Questions tagged [vasicek]
The vasicek tag has no usage guidance.
82
questions
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44 views
How to obtain the discount curve under Vasicek interest rate for discounting cash flow?
Suppose that the spot rate is governed by a Vasicek model. We know that there is an analytical solution for the zero-coupon bond.
I guess the discount curve is constructed by the Yield curve in which ...
1
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0answers
49 views
Distribution and Analytical solution of a GBM with stochastic interest rate?
We model the exchange rate $S_t$ with a geometric Brownian motion and the USD and EUR interest rates $r_u$ and $r_e$ each according to the Vasicek model. Under the domestic equivalent martingale ...
1
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2answers
105 views
Trouble Calibrating a Vasicek Model
I have simulated some data according to a Vasicek process and I am then trying to apply ordinary least squares (OLS) regression analysis to see how accurate the estimated model parameters are from the ...
3
votes
2answers
212 views
How to determine the risk premium from the Vasicek one factor model?
The short rate under the Vasicek one factor model under the real-world measure $\mathbb{P} $ follows :
$$ dr(t)=(a\theta - (a+\lambda \sigma)r(t))dt + \sigma dW(t),$$
$$ r(0)=r_0 $$
where $ \lambda $ ...
2
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0answers
62 views
Why does adding a negative risk premium to the short rate avoid the occurrence of inverse yield curves?
I am reading about the Vasicek One Factor short rate model and how to implement a change in measure from a risk-neutral to real-world measure, when I came across this comment:
Adding a negative risk ...
1
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0answers
71 views
Changing order of integration on stochastic term in Vasicek [closed]
This question is in relation to the vasicek model, where i am trying to find the solution.
I have this term: $-\int_{t}^{T} \sigma \int_{t}^{s} e^{-\kappa(s-u)} d W(u) d s$
I need to change the ...
1
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0answers
44 views
Change of numeraire to the forward measure in the Vasicek model
I am working through the Brigo/Mercurio book on Interest Rate Models (Second Edition) and I am having some trouble with the change of numeraire in chapter 3.2.1, page 59 to be exact, formula 3.9. It ...
1
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1answer
40 views
Why can a two-factor interest rate model not be used to value a coupon bearing bond as the sum of options on ZCBs
I am currently reading some notes which state that
For one-factor models, the value of a European option on a coupon bond can be calculated as the sum of European options on zero-coupon bonds (ZCBs). ...
1
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2answers
147 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*}
\...
3
votes
1answer
129 views
Pricing Call Option on Coupon Bond under Vasicek
Consider a the Vascicek model, and let A and B denote the functions such that $P(t,T)=\exp(A(t,T)-B(t,T)r(t))$. We now look at a coupon bond that makes deterministic payments $\alpha_1,...,\alpha_N$ ...
0
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0answers
95 views
Vasicek Model With Jumps
I'm trying to calibrate a mean-reverting, jump diffusion model using the outline provided on page 11 here:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.3489&rep=rep1&type=pdf
...
1
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1answer
123 views
Simulating exponential Vasicek/Ornstein-Uhlenbeck
I am trying to simulate commodity prices using the exponential Vasicek/Ornstein-Uhlenbeck model from Schwartz 1997 p. 926 Equation (1). I am using the closed form solution from Vega 2018 p. 5 Equation ...
0
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2answers
171 views
Vasicek Model (estimation of parameters)
I have a question concerning the "choice" of parameters for the Vasicek model (formula below).
Consider me as a moron with below average level in maths haha.
What I've done is basically run ...
1
vote
1answer
72 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 ...
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0answers
134 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
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1answer
142 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
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1answer
190 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
55 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
233 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
48 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
36 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
78 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
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1answer
109 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
81 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
66 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 ...
2
votes
1answer
208 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
votes
1answer
195 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 ...
1
vote
1answer
254 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
546 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
760 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
108 views
Derive the discount bond prices of the Vasicek model by the PDE approach
The question is shown above.
Anyone can help me?
0
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1answer
85 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
149 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
110 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
65 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
554 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
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1answer
428 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
203 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
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1answer
211 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
422 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
60 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"...
4
votes
1answer
411 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
148 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
434 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
207 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 ...
2
votes
1answer
763 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 $$\mathbb{E}^{\mathbb{Q}}\left[e^{−\int\...
2
votes
1answer
207 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ 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
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0answers
219 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
996 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
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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 ...
