Questions tagged [vasicek]
The vasicek tag has no usage guidance.
73
questions
1
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1answer
62 views
Affine term structure for CDS
in papres such as https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2686284 (Exploring Mispricing in the Term Structure of CDS Spreads by Robert A. Jarrow, Haitao Li, Xiaoxia Ye, and May Hu) a ...
1
vote
0answers
42 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 ...
0
votes
1answer
55 views
Monte Carlo price of European option on ZCB under Vasicek short rate
I'm trying to replicate the analytical result from the closed form Vasicek formula for European options on zero-coupon bonds using Monte-Carlo simulation.
The interest rate paths I've simulated seem ...
0
votes
1answer
94 views
Hazard rate and Term structure model
About the paper of Pan and Singleton 2008 “Default and Recovery Implicit in the Term Structure of Sovereign CDS Spreads”,
once the lambdas (hazard rates) for the different tenors of the term structure ...
0
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0answers
32 views
How do I hedge two/three zero coupon bonds with different maturity under Vasicek short rate model?
I am working on the case that I need to hedge two bonds with different maturites under Vasicek model, which is
\begin{equation}
dr_t=a(b-r_t)dt+\sigma dW^Q_t
\end{equation}
and I know how to price the ...
0
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0answers
52 views
Calibration and comparison of the Vasicek model and Ho-Lee model
I would like to calibrate the Hoo-Lee model and the Vasicek model to a historical interest rate series and compare the interest rate development of both models with the historical interest rate series....
0
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0answers
37 views
Multi-period Basel/Vasicek formula
I need to apply Basel/Vasicek formula to a 20-years horizon, both from a 20-years cumulative perspective and year-on-year basis.
Please find below the formula of the Basel Capital (ie. unexpected loss)...
0
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0answers
28 views
Vasicek credit loss for real portfolio
Consider the Vasicek limiting distribution for losses of a loans portfolio.
Now, consider a real portfolio, made of 10 loans each with a different rating class; eg:
LN#1 - rating A+
LN#2 - rating BB
...
1
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0answers
39 views
Difference between Vasicek and Gordy models
I'm trying to understand what Gordy [1] added to Vasicek [2] model (the core of the IRB formula of Basel Accords).
Is it correct to say the Vasicek shows that the portfolio loss conditional on $Y$ ...
0
votes
1answer
80 views
Vasicek model - Bond price and volatility
Why does the bond price under the Vasicek model increase as the rate volatility increases? What is the intuition behind this?
2
votes
1answer
50 views
Choosing which interest rate model to go with?
I've been assigned with the task of modelling zero rate curve. I did it with two models: Vasicek and CIR. Looking at the two curves produced, I can see that one is closer to the observed curve than ...
1
vote
1answer
59 views
Distribution and parameters for the amount at time T of Bond
An investor follows the following investment strategy from time t to time
T: buys a 10-year zero coupon bond, holds it for a time-length dt, sells it
and buys a new 10-year ZCB with the proceeds. The ...
0
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0answers
52 views
Option pricing PDE Black Scholes one-factor Hull-White (or Vasicek) model
I am trying to find the option pricing PDE of the Black Scholes one-factor Hull-White (or Vasicek) model using a self-financing portfolio strategy. The system is as following
\begin{equation*}
\begin{...
0
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0answers
39 views
calibration of correlation in vasicek model
how can I calibrate the correlation by numerical integration of the normal bivariate distribution assuming that the standardized asset returns of two firms are described by the single-factor Vasicek ...
0
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0answers
16 views
Python Panel Data Regression for OUP Calibration
I have a model for predicting stock returns that classifies stocks as overbought or oversold, kind of like an RSI. It follows an OUP and I am curious about my $\mu$, $\sigma$, and $\kappa$ parameters, ...
2
votes
1answer
62 views
Exact solution stock price with Vasicek interest rate model
Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$
$$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$
$$dr(t)=\kappa(\theta-r(...
0
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1answer
81 views
Black & Scholes under stochastic interest rate (Vasicek) [closed]
I'm a beginner in Quantitative finance and I'd like to ask you for help about this exercise. I have to price a put option on a risky asset by working under stochastic interest rate, so I have to ...
0
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0answers
133 views
Frye Jacobs LGD function
I have a regression model that predicts the PiT (Point in Time) default rate (PD). Can we use this PiT
PD in the Frye Jacobs LGD function for making LGD forward looking ?
In addition, is the ...
0
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1answer
110 views
Vasicek Model, zero coupon bond question [closed]
I am trying to solve questions in the Vasicek model. Can anyone help me to solve this question...
In the Vasicek model with parameters $\theta = 0.08$, $k$ = 2.5, $\sigma = 0.2$, assuming to be ...
1
vote
1answer
347 views
Vasicek Model Parameters Estimation
I'm currently trying to estimate the market price of risk (lambda) in the Vasicek Model, and am running into difficulties.
Using the Excel Solver tool and the Maximum Likelihood Estimation method ...
4
votes
1answer
447 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?
1
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0answers
68 views
Derive the discount bond prices of the Vasicek model by the PDE approach
The question is shown above.
Anyone can help me?
0
votes
1answer
72 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—...
3
votes
1answer
111 views
Aggregation of $\rho$ and $p$ for a vasicek model
I'm currently facing the problem of how properly (analytically) adjust the parameters of an aggregated Vasicek (2002) loss distribution so that it has the same expected loss and 99% quantile as the ...
2
votes
0answers
69 views
State price deflator in the Vasicek model
I am trying to implement a simple bond pricing model using state price deflators in a Vasicek model. I am simulating paths of the processes
$$\mathrm{d}r^{P} =\kappa^{P}(\theta^P - r^P(t))\mathrm{d}t ...
1
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0answers
59 views
Can I use 1M Libor monthly data to estimate Vasicek parameters and use them to price quarterly swap
I am working on CVA algorithm. I am using vasicek model to evolve short rate. At hand, I am supposed to value a fixed to floating IRS quarterly paying.
Can I use 1M Libor as surrogate for short ...
4
votes
1answer
330 views
Bond-price dynamics in the Vasicek model
Hello I am studying about interest rate modeling
There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable ...
0
votes
1answer
232 views
Vasicek model and spot interest rate parametrised by reversion rate
By solving an SDE I want to derive the analytical results for mean and variance of the process of extended Vasicek model.
$$
dr(t) = \left(\eta - \gamma r(t) \right)dt + c dX(t)
$$
where $\gamma$ ...
2
votes
0answers
161 views
Term structure equation in the Vasicek model
Consider the SDE $$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$ Let $$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r).$$ (F can be interpreted as price of a zero coupon bond with ...
0
votes
1answer
152 views
Vasicek and Extended Vasicek Model
I want to ask about basic reasoning in Vasicek and Extended Vasicek model.
Why $P(T,T) = 1$ for non arbitrage model? Can we place $P(T,T) = 10$ or other numbers? Is it correlated with The Law of ...
2
votes
0answers
281 views
Need to solve the stochastic differential equation of Vasicek Model
How to solve the stochastic differential equation of the Vasicek model for the analysis of credit risk? I search in the article "The Distribution of loan portfolio value" (Vasicek) but he doesn't ...
1
vote
1answer
52 views
Finding B(t) in the Vasicek model relating to the bond equation, more specifcally from the initial condition
In the Vasicek model for derving bond prices, we have the ODE $$\frac{dB}{dt}=\gamma B-1$$ which gives rise to the general solution $$B(t)=C_1 e^{\gamma t}+C_2$$My problem is that we have the "initial"...
3
votes
1answer
272 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
141 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
348 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
147 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
542 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 $$\mathbb{E}^{\mathbb{Q}}\left[e^{−\int\...
2
votes
1answer
185 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ 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
137 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
$$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-...
4
votes
3answers
380 views
What is the purpose of short rate models?
Just venturing into quantitative finance and studying short rate models (Vasicek, CIR, Hull-White etc.). Wanted to ask a very simple intuitive question. How would a practitioner use these models? I ...
1
vote
1answer
1k views
Hull-White Extension of Vasicek Model
I am reading the book Interest Rate Models by Brigo and Mercurio and try to understand the Hull White Model Extended Vasicek Model. They start off by defining the instantaneous short-rate process ...
1
vote
1answer
199 views
Term structure used in Geometric Brownian Motions under Risk Neutral Measure?
When using a GBM under a risk-neutral measure to simulate stock prices, we have to use the risk-free interest rate, but how exactly do you determine what interest rate to use?
I have used the Vasicek ...
0
votes
1answer
45 views
compute r(t) in Vasiceck model, what is $e^{at}r$
I know how to solve the exercise using the hint. But I do not understand where the hint is coming from. Is it just continous compounding?
Can anybody explain $f(t,r) = e^{at}r$?
What does it stand ...
0
votes
1answer
110 views
time step choice impact in Vasicek model simulations
I am trying to make some computations using Vasicek short rate model. Especially I a trying to compare exact expectation(obtained with the formula) and the expectation from Monte Carlo simulation.
...
5
votes
1answer
3k views
Vasicek model calibration
I am trying to calibrate Vasicek model, i.e. to determine the parameters $\kappa, \mu, \bar{\mu}$ and $\sigma$ where the process dynamics are given through
$$ dr_t=\kappa\left( \mu - r_t\right) dt+\...
0
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0answers
454 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:
$$ dr_t=\alpha \left( \bar{\mu} - r_t\right) dt+\sigma d W^{\mathbb{P}}(t), $$
$$ dr_t=\...
0
votes
1answer
239 views
Help evaluating covariance integral when deriving vasiceks model
Im working through a solution to evaluating pricing for Vasiceks model
However i dont understand the u∧t terms and how that behaves under the integrals...any help??
Cheers
2
votes
1answer
480 views
Difference between the Basel IRB and the Vasicek formula
The well known Basel IRB formula is as follows:
$${\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 ...
2
votes
0answers
139 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:
$...
2
votes
0answers
176 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:
...
