Questions tagged [vasicek]

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Mean reversion parameter estimation in Hull White - Vasicek extension [closed]

I'm fitting the Hull White - Vasicek extension model (1994a) for one-factor short interest rate as shown in Brigo Mercurio book. Short rate dynamics: $$ dr(t) = [\theta(t) + ar(t)]dt + \sigma dW(t) $$ ...
2 votes
1 answer
373 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 ...
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0 answers
44 views

The PDE of an Interest Rate Swap - ImplicitFinite Difference Method

Context: I'm stuggling to implement an implicit finite difference scheme on an Interest Rate Swap (IRS) with a single factor short rate model: I was following the textbook(Daniel J. Duffy - Finite ...
2 votes
0 answers
97 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
51 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 ...
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Using short-term interest rate models in practice [duplicate]

I come from a high-dimensional statistics background, and have started venturing in the world of continuous-time stochastic processes, SDEs and so on. I have invested in the following book and by ...
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3 votes
0 answers
36 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 ...
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1 vote
3 answers
283 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)....
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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
191 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 ...
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2 votes
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67 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
30 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 ...
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53 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_{\...
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1 vote
1 answer
148 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 ...
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5 votes
0 answers
126 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 ...
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1 vote
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77 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 ...
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1 vote
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70 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 ...
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1 vote
2 answers
755 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 ...
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3 votes
2 answers
400 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
71 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
105 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 ...
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2 votes
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135 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
59 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). ...
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1 vote
2 answers
180 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*} \...
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3 votes
1 answer
199 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
294 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 ...
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2 answers
439 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
95 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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1 vote
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402 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 ...
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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 ...
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1 answer
295 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 ...
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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
185 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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1 answer
147 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
129 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
80 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
337 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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0 votes
1 answer
270 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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1 vote
1 answer
448 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
1 answer
741 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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4 votes
1 answer
1k 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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1 vote
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226 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
1 answer
103 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
1 answer
277 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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2 votes
0 answers
130 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 vote
0 answers
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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
1 answer
729 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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1 answer
588 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
0 answers
236 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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1 answer
307 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 ...