a Brownian motion from a Gaussian process
Let $X$ a Gaussian process defined by
$$
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]
$...
4
votes
2answers
596 views
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
$$dr(t) = a(b-r(t))dt +...
1
vote
1answer
367 views
Timesteps in Vasicek model
When simulating stocks one can easily use GBM with only one random variable per simulation to create a new stock price in say 5 years, you don't need to create the whole asset paths if you don't need ...
2
votes
1answer
98 views
simulation and timestep
Suppose I have a stochastic process i.e. a Vasicek process with parameteres estimated with monthly (RW measure) data and want simulate the process using a daily timestep. Is this a good practice?
2
votes
0answers
53 views
MLE for two independent factor CIR
Following the maximun likelihood estimation as done in Klavidko I would like to generalize this to more independent factors . In first istance I would use the transition function at time t as a sum ...
2
votes
1answer
2k views
Pricing a zero with Vasicek model
I'm trying to understand bond pricing with the Vasicek interest rate model. I'm using McDonald's book for this purpose (not homework).
Recall that Vasicek dynamics are
\begin{equation*}
\mathrm{d}...
2
votes
2answers
239 views
What is the reasoning to derive this financial model called the Vasicek Model?
The model specifies that the instantaneous interest rate follows the stochastic differential equation
$$\mathrm{d}r_t = a(b-r_t)\: \mathrm{d}t + \sigma \: \mathrm{d}W_t$$
where $W_{t}$ is a Wiener ...
5
votes
1answer
341 views
From $AR(p)$ to SDE
Let the Vasicek model to be
$$\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}$$
if you let $\Delta t=1$, it is easy to see by ...
6
votes
3answers
3k views
Is Vasicek risk neutral?
I am a bit new to this, and am trying to understand the concepts of the risk neutrality in interest-rate models.
What I can't seem to understand is why the Vasicek model is risk-neutral? Following ...
4
votes
1answer
582 views
How to design back-testing (validation) for such modified Vasicek model?
Consider a classical Black Scholes model ,
$$\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-...
4
votes
2answers
1k views
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 ...
