Questions tagged [vasicek]

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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 ...
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36 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 $Y$ ...
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66 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?
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48 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 ...
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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 ...
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41 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{...
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34 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 ...
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15 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, ...
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53 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(...
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61 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 ...
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96 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 ...
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1answer
100 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 ...
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1answer
301 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 ...
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380 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?
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53 views

Derive the discount bond prices of the Vasicek model by the PDE approach

The question is shown above. Anyone can help me?
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68 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—...
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1answer
101 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 ...
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59 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 ...
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58 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 ...
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283 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 ...
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208 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 $\gamma$ ...
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154 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 ...
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142 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 ...
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239 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 ...
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1answer
50 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"...
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241 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 ...
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1answer
139 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
323 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 ...
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141 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 ...
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1answer
495 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
178 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)...
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121 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^{-...
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238 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 ...
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987 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
192 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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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 ...
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1answer
109 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. ...
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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+\...
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436 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=\...
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224 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
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1answer
452 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 ...
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136 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: $...
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172 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: ...
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1answer
87 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 \...
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1answer
129 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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819 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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189 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 ...
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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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1answer
774 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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71 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 $V$. ...