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11 votes

Two papers - two different solutions of the Ornstein-Uhlenbeck process

Note that the Ito integral of a deterministic integrand $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ is normally distributed \begin{equation} \int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...
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8 votes

What is the purpose of short rate models?

Short rate models were first used in the 1970s and 1980s to try to fit and explain the term structure of interest rates - they went beyond simple parametric shapes (polynomials and exponential forms). ...
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  • 2,069
7 votes

Why isn't the Vasicek model arbitrage-free?

Short rate models are broadly divided into equilibrium models and no-arbitrage models. The models from Vasicek, Dothan and Cox, Ingersoll and Ross are examples of equilibrium short rate models. The ...
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  • 13.9k
6 votes
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Vasicek short rate: Risk-neutral measure into real-world measure

Vasnicek by itself does not specify what form the change of measure should be and how you should parameterise the market price of risk. A very natural parameterisation is affine in the factor, i.e., ...
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  • 1,028
5 votes
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Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model

Apply the Ito product rule, noting the cov of a deterministic and stochastic term is zero: $$\begin{align} d\left(e^{at}r_t\right)&=e^{at} dr_t+r_t de^{at} \\[6pt] &=e^{at} dr_t+r_t e^{at} d(...
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5 votes
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What is the reasoning to derive this financial model called the Vasicek Model?

The original Vasicek paper is "An equilibrium model of the term structure". If you google for it, you'll find it and you can read in his own words his motivation for developing it. In particular, what ...
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  • 794
5 votes
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Vasicek model - Bond price and volatility

Intuition is just that the bond price by definition is a convex function of the rates, and the expectation of a convex function increases with volatility. Note that this result is model independent.
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  • 1,662
4 votes
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How to find the transition distribution functions of these two processes?

We consider the first one, that is, $X_t = X_s + \mu (t-s) + \sigma (W_t-W_s)$, for $t>s$. Then, \begin{align*} P(X_t \le y \mid X_s) &= P(X_t-\mu(t-s)-X_s \le y-\mu(t-s)-X_s \mid X_s)\\ &=...
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  • 20.5k
4 votes

What is the purpose of short rate models?

I might get down-voted for this, but in my opinion, short-rate models are not very useful for any practical pricing problems in today's finance. Even for simple vanilla rate derivatives (i.e. Caplet ...
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  • 5,191
4 votes
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Choosing which interest rate model to go with?

Calibrate to many observed curves, over all kinds of shapes: flat, normal, inverted, and humped, and measure and compare the model fitting errors. If you can't find all the shapes in history, make ...
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  • 5,038
4 votes
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Pricing Call Option on Coupon Bond under Vasicek

It seems to me what you want to prove is the Jamshidian's trick. We know that the function $\Bbb R \ni r \to \exp(A(t,T)-B(t,T)r)$ is monotone and if $B(t,T) \neq 0$ (If my memory is good, normally, $...
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  • 486
4 votes

How to determine the risk premium from the Vasicek one factor model?

I recommend two papers that should help you with this exercise. The first is "Kalman Filtering of Generalized Vasicek Term Structure Models." This paper provides a general framework for ...
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  • 10.9k
3 votes

What is the purpose of short rate models?

Long story short, the main reason of a short rate model is to provide an analytical solution for the zero coupon bond $P(t, T)$, given by the following expectation: $$ P(t, T) = E_t^Q \left[ \exp \...
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  • 711
3 votes

Timesteps in Vasicek model

Yes you can! Any SDE that has an analytic solution can be simulated exactly. The vasicek model has dynamics $dr=a(b-r)dt+\sigma dW_t$. By Ito's lemma, $$d\left(e^{at}r\right)=e^{at}\left(a(b-r)dt+\...
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  • 1,329
3 votes

How to price a stock under Q and stochastic interest rates?

The derivation in Appendix A of the paper Valuation of Equity-Indexed Annuities under Stochastic Interest Rates that you mentioned is Wrong: the Girsanov transformation is applied to an $n$-...
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  • 20.5k
3 votes
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Difference between the Basel IRB and the Vasicek formula

The paper continues "The quantity p(Y) provides the loan default probability under the given scenario." But the default probability is 0.001, not 0.999 as in the IRB version. So G(0.999) = -G(1 - 0....
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  • 66
3 votes
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LIBOR rates from Vasicek/Hull-White model?

In practice, you can calibrate to either 1 month libor or 3 month libor, but not both. That's because there's a basis swap between 1 month libor and 3 month libor that can't be explained by your ...
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  • 14.1k
3 votes
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Vasicek model: joint simulation with discount factor

Although it's been a long time this question has been asked, I'd like to propose an answer in case someone was looking for the same thing. First, I think there's a confusion between $P(t,T)$ and $DF(t,...
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  • 316
3 votes

Bond-price dynamics in the Vasicek model

You know the bond price formula takes this form: $P \left( t, T \right)= A \left( t, T \right) e^{ -r_{t} B \left(t, T \right) }$ Now apply Ito's lemma, so you will get after some manipulation: $\...
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3 votes
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Aggregation of $\rho$ and $p$ for a vasicek model

You can first compute the average PD - few choices would be: Simple average of the individual PDs Exposure weighted average of the PDs If the PDs range is too large, then you might want to bucket ...
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3 votes
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Affine Structure Resolution for the Vasicek model

We begin with the equation $1+B_t(t,T)-kB(t,T) = 0 \quad(1)$ \begin{align} (1) & \iff e^{-kt}+e^{-kt}B_t(t,T)+(-k)e^{-kt}B(t,T) = 0 \\ & \iff e^{-kt}+ \frac{\partial}{\partial t}\left(e^{-kt}B(...
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  • 486
3 votes
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Why can a two-factor interest rate model not be used to value a coupon bearing bond as the sum of options on ZCBs

On a conceptual level an option on a coupon bonds is an option on a sum of the coupons (and principal), and we are comparing it to the sum of the options on coupons. In a one-factor model all coupons/...
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  • 910
2 votes
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How to show that the exponential Vasicek model is not an affine term-structure model?

Here is a general proof for all parameters in an open domain. $$dr = adt+bdW:=r\big(k(\theta-x)+\frac12\sigma^2\big)dt+\sigma rdW.$$ Let $$u(r(s),s):=e^{-\int_t^sr}B(r(s),s,T)=:\phi(s) B.$$ Then $$u(...
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  • 2,471
2 votes
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How to find the transition distribution functions of these two processes?

Here is a derivation for the Ornstein-Uhlenbeck process. Solution to the SDE $$dX_t = \theta(\mu-X_t) dt + \sigma dW_t$$ subject to the initial condition $X_0=x$ has the form $$X_t= \mu + (x - \mu)e^{-...
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  • 4,910
2 votes

Help evaluating covariance integral when deriving vasiceks model

that symbol means "the min of". So for example, if: $s<t$, then $s$ ^ t = s. If you look in any book for the Covariance of a BM, you will see that same symbol and how to work with it. Cheers.
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  • 641
2 votes
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Vasicek model problem

For starters, the short rate model you mention in equation (1) is Cox-Ingersoll-Ross while the bond price in equations (2)-(4) correspond to the Vacisek model. So there is a problem somewhere, I would ...
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  • 14k
2 votes
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Hull-White Extension of Vasicek Model

$\theta(t) - a(t) r(t)$ is the risk neutral drift. The Hull & White models posits the dynamics $dr(t) = (\theta(t) - a(t) r(t)) dt + \sigma dW(t)$ under the risk neutral measure $P$ and then ...
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2 votes

Term structure used in Geometric Brownian Motions under Risk Neutral Measure?

It should be time dependent and set to the spot forward rate $= -\frac{\partial}{\partial t} \ln(\text{discount}(t))$ when simulating in continuous time. When discretizing the simulation use the ...
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