Questions tagged [vasicek]
The Vasicek model is a 1-factor short-rate model.
103
questions
0
votes
0
answers
34
views
Vasicek model calibration to bond prices or rates (no swaptions)
I need to calibrate Vasicek's model $dr_{t} = a(\theta - r_{t})dt + \sigma dW_{t}$ in a market with no swaptions. I was thinking in estimating $\sigma$ with historic data, but I'm in the doubt with ...
0
votes
1
answer
59
views
Are instantaneous short rates compatible across models?
If I calibrate the Vasicek's yield curve to the Nelson-Siegel's (NS) yield curve, can I assume that $r_V(0) = r_{NS}(0) = \beta_0 + \beta_1$ or not?
NS short rate:
$r_{NS}(S) = β_0 + β_1 e^{-S/\tau} + ...
0
votes
0
answers
104
views
Simulating Hull-White Model in Python
I first simulated the short rate in the Vasicek model using the following code, which is equivalent to simulating the following normal distribution $r_{t} \sim N\left(r_{0}e^{-at} + b\left(1-e^{-at}\...
0
votes
1
answer
64
views
Half-life of short rate
The SDE for the short rate r(t) in the Vasicek model is given by:
$$ d(r) = k(r^* - r)dt + \sigma dW $$
The deterministic part of the above SDE is the following ODE
$$ d(r) = k(r^* - r)dt, $$
where $k$...
0
votes
0
answers
33
views
Understanding the application of Asset-Correlation to credit risk models
Suppose we have a portfolio of $n$ credits. In order the estimate the Portfolio Value at Risk (99,9) we use a standard vasicek model with the Ability to pay variable $A_i=\sqrt{\rho}x+\sqrt{1-\rho}z_i$...
2
votes
0
answers
45
views
ATM cap prices in Vasicek model (Filipovic)
I am trying to replicate the ATM cap prices in table 7.1 (see bottom of this post) from Filipovic's book "Term Structure Models - A Graduate Course" which assume the Vasicek model and uses ...
3
votes
0
answers
86
views
How to calibrate an O-U process based on historical data?
Background: I have been working on my master thesis project for the last few months and gave the final presentation on the 2023-06-01. As a part of the master thesis project, I did a complete ...
2
votes
0
answers
119
views
Bond-pricing under the Vasicek short rate model
I'm currently studying the Vasicek model of the short interest rate
$$dr_t=a(\mu-r_t)dt+\sigma dW_t$$
I know how to solve this stochastic differential equation (SDE) and how to find expectation and ...
2
votes
0
answers
125
views
Expected value and variance of the short rate under the Vasicek model
Would be grateful for any assistance.
Below are the expected value and variance of the integral of the short rate under the Vasicek model (https://www.researchgate.net/publication/41448002):
$E\left[ \...
0
votes
0
answers
39
views
Likelihood of least squares estimates of Vasicek model
I want to compare some short-rate models based on likelihood and AIC. I will use least squares estimates.
Let's take the Vasicek model as an example and its discretized version:
$$
dr = \alpha(\beta-r)...
0
votes
0
answers
160
views
Are my fitted Vasicek model parameters market consistent or realistic?
In view of this question I asked some time ago, I tried to calibrate a Vasicek model to some cap volatilities, given as follows. I consider the maturities (in years)
$$
0.5,1,2,3,4,5,7,10,15,20
$$
and ...
0
votes
0
answers
59
views
Simulating of short rate model
I'm trying to simulate the risk factor of PFE from the interest rate model.
For example, under Vasicek model :
$$dr_t = k(\theta-r_t)dt + {\sigma}dW_t$$
with the analytic solution, we can simulate N ...
2
votes
0
answers
316
views
How does one calibrate a Vasicek model to actual cap prices?
I am trying to calibrate a Vasicek model given by
$$
dr(t) = k[\theta - r(t)] dt + \sigma dW(t), \quad r(0) = r_0
$$
where $k, \theta, \sigma, r_0 > 0$. I am using the book by Brigo and ...
2
votes
1
answer
662
views
Problem with pricing a call option using the Monte Carlo Vasicek model
I am trying to price a call option on a zero coupon under the Vasicek Model using Monte Carlo method:
$$C_0 = B(0,\theta) \ \mathbb{E}^{\mathbb{Q}_T}[(B(\theta,T)-K)^{+}]$$
The problem is that the ...
2
votes
0
answers
136
views
Vasicek Model: smile dynamics
I have come across the statement that the Vasicek model cannot be used to price skew / smile sensitive products: i.e. it cannot be calibrated to replicate a skew or smile. Why is that?
My guess is ...
1
vote
0
answers
59
views
Is there a closed form solution to the following system of SDEs?
Suppose we have the system
\begin{align}
dr_t=\alpha_r(x_t-r_t)dt+\sigma_rdW_t^r\\
dx_t=\alpha_x(\bar{x}-x_t)dt+\sigma_xdW_t^x\\
\end{align}
As this system is affine, I believe there should be an easy ...
3
votes
0
answers
48
views
Is the G2++ model apt to use when one needs estimates of longer term refinance rates for mortgages and can the model be created with Monte Carlo?
I am currently in the process of developing an interest rate model that would be used to price mortgage-backed securities and develop an OAS estimate. Referring to Brigo and Mercurio (2006) I'm ...
1
vote
3
answers
322
views
Square root specification of parameters in factor models
The following formulation is from Vasicek and refers to the cond. probability of the loss of a loan (equ. 3 in the reference):
$$p(Y)=\Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho}\,Y}{\sqrt{1-\rho}}\right)....
1
vote
0
answers
195
views
Differential vs. derivative in the Vasicek model [closed]
Can anyone help me in understanding how we get the line I have marked with a red arrow?
I guess I have trouble in understanding the difference between differentials and derivatives, i.e. what is the ...
2
votes
1
answer
337
views
Obtaining the dynamics of the Vasicek model using Itô
Consider the following expression for the short-term interest rate
$$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$
which is ...
2
votes
0
answers
135
views
Zero coupon price using Vasiceks model under the Real-world P measure model
I'm wondering if there is a way to work out the formula for the price of the zero-coupon bond using the Vasicek's model (P measure). I have tried to find reference on it but could not, I don't know if ...
1
vote
0
answers
35
views
Including Exogeneous variables in short rate models
I am trying to use short rate models (e.g. Vasicek, CIR or Hull-White) to forecast next one or two months yield curve. In this context, is there a way that I can include some exogenous economic or ...
0
votes
0
answers
56
views
Vasicek process with no reversion
I want to prove that as the speed of reversion $\lambda$ in the Vasicek process $dR=\lambda(R_{\infty}−R)dt + \sigma d\beta$ approaches 0, the expected long rate
$$r(0,T)=R_{\infty}+\frac{(R(0)-R_{\...
1
vote
1
answer
239
views
How do I estimate the factor sensitivity in a Vasicek Single Factor Model?
I understand the formula of an asset return for an obligor i is given by the following:
$$A_i = \sqrt{w_i}*Z + \sqrt{1-w_i}*\epsilon_i
$$
My question is - How do I calculate $w_i$? I have the PD, LGD ...
5
votes
0
answers
184
views
Why Vasicek model on a tree is a bad choice for pricing American option on credit prepayment?
I have an American option on a credit prepayment, i.e. the holder of the option can prepay the remaining credit if the interest rate falls below the initial strike. The pricing of this option was done ...
1
vote
0
answers
101
views
How to obtain the discount curve under Vasicek interest rate for discounting cash flow?
Suppose that the spot rate is governed by a Vasicek model. We know that there is an analytical solution for the zero-coupon bond.
I guess the discount curve is constructed by the Yield curve in which ...
1
vote
0
answers
84
views
Distribution and Analytical solution of a GBM with stochastic interest rate?
We model the exchange rate $S_t$ with a geometric Brownian motion and the USD and EUR interest rates $r_u$ and $r_e$ each according to the Vasicek model. Under the domestic equivalent martingale ...
1
vote
2
answers
1k
views
Trouble Calibrating a Vasicek Model
I have simulated some data according to a Vasicek process and I am then trying to apply ordinary least squares (OLS) regression analysis to see how accurate the estimated model parameters are from the ...
3
votes
2
answers
577
views
How to determine the risk premium from the Vasicek one factor model?
The short rate under the Vasicek one factor model under the real-world measure $\mathbb{P} $ follows :
$$ dr(t)=(a\theta - (a+\lambda \sigma)r(t))dt + \sigma dW(t),$$
$$ r(0)=r_0 $$
where $ \lambda $ ...
2
votes
0
answers
82
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
vote
0
answers
182
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 ...
3
votes
1
answer
336
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
vote
1
answer
85
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
vote
2
answers
243
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
1
answer
266
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$ ...
1
vote
1
answer
463
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 ...
1
vote
2
answers
773
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
1
answer
114
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
0
answers
794
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
1
answer
408
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
1
answer
477
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
votes
0
answers
75
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)...
1
vote
0
answers
336
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
1
answer
218
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
1
answer
176
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
1
answer
113
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
1
answer
473
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
1
answer
329
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
1
answer
606
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 ...
2
votes
1
answer
939
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 ...